## Standards

**1**

Implementation of Texas Essential Knowledge and Skills for Mathematics, Elementary, Adopted 2012.

**1**

Grade 1

**1.**

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

*For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."***1.**

(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

**1.**

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

*For example, describe a context in which a total number of objects can be expressed as 5 × 7.***1.**

Know there is a complex number *i* such that *i*² = –1, and every complex number has the form *a + bi* with a and *b* real.

**1.**

Know there is a complex number *i* such that *i*² = –1, and every complex number has the form *a + bi* with a and *b* real.

**1.**

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

*For example, we define 5*^{1/3}to be the cube root of 5 because we want (5^{1/3})³ = 5(^{1/3})³ to hold, so (5^{1/3})³ must equal 5.**1.**

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

**1.**

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

**1.**

Count to 100 by ones and by tens.

**1.**

Make sense of problems and persevere in solving them.

**1.**

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**1.**

Interpret key features of an expression (i.e., terms, factors, and coefficients).

**1.**

Solves mathematical problems.

**1.**

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, │v│, ││v││), including the use of eigen-values and eigen-vectors.

**1.**

Estimate limits from graphs or tables.

**1.**

Express sequences and series using recursive and explicit formulas.

**1.**

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

**1.**

Create algebraic models for application-based problems by developing and solving equations and inequalities, including those involving direct, inverse, and joint variation. *Example: The amount of sales tax on a new car is directly proportional to the purchase price of the car. If the sales tax on a $20,500 car is $1,600, what is the purchase price of a new car that has a sales tax of $3,200? Answer: The purchase price of the new car is $41,000.*

**1.**

Analyze topics from elementary number theory, including perfect numbers and prime numbers, to determine properties of integers.

**1.**

Critique ancient numeration systems and applications, including astronomy and the development and use of money and calendars.

**1.**

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

**1.**

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.**

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

*For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction*^{1/2}/_{1/4}miles per hour, equivalently 2 miles per hour.**1.0**

Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events.

**1.0**

Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:

**10.**

State and apply the formal definition of a derivative.

**10.**

Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

**10.**

Understand ordering and absolute value of rational numbers.

**10.**

(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

**10.**

Add up to four two-digit numbers using strategies based on place value and properties of operations.

**10.**

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

**10.**

Analyze and solve pairs of simultaneous linear equations.

**10.**

(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Solve matrix application problems using reduced row echelon form.

**10.**

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**10.**

Critique measurements in terms of precision, accuracy, and approximate error. *Example: Determine whether one candidate has a significant lead over another candidate when given their current standings in a poll and the margin of error.*

**10.**

Graph the solution point of an equation and the solution set of an inequality in one variable on a horizontal number line. For inequalities, be able to interpret and write the solution set in a variety of ways (e.g., set notation).

**10.**

(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

**10.**

(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

**10.**

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

**10.**

Use place value understanding to round whole numbers to the nearest 10 or 100.

**10.**

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

**10.**

Use vertex-coloring techniques and matching techniques to solve application-based problems. *Example: Use graph-coloring techniques to color a map of the western states of the United States so no adjacent states are the same color, including determining the minimum number of colors needed and why no fewer colors may be used.*

**10.**

Prove theorems about triangles. *Theorems include measures of interior angles of a triangle sum to 180º, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point.*

**10.**

Determine the mathematical impact of the ancient Greeks, including Archimedes, Eratosthenes, Euclid, Hypatia, Pythagoras, and the Pythagorean Society. *Example: Use Euclid’s proposition to inscribe a regular hexagon within a circle.*

**10.**

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

**10.**

Prove polynomial identities and use them to describe numerical relationships. *For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.*

**10.0**

Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.

**10.0**

Students know Newton’s method for approximating the zeros of a function.

**10a.**

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

*For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?***10a.**

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

**10a.**

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

*For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.***10a.**

10 can be thought of as a bundle of ten ones — called a "ten."

**10b.**

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

**10b.**

Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

*For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.***10b.**

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

*For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.***10b.**

Write, interpret, and explain statements of order for rational numbers in real-world contexts.

*For example, write -3 °C > -7 °C to express the fact that -3 °C is warmer than -7 °C.***10c.**

Solve real-world and mathematical problems leading to two linear equations in two variables.

*For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.***10c.**

Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

*For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.***10c.**

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**10d.**

Distinguish comparisons of absolute value from statements about order.

*For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.***1.1**

Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.

**1.1**

The student will

- count forward orally by ones to 110, starting at any number between 0 and 110;
- write the numerals 0 to 110 in sequence and out-of-sequence;
- count backward orally by ones when given any number between 1 and 30; and
- count forward orally by ones, twos, fives, and tens to determine the total number of objects to 110.

**11.**

Justify when linear equations in one variable will yield one solution, infinitely many solutions, or no solution.

**11.**

Use ratios of perimeters, areas, and volumes of similar figures to solve applied problems. *Example: Use a blueprint or scale drawing of a house to determine the amount of carpet to be purchased.*

**11.**

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

**11.**

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

*For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)***11.**

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

**11.**

Solve application-based logic problems using Venn diagrams, truth tables, and matrices.

**11.**

(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

**11.**

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**11.**Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

**11.**Verify the Binomial Theorem by mathematical induction or by a combinatorial argument.

**11.**(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

**11.**Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

**11.**Describe the development of mathematical tools and their applications. *Examples: Use knotted ropes for counting; Napier’s bones for multiplication; a slide rule for multiplying and calculating values of trigonometric, exponential, and logarithmic functions; and a graphing calculator for analyzing functions graphically and numerically.*

**11.**(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

**11.**Apply differentiation rules to sums, products, quotients, and powers of functions.

**11.**Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

**11.**Prove theorems about parallelograms. *Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals.*

**11.**Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**11.**(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

**11.**Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

**11.**(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Understand the importance of using complex numbers in graphing functions on the Cartesian or complex plane.

**1.10**The student will use nonstandard units to measure and compare length, weight, and volume.

**11.0**Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.

**11.0**Students compute the variance and the standard deviation of a distribution of data.

**1.11**The student will

- identify, trace, describe, and sort plane figures (triangles, squares, rectangles, and circles) according to number of sides, vertices, and angles; and
- identify and describe representations of circles, squares, rectangles, and triangles in different environments, regardless of orientation, and explain reasoning.

**1.1.1.1**Use place value to describe whole numbers between 10 and 100 in terms of tens and ones.
*For example*: Recognize the numbers 21 to 29 as 2 tens and a particular number of ones.

**1.1.1.2**Read, write and represent whole numbers up to 120. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

**1.1.1.3**Count, with and without objects, forward and backward from any given number up to 120.

**1.1.1.4**Find a number that is 10 more or 10 less than a given number.
*For example*: Using a hundred grid, find the number that is 10 more than 27.

**1.1.1.5**Compare and order whole numbers up to 120.

**1.1.1.6**Use words to describe the relative size of numbers.
*For example*: Use the words equal to, not equal to, more than, less than, fewer than, is about, and is nearly to describe numbers.

**1.1.1.7**Use counting and comparison skills to create and analyze bar graphs and tally charts.
*For example*: Make a bar graph of students' birthday months and count to compare the number in each month.

**1.12**The student will

- collect, organize, and represent various forms of data using tables, picture graphs, and object graphs; and
- read and interpret data displayed in tables, picture graphs, and object graphs, using the vocabulary more, less, fewer, greater than, less than, and equal to.

**1.1.2.1**Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.

**1.1.2.2**Compose and decompose numbers up to 12 with an emphasis on making ten.
*For example*: Given 3 blocks, 7 more blocks are needed to make 10.

**1.1.2.3**Recognize the relationship between counting and addition and subtraction. Skip count by 2s, 5s, and 10s.

**1.13**The student will sort and classify concrete objects according to one or two attributes.

**1.14**The student will identify, describe, extend, create, and transfer growing and repeating patterns.

**1.15**The student will demonstrate an understanding of equality through the use of the equal symbol.

**1.2**The student, given up to 110 objects, will

- group a collection into tens and ones and write the corresponding numeral;
- compare two numbers between 0 and 110 represented pictorially or with concrete objects, using the words greater than, less than or equal to; and
- order three or fewer sets from least to greatest and greatest to least.

**1.2**Students use graphical calculators to verify and estimate limits.

**12.**Fluently add and subtract within 5.

**12.**Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

**12.**Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

**12.**Interpret expressions that represent a quantity in terms of its context.

**12.**Interpret expressions that represent a quantity in terms of its context.

**12.**Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. *Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. *

**12.**Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

**12.**Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

*For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.***12.**Write and evaluate numerical expressions involving whole-number exponents.

**12.**Know and apply the Binomial Theorem for the expansion of (x + y)

^{n}in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.**12.**Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. *(Extend to infinite geometric series.) Example: Calculate mortgage payments.*

**12.**Use combinatorial reasoning and counting techniques to solve application-based problems. Example: *Determine the probability of a safe opening on the first attempt given the combination uses the digits 2, 4, 6, and 8 with the order unknown.*

**12.**Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Use function notation, where appropriate.

**12.**Create equations and inequalities in one variable, and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

**12.**Use the chain rule and implicit differentiation.

**12.**Create a model of a set of data by estimating the equation of a curve of best fit from tables of values or scatter plots. *Create models of election results as a function of population change, inflation or employment rate as a function of time, cholesterol density as a function of age or weight of a person.*

**12.**Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

*For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.***12.**Calculate the limit of a sequence, of a function, and of an infinite series.

**12.**Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

**12.**Summarize the history of probability, including the works of Blaise Pascal; Pierre de Fermat; Abraham de Moivre; and Pierre-Simon, marquis de Laplace. *Example: Discuss the impact of probability on gaming, economics, and insurance.*

**12.**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**12.0**Students use differentiation to solve related rate problems in a variety of pure and applied contexts.

**12.0**Students find the line of best fit to a given distribution of data by using least squares regression.

**1.2.1.1**Create simple patterns using objects, pictures, numbers and rules. Identify possible rules to complete or extend patterns. Patterns may be repeating, growing or shrinking. Calculators can be used to create and explore patterns.
*For example*: Describe rules that can be used to extend the pattern 2, 4, 6, 8, □, □, □ and complete the pattern 33, 43, □, 63, □, 83 or 20, □, □, 17.

**1.2.2.1**Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.
*For example*: One way to represent the number of toys that a child has left after giving away 4 of 6 toys is to begin with a stack of 6 connecting cubes and then break off 4 cubes.

**1.2.2.2**Determine if equations involving addition and subtraction are true.
*For example*: Determine if the following number sentences are true or false
7 = 7
7 = 8 – 1
5 + 2 = 2 + 5
4 + 1 = 5 + 2.

**1.2.2.3**Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:
2 + 4 =
3 + = 7
5 = – 3.

**1.2.2.4**Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
*For example*: 5 + 3 = 8 could be used to represent a situation in which 5 red balloons are combined with 3 blue balloons to make 8 total balloons.

**12a.**Predict probabilities given a frequency distribution.

**12a.**Interpret parts of an expression such as terms, factors, and coefficients.

**12a.**Interpret parts of an expression such as terms, factors, and coefficients.

**12b.**Interpret complicated expressions by viewing one or more of their parts as a single entity. *Example: Interpret P(1+r)

^{n}as the product of P and a factor not depending on P.***12b.**Interpret complicated expressions by viewing one or more of their parts as a single entity. *Example: Interpret P(1+r)

^{n}as the product of P and a factor not depending on P.***1.3**Students prove and use special limits, such as the limits of (sin(*x*))/*x* and (1−cos(*x*))/*x* as *x* tends to 0.

**1.3**The student, given an ordered set of ten objects and/or pictures, will indicate the ordinal position of each object, first through tenth.

**13.**Use the structure of an expression to identify ways to rewrite it.

*For example, see x*^{4}— y^{4}as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).**13.**Use the structure of an expression to identify ways to rewrite it.

*For example, see x*^{4}— y^{4}as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).**13.**Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

*For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?***13.**Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

**13.**Write, read, and evaluate expressions in which letters stand for numbers.

**13.**Describe the relationship between differentiability and continuity.

**13.**Write rational expressions in simplest form.

**13.**(+) Know and apply the Binomial Theorem for the xpansion of *(x + y)

^{n}* in powers of *x* and *y* for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)**13.**Use the laws of Boolean Algebra to describe true/false circuits. Simplify Boolean expressions using the relationships between conjunction, disjunction, and negation operations.

**13.**Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

*For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.***13.**Analyze election data to compare election methods and voting apportionment, including determining strength within specific groups.

**13.**Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

**13.**Explain why addition and subtraction strategies work, using place value and the properties of operations.

**13.**Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or

**13.**Compare and contrast a function and a relation. Use appropriate strategies to assess whether a given situation represents a function or a relation (e.g,. the vertical line test).

**13.**Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

**13.**Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

**13.**Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

**13.**Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

**13.0**Students know what the correlation coefficient of two variables means and are familiar with the coefficient’s properties.

**13.0**Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.

**1.3.1.1**Describe characteristics of two- and three-dimensional objects, such as triangles, squares, rectangles, circles, rectangular prisms, cylinders, cones and spheres.

**1.3.1.2**Compose (combine) and decompose (take apart) two- and three-dimensional figures such as triangles, squares, rectangles, circles, rectangular prisms and cylinders.

**1.3.2.1**Measure the length of an object in terms of multiple copies of another object.

**1.3.2.2**Tell time to the hour and half-hour.

**1.3.2.3**Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.

**13a.**Write expressions that record operations with numbers and with letters standing for numbers.

*For example, express the calculation "Subtract y from 5" as 5 - y.***13b.**Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

*For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.***13c.**Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

*For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.***1.4**The student will

- represent and solve practical problems involving equal sharing with two or four sharers; and
- represent and name fractions for halves and fourths, using models.

**14.**Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. *Example, calculate mortgage payments.*

**14.**Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**14.**Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. *Example, calculate mortgage payments.*

**14.**Use logic symbols to write truth tables.

**14.**Represent constraints by equations or inequalities, and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context. *Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods.*

**14.**Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

**14.**Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

**14.**Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

**14.**Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

**14.**(+) Represent a system of linear equations as a single matrix equation in a vector variable.

**14.**Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

**14.**Decompose a rational function into partial fractions.

**14.**Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

**14.**Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

**14.**Understand a fraction as a number on the number line; represent fractions on a number line diagram.

**14.**Apply the properties of operations to generate equivalent expressions.

*For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.***14.**Verify experimentally the properties of dilations given by a center and a scale factor.

**14.0**Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.

**14.0**Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

**14a.**Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

**14a.**Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

*For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)***14a.**Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

**14a.**A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged.

**14b.**Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

**14b.**Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

**14b.**Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.

**14b.**The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

**14c.**Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

**14d.**Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

**1.5**The student, given a familiar problem situation involving magnitude, will

- select a reasonable order of magnitude from three given quantities: a one-digit numeral, a two-digit numeral, and a three-digit numeral (e.g., 5, 50, 500); and
- explain the reasonableness of the choice.

**15.**Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. *Example:Graph x² – 6x + y² – 12y + 41 = 0 or y² – 4x + 2y + 5 = 0.*

**15.**Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. *Example, directly compare the heights of two children and describe one child as taller/shorter.*

**15.**Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

*For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.***15.**Interpret multiplication as scaling (resizing), by:

**15.**Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

**15.**Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

**15.**Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

**15.**Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

**15.**Order three objects by length; compare the lengths of two objects indirectly by using a third object.

**15.**Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

**15.**Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

**15.**Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

**15.**Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

**15.**Determine asymptotes and holes of rational functions, explain how each was found, and relate these behaviors to continuity.

**15.**Define a derivative and explain the purpose/utility of the derivative.

**15.**Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. *Example: Rearrange Ohm’s law V = IR to highlight resistance R.*

**15.**Reduce the degree of either the numerator or denominator of a rational function by using partial fraction decomposition or partial fraction expansion.

**15.**Determine the rate of change of a linear function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Use the rate of change to determine if two lines are parallel, perpendicular, or neither.

**15.0**Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.

**15.0**Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.

**15a.**Understand a fraction a/b as a multiple of 1/b.

*For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).***15a.**Formulate equations of conic sections from their determining characteristics. *Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4.*

**15a.**Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

**15a.**Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

**15b.**Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

*For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)***15b.**Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

**15b.**Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

**15c.**Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.

*For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?***15c.**Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

**15d.**Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or

**1.6**The student will create and solve single-step story and picture problems using addition and subtraction within 20.

**16.**Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

**16.**Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

**16.**Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

**16.**Estimate lengths using units of inches, feet, centimeters, and meters.

**16.**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

**16.**Add, subtract, multiply and divide rational expressions.

**16.**Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.

**16.**Know and apply the Remainder Theorem: For a polynomial *p(x)* and a number *a*, the remainder on division by *x – a* is *p(a)*, so *p(a)* = 0 if and only if *(x – a)* is a factor of *p(x)*.

**16.**Know and apply the Remainder Theorem: For a polynomial *p(x)* and a number *a*, the remainder on division by *x – a* is *p(a)*, so *p(a)* = 0 if and only if *(x – a)* is a factor of *p(x)*.

**16.**Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

**16.**For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *(Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine odd, even, neither.)*

**16.**Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

**16.**Verify experimentally the properties of rotations, reflections, and translations:

**16.**Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

**16.**Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.

**16.**Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

**16.**(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

**16.**Apply the derivative as a rate of change in varied contexts, including velocity, speed, and acceleration.

**16.0**Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

**16.0**Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution.

**16a.**Lines are taken to lines, and line segments to line segments of the same length.

**16b.**Angles are taken to angles of the same measure.

**16c.**Parallel lines are taken to parallel lines.

**1.7**The student will

- recognize and describe with fluency part-whole relationships for numbers up to 10; and
- demonstrate fluency with addition and subtraction within 10.

**17.**Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. *

**17.**Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

**17.**Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

**17.**Create and graph the equation of a linear function given the rate of change and y-intercept. Compare and contrast up to three linear functions written in a various forms (i.e., point-slope, slope-intercept, standard form).

**17.**Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

**17.**Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

**17.**Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

**17.**Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

**17.**Apply the derivative to find tangent lines and normal lines to given curves at given points.

**17.**Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

**17.**Tell and write time in hours and half-hours using analog and digital clocks.

**17.**(+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. Understand Law of Sines = *2r*, where r is the radius of the circumscribed circle of the triangle. Apply the Law of Tangents.

**17.**Solve polynomial and rational inequalities. Relate results to the behavior of the graphs.

**17.**Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

**17.**Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

**17.**Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

**17.**Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

**17.**Use decimal notation for fractions with denominators 10 or 100.

*For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.***17.0**Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.

**17.0**Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.

**17a.**Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

*For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.***17b.**Interpret division of a whole number by a unit fraction, and compute such quotients.

*For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.***17c.**Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

*For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?***1.8**The student will determine the value of a collection of like coins (pennies, nickels, or dimes) whose total value is 100 cents or less.

**18.**Given two points, a graph, a table of values, a mapping, or a real-world context determine the linear function that models this information. Fluently convert between the point-slope, slope-intercept, and standard form of a line.

**18.**Prove polynomial identities and use them to describe numerical relationships. *Example: The polynomial identity (x² + y²)2 = (x² — y²)² + (2xy)² can be used to generate Pythagorean triples.*

**18.**Prove polynomial identities and use them to describe numerical relationships. *Example: The polynomial identity (x² + y²)2 = (x² — y²)² + (2xy)² can be used to generate Pythagorean triples.*

**18.**Solve quadratic equations in one variable.

**18.**Apply Euler’s and deMoivre’s formulas as links between complex numbers and trigonometry.

**18.**Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

**18.**Predict and explain the relationships between functions and their derivatives.

**18.**Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

**18.**Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

**18.**Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

**18.**Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

**18.**Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs.

*For example, draw a bar graph in which each square in the bar graph might represent 5 pets.***18.**Find the composite of two given functions and find the inverse of a given function. Extend this concept to discuss the identity function f(x) = x.

**18.**Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

*For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.***18.**Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

**18.**Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**18.**Correctly name shapes regardless of their orientations or overall size.

**18.**Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

**18.0**Students determine the P-value for a statistic for a simple random sample from a normal distribution.

**18.0**Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.

**18a.**Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)² = q* that has the same solutions. Derive the quadratic formula from this form.

**18a.**Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

**18b.**Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square and the quadratic formula, and factoring as appropriate to the initial form of the equation.

**18b.**Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

**18c.**(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

**18d.**Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

**1.9**The student will investigate the passage of time and

- tell time to the hour and half-hour, using analog and digital clocks; and
- read and interpret a calendar.

**19.**Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table.

*For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …***19.**Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

*For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.***19.**Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

**19.**Create and identify the parent function for linear and quadratic functions in the Coordinate Plane.

**19.**Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").

**19.**Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

*For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.***19.**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

**19.**Rewrite simple rational expressions in different forms; write *a(x)/b(x)* in the form *q(x) + r(x)/b(x)*, where *a(x)*, *b(x)*, *q(x)*, and *r(x)* are polynomials with the degree of r(x) less than the degree of *b(x)*, using inspection, long division, or for the more complicated examples, a computer algebra system.

**19.**Rewrite simple rational expressions in different forms; write *a(x)/b(x)* in the form *q(x) + r(x)/b(x)*, where *a(x)*, *b(x)*, *q(x)*, and *r(x)* are polynomials with the degree of r(x) less than the degree of *b(x)*, using inspection, long division, or for the more complicated examples, a computer algebra system.

**19.**Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

**19.**Simplify complex algebraic fractions (with/without variable expressions and integer exponents) to include expressing (f(x + h) - f(x))/h as a single simplified fraction when f(x) = 1/1 - x for example.

**19.**(+) Compose functions. *Example: If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.*

**19.**Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles.

**19.**Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

**19.**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

**19.**Model rates of change to solve related rate problems.

**19.**Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

**19.0**Students are familiar with the chi-square distribution and chi-square test and understand their uses.

**19.0**Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square.

**1.a**The provisions of §§111.2-111.7 of this subchapter shall be implemented by school districts.

**1a.**Determine relationships among mathematical achievements of ancient peoples, including the Sumerians, Babylonians, Egyptians, Mesopotamians, Chinese, Aztecs, and Incas.

**1.A.1**Identify patterns found in real-world and mathematical situations.

**1.A.1.1**Identify, create, complete, and extend repeating, growing, and shrinking patterns with quantity, numbers, or shapes in a variety of real-world and mathematical contexts.

**1.ATO.1**Solve real-world/story problems using addition (as a joining action and as a part- part-whole action) and subtraction (as a separation action, finding parts of the whole, and as a comparison) through 20 with unknowns in all positions.

**1.ATO.2**Solve real-world/story problems that include three whole number addends whose sum is less than or equal to 20.

**1.ATO.3**Apply Commutative and Associative Properties of Addition to find the sum (through 20) of two or three addends.

**1.ATO.4**Understand subtraction as an unknown addend problem.

**1.ATO.5**Recognize how counting relates to addition and subtraction.

**1.ATO.6**Demonstrate:

**1.ATO.6.a**addition and subtraction through 20;

**1.ATO.6.b**fluency with addition and related subtraction facts through 10.

**1.ATO.7**Understand the meaning of the equal sign as a relationship between two quantities (sameness) and determine if equations involving addition and subtraction are true.

**1.ATO.8**Determine the missing number in addition and subtraction equations within 20.

**1.ATO.9**Create, extend and explain using pictures and words for:

**1.ATO.9.a**repeating patterns (e.g., AB, AAB, ABB, and ABC type patterns);

**1.ATO.9.b**growing patterns (between 2 and 4 terms/figures).

**1.b**No later than August 31, 2013, the commissioner of education shall determine whether instructional materials funding has been made available to Texas public schools for materials that cover the essential knowledge and skills for mathematics as adopted in §§111.2-111.7 of this subchapter.

**1b.**Explain origins of the Hindu-Arabic numeration system. *Example: Perform addition and subtraction in both the Hindu-Arabic and the Roman numeration systems to compare place value and place holders.*

**1.c**If the commissioner makes the determination that instructional materials funding has been made available under subsection (b) of this section, §§111.2-111.7 of this subchapter shall be implemented beginning with the 2014-2015 school year and apply to the 2014-2015 and subsequent school years.

**1.CA.1**Demonstrate fluency with addition facts and the corresponding subtraction facts within 20. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Understand the role of 0 in addition and subtraction.

**1.CA.2**Solve real-world problems involving addition and subtraction within 20 in situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all parts of the addition or subtraction problem (e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem).

**1.CA.3**Create a real-world problem to represent a given equation involving addition and subtraction within 20.

**1.CA.4**Solve real-world problems that call for addition of three whole numbers whose sum is within 20 (e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem).

**1.CA.5**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; describe the strategy and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones, and that sometimes it is necessary to compose a ten.

**1.CA.6**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false (e.g., Which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2).

**1.CA.7**Create, extend, and give an appropriate rule for number patterns using addition within 100.

**1.CC**Counting and Cardinality

**1.CC.1**Skip count by 2s and 5s.

**1.CC.2**Use ordinal numbers correctly when identifying object position (e.g., first, second, third, etc.).

**1.CC.3**Order numbers from 1-100. Demonstrate ability in counting forward and backward.

**1.CC.4**Count a large quantity of objects by grouping into 10s and counting by 10s and 1s to find the quantity.

**1.CC.5**Use the symbols for greater than, less than or equal to when comparing two numbers or groups of objects.

**1.CC.6**Estimate how many and how much in a given set to 20 and then verify estimate by counting.

**1: Cluster: (ID 1)**Reason with shapes and their attributes.

**1: Cluster: (ID 10)**Represent and interpret data.

**1: Cluster: (ID 1062)**Reason with shapes and their attributes.

**1: Cluster: (ID 1066)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 1069)**Work with time and money.

**1: Cluster: (ID 1071)**Represent and interpret data.

**1: Cluster: (ID 1073)**Extend the counting sequence.

**1: Cluster: (ID 1075)**Understand place value.

**1: Cluster: (ID 1081)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 1085)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 1088)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 1091)**Add and subtract using numbers up to 20.

**1: Cluster: (ID 1094)**Work with addition and subtraction equations.

**1: Cluster: (ID 11507)**Reason with shapes and their attributes.

**1: Cluster: (ID 11511)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 11514)**Tell and write time.

**1: Cluster: (ID 11516)**Represent and interpret data.

**1: Cluster: (ID 11518)**Extend the counting sequence.

**1: Cluster: (ID 11520)**Understand place value.

**1: Cluster: (ID 11526)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 11533)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 11536)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 11539)**Add and subtract within 20.

**1: Cluster: (ID 11542)**Work with addition and subtraction equations.

**1: Cluster: (ID 12)**Extend the counting sequence.

**1: Cluster: (ID 14)**Understand place value.

**1: Cluster: (ID 16994)**Reason with shapes and their attributes

**1: Cluster: (ID 16998)**Measure lengths indirectly and by iterating length units

**1: Cluster: (ID 17001)**Tell and write time with respect to a clock and a calendar

**1: Cluster: (ID 17003)**Represent and interpret data

**1: Cluster: (ID 17005)**Extend the counting sequence

**1: Cluster: (ID 17007)**Understand place value

**1: Cluster: (ID 17013)**Use place value understanding and properties of operations to add and subtract

**1: Cluster: (ID 17017)**Represent and solve problems involving addition and subtraction

**1: Cluster: (ID 17020)**Understand and apply properties of operations and the relationship between addition and subtraction

**1: Cluster: (ID 17023)**Add and subtract within 20

**1: Cluster: (ID 17026)**Work with addition and subtraction equations

**1: Cluster: (ID 20)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 22546)**Reason with shapes and their attributes.

**1: Cluster: (ID 22550)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 22553)**Tell and write time and money.

**1: Cluster: (ID 22555)**Represent and interpret data.

**1: Cluster: (ID 22557)**Extend the counting sequence.

**1: Cluster: (ID 22559)**Understand place value.

**1: Cluster: (ID 22565)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 22569)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 22572)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 22575)**Add and subtract within 20.

**1: Cluster: (ID 22579)**Work with addition and subtraction equations.

**1: Cluster: (ID 23265)**Reason with shapes and their attributes.

**1: Cluster: (ID 23269)**Measure lengths.

**1: Cluster: (ID 23272)**Build understanding of time and money.

**1: Cluster: (ID 23274)**Represent and interpret data.

**1: Cluster: (ID 23276)**Extend and recognize patterns in the counting sequence.

**1: Cluster: (ID 23279)**Understand place value.

**1: Cluster: (ID 23282)**Use place value understanding and properties of operations.

**1: Cluster: (ID 23286)**Represent and solve problems

**1: Cluster: (ID 23289)**Understand and apply the properties of operations.

**1: Cluster: (ID 23292)**Add and subtract within 20.

**1: Cluster: (ID 23295)**Analyze addition and subtraction equations within 20.

**1: Cluster: (ID 23806)**Reason with shapes and solids and their attributes (squares, circles, triangles, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons, cubes, spheres, cylinders, cones, triangular prisms, and rectangular prisms).

**1: Cluster: (ID 23810)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 23813)**Work with time.

**1: Cluster: (ID 23815)**Represent and interpret data.

**1: Cluster: (ID 23817)**Extend the counting sequence.

**1: Cluster: (ID 23819)**Understand place value.

**1: Cluster: (ID 23825)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 23831)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 23834)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 23837)**Add and subtract within 20.

**1: Cluster: (ID 23840)**Work with addition and subtraction equations.

**1: Cluster: (ID 24)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 24459)**Reason with shapes and their attributes.

**1: Cluster: (ID 24463)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 24466)**Work with time and money.

**1: Cluster: (ID 24468)**Represent and interpret data.

**1: Cluster: (ID 2447)**Reason with shapes and their attributes

**1: Cluster: (ID 24470)**Extend the counting sequence.

**1: Cluster: (ID 24472)**Understand place value.

**1: Cluster: (ID 24475)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 24479)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 24482)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 24485)**Add and subtract within 20.

**1: Cluster: (ID 24488)**Work with addition and subtraction equations.

**1: Cluster: (ID 2451)**Measure lengths indirectly and by iterating length units

**1: Cluster: (ID 2454)**Work with time and money

**1: Cluster: (ID 2456)**Represent and interpret data

**1: Cluster: (ID 2458)**Extend the counting sequence

**1: Cluster: (ID 2460)**Understand place value

**1: Cluster: (ID 2463)**Use place value understanding and properties of operations to add and subtract

**1: Cluster: (ID 2467)**Represent and solve problems involving addition and subtraction

**1: Cluster: (ID 2470)**Understand and apply properties of operations and the relationship between addition and subtraction

**1: Cluster: (ID 2473)**Add and subtract within 20

**1: Cluster: (ID 2476)**Work with addition and subtraction equations

**1: Cluster: (ID 27)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 30)**Add and subtract within 20.

**1: Cluster: (ID 30044)**Reason with shapes and their attributes

**1: Cluster: (ID 30050)**Measure lengths indirectly and by iterating length units

**1: Cluster: (ID 30055)**Represent and interpret data

**1: Cluster: (ID 30057)**Extend the counting sequence

**1: Cluster: (ID 30059)**Understand place value

**1: Cluster: (ID 30065)**Use place value understanding and properties of operations to add and subtract

**1: Cluster: (ID 30069)**Represent and solve problems involving addition and subtraction within 20

**1: Cluster: (ID 30072)**Understand and apply properties of operations and the relationship between addition and subtraction

**1: Cluster: (ID 30078)**Work with addition and subtraction equations

**1: Cluster: (ID 32379)**Reason with shapes and their attributes.

**1: Cluster: (ID 32383)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 32386)**Tell and write time.

**1: Cluster: (ID 32388)**Represent and interpret data.

**1: Cluster: (ID 32390)**Extend the counting sequence.

**1: Cluster: (ID 32392)**Understand place value.

**1: Cluster: (ID 32398)**Use place value understanding and properties of operations to add and subtract.

**1: Cluster: (ID 32402)**Represent and solve problems involving addition and subtraction.

**1: Cluster: (ID 32405)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: Cluster: (ID 32408)**Add and subtract within 20.

**1: Cluster: (ID 32411)**Work with addition and subtraction equations.

**1: Cluster: (ID 33)**Work with addition and subtraction equations.

**1: Cluster: (ID 35661)**Know ordinal names and counting flexibility.

**1: Cluster: (ID 35665)**Count to tell the number of objects.

**1: Cluster: (ID 35667)**Compare numbers.

**1: Cluster: (ID 35672)**Identify and continue patterns.

**1: Cluster: (ID 39751)**Work with money

**1: Cluster: (ID 41267)**Identify and count money.

**1: Cluster: (ID 44004)**Identify the value of coins

**1: Cluster: (ID 5)**Measure lengths indirectly and by iterating length units.

**1: Cluster: (ID 8)**Tell and write time.

**1: Curricular Indicator: (ID 40381)**No additional indicator(s) at this level. Mastery is expected at previous grade levels.

**1: Curricular Indicator: (ID 40388)**No additional indicator(s) at this level.

**1.d**If the commissioner does not make the determination that instructional materials funding has been made available under subsection (b) of this section, the commissioner shall determine no later than August 31 of each subsequent school year whether instructional materials funding has been made available. If the commissioner determines that instructional materials funding has been made available, the commissioner shall notify the State Board of Education and school districts that §§111.2-111.7 of this subchapter shall be implemented for the following school year.

**1.D.1**Collect, organize, and interpret categorical and numerical data.

**1.D.1.1**Collect, sort, and organize data in up to three categories using representations (e.g., tally marks, tables, Venn diagrams).

**1.D.1.2**Use data to create picture and bar-type graphs to demonstrate one-to-one correspondence.

**1.D.1.3**Draw conclusions from picture and bar-type graphs.

**1.DA.1**Organize and interpret data with up to three choices (What is your favorite fruit? apples, bananas, oranges); ask and answer questions about the total number of data points, how many in each choice, and how many more or less in one choice compared to another.

**1: Domain: (ID 35102)**Operations & Algebraic Thinking

**1: Domain: (ID 35116)**Number & Operations in Base Ten

**1: Domain: (ID 35128)**Measurement & Data

**1: Domain: (ID 35140)**Geometry

**1: Domain: (ID 35392)**Operations and Algebraic Thinking

**1: Domain: (ID 35406)**Number and Operations in Base Ten

**1: Domain: (ID 35418)**Measurement and Data

**1: Domain: (ID 35430)**Geometry

**1: Domain: (ID 35790)**Operations & Algebraic Thinking

**1: Domain: (ID 35804)**Number & Operations in Base Ten

**1: Domain: (ID 35815)**Measurement & Data

**1: Domain: (ID 35827)**Geometry

**1: Domain: (ID 36011)**Operations and Algebraic Thinking

**1: Domain: (ID 36012)**Number and Operations in Base Ten

**1: Domain: (ID 36013)**Measurement and Data

**1: Domain: (ID 36014)**Geometry

**1: Domain: (ID 36504)**Operations & Algebraic Thinking

**1: Domain: (ID 36518)**Number & Operations in Base Ten

**1: Domain: (ID 36530)**Measurement & Data

**1: Domain: (ID 36542)**Geometry

**1: Domain: (ID 36594)**Operations & Algebraic Thinking

**1: Domain: (ID 36608)**Number & Operations in Base Ten

**1: Domain: (ID 36620)**Measurement & Data

**1: Domain: (ID 36632)**Geometry

**1: Domain: (ID 36774)**Operations and Algebraic Thinking

**1: Domain: (ID 36775)**Number and Operations in Base Ten

**1: Domain: (ID 36776)**Measurement and Data

**1: Domain: (ID 36777)**Geometry

**1: Domain: (ID 37894)**Operations & Algebraic Thinking

**1: Domain: (ID 37896)**Number & Operations in Base Ten

**1: Domain: (ID 37897)**Measurement & Data

**1: Domain: (ID 37898)**Geometry

**1: Domain: (ID 37994)**Operations & Algebraic Thinking

**1: Domain: (ID 38008)**Number & Operations in Base Ten

**1: Domain: (ID 38020)**Measurement & Data

**1: Domain: (ID 38032)**Geometry

**1: Domain: (ID 38084)**Operations & Algebraic Thinking

**1: Domain: (ID 38098)**Number & Operations in Base Ten

**1: Domain: (ID 38110)**Measurement & Data

**1: Domain: (ID 38122)**Geometry

**1: Domain: (ID 38571)**Operations & Algebraic Thinking

**1: Domain: (ID 38585)**Number & Operations in Base Ten

**1: Domain: (ID 38597)**Measurement & Data

**1: Domain: (ID 38609)**Geometry

**1: Domain: (ID 38779)**Operations & Algebraic Thinking

**1: Domain: (ID 38793)**Number & Operations in Base Ten

**1: Domain: (ID 38805)**Measurement & Data

**1: Domain: (ID 38817)**Geometry

**1: Domain: (ID 38846)**Operations and Algebraic Thinking

**1: Domain: (ID 38859)**Number and Operations in Base Ten

**1: Domain: (ID 38870)**Measurement and Data

**1: Domain: (ID 38881)**Geometry

**1: Domain: (ID 38984)**Operations & Algebraic Thinking

**1: Domain: (ID 38998)**Number & Operations in Base Ten

**1: Domain: (ID 39010)**Measurement & Data

**1: Domain: (ID 39022)**Geometry

**1: Domain: (ID 39051)**Operations & Algebraic Thinking

**1: Domain: (ID 39064)**Number & Operations in Base Ten

**1: Domain: (ID 39075)**Measurement & Data

**1: Domain: (ID 39086)**Geometry

**1: Domain: (ID 39405)**Operations and Algebraic Thinking

**1: Domain: (ID 39411)**Number and Operations in Base Ten

**1: Domain: (ID 39429)**Measurement and Data

**1: Domain: (ID 39436)**Geometry

**1: Domain: (ID 39482)**Operations & Algebraic Thinking

**1: Domain: (ID 39496)**Number & Operations in Base Ten

**1: Domain: (ID 39508)**Measurement & Data

**1: Domain: (ID 39520)**Geometry

**1: Domain: (ID 40009)**Operations and Algebraic Thinking

**1: Domain: (ID 40010)**Number and Operations in Base Ten

**1: Domain: (ID 40011)**Measurement and Data

**1: Domain: (ID 40012)**Geometry

**1: Domain: (ID 40216)**Number Sense

**1: Domain: (ID 40217)**Number Sense and Operations in Base Ten

**1: Domain: (ID 40218)**Relationships and Algebraic Thinking

**1: Domain: (ID 40219)**Geometry and Measurement

**1: Domain: (ID 40220)**Data and Statistics

**1: Domain: (ID 40289)**Operations & Algebraic Thinking

**1: Domain: (ID 40303)**Number & Operations in Base Ten

**1: Domain: (ID 40315)**Measurement & Data

**1: Domain: (ID 40327)**Geometry

**1: Domain: (ID 40610)**Operations & Algebraic Thinking

**1: Domain: (ID 40624)**Number & Operations in Base Ten

**1: Domain: (ID 40636)**Measurement & Data

**1: Domain: (ID 40648)**Geometry

**1: Domain: (ID 40700)**Operations & Algebraic Thinking

**1: Domain: (ID 40714)**Number & Operations in Base Ten

**1: Domain: (ID 40726)**Measurement & Data

**1: Domain: (ID 40738)**Geometry

**1: Domain: (ID 40780)**Operations & Algebraic Thinking

**1: Domain: (ID 40782)**Number & Operations in Base Ten

**1: Domain: (ID 40783)**Measurement & Data

**1: Domain: (ID 40784)**Geometry

**1: Domain: (ID 40880)**Operations & Algebraic Thinking

**1: Domain: (ID 40894)**Number & Operations in Base Ten

**1: Domain: (ID 40906)**Measurement & Data

**1: Domain: (ID 40918)**Geometry

**1: Domain: (ID 41144)**Measurement and Data

**1: Domain: (ID 41145)**Geometry

**1: Domain: (ID 41146)**Operations and Algebraic Thinking

**1: Domain: (ID 41147)**Number and Operations in Base Ten

**1: Domain: (ID 41312)**Operations and Algebraic Thinking

**1: Domain: (ID 41313)**Number and Operations in Base Ten

**1: Domain: (ID 41314)**Measurement and Data

**1: Domain: (ID 41315)**Geometry

**1: Domain: (ID 41456)**Operations and Algebraic Thinking

**1: Domain: (ID 41457)**Number and Operations in Base 10

**1: Domain: (ID 41458)**Measurement and Data

**1: Domain: (ID 41459)**Geometry

**1: Domain: (ID 41769)**Operations & Algebraic Thinking

**1: Domain: (ID 41783)**Number & Operations in Base Ten

**1: Domain: (ID 41795)**Measurement & Data

**1: Domain: (ID 41807)**Geometry

**1: Domain: (ID 41883)**Numbers & Operations in Base Ten

**1: Domain: (ID 41884)**Operations and Algebraic Thinking

**1: Domain: (ID 41886)**Geometry

**1: Domain: (ID 41888)**Measurement and Data

**1: Domain: (ID 41986)**Operations & Algebraic Thinking

**1: Domain: (ID 42000)**Number & Operations in Base Ten

**1: Domain: (ID 42012)**Measurement & Data

**1: Domain: (ID 42024)**Geometry

**1: Domain: (ID 42350)**Operations & Algebraic Thinking

**1: Domain: (ID 42362)**Number & Operations in Base Ten

**1: Domain: (ID 42373)**Measurement & Data

**1: Domain: (ID 42384)**Geometry

**1: Domain: (ID 42960)**Operations and Algebraic Thinking

**1: Domain: (ID 42961)**Number and Operations in Base Ten

**1: Domain: (ID 42962)**Measurement and Data

**1: Domain: (ID 42963)**Geometry

**1: Domain: (ID 44044)**Operations & Algebraic Thinking

**1: Domain: (ID 44058)**Number & Operations in Base Ten

**1: Domain: (ID 44070)**Measurement & Data

**1: Domain: (ID 44082)**Geometry

**1: Domain: (ID 44300)**Operations & Algebraic Thinking

**1: Domain: (ID 44314)**Number & Operations in Base Ten

**1: Domain: (ID 44326)**Measurement & Data

**1: Domain: (ID 44338)**Geometry

**1: Domain: (ID 44531)**Operations and Algebraic Thinking

**1: Domain: (ID 44532)**Number and Operations in Base Ten

**1: Domain: (ID 44533)**Measurement and Data

**1: Domain: (ID 44534)**Geometry

**1: Domain: (ID 44661)**Operations & Algebraic Thinking

**1: Domain: (ID 44675)**Number & Operations in Base Ten

**1: Domain: (ID 44687)**Measurement & Data

**1: Domain: (ID 44699)**Geometry

**1: Domain: (ID 44751)**Operations & Algebraic Thinking

**1: Domain: (ID 44765)**Number & Operations in Base Ten

**1: Domain: (ID 44777)**Measurement & Data

**1: Domain: (ID 44789)**Geometry

**1.DS.A**Represent and interpret data.

**1.DS.A.1**Collect, organize and represent data with up to three categories.

**1.DS.A.2**Draw conclusions from object graphs, picture graphs, T-charts and tallies.

**1.G**Grade 1 - Geometry

**1.G**Geometry

**1.G**Geometry

**1.G**Geometry

**1.G**Geometry

**1.G.1**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

**1.G.1**Distinguish between defining attributes (*e.g. triangles are closed and three-sided*) versus non-defining attributes (*e.g. color, orientation, overall size*); build and draw shapes that possess defining attributes.

**1.G.1**Distinguish between a two-dimensional shape’s defining (e.g., number of sides) and non-defining attributes (e.g., color).

**1.G.1**Distinguish between defining attributes, e.g., triangles are closed and three-sided, versus non-defining attributes, e.g., color, orientation, overall size; build and draw shapes that possess defining attributes.

**1.G.1**Distinguish between defining attributes versus non-defining attributes.
Use defining attributes to build and draw two-dimensional shapes (squares, circles, triangles, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons).

**1.G.1**Identify objects as two-dimensional or three-dimensional. Classify and sort two-dimensional and three-dimensional objects by shape, size, roundness and other attributes. Describe how two-dimensional shapes make up the faces of three-dimensional objects.

**1.G.1**Distinguish between defining attributes (*for example, triangles are closed and three-sided*) versus non-defining attributes (*for example, color, orientation, overall size*); build and draw shapes that possess defining attributes.

**1.G.1**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes. Identify shapes that have non-defining attributes (e.g., color, orientation, overall size). Build and draw shapes given specified attributes.

**1.G.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter- circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. Students do not need to learn formal names such as “right rectangular prism.”

**1.G.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. Students do not need to learn formal names such as "right rectangular prism."

**1.G.2**Compose (put together) two-dimensional or three-dimensional shapes to create a larger, composite shape, and compose new shapes from the composite shape.

**1.G.2**Distinguish between defining attributes of two- and three-dimensional shapes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size). Create and draw two-dimensional shapes with defining attributes.

**1.G.2**Combine two-dimensional shapes (i.e., square, rectangle, triangle, hexagon, rhombus, and trapezoid) or three-dimensional shapes (i.e., cube, rectangular prism, cone, and cylinder) in more than one way to form a composite shape.

**1.G.2**Compose shapes.

**1.G.2**Compose a new shape or solid from two-dimensional shapes and/or three- dimensional solids (squares, circles, triangles, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons, cubes, spheres, cylinders, cones, triangular prisms, and rectangular prisms).

**1.G.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

**1.G.2.a**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, halfcircles, and quarter-circles) to create a composite shape, and compose new shapes from the composite shape.

**1.G.2.b**Compose three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. First grade students do not need to learn formal names such as "right rectangular prism."

**1.G.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases *half of*, *fourth of*, and *quarter of*. Describe the whole as two of or four of the shares in real-world contexts. Understand for these examples that decomposing into more equal shares creates smaller shares.

**1.G.3**Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of. Describe the whole as two or four of the shares. Understand that, for these examples, decomposing into more equal shares creates smaller shares.

**1.G.3**Partition two-dimensional shapes (i.e., square, rectangle, circle) into two or four equal parts.

**1.G.3**Partition circles and rectangles into two and four equal shares. Describe the shares using the words, halves, fourths, and quarters and phrases half of, fourth of and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that decomposing (break apart) into more equal shares creates smaller shares.

**1.G.3**Use two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

**1.G.3**Partition circles and rectangles into two equal shares.
Describe the shares using the word halves, and use the phrase half of.
Describe the whole as two of the shares.

**1.G.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words *halves, fourths*, and *quarters*, and use the phrases *half of, fourth of*, and *quarter of*. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

**1.G.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words
halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.

**1.G.4**Partition circles and rectangles into two and four equal parts; describe the parts using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of, the parts. Understand for partitioning circles and rectangles into two and four equal parts that decomposing into equal parts creates smaller parts.

**1.G.4**Identify and name two-dimensional shapes (i.e., square, rectangle, triangle, hexagon, rhombus, trapezoid, and circle).

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason about shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A**Reason with shapes and their attributes.

**1.G.A.1**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

**1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1**Distinguish between attributes that define a shape (e.g., number of sides and vertices) versus attributes that do not define the shape (e.g., color, orientation, overall size); build and draw two-dimensional shapes to possess defining attributes.

**1.G.A.1****1.G.A.1**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes.

**1.G.A.1**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes that possess defining attributes.

**1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1**Distinguish between defining attributes (triangles are closed and 3 sided) versus non-defining attributes (color, orientation, overall size) for two-dimensional shapes; build and draw shapes that possess defining attributes.

**1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1****1.G.A.1**Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes that possess defining attributes.

**1.G.A.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

**1.G.A.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

**1.G.A.2**Compose two-dimensional shapes or three-dimensional shapes to create a composite shape.

**1.G.A.2**Create a composite shape and use the composite shape to make new shapes by using two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, rectangular prisms, cones, and cylinders).

**1.G.A.2****1.G.A.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) and three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

**1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.Students do not need to learn formal names such as “right rectangular prism.”

**1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2****1.G.A.2**Compose and Identify regular and irregular two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) and compose three-dimensional shapes (cubes, spheres, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to master formal names such as “right rectangular prism.”)

**1.G.A.2****1.G.A.2****1.G.A.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

**1.G.A.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

**1.G.A.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

**1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words *halves, fourths*, and *quarters*, and use the phrases *half of, fourth of*, and *quarter of*. Describe the whole as *two of*, or *four of* the shares. Understand for these examples that partitioning into more equal shares creates smaller shares.

**1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3****1.G.A.3**Partition circles and rectangles into two and four equal shares, describe the shares using the words

*halves*,*fourths*, and*quarters*, and use the phrases*half of*,*fourth of*, and*quarter of*. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.**1.G.A.3****1.GM.1**Recognize, compose, and decompose two- and three-dimensional shapes.

**1.GM.1.1**Identify trapezoids and hexagons by pointing to the shape when given the name.

**1.GM.1.2**Compose and decompose larger shapes using smaller two-dimensional shapes.

**1.GM.1.3**Compose structures with three-dimensional shapes.

**1.GM.1.4**Recognize three-dimensional shapes such as cubes, cones, cylinders, and spheres.

**1.GM.2**Select and use nonstandard and standard units to describe length and volume/capacity.

**1.GM.2.1**Use nonstandard and standard measuring tools to measure the length of objects to reinforce the continuous nature of linear measurement.

**1.GM.2.2**Illustrate that the length of an object is the number of same-size units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other.

**1.GM.2.3**Measure the same object/distance with units of two different lengths and describe how and why the measurements differ.

**1.GM.2.4**Describe a length to the nearest whole unit using a number and a unit.

**1.GM.2.5**Use standard and nonstandard tools to identify volume/capacity. Compare and sort containers that hold more, less, or the same amount.

**1.GM.3**Tell time to the half and full hour.

**1.GM.3.1**Tell time to the hour and half-hour (analog and digital).

**1.GM.A**Reason with shapes and their attributes.

**1.GM.A.1**Distinguish between defining attributes versus non-defining attributes; build and draw shapes that possess defining attributes.

**1.GM.A.2**Compose and decompose two- and three-dimensional shapes to build an understanding of part-whole relationships and the properties of the original and composite shapes.

**1.GM.A.3**Recognize two- and three-dimensional shapes from different perspectives and orientations.

**1.GM.A.4**Partition circles and rectangles into two or four equal shares, and describe the shares and the wholes verbally.

**1.GM.B**Measure lengths in non-standard units.

**1.GM.B.5**Order three or more objects by length.

**1.GM.B.6**Compare the lengths of two objects indirectly by using a third object.

**1.GM.B.7**Demonstrate the ability to measure length or distance using objects.

**1.GM.C**Work with time and money.

**1.GM.C.8**Tell and write time in hours and half-hours using analog and digital clocks.

**1.GM.C.9**Know the value of a penny, nickel, dime and quarter.

**1: Grade Level: (ID 35115)**Grade 1

**1: Grade Level: (ID 35405)**Grade 1

**1: Grade Level: (ID 35730)**Grade 1

**1: Grade Level: (ID 35803)**Grade 1

**1: Grade Level: (ID 36010)**Grade 1

**1: Grade Level: (ID 36517)**Grade 1

**1: Grade Level: (ID 36607)**Grade 1

**1: Grade Level: (ID 36773)**Grade 1

**1: Grade Level: (ID 37895)**Grade 1

**1: Grade Level: (ID 38007)**Grade 1

**1: Grade Level: (ID 38097)**Grade 1

**1: Grade Level: (ID 38149)**Grade 1

**1: Grade Level: (ID 38584)**Grade 1

**1: Grade Level: (ID 38706)**Grade 1

**1: Grade Level: (ID 38792)**Grade 1

**1: Grade Level: (ID 38858)**Grade 1

**1: Grade Level: (ID 38997)**Grade 1

**1: Grade Level: (ID 39063)**Grade 1

**1: Grade Level: (ID 39391)**Grade 1

**1: Grade Level: (ID 39495)**Grade 1

**1: Grade Level: (ID 39707)**Grade 1

**1: Grade Level: (ID 40008)**Grade 1

**1: Grade Level: (ID 40215)**Grade 1

**1: Grade Level: (ID 40302)**Grade 1

**1: Grade Level: (ID 40567)**Grade 1

**1: Grade Level: (ID 40623)**Grade 1

**1: Grade Level: (ID 40713)**Grade 1

**1: Grade Level: (ID 40781)**Grade 1

**1: Grade Level: (ID 40893)**Grade 1

**1: Grade Level: (ID 40955)**Grade 1

**1: Grade Level: (ID 41143)**Grade 1

**1: Grade Level: (ID 41311)**Grade 1

**1: Grade Level: (ID 41455)**Grade 1

**1: Grade Level: (ID 41708)**Grade 1

**1: Grade Level: (ID 41782)**Grade 1

**1: Grade Level: (ID 41881)**Grade 1

**1: Grade Level: (ID 41999)**Grade 1

**1: Grade Level: (ID 42361)**Grade 1

**1: Grade Level: (ID 42959)**Grade 1

**1: Grade Level: (ID 44003)**Grade 1

**1: Grade Level: (ID 44057)**Grade 1

**1: Grade Level: (ID 44120)**Grade 1

**1: Grade Level: (ID 44313)**Grade 1

**1: Grade Level: (ID 44530)**Grade 1

**1: Grade Level: (ID 44674)**Grade 1

**1: Grade Level: (ID 44764)**Grade 1

**1: : (ID 36961)**Grade 1

**1: : (ID 42112)**Grade 1

**1: : (ID 42113)**Number Sense and Base Ten

**1: : (ID 42114)**Algebraic Thinking and Operations

**1: : (ID 42115)**Geometry

**1: : (ID 42116)**Measurement and Data Analysis

**1: : (ID 7016)**Reason with shapes and their attributes.

**1: : (ID 7020)**Measure lengths indirectly and by iterating length units.

**1: : (ID 7023)**Tell and write time.

**1: : (ID 7025)**Represent and interpret data.

**1: : (ID 7027)**Extend the counting sequence.

**1: : (ID 7029)**Understand place value.

**1: : (ID 7035)**Use place value understanding and properties of operations to add and subtract.

**1: : (ID 7039)**Represent and solve problems involving addition and subtraction.

**1: : (ID 7042)**Understand and apply properties of operations and the relationship between addition and subtraction.

**1: : (ID 7045)**Add and subtract within 20.

**1: : (ID 7050)**Work with addition and subtraction equations.

**1.M.1**Use direct comparison or a nonstandard unit to compare and order objects according to length, area, capacity, weight, and temperature.

**1.M.2**Tell and write time to the nearest half-hour and relate time to events (before/after, shorter/longer) using analog clocks. Understand how to read hours and minutes using digital clocks.

**1.M.3**Find the value of a collection of pennies, nickels, and dimes.

**1.MD**Grade 1 - Measurement and Data

**1.MD**Measurement and Data

**1.MD**Measurement and Data

**1.MD**Measurement and Data

**1.MD**Measurement and Data

**1.MD.1**Order three objects by length; compare the lengths of two objects indirectly by using a third object.

**1.MD.1**Order three objects by length; compare the lengths of two objects indirectly by using a third object.

**1.MD.1****1.MD.1**Measure and compare three objects using standard or non-standard units.

**1.MD.1**Order three objects by length.
Compare the lengths of two objects indirectly by using a third object.

**1.MD.1****1.MD.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. *Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.*

**1.MD.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

**1.MD.2**Demonstrate understanding that the length measurement of an object is the number of same-size length units that span the object with no gaps or overlaps.
Measure and express the length of an object using whole non-standards units.

**1.MD.2**Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. *Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.*

**1.MD.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

**1.MD.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. *Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.*

**1.MD.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.3**Work with time and money.

**1.MD.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.3**Tell and write time to the hour and half-hour (including o’clock and half past) using analog and digital clocks.

**1.MD.3**Tell and write time in half hours using both analog and digital clocks.

**1.MD.3.a**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.3.a**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.3.b**Identify pennies and dimes by name and value.

**1.MD.3.b**Identify the days of the week, the number of days in a week, and the number of weeks in each month.

**1.MD.4**Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

**1.MD.4**Organize, represent, and interpret data with up to three categories.
Ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

**1.MD.4**Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

**1.MD.4**Read a calendar distinguishing yesterday, today and tomorrow. Read and write a date.

**1.MD.4****1.MD.4****1.MD.5**Recognize and read money symbols including $ and ¢.

**1.MD.5**Identify and tell the value of a dollar bill, quarter, dime, nickel, and penny.

**1.MD.5**Identify the values of pennies, nickels, dimes and quarters and know their comparative values. (*For example, a dime is of greater value than a nickel.*) Use appropriate notation to designate a coin’s value. (*For example, 5¢.*)

**1.MD.5.a**Identify the value of all U.S. coins (penny, nickel, dime, quarter, half-dollar, and dollar coins). Use appropriate cent and dollar notation (e.g., 25¢, $1).

**1.MD.5b**Know the comparative values of all U.S. coins (e.g., a dime is of greater value than a nickel).

**1.MD.5.c**Count like U.S. coins up to the equivalent of a dollar.

**1.MD.5.d**Find the equivalent value for all greater value U.S. coins using like value smaller coins (e.g., 5 pennies equal 1 nickel; 10 pennies equal dime, but not 1 nickel and 5 pennies equal 1 dime).

**1.MD.6**Count and tell the value of combinations of dimes and pennies up to one dollar.

**1.MD.6**Identify values of coins (e.g., nickel = 5 cents, quarter = 25 cents). Identify equivalent values of coins up to $1 (e.g., 5 pennies = 1 nickel, 5 nickels = 1 quarter).

**1.MD.7**Organize, represent and interpret data with up to three categories. Ask and answer comparison and quantity questions about the data.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A**Measure lengths indirectly and by iterating length units.

**1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1**Order three objects from a variety of cultural contexts, including those of Montana American Indians, by length and compare the lengths of two objects indirectly by using a third object.

**1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1**Order three objects by length. Compare the lengths of two objects indirectly by using a third object. *For example, to compare indirectly the heights of Bill and Susan: if Bill is taller than mother and mother is taller than Susan, then Bill is taller than Susan.*

**1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MD.A.1****1.MDA.1**Order three objects by length using indirect comparison.

**1.MD.A.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

**1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

**1.MD.A.2**Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

**1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MD.A.2**Measure the length of an object using non-standard units and express this length as a whole number of units.

**1.MD.A.2****1.MD.A.2****1.MD.A.2****1.MDA.2**Use nonstandard physical models to show the length of an object as the number of same size units of length with no gaps or overlaps.

**1.MDA.3**Use analog and digital clocks to tell and record time to the hour and half hour.

**1.MDA.4**Collect, organize, and represent data with up to 3 categories using object graphs, picture graphs, t-charts and tallies.

**1.MDA.5**Draw conclusions from given object graphs, picture graphs, t-charts, tallies, and bar graphs.

**1.MDA.6**Identify a penny, nickel, dime and quarter and write the coin values using a ȼ symbol.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Work with time and money.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Work with time and money.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Work with Time and Money

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B**Tell and write time.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3a**Tell and write time in hours and half-hours using analog and digital clocks.

**1.MD.B.3b**Identify coins by name and value (pennies, nickels, dimes and quarters).

**1.MD.B.4**Count the value of a set of like coins less than one dollar using the ¢ symbol only.

**1.MD.B.5**Identify nickels and understand that five pennies can be thought of as a nickel. Identify dimes and understand ten pennies can be thought of as a dime. Count the value of a set of coins comprised of pennies, nickels, and dimes.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C**Represent and interpret data.

**1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.4****1.MD.C.5**Organize, represent, and interpret data with up to three categories. Ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

**1.MD.D**Work with money.

**1.MD.D**Work with money.

**1.MD.D.5**Identify the values of all U.S. coins and know their comparative values (e.g., a dime is of greater value than a nickel). Find equivalent values (e.g., a nickel is equivalent to five pennies). Use appropriate notation (e.g., 69¢). Use the values of coins in the solutions of problems (up to 100¢).

**1.MD.D.5**Determine the value of a collection of coins up to 50 cents. (Pennies, nickels, dimes, and quarters in isolation; not to include a combination of different coins.)

**1.MP**Mathematical Practices

**1.MP.1**Make sense of problems and persevere in solving them.

**1.MP.2**Reason abstractly and quantitatively.

**1.MP.3**Construct viable arguments and critique the reasoning of others.

**1.MP.4**Model with mathematics.

**1.MP.5**Use appropriate tools strategically.

**1.MP.6**Attend to precision.

**1.MP.7**Look for and make use of structure.

**1.MP.8**Look for and express regularity in repeated reasoning.

**1.N.1**Count, compare, and represent whole numbers up to 100, with an emphasis on groups of tens and ones.

**1.N.1.1**Recognize numbers to 20 without counting (subitize) the quantity of structured arrangements.

**1.N.1.2**Use concrete representations to describe whole numbers between 10 and 100 in terms of tens and ones.

**1.N.1.3**Read, write, discuss, and represent whole numbers up to 100. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

**1.N.1.4**Count forward, with and without objects, from any given number up to 100 by 1s, 2s, 5s and 10s.

**1.N.1.5**Find a number that is 10 more or 10 less than a given number up to 100.

**1.N.1.6**Compare and order whole numbers from 0 to 100.

**1.N.1.7**Use knowledge of number relationships to locate the position of a given whole number on an open number line up to 20.

**1.N.1.8**Use objects to represent and use words to describe the relative size of numbers, such as more than, less than, and equal to.

**1.N.2**Solve addition and subtraction problems up to 10 in real-world and mathematical contexts.

**1.N.2.1**Represent and solve real-world and mathematical problems using addition and subtraction up to ten.

**1.N.2.2**Determine if equations involving addition and subtraction are true.

**1.N.2.3**Demonstrate fluency with basic addition facts and related subtraction facts up to 10.

**1.N.3**Develop foundational ideas for fractions.

**1.N.3.1**Partition a regular polygon using physical models and recognize when those parts are equal.

**1.N.3.2**Partition (fair share) sets of objects into equal groupings.

**1.N.4**Identify coins and their values.

**1.N.4.1**Identifying pennies, nickels, dimes, and quarters by name and value.

**1.N.4.2**Write a number with the cent symbol to describe the value of a coin.

**1.N.4.3**Determine the value of a collection of pennies, nickels, or dimes up to one dollar counting by ones, fives, or tens.

**1.NBT**Numbers and Operations in Base Ten

**1.NBT**Number and Operations in Base Ten

**1.NBT**Number and Operations in Base Ten

**1.NBT**Grade 1 - Number and Operations in Base Ten

**1.NBT**Number and Operations in Base Ten

**1.NBT.1**Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

**1.NBT.1**Count to 120 (recognizing growth and repeating patterns), starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

**1.NBT.1**Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

**1.NBT.1**Count to 120. In this range, read, write and order numerals and represent a number of objects with a written numeral.

**1.NBT.1**Count forward and backward within 120, starting at any given number.
Read and write numerals within 120.
Represent a number of objects up to 120 with a written numeral.

**1.NBT.1**Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

**1.NBT.2**Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

**1.NBT.2**Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: 10 can be thought of as a bundle of ten ones — called a “ten;” the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones; and the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**1.NBT.2**Model and identify place value positions of two digit numbers. Include:

**1.NBT.2**Demonstrate understanding that the two digits of a two-digit number represent amounts of tens and ones, including:

**1.NBT.2**Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

**1.NBT.2****1.NBT.2.a**10 can be thought of as a bundle of ten ones — called a “ten.”

**1.NBT.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.2.a**10 can be thought of as a bundle of ten ones, called a "ten."

**1.NBT.2.a**10 can be thought of as a bundle of ten ones, called a "ten".

**1.NBT.2.a**10 can be thought of as a grouping of ten ones—called a “ten.”

**1.NBT.2.b**The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight or nine ones.

**1.NBT.2.b**The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

**1.NBT.2.b**The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

**1.NBT.2.b**The numbers from 11 to 19 are composed of a ten and additional ones.

**1.NBT.2.b****1.NBT.2.c**The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**1.NBT.2.c**The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90, refer to one, two, three, four, five, six, seven, eight or nine tens (and 0 ones).

**1.NBT.2.c**Multiples of 10 up to 90 represent a number of tens and 0 ones.

**1.NBT.2.c**The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**1.NBT.2.c****1.NBT.2.d**Show flexibility in composing and decomposing tens and ones (*e.g. 20 can be composed from 2 tens or 1 ten and 10 ones, or 20 ones.*)

**1.NBT.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols > , = , and < .

**1.NBT.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and < .

**1.NBT.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <.>

**1.NBT.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the relational symbols > ,

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**1.NBT.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and < .

**1.NBT.4**Add within 100, including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; record the strategy with a written numerical method (drawings and, when appropriate, equations) and explain the reasoning used. Understand that when adding two-digit numbers, tens are added to tens; ones are added to ones; and sometimes it is necessary to compose a ten.

**1.NBT.4**Add using numbers up to 100 including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10. Use:

- concrete models or drawings and strategies based on place value
- properties of operations
- and/or relationship between addition and subtraction.

**1.NBT.4**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens to tens and ones to ones, and that it is sometimes necessary to compose a ten.

**1.NBT.4**Demonstrate understanding of place value when adding two-digit numbers within 100.

**1.NBT.4**Add within 100 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used including:

**1.NBT.4**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**1.NBT.4.a**Add a two-digit number and a one-digit number.

**1.NBT.4.a**Adding a two-digit number and a one-digit number

**1.NBT.4.b**Add a two-digit number and a multiple of 10.
Use concrete models or drawing strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to add and subtract within 100.
Relate the strategy to a written method and explain the reasoning used.

**1.NBT.4.b**Adding a two-digit number and a multiple of 10

**1.NBT.4.c**Understanding that when adding two-digit numbers, combine like base-ten units such as tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**1.NBT.5**Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

**1.NBT.5**Mentally add or subtract 10 to or from a given two-digit number. Explain the reasoning used.

**1.NBT.5**Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

**1.NBT.5****1.NBT.5****1.NBT.5****1.NBT.6**Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.6**Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to subtract multiples of 10 in the range of 10-90 from multiples of 10 in the same range resulting in a positive or zero difference.
Use a written method to explain the strategy.

**1.NBT.6**Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.6**Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.6**Subtract multiples of 10 in the range 10 to 90 from multiples of 10 in the range 10 to 90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.6**Subtract multiples of 10 up to 100. Use:

- concrete models or drawings
- strategies based on place value
- properties of operations
- and/or the relationship between addition and subtraction.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A**Understand place value of two-digit numbers.

**1.NBT.A**Extend the counting sequence.

**1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1**Count to 120 by 1's, 2's, and 10's starting at any number less than 100. In this range, read and write numerals and represent a number of objects with a written numeral.

**1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1**In the range of 0 - 120

**1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1****1.NBT.A.1**Count to 120, starting at any number. Read and write numerals to 120 and represent a number of objects with a written numeral. Count backward from 20.

**1.NBT.A.1****1.NBT.A.1**Understand that 10 can be thought of as a bundle of 10 ones – called a “ten”.

**1.NBT.A.1****1.NBT.A.1****1.NBT.A.1.a**Count on from any given number.

**1.NBT.A.1.b**Read and write numerals.

**1.NBT.A.1.c**Represent a number of objects with a written numeral.

**1.NBT.A.2**Understand two-digit numbers are composed of ten(s) and one(s).

**1.NBT.A.3**Compare two two-digit numbers using the symbols >, = or < .

**1.NBT.A.4**Count by 10s to 120 starting at any number.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Use place value understanding to add and subtract.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B**Understand place value.

**1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2**Know that the digits of a two-digit number represent groups of tens and ones (e.g., 39 can be represented as 39 ones, 2 tens and 19 ones, or 3 tens and 9 ones).

**1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2****1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones–-called a “ten.”

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones—called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2a**10 can be thought of as a bundle of ten ones — called a "ten."

**1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2.b****1.NBT.B.2b****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2.c****1.NBT.B.2c****1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and < .

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on the meanings of the digits in each place and use the symbols >, =, and < to show the relationship.

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols \(>\), \(=\), and \(<\).

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

**1.NBT.B.3**Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.>

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Understand subtraction as an unknown-addend problem.

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.*

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples.

Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

**1.NBT.B.5**Add within 100.

**1.NBT.B.6**Calculate 10 more or 10 less than a given number mentally without having to count.

**1.NBT.B.7**Add or subtract a multiple of 10 from another two-digit number, and justify the solution.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C**Use place value understanding and properties of operations to add and subtract.

**1.NBT.C.4**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings, and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**1.NBT.C.4****1.NBT.C.4****1.NBT.C.4**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10.

**1.NBT.C.4****1.NBT.C.4**Add and subtract within 100.

**1.NBT.C.4****1.NBT.C.4**Demonstrate understanding of addition within 100, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form. See Table 1.

**1.NBT.C.4****1.NBT.C.4**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models (e.g., base ten blocks) or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4****1.NBT.C.4**Add a two-digit number to a one-digit number and a two-digit number to a multiple of ten (within 100). Use concrete models, drawings, strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to explain the reasoning used.

**1.NBT.C.4.a**Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a number sentence; justify the reasoning used with a written explanation.

**1.NBT.C.4.a**Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.C.4.b**Understand that in adding two-digit numbers (sums within 100) add tens and tens, ones and ones; and
sometimes it is necessary to compose a ten.

**1.NBT.C.4.b**Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

**1.NBT.C.5**Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. Identify arithmetic patterns of 10 more and 10 less than using strategies based on place value.

**1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5**Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.

**1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5**Mentally find 10 more or 10 less than a given two-digit number without having to count by ones and explain the reasoning used.

**1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.5****1.NBT.C.6**Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6**Subtract multiples of 10 from multiples of 10 in the range 10-90 using concrete models, drawings, strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

**1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6**Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

**1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6****1.NBT.C.6**Subtract multiples of 10 in the range of 10 to 90 (positive or zero differences), using objects or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form.

**1.NBT.C.6****1.NBT.C.6****1.NS.1**Count to at least 120 by ones, fives, and tens from any given number. In this range, read and write numerals and represent a number of objects with a written numeral.

**1.NS.2**Understand that 10 can be thought of as a group of ten ones — called a “ten." Understand that the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. Understand that the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

**1.NS.3**Match the ordinal numbers first, second, third, etc., with an ordered set up to 10 items.

**1.NS.4**Use place value understanding to compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols > , = , and < .

**1.NS.5**Find mentally 10 more or 10 less than a given two-digit number without having to count, and explain the thinking process used to get the answer.

**1.NS.6**Show equivalent forms of whole numbers as groups of tens and ones, and understand that the individual digits of a two-digit number represent amounts of tens and ones.

**1.NS.A**Understand and use numbers up to 120.

**1.NS.A.1**Count to 120, starting at any number less than 120.

**1.NS.A.2**Read and write numerals and represent a number of objects with a written numeral.

**1.NS.A.3**Count backward from a given number between 20 and 1.

**1.NS.A.4**Count by 5s to 100 starting at any multiple of five.

**1.NSBT.1**Extend the number sequence to:

**1.NSBT.1.a**count forward by ones to 120 starting at any number;

**1.NSBT.1.b**count by fives and tens to 100, starting at any number;

**1.NSBT.1.c**read, write and represent numbers to 100 using concrete models, standard form, and equations in expanded form;

**1.NSBT.1.d**read and write in word form numbers zero through nineteen, and multiples of ten through ninety.

**1.NSBT.2**Understand place value through 99 by demonstrating that:

**1.NSBT.2.a**ten ones can be thought of as a bundle (group) called a “ten”;

**1.NSBT.2.b**the tens digit in a two-digit number represents the number of tens and the ones digit represents the number of ones;

**1.NSBT.2.c**two-digit numbers can be decomposed in a variety of ways (e.g., 52 can be decomposed as 5 tens and 2 ones or 4 tens and 12 ones, etc.) and record the decomposition as an equation.

**1.NSBT.3**Compare two two-digit numbers based on the meanings of the tens and ones digits, using the words *greater than, equal to, or less than*.

**1.NSBT.4**Add through 99 using concrete models, drawings, and strategies based on place value to:

**1.NSBT.4.a**add a two-digit number and a one-digit number, understanding that sometimes it is necessary to compose a ten (regroup);

**1.NSBT.4.b**add a two-digit number and a multiple of 10.

**1.NSBT.5**Determine the number that is 10 more or 10 less than a given number through 99 and explain the reasoning verbally and with multiple representations, including concrete models.

**1.NSBT.6**Subtract a multiple of 10 from a larger multiple of 10, both in the range 10 to 90, using concrete models, drawings, and strategies based on place value.

**1.OA**Operations and Algebraic Thinking

**1.OA**Operations and Algebraic Thinking

**1.OA**Operations and Algebraic Thinking

**1.OA**Grade 1 - Operations and Algebraic Thinking

**1.OA**Operations and Algebraic Thinking

**1.OA.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, (*e.g. by using objects, drawings, and situation equations and/or solution equations with a symbol for the unknown number to represent the problem.*)

**1.OA.1**Use addition and subtraction strategies to solve word problems (using numbers up to 20), involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions, using a number line (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.

**1.OA.1**Use strategies to add and subtract within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

**1.OA.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. *For example, use objects, drawings, and equations with a symbol for the unknown number to represent the problem.*

**1.OA.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, (*e.g. by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.*)

**1.OA.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20. *For example, use objects, drawings, and equations with a symbol for the unknown number to represent the problem.*

**1.OA.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20

**1.OA.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.

**1.OA.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Drawings need not show details, but should show the mathematics in the problem.

**1.OA.3**Apply (not necessary to name) properties of operations as strategies to add and subtract.* Examples: 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) To add 0 to any number, the answer is that number 7 + 0 = 7 (Additive identity property of 0). Students need not use formal terms for these properties. *

**1.OA.3**Apply properties of operations as strategies to add and subtract. *For example, if 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (Commutative Property of Addition); to add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (Associative Property of Addition).* Students need not use formal terms for these properties.

**1.OA.3**Apply properties of operations as strategies to add and subtract. (Students need not know the name of the property.) *For example: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (Commutative property of addition). To add 2 + 6 + 4, the second two numbers can be added to make a ten, so
2 + 6 + 4 = 2 + 10 = 12 (Associative property of addition). Demonstrate that when adding zero to any number, the quantity does not change (Identity property of addition).*

**1.OA.3**Apply properties of operations as strategies to add and subtract.* For example: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 =12. (Associative property of addition.)* First grade students need not use formal terms for these properties.

**1.OA.3**Apply properties of operations as strategies to add and subtract. *Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)*

**1.OA.3**Apply properties of operations as strategies to add and subtract.

**1.OA.4**Demonstrate understanding of subtraction as an unknown-addend problem.

**1.OA.4**Understand subtraction as an unknown-addend problem. *For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.4**Understand subtraction as an unknown-addend problem. *For example, subtract 10 − 8 by finding the number that makes 10 when added to 8.*

**1.OA.4**Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

**1.OA.4**Understand subtraction as an unknown-addend problem. *For example, subtract 10 − 8 by finding the number that makes 10 when added to 8.*

**1.OA.4**Understand subtraction as an unknown-addend problem. *For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.*

**1.OA.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.5**Relate counting to addition and subtraction, e.g., by counting on 2 to add 2.

**1.OA.5**Relate counting to addition and subtraction. *For example, by counting on 2 to add 2.*

**1.OA.5**Relate counting to addition and subtraction.

**1.OA.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.5**Relate counting to addition and subtraction (*e.g. by counting on 2 to add 2, counting back 1 to subtract 1*).

**1.OA.6**Add and subtract within 20.

**1.OA.6**Add and subtract using numbers up to 20, demonstrating fluency for addition and subtraction up to 10. Use strategies such as

- counting on
- making ten (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14)
- decomposing a number leading to a ten (13 - 4 = 13 - 3 - 1 = 10 - 1 = 9)
- using the relationship between addition and subtraction, such as fact families, (8 + 4 = 12 and 12 - 8 = 4)
- creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.6**Add and subtract within 20, demonstrating fluency (efficiently, accurately, and flexibly) for addition and subtraction within 10. Use mental strategies such as counting on; making ten (*e.g. 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g. 13 − 4 = 13 − 3 − 1 = 10 − 1 = 9*); using the relationship between addition and subtraction (*e.g. knowing that 8 + 4 = 12, one knows 12 − 8 = 4*); and creating equivalent but easier or known sums (*e.g. adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13*).

**1.OA.6**Add and subtract within 20, demonstrating fluency with various strategies for addition and subtraction within 10. Strategies may include counting on; making ten, e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; decomposing a number leading to a ten, e.g., 13 − 4 = 13 − 3 − 1 = 10 − 1 = 9; using the relationship between addition and subtraction, e.g., knowing that 8 + 4 = 12, one knows 12 − 8 = 4; and creating equivalent but easier or known sums, e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13.

**1.OA.6**Use strategies to add and subtract within 20.
Fluently add and subtract within 10.

**1.OA.6.a**Use strategies such as counting on; making ten (*for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14*); decomposing a number leading to a ten (*for example, 13 - 4 = 13 – 3 – 1 = 10 – 1 = 9*); using the relationship between addition and subtraction (*for example, knowing that 8 + 4 = 12, one knows 12 – 8 = 4*); and creating equivalent but easier or known sums (*for example, adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13*).

**1.OA.6.b**By the end of Grade 1, demonstrate fluency for addition and subtraction within 10.

**1.OA.7**Understand the meaning of the equal sign, and determine whether equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

**1.OA.7**Understand the meaning of the equal sign (the value is the same on both sides of the equal sign), and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6; 7 = 8 − 1; 5 + 2 = 2 + 5; 4 + 1 = 3 + 2; 7 − 1 = 4; 5 + 4 = 7 − 2.*

**1.OA.7**Understand the meaning of the equal sign (e.g., read equal sign as “same as”) and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2).*

**1.OA.7**Demonstrate understanding of the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

**1.OA.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

**1.OA.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6; 7 = 8 – 1; 5 + 2 = 2 + 5; 4 + 1 = 5 + 2.*

**1.OA.8**Determine the unknown whole number in an addition or subtraction equation. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 6 + 6 = ?, 5 = ? - 3.*

**1.OA.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?*

**1.OA.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations: 8 + □ = 11; 5 = □ − 3; 6 + 6 = □.

**1.OA.8**Using related equations, Determine the unknown whole number in an addition or subtraction equation. *For example, determine the unknown number that makes the equation true in each of the equations ∎ − 3 = 7; 7 + 3 = ∎.*

**1.OA.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = D – 3, 6 + 6 = D.*

**1.OA.8**Determine the unknown whole number in an addition or subtraction equation that uses three whole numbers.

**1.OA.9**Identify, continue and label patterns (e.g., aabb, abab). Create patterns using number, shape, size, rhythm or color.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A**Represent and solve problems involving addition and subtraction.

**1.OA.A.1**Use addition and subtraction within 20 to solve word problems within a cultural context, including those of Montana American Indians, involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations (number sentences) with a symbol for the unknown number to represent the problem.

**1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1**Use addition and subtraction within 20 to solve word problems with unknowns in all positions (e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem). See Table 1.

**1.OA.A.1****1.OA.A.1****1.OA.A.1**Add and subtract within 20 to solve contextual problems, with unknowns in all positions, involving situations of *add to, take from, put together/take apart*, and *compare*. Use objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.1****1.OA.A.1****1.OA.A.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.See Glossary, Table 1.

**1.OA.A.1****1.OA.A.1**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, (e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem).

**1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.1****1.OA.A.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.2**Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2**Solve word problems within a cultural context, including those of Montana American Indians, that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2**Add three whole numbers whose sum is within 20 to solve contextual problems using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.A.2****1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply commutative, associative, and additive identity properties of operations as strategies to add. (Students need not use formal terms for these properties.) Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 8 + 0 = 8 (Additive Identity property)

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations to add. *For example, when adding numbers order does not matter. If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (Commutative property of addition). To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (Associative property of addition). When adding zero to a number, the result is the same number (Identity property of zero for addition).*

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations (commutative and associative properties of addition) as strategies to add and subtract within 20. (Students need not use formal terms for these properties.)

**1.OA.B.3**Apply properties of operations to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations (additive identity, commutative, and associative) as strategies to add and subtract. (Students need not use formal terms for these properties.)

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.Students need not use formal terms for these properties. Examples: If \(8 + 3 = 11\) is known, then \(3 + 8 = 11\) is also known. (Commutative property of addition.) To add \(2 + 6 + 4\), the second two numbers can be added to make a ten, so \(2 + 6 + 4 = 2 + 10 = 12\). (Associative property of addition.)

**1.OA.B.3**Apply properties of operations as strategies to add and subtract.

**1.OA.B.4**Understand subtraction as an unknown-addend problem.

*For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.***1.OA.B.4**Understand subtraction as an unknown-addend problem.

*For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.***1.OA.B.4**Understand subtraction as an unknown-addend problem. For example, subtract \(10 - 8\) by finding the number that makes 10 when added to 8.

**1.OA.B.4**Understand subtraction as an unknown-addend problem.

*For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.***1.OA.B.4**Understand subtraction as an unknown-addend problem. *For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4**Understand subtraction as an unknown-addend problem. *For example, to solve 10 – 8 = , a student can use 8 + = 10.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4**Understand subtraction as an unknown-addend problem. *For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4**Understand subtraction as an unknown-addend problem within 20 (e.g., subtract 10 - 8 by finding the number that makes 10 when added to 8).

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.B.4***For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.*

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C**Add and subtract within 20.

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Understand counting on as addition and counting back as subtraction e.g. 5, (6,7,8) means 5 + 3 and 5, (4,3,2) means 5-3.

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Add and subtract within 20 using strategies such as counting on, counting back, making 10, using fact families and related known facts, and composing/decomposing numbers with an emphasis on making ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9 or adding 6 + 7 by creating the known equivalent 6 + 4 + 3 = 10 + 3 = 13).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.5**Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

**1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use mental strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.C.6****1.OA.C.6****1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., \(8 + 6 = 8 + 2 + 4 = 10 + 4 = 14\)); decomposing a number leading to a ten (e.g., \(13 - 4 = 13 - 3 - 1 = 10 - 1 = 9\)); using the relationship between addition and subtraction (e.g., knowing that \(8 + 4 = 12\), one knows \(12 - 8 = 4\)); and creating equivalent but easier or known sums (e.g., adding \(6 + 7\) by creating the known equivalent \(6 + 6 + 1 = 12 + 1 = 13\)).

**1.OA.C.6****1.OA.C.6**Fluently add and subtract within 10.

**1.OA.C.6**Fluently add and subtract within 20 using mental strategies. By the end of 1st grade, know from memory all sums up to 10.

**1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.C.6**Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use mental strategies such as counting on; making 10 (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a 10 (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

**1.OA.C.6****1.OA.C.6****1.OA.C.6****1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D**Work with addition and subtraction equations.

**1.OA.D.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

*For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.***1.OA.D.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

*For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.***1.OA.D.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

*For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.***1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

**1.OA.D.7****1.OA.D.7**Understand the meaning of the equal sign (e.g., 6 = 6; 5 + 2 = 4 + 3; 7 = 8 - 1). Determine if equations involving addition and subtraction are true or false.

**1.OA.D.7****1.OA.D.7****1.OA.D.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. *For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.*

**1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7**Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? \(6 = 6\), \(7 = 8 - 1\), \(5 + 2 = 2 + 5\), \(4 + 1 = 5 + 2\).

**1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.7****1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation, with the unknown in any position (e.g., 8 + ? = 11, 5 = ? - 3, 6 + 6 = ?).

**1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

*For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.***1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

*For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.***1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

*For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.***1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = - 3, 6 + 6 = .*

**1.OA.D.8****1.OA.D.8****1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations \(8 + ? = 11\), \(5 = \boxvoid - 3\), \(6 + 6 = \boxvoid\).

**1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. *For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = - 3, 6 + 6 = .*

**1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.OA.D.8****1.RA.A**Represent and solve problems involving addition and subtraction.

**1.RA.A.1**Use addition and subtraction within 20 to solve problems.

**1.RA.A.2**Solve problems that call for addition of three whole numbers whose sum is within 20.

**1.RA.A.3**Develop the meaning of the equal sign and determine if equations involving addition and subtraction are true or false.

**1.RA.A.4**Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

**1.RA.B**Understand and apply properties of operations and the relationship between addition and subtraction.

**1.RA.B.5**Use properties as strategies to add and subtract.

**1.RA.B.6**Demonstrate that subtraction can be solved as an unknown-addend problem.

**1.RA.C**Add and subtract within 20.

**1.RA.C.7**Add and subtract within 20.

**1.RA.C.8**Demonstrate fluency with addition and subtraction within 10.

**1.SMP**Grade 1 - Standards for Mathematical Practice

**1.SMP.1**Make sense of problems and persevere in solving them.

**1.SMP.2**Reason abstractly and quantitatively.

**1.SMP.3**Construct viable arguments and critique the reasoning of others.

**1.SMP.4**Model with mathematics.

**1.SMP.5**Use appropriate tools strategically.

**1.SMP.6**Attend to precision.

**1.SMP.7**Look for and make use of structure.

**1.SMP.8**Look for and express regularity in repeated reasoning.

**1: Standard: (ID 39548)**Count, compare and represent whole numbers up to 120, with an emphasis on groups of tens and ones.

**1: Standard: (ID 39550)**Use a variety of models and strategies to solve addition and subtraction problems in real-world and mathematical contexts.

**1: Standard: (ID 39588)**Recognize and create patterns; use rules to describe patterns.

**1: Standard: (ID 39590)**Use number sentences involving addition and subtraction basic facts to represent and solve real-world and mathematical problems; create real-world situations corresponding to number sentences.

**1: Standard: (ID 39635)**Describe characteristics of basic shapes. Use basic shapes to compose and decompose other objects in various contexts.

**1: Standard: (ID 39636)**Use basic concepts of measurement in real-world and mathematical situations involving length, time and money.

**1: Strand: (ID 38169)**Number Sense

**1: Strand: (ID 38171)**Computation and Algebraic Thinking

**1: Strand: (ID 38174)**Geometry

**1: Strand: (ID 38176)**Measurement

**1: Strand: (ID 38178)**Data Analysis

**1: Strand: (ID 39708)**Number & Operation

**1: Strand: (ID 39709)**Algebra

**1: Strand: (ID 39710)**Geometry & Measurement

**1: Strand: (ID 41709)**Number & Operations (N)

**1: Strand: (ID 41710)**Algebraic Reasoning & Algebra (A)

**1: Strand: (ID 41711)**Geometry & Measurement (GM)

**1: Strand: (ID 41712)**Data & Probability (D)

**1: Strand: (ID 44124)**Number and Number Sense

**1: Strand: (ID 44127)**Computation and Estimation

**1: Strand: (ID 44128)**Measurement and Geometry

**1: Strand: (ID 44129)**Probability and Statistics

**1: Strand: (ID 44130)**Patterns, Functions, and Algebra

**2**Grade 2

**2**Kindergarten, Adopted 2012.

**2.**Evaluate and apply formulas for arithmetic and geometric sequences and series.

**2.**Estimate numerical derivatives from graphs or tables of data.

**2.**Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

**2.**Rewrite expressions involving radicals and rational exponents using the properties of exponents.

**2.**Reason abstractly and quantitatively.

**2.**Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

*For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.***2.**Solve application-based problems by developing and solving systems of linear equations and inequalities.

**2.**Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).

*For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.***2.**Models and represents mathematical problems.

**2.**Understand the concept of a unit rate a/b associated with a ratio a:b with b â‰ 0, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."

**2.**Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

**2.**Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

**2.****2.**Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

**2.**Determine characteristics of sequences, including the Fibonacci sequence, the triangular numbers, and pentagonal numbers. *Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula to find the nth triangular number.*

**2.**Create expressions that can be modeled by a real-world context.

**2.**(+) Solve problems involving velocity and other quantities that can be represented by vectors, including navigation (e.g., airplane, aerospace, oceanic).

**2.**Analyze mathematical relationships in music to interpret frequencies of musical notes and to compare mathematical structures of various musical instruments. *Examples: Compare frequencies of notes exactly one octave apart on the musical scale; using frequencies and wave patterns of middle C, E above middle C, and G above middle C to explain why the C major chord is harmonious.*

**2.**Use the relation *i*² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

**2.**Recognize and represent proportional relationships between quantities.

**2.**Use the relation *i*² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

**2.**(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. *Example: (-1 + √3 i)³ = 8 because (-1 + √3 i) has modulus 2 and argument 120°.*

**2.**Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

*For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.***2.0**Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

**2.0**Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

**20.**Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

*For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.***20.**Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

*For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.***20.**Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

**20.**Determine the inverse of a function and a relation.

**20.****20.**Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

**20.**Find the possible rational roots using the Rational Root Theorem.

**20.**Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

**20.**Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length).

**20.**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**20.**Solve optimization problems.

**20.**Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

**20.**Compare the properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. (Limited to linear and quadratic functions only.)

**20.**Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

*For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.***20.**Recognize area as an attribute of plane figures and understand concepts of area measurement.

**20.**Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

**20.**Explain and use the relationship between the sine and cosine of complementary angles.

**20.0**Students compute the integrals of trigonometric functions by using the techniques noted above.

**20a.**A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.

**20a.**A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.

**20b.**A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

**20b.**A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

**2.1**The student will

- read, write, and identify the place and value of each digit in a three-digit numeral, with and without models;
- identify the number that is 10 more, 10 less, 100 more, and 100 less than a given number up to 999;
- compare and order whole numbers between 0 and 999; and
- round two-digit numbers to the nearest ten.

**21.**Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

**21.**Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. *Example: Find the points of intersection between the line y = –3x and the circle x² + y² = 3.*

**21.****21.**Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

**21.**Explain a proof of the Pythagorean Theorem and its converse.

**21.**Describe the following characteristics of linear and quadratic parent functions by inspection: domain/range, increasing/decreasing intervals, intercepts, symmetry, and asymptotic behavior. Identify each characteristic in set notation or words, where appropriate.

**21.**Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

**21.**Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

**21.**Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

**21.**Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

**21.**Find the zeros of polynomial functions by synthetic division and the Factor Theorem.

**21.**State and apply the First and Second Fundamental Theorem of Calculus.

**21.**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

**21.**Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

**21.**Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

*For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.***21.**(+) Verify by composition that one function is the inverse of another.

**21.**Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

**2.10**The student will

- determine past and future days of the week; and
- identify specific days and dates on a given calendar.

**21.0**Students understand the algorithms involved in Simpson’s rule and Newton’s method. They use calculators or computers or both to approximate integrals numerically.

**2.11**The student will read temperature to the nearest 10 degrees.

**2.1.1.1**Read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

**2.1.1.2**Use place value to describe whole numbers between 10 and 1000 in terms of hundreds, tens and ones. Know that 100 is 10 tens, and 1000 is 10 hundreds.
*For example*: Writing 853 is a shorter way of writing 8 hundreds + 5 tens + 3 ones.

**2.1.1.3**Find 10 more or 10 less than a given three-digit number. Find 100 more or 100 less than a given three-digit number.
*For example*: Find the number that is 10 less than 382 and the number that is 100 more than 382.

**2.1.1.4**Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.
*For example*: If there are 17 students in the class and granola bars come 10 to a box, you need to buy 20 bars (2 boxes) in order to have enough bars for everyone.

**2.1.1.5**Compare and order whole numbers up to 1000.

**2.12**The student will

- draw a line of symmetry in a figure; and
- identify and create figures with at least one line of symmetry.

**2.1.2.1**Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the commutative and associative properties. Use the relationship between addition and subtraction to generate basic facts.
*For example*: Use the associative property to make tens when adding
5 + 8 = (3 + 2) + 8 = 3 + (2 + 8) = 3 + 10 = 13.

**2.1.2.2**Demonstrate fluency with basic addition facts and related subtraction facts.

**2.1.2.3**Estimate sums and differences up to 100.
*For example*: Know that 23 + 48 is about 70.

**2.1.2.4**Use mental strategies and algorithms based on knowledge of place value and equality to add and subtract two-digit numbers. Strategies may include decomposition, expanded notation, and partial sums and differences.
*For example*: Using decomposition, 78 + 42, can be thought of as:
78 + 2 + 20 + 20 = 80 + 20 + 20 = 100 + 20 = 120
and using expanded notation, 34 - 21 can be thought of as:
30 + 4 – 20 – 1 = 30 – 20 + 4 – 1 = 10 + 3 = 13.

**2.1.2.5**Solve real-world and mathematical addition and subtraction problems involving whole numbers with up to 2 digits.

**2.1.2.6**Use addition and subtraction to create and obtain information from tables, bar graphs and tally charts.

**2.13**The student will identify, describe, compare, and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms).

**2.14**The student will use data from probability experiments to predict outcomes when the experiment is repeated.

**2.15**The student will

- collect, organize, and represent data in pictographs and bar graphs; and
- read and interpret data represented in pictographs and bar graphs.

**2.16**The student will identify, describe, create, extend, and transfer patterns found in objects, pictures, and numbers.

**2.17**The student will demonstrate an understanding of equality through the use of the equal symbol and the use of the not equal symbol.

**2.2**The student will

- count forward by twos, fives, and tens to 120, starting at various multiples of 2, 5, or 10;
- count backward by tens from 120; and
- use objects to determine whether a number is even or odd.

**22.**Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.*Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods.*

**22.**Compose simple shapes to form larger shapes.*Example: "Can you join these two triangles with full sides touching to make a rectangle?"*

**22.**(+) Prove the Law of Sines and the Law of Cosines and use them to solve problems.

**22.**Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

**22.**Relate area to the operations of multiplication and addition.

**22.**Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.

*For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.***22.**(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

**22.**Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

**22.**Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

**22.**Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

*For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.***22.**Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

**22.**Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

**22.**Graph a system of two functions, f(x) and g(x), on the same Coordinate Plane by hand for simple cases, and with technology for complicated cases. Explain the relationship between the point(s) of intersection and the solution to the system. Determine the solution(s) using technology, a tables of values, substitution, or successive approximations. (Limited to linear and quadratic functions only.)

**22.**Graph and solve quadratic inequalities.

**22.**Apply the power rule and u-substitution to evaluate indefinite integrals.

**22.**Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.*Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods.*

**22.0**Students understand improper integrals as limits of definite integrals.

**2.2.1.1**Identify, create and describe simple number patterns involving repeated addition or subtraction, skip counting and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts.
*For example*: Skip count by 5s beginning at 3 to create the pattern
3, 8, 13, 18, … .
*Another example*: Collecting 7 empty milk cartons each day for 5 days will generate the pattern 7, 14, 21, 28, 35, resulting in a total of 35 milk cartons.

**2.2.2.1**Understand how to interpret number sentences involving addition, subtraction and unknowns represented by letters. Use objects and number lines and create real-world situations to represent number sentences.
*For example*: One way to represent n + 16 = 19 is by comparing a stack of 16 connecting cubes to a stack of 19 connecting cubes; 24 = a + b can be represented by a situation involving a birthday party attended by a total of 24 boys and girls.

**2.2.2.2**Use number sentences involving addition, subtraction, and unknowns to represent given problem situations. Use number sense and properties of addition and subtraction to find values for the unknowns that make the number sentences true.
*For example*: How many more players are needed if a soccer team requires 11 players and so far only 6 players have arrived? This situation can be represented by the number sentence 11 – 6 = p or by the number sentence 6 + p = 11.

**22a.**Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

**22a.**Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

**22b.**Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

**22b.**Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

**22c.**Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

**22c.**Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

**22d.**Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

**2.3**The student will

- count and identify the ordinal positions first through twentieth, using an ordered set of objects; and
- write the ordinal numbers 1st through 20th.

**23.**Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

**23.**Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

**23.**Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

**23.**Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. *Example: Rearrange Ohm’s law V = IR to highlight resistance R.*

**23.**Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

**23.**(+) Produce an invertible function from a non-invertible function by restricting the domain.

**23.**Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

**23.**Explain why the x-coordinates of the points where the graphs of the equations *y = f(x)* and *y = g(x)* intersect are the solutions of the equation *f(x) = g(x)*; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f(x)* and/or *g(x)* are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

**23.**Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. *Example: Rearrange Ohm’s law V = IR to highlight resistance R.*

**23.**Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

**23.**With accuracy, graph the solutions to a linear inequality in two variables as a half-plane, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes on the same Coordinate Plane. Construct graphs of linear inequalities and systems of linear inequalities without technology. Use appropriate strategies to verify points that may or may not belong to the solution set.

**23.**Demonstrate and explain the differences between average and instantaneous rates of change.

**23.**Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

**23.**(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

**23.**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

**23.0**Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.

**2.3.1.1**Describe, compare, and classify two- and three-dimensional figures according to number and shape of faces, and the number of sides, edges and vertices (corners).

**2.3.1.2**Identify and name basic two- and three-dimensional shapes, such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, rectangular prisms, cones, cylinders and spheres.
*For example*: Use a drawing program to show several ways that a rectangle can be decomposed into exactly three triangles.

**2.3.2.1**Understand the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.
*For example*: It will take more paper clips than whiteboard markers to measure the length of a table.

**2.3.2.2**Demonstrate an understanding of the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest centimeter or inch.
*For example*: Draw a line segment that is 3 inches long.

**2.3.3.1**Tell time to the quarter-hour and distinguish between a.m. and p.m.

**2.3.3.2**Identify pennies, nickels, dimes and quarters. Find the value of a group of coins and determine combinations of coins that equal a given amount.
*For example*: 50 cents can be made up of 2 quarters, or 4 dimes and 2 nickels, or many other combinations.

**23a.**An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.

**23a.**Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

*For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.***23b.**An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

**23b.**Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

*For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?***2.4**The student will

- name and write fractions represented by a set, region, or length model for halves, fourths, eighths, thirds, and sixths;
- represent fractional parts with models and with symbols; and
- compare the unit fractions for halves, fourths, eighths, thirds, and sixths, with models.

**24.**Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

**24.**Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

**24.**Identify real-world contexts that can be modeled by a system of inequalities in two variables. (Limited to three inequalities.)

**24.**Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

**24.**Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

**24.**Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

**24.**Prove that all circles are similar.

**24.**Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

**24.**Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

**24.**Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

**24.**(+) Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.

**24.**Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

**24.**Apply differentiation techniques to curve sketching

**24.**Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

**24.**Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

**24.0**Students understand and can compute the radius (interval) of the convergence of power series.

**24a.**Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

**24b.**Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

**24c.**Design and use a simulation to generate frequencies for compound events.

*For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?***2.5**The student will

- recognize and use the relationships between addition and subtraction to solve single-step practical problems, with whole numbers to 20; and
- demonstrate fluency with addition and subtraction within 20.

**25**Implementation of Texas Essential Knowledge and Skills for Mathematics, Middle School, Adopted 2012.

**25.**Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

**25.**Apply Rolle’s Theorem and the Mean Value Theorem and their geometric consequences.

**25.**Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

**25.**Identify and describe relationships among inscribed angles, radii, and chords. *Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.*

**25.**Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

*For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.***25.**Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If *f* is a function and *x* is an element of its domain, then *f(x)* denotes the output of *f* corresponding to the input *x*. The graph of *f* is the graph of the equation *y = f(x)*.

**25.**Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

*For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.***25.**Compose functions. *For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.*

**25.**Identify when systems of equations and inequalities have constraints.

**25.**Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

*For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.***25.**Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

**25.**Compare effects of parameter changes on graphs of transcendental functions. *Example: Explain the relationship of the graph y = e

^{x-2}to the graph y = e^{x}.***25.**Recognize when the quadratic formula gives complex solutions, and write them as *a ± bi* for real numbers a and *b*.

**25.**Recognize when the quadratic formula gives complex solutions, and write them as *a ± bi* for real numbers a and *b*.

**25.0**Students differentiate and integrate the terms of a power series in order to form new series from known ones.

**25.a**The provisions of §§111.26-111.28 of this subchapter shall be implemented by school districts.

**25.b**No later than August 31, 2013, the commissioner of education shall determine whether instructional materials funding has been made available to Texas public schools for materials that cover the essential knowledge and skills for mathematics as adopted in §§111.26-111.28 of this subchapter.

**25.c**If the commissioner makes the determination that instructional materials funding has been made available under subsection (b) of this section, §§111.26-111.28 of this subchapter shall be implemented beginning with the 2014-2015 school year and apply to the 2014-2015 and subsequent school years.

**25.d**If the commissioner does not make the determination that instructional materials funding has been made available under subsection (b) of this section, the commissioner shall determine no later than August 31 of each subsequent school year whether instructional materials funding has been made available. If the commissioner determines that instructional materials funding has been made available, the commissioner shall notify the State Board of Education and school districts that §§111.26-111.28 of this subchapter shall be implemented for the following school year.

**2.6**The student will

- estimate sums and differences;
- determine sums and differences, using various methods; and
- create and solve single-step and two-step practical problems involving addition and subtraction.

**26**Grade 6, Adopted 2012.

**26.**Verify by composition that one function is the inverse of another.

**26.**Perform simple translations on linear functions given in a variety of forms (e.g., two points, a graph, a table of values, a mapping, slope-intercept form, or standard form). Explain the impact on the parent function when the slope is greater than one or less than one and the effect of increasing/decreasing the y-intercept.

**26.**Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

**26.**Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses.

**26.**Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

**26.**Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

**26.**Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

**26.**Classify two-dimensional figures in a hierarchy based on properties.

**26.**Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

**26.**Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

**26.**Identify and apply local linear approximations.

**26.**(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

**26.**(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

**26.0**Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.

**26.a**Introduction.

**26.a.1**The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

**26.a.2**The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**26.a.3**The primary focal areas in Grade 6 are number and operations; proportionality; expressions, equations, and relationships; and measurement and data. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use concepts of proportionality to explore, develop, and communicate mathematical relationships. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other. Students connect verbal, numeric, graphic, and symbolic representations of relationships, including equations and inequalities. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, and reasoning to draw conclusions, evaluate arguments, and make recommendations. While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.

**26.a.4**Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples.

**26.b**Knowledge and skills.

**26.b.1**Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

**26.b.10**Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to solve problems. The student is expected to:

**26.b.10.A**model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts; and

**26.b.10.B**determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

**26.b.11**Measurement and data. The student applies mathematical process standards to use coordinate geometry to identify locations on a plane. The student is expected to graph points in all four quadrants using ordered pairs of rational numbers.

**26.b.12**Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to analyze problems. The student is expected to:

**26.b.12.A**represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots;

**26.b.12.B**use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution;

**26.b.12.C**summarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution; and

**26.b.12.D**summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution.

**26.b.13**Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to solve problems. The student is expected to:

**26.b.13.A**interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots; and

**26.b.13.B**distinguish between situations that yield data with and without variability.

**26.b.14**Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:

**26.b.14.A**compare the features and costs of a checking account and a debit card offered by different local financial institutions;

**26.b.14.B**distinguish between debit cards and credit cards;

**26.b.14.C**balance a check register that includes deposits, withdrawals, and transfers;

**26.b.14.D**explain why it is important to establish a positive credit history;

**26.b.14.E**describe the information in a credit report and how long it is retained;

**26.b.14.F**describe the value of credit reports to borrowers and to lenders;

**26.b.14.G**explain various methods to pay for college, including through savings, grants, scholarships, student loans, and work-study; and

**26.b.14.H**compare the annual salary of several occupations requiring various levels of post-secondary education or vocational training and calculate the effects of the different annual salaries on lifetime income.

**26.b.1.A**apply mathematics to problems arising in everyday life, society, and the workplace;

**26.b.1.B**use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

**26.b.1.C**select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

**26.b.1.D**communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

**26.b.1.E**create and use representations to organize, record, and communicate mathematical ideas;

**26.b.1.F**analyze mathematical relationships to connect and communicate mathematical ideas; and

**26.b.1.G**display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**26.b.2**Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to:

**26.b.2.A**classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers;

**26.b.2.B**identify a number, its opposite, and its absolute value;

**26.b.2.C**locate, compare, and order integers and rational numbers using a number line;

**26.b.2.D**order a set of rational numbers arising from mathematical and real-world contexts; and

**26.b.2.E**extend representations for division to include fraction notation such as $a/b$ represents the same number as $a ÷ b$ where b ≠ 0.

**26.b.3**Number and operations. The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. The student is expected to:

**26.b.3.A**recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values;

**26.b.3.B**determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one;

**26.b.3.C**represent integer operations with concrete models and connect the actions with the models to standardized algorithms;

**26.b.3.D**add, subtract, multiply, and divide integers fluently; and

**26.b.3.E**multiply and divide positive rational numbers fluently.

**26.b.4**Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:

**26.b.4.A**compare two rules verbally, numerically, graphically, and symbolically in the form of $y = ax$ or$ y = x + a$ in order to differentiate between additive and multiplicative relationships;

**26.b.4.B**apply qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates;

**26.b.4.C**give examples of ratios as multiplicative comparisons of two quantities describing the same attribute;

**26.b.4.D**give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients;

**26.b.4.E**represent ratios and percents with concrete models, fractions, and decimals;

**26.b.4.F**represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers;

**26.b.4.G**generate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money; and

**26.b.4.H**convert units within a measurement system, including the use of proportions and unit rates.

**26.b.5**Proportionality. The student applies mathematical process standards to solve problems involving proportional relationships. The student is expected to:

**26.b.5.A**represent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions;

**26.b.5.B**solve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models; and

**26.b.5.C**use equivalent fractions, decimals, and percents to show equal parts of the same whole.

**26.b.6**Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to:

**26.b.6.A**identify independent and dependent quantities from tables and graphs;

**26.b.6.B**write an equation that represents the relationship between independent and dependent quantities from a table; and

**26.b.6.C**represent a given situation using verbal descriptions, tables, graphs, and equations in the form $y = kx$ or $y = x + b$.

**26.b.7**Expressions, equations, and relationships. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:

**26.b.7.A**generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;

**26.b.7.B**distinguish between expressions and equations verbally, numerically, and algebraically;

**26.b.7.C**determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations; and

**26.b.7.D**generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

**26.b.8**Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to represent relationships and solve problems. The student is expected to:

**26.b.8.A**extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle;

**26.b.8.B**model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes;

**26.b.8.C**write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers; and

**26.b.8.D**determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.

**26.b.9**Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to represent situations. The student is expected to:

**26.b.9.A**write one-variable, one-step equations and inequalities to represent constraints or conditions within problems;

**26.b.9.B**represent solutions for one-variable, one-step equations and inequalities on number lines; and

**26.b.9.C**write corresponding real-world problems given one-variable, one-step equations or inequalities.

**2.7**The student will

- count and compare a collection of pennies, nickels, dimes, and quarters whose total value is $2.00 or less; and
- use the cent symbol, dollar symbol, and decimal point to write a value of money.

**27**Grade 7, Adopted 2012.

**27.**Explain why the x-coordinates of the points where the graphs of the equations *y = f(x)* and *y = g(x)* intersect are the solutions of the equation* f(x) = g(x)*; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f(x)* and/or *g(x)* are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

**27.**Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

**27.**Explain why the x-coordinates of the points where the graphs of the equations *y = f(x)* and *y = g(x)* intersect are the solutions of the equation* f(x) = g(x)*; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f(x)* and/or *g(x)* are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

**27.**(+) Construct a tangent line from a point outside a given circle to the circle.

**27.**Use the sum, difference, and half-angle identities to find the exact value of a trigonometric function.

**27.**Analyze curves with attention to non-decreasing functions (monotonicity) and concavity.

**27.**Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

*For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.***27.**Given the graph of function in the form f(x)+ k, kf(x), f(kx), or f(x + k), where k belongs to the set of integers, identify the domain/range, increasing/decreasing intervals, intercepts, symmetry, and asymptotic behavior, where appropriate. Identify each characteristic in set notation or as an inequality, where appropriate. (Limited to linear and quadratic functions only.)

**27.**Read values of an inverse function from a graph or a table, given that the function has an inverse.

**27.**Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

**27.**Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. *Example: The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥1.*

**27.0**Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.

**27.a**Introduction.

**27.a.1**The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

**27.a.2**The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**27.a.3**The primary focal areas in Grade 7 are number and operations; proportionality; expressions, equations, and relationships; and measurement and data. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use concepts of proportionality to explore, develop, and communicate mathematical relationships, including number, geometry and measurement, and statistics and probability. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other. Students connect verbal, numeric, graphic, and symbolic representations of relationships, including equations and inequalities. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, and reasoning to draw conclusions, evaluate arguments, and make recommendations. While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.

**27.a.4**Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples.

**27.b**Knowledge and skills.

**27.b.1**Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

**27.b.10**Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations and inequalities to represent situations. The student is expected to:

**27.b.10.A**write one-variable, two-step equations and inequalities to represent constraints or conditions within problems;

**27.b.10.B**represent solutions for one-variable, two-step equations and inequalities on number lines; and

**27.b.10.C**write a corresponding real-world problem given a one-variable, two-step equation or inequality.

**27.b.11**Expressions, equations, and relationships. The student applies mathematical process standards to solve one-variable equations and inequalities. The student is expected to:

**27.b.11.A**model and solve one-variable, two-step equations and inequalities;

**27.b.11.B**determine if the given value(s) make(s) one-variable, two-step equations and inequalities true; and

**27.b.11.C**write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.

**27.b.12**Measurement and data. The student applies mathematical process standards to use statistical representations to analyze data. The student is expected to:

**27.b.12.A**compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads;

**27.b.12.B**use data from a random sample to make inferences about a population; and

**27.b.12.C**compare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations.

**27.b.13**Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:

**27.b.13.A**calculate the sales tax for a given purchase and calculate income tax for earned wages;

**27.b.13.B**identify the components of a personal budget, including income; planned savings for college, retirement, and emergencies; taxes; and fixed and variable expenses, and calculate what percentage each category comprises of the total budget;

**27.b.13.C**create and organize a financial assets and liabilities record and construct a net worth statement;

**27.b.13.D**use a family budget estimator to determine the minimum household budget and average hourly wage needed for a family to meet its basic needs in the student's city or another large city nearby;

**27.b.13.E**calculate and compare simple interest and compound interest earnings; and

**27.b.13.F**analyze and compare monetary incentives, including sales, rebates, and coupons.

**27.b.1.A**apply mathematics to problems arising in everyday life, society, and the workplace;

**27.b.1.B**use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

**27.b.1.C**select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

**27.b.1.D**communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

**27.b.1.E**create and use representations to organize, record, and communicate mathematical ideas;

**27.b.1.F**analyze mathematical relationships to connect and communicate mathematical ideas; and

**27.b.1.G**display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**27.b.2**Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers.

**27.b.3**Number and operations. The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. The student is expected to:

**27.b.3.A**add, subtract, multiply, and divide rational numbers fluently; and

**27.b.3.B**apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.

**27.b.4**Proportionality. The student applies mathematical process standards to represent and solve problems involving proportional relationships. The student is expected to:

**27.b.4.A**represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including $d = rt$;

**27.b.4.B**calculate unit rates from rates in mathematical and real-world problems;

**27.b.4.C**determine the constant of proportionality $(k = y/x)$ within mathematical and real-world problems;

**27.b.4.D**solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems; and

**27.b.4.E**convert between measurement systems, including the use of proportions and the use of unit rates.

**27.b.5**Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to:

**27.b.5.A**generalize the critical attributes of similarity, including ratios within and between similar shapes;

**27.b.5.B**describe π as the ratio of the circumference of a circle to its diameter; and

**27.b.5.C**solve mathematical and real-world problems involving similar shape and scale drawings.

**27.b.6**Proportionality. The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships. The student is expected to:

**27.b.6.A**represent sample spaces for simple and compound events using lists and tree diagrams;

**27.b.6.B**select and use different simulations to represent simple and compound events with and without technology;

**27.b.6.C**make predictions and determine solutions using experimental data for simple and compound events;

**27.b.6.D**make predictions and determine solutions using theoretical probability for simple and compound events;

**27.b.6.E**find the probabilities of a simple event and its complement and describe the relationship between the two;

**27.b.6.F**use data from a random sample to make inferences about a population;

**27.b.6.G**solve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents;

**27.b.6.H**solve problems using qualitative and quantitative predictions and comparisons from simple experiments; and

**27.b.6.I**determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces.

**27.b.7**Expressions, equations, and relationships. The student applies mathematical process standards to represent linear relationships using multiple representations. The student is expected to represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form $y = mx + b$.

**27.b.8**Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume. The student is expected to:

**27.b.8.A**model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas;

**27.b.8.B**explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas; and

**27.b.8.C**use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas.

**27.b.9**Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems. The student is expected to:

**27.b.9.A**solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids;

**27.b.9.B**determine the circumference and area of circles;

**27.b.9.C**determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles; and

**27.b.9.D**solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net.

**2.8**The student will estimate and measure

- length to the nearest inch; and
- weight to the nearest pound.

**28**Grade 8, Adopted 2012.

**28.**Produce an invertible function from a non-invertible function by restricting the domain.

**28.**Apply integration to solve problems of area.

**28.**Identify and graph real-world contexts that can be modeled by a quadratic equation.

**28.**Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

*For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?***28.**Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

**28.**Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

**28.**Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. *Example: Graph x² – 6x + y² – 12y + 41 = 0 or y² – 4x + 2y + 5 = 0.*

**28.**Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. *Example: Graph x² – 6x + y² – 12y + 41 = 0 or y² – 4x + 2y + 5 = 0.*

**28.**Utilize parametric equations by graphing and by converting to rectangular form.

**28.**For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. *Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

**28.**Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

**28.a**Introduction.

**28a.**Formulate equations of conic sections from their determining characteristics. *Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4.*

**28a.**Formulate equations of conic sections from their determining characteristics. *Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4.*

**28a.**Solve application-based problems involving parametric equations.

**28.a.1**The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

**28.a.2**The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**28.a.3**The primary focal areas in Grade 8 are proportionality; expressions, equations, relationships, and foundations of functions; and measurement and data. Students use concepts, algorithms, and properties of real numbers to explore mathematical relationships and to describe increasingly complex situations. Students use concepts of proportionality to explore, develop, and communicate mathematical relationships. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other. Students connect verbal, numeric, graphic, and symbolic representations of relationships, including equations and inequalities. Students begin to develop an understanding of functional relationships. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, and reasoning to draw conclusions, evaluate arguments, and make recommendations. While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.

**28.a.4**Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples.

**28.b**Knowledge and skills.

**28b.**Solve applied problems that include sequences with recurrence relations.

**28.b.1**Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

**28.b.10**Two-dimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts. The student is expected to:

**28.b.10.A**generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two-dimensional shapes on a coordinate plane;

**28.b.10.B**differentiate between transformations that preserve congruence and those that do not;

**28.b.10.C**explain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation; and

**28.b.10.D**model the effect on linear and area measurements of dilated two-dimensional shapes.

**28.b.11**Measurement and data. The student applies mathematical process standards to use statistical procedures to describe data. The student is expected to:

**28.b.11.A**construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data;

**28.b.11.B**determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points; and

**28.b.11.C**simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected.

**28.b.12**Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:

**28.b.12.A**solve real-world problems comparing how interest rate and loan length affect the cost of credit;

**28.b.12.B**calculate the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest and over different periods using an online calculator;

**28.b.12.C**explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time;

**28.b.12.D**calculate and compare simple interest and compound interest earnings;

**28.b.12.E**identify and explain the advantages and disadvantages of different payment methods;

**28.b.12.F**analyze situations to determine if they represent financially responsible decisions and identify the benefits of financial responsibility and the costs of financial irresponsibility; and

**28.b.12.G**estimate the cost of a two-year and four-year college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college.

**28.b.1.A**apply mathematics to problems arising in everyday life, society, and the workplace;

**28.b.1.B**use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

**28.b.1.C**select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

**28.b.1.D**communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

**28.b.1.E**create and use representations to organize, record, and communicate mathematical ideas;

**28.b.1.F**analyze mathematical relationships to connect and communicate mathematical ideas; and

**28.b.1.G**display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**28.b.2**Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to:

**28.b.2.A**extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers;

**28.b.2.B**approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line;

**28.b.2.C**convert between standard decimal notation and scientific notation; and

**28.b.2.D**order a set of real numbers arising from mathematical and real-world contexts.

**28.b.3**Proportionality. The student applies mathematical process standards to use proportional relationships to describe dilations. The student is expected to:

**28.b.3.A**generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation;

**28.b.3.B**compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane; and

**28.b.3.C**use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.

**28.b.4**Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:

**28.b.4.A**use similar right triangles to develop an understanding that slope, $m$, given as the rate comparing the change in $y$-values to the change in $x$-values, $(y_2 - y_1)/(x_2 - x_1)$, is the same for any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the same line;

**28.b.4.B**graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship; and

**28.b.4.C**use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

**28.b.5**Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. The student is expected to:

**28.b.5.A**represent linear proportional situations with tables, graphs, and equations in the form of $y = kx$;

**28.b.5.B**represent linear non-proportional situations with tables, graphs, and equations in the form of $y = mx + b$, where$ b \not = 0$;

**28.b.5.C**contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation;

**28.b.5.D**use a trend line that approximates the linear relationship between bivariate sets of data to make predictions;

**28.b.5.E**solve problems involving direct variation;

**28.b.5.F**distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form $y = kx$ or $y = mx + b$, where $b \not = 0$;

**28.b.5.G**identify functions using sets of ordered pairs, tables, mappings, and graphs;

**28.b.5.H**identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems; and

**28.b.5.I**write an equation in the form $y = mx + b$ to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

**28.b.6**Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to:

**28.b.6.A**describe the volume formula $V = Bh$ of a cylinder in terms of its base area and its height;

**28.b.6.B**model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas; and

**28.b.6.C**use models and diagrams to explain the Pythagorean theorem.

**28.b.7**Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to:

**28.b.7.A**solve problems involving the volume of cylinders, cones, and spheres;

**28.b.7.B**use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders;

**28.b.7.C**use the Pythagorean Theorem and its converse to solve problems; and

**28.b.7.D**determine the distance between two points on a coordinate plane using the Pythagorean Theorem.

**28.b.8**Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations or inequalities in problem situations. The student is expected to:

**28.b.8.A**write one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants;

**28.b.8.B**write a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants;

**28.b.8.C**model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants; and

**28.b.8.D**use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

**28.b.9**Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of $x$ and y that simultaneously satisfy two linear equations in the form $y = mx + b$ from the intersections of the graphed equations.

**2.9**The student will tell time and write time to the nearest five minutes, using analog and digital clocks.

**29.**Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *Example, If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

**29.**Summarize numerical data sets in relation to their context, such as by:

**29.**Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

**29.**Solve quadratic equations in standard form by factoring, graphing, tables, and the Quadratic Formula. Know when the Quadratic Formula might yield complex solutions and the location of the solutions in relationship to the x-axis. Know suitable alternatives for the terminology “solution of a quadratic” and when each is appropriate to use.

**29.**Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

**29.**(+) Use special triangles to determine geometrically the values of sine, cosine, and tangent π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.

**29.**Utilize integrals to model and find solutions to real-world problems such as calculating displacement and total distance traveled.

**29.**Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *Example: If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

**29.**Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. *Example, If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

**29a.**Reporting the number of observations.

**29b.**Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

**29c.**Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

**29d.**Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

**2.a**Introduction.

**2a.**Determine lengths of strings necessary to produce harmonic tones as in Pythagorean tuning.

**2a.**Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

**2.a.1****2.A.1**Describe the relationship found in patterns to solve real-world and mathematical problems.

**2.A.1.1**Represent, create, describe, complete, and extend growing and shrinking patterns with quantity and numbers in a variety of real-world and mathematical contexts.

**2.A.1.2**Represent and describe repeating patterns involving shapes in a variety of contexts.

**2.a.2****2.A.2**Use number sentences involving unknowns to represent and solve real-world and mathematical problems.

**2.A.2.1**Use objects and number lines to represent number sentences.

**2.A.2.2**Generate real-world situations to represent number sentences and vice versa.

**2.A.2.3**Apply commutative and identity properties and number sense to find values for unknowns that make number sentences involving addition and subtraction true or false.

**2.a.3**For students to become fluent in mathematics, students must develop a robust sense of number. The National Research Council's report, "Adding It Up," defines procedural fluency as "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately." As students develop procedural fluency, they must also realize that true problem solving may take time, effort, and perseverance. Students in Kindergarten are expected to perform their work without the use of calculators.

**2.a.4**The primary focal areas in Kindergarten are understanding counting and cardinality, understanding addition as joining and subtraction as separating, and comparing objects by measurable attributes.

**2.a.4.A**Students develop number and operations through several fundamental concepts. Students know number names and the counting sequence. Counting and cardinality lay a solid foundation for number. Students apply the principles of counting to make the connection between numbers and quantities.

**2.a.4.B**Students use meanings of numbers to create strategies for solving problems and responding to practical situations involving addition and subtraction.

**2.a.4.C**Students identify characteristics of objects that can be measured and directly compare objects according to these measurable attributes.

**2.a.5****2.ATO.1**Solve one- and two-step real-world/story problems using addition (as a joining action and as a part-part-whole action) and subtraction (as a separation action, finding parts of the whole, and as a comparison) through 99 with unknowns in all positions.

**2.ATO.2**Demonstrate fluency with addition and related subtraction facts through 20.

**2.ATO.3**Determine whether a number through 20 is odd or even using pairings of objects, counting by twos, or finding two equal addends to represent the number (e.g., 3 + 3 = 6).

**2.ATO.4**Use repeated addition to find the total number of objects arranged in a rectangular array with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

**2.b**Knowledge and skills.

**2b.**Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

**2.b.1****2.b.1.A**apply mathematics to problems arising in everyday life, society, and the workplace;

**2.b.1.B****2.b.1.C****2.b.1.D****2.b.1.E**create and use representations to organize, record, and communicate mathematical ideas;

**2.b.1.F**analyze mathematical relationships to connect and communicate mathematical ideas; and

**2.b.1.G****2.b.2**Number and operations. The student applies mathematical process standards to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system. The student is expected to:

**2.b.2.A**count forward and backward to at least 20 with and without objects;

**2.b.2.B**read, write, and represent whole numbers from 0 to at least 20 with and without objects or pictures;

**2.b.2.C**count a set of objects up to at least 20 and demonstrate that the last number said tells the number of objects in the set regardless of their arrangement or order;

**2.b.2.D**recognize instantly the quantity of a small group of objects in organized and random arrangements;

**2.b.2.E**generate a set using concrete and pictorial models that represents a number that is more than, less than, and equal to a given number up to 20;

**2.b.2.F**generate a number that is one more than or one less than another number up to at least 20;

**2.b.2.G**compare sets of objects up to at least 20 in each set using comparative language;

**2.b.2.H**use comparative language to describe two numbers up to 20 presented as written numerals; and

**2.b.2.I**compose and decompose numbers up to 10 with objects and pictures.

**2.b.3**Number and operations. The student applies mathematical process standards to develop an understanding of addition and subtraction situations in order to solve problems. The student is expected to:

**2.b.3.A**model the action of joining to represent addition and the action of separating to represent subtraction;

**2.b.3.B**solve word problems using objects and drawings to find sums up to 10 and differences within 10; and

**2.b.3.C**explain the strategies used to solve problems involving adding and subtracting within 10 using spoken words, concrete and pictorial models, and number sentences.

**2.b.4**Number and operations. The student applies mathematical process standards to identify coins in order to recognize the need for monetary transactions. The student is expected to identify U.S. coins by name, including pennies, nickels, dimes, and quarters.

**2.b.5**Algebraic reasoning. The student applies mathematical process standards to identify the pattern in the number word list. The student is expected to recite numbers up to at least 100 by ones and tens beginning with any given number.

**2.b.6**Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties. The student is expected to:

**2.b.6.A**identify two-dimensional shapes, including circles, triangles, rectangles, and squares as special rectangles;

**2.b.6.B**identify three-dimensional solids, including cylinders, cones, spheres, and cubes, in the real world;

**2.b.6.C**identify two-dimensional components of three-dimensional objects;

**2.b.6.D**identify attributes of two-dimensional shapes using informal and formal geometric language interchangeably;

**2.b.6.E**classify and sort a variety of regular and irregular two- and three-dimensional figures regardless of orientation or size; and

**2.b.6.F**create two-dimensional shapes using a variety of materials and drawings.

**2.b.7**Geometry and measurement. The student applies mathematical process standards to directly compare measurable attributes. The student is expected to:

**2.b.7.A**give an example of a measurable attribute of a given object, including length, capacity, and weight; and

**2.b.7.B**compare two objects with a common measurable attribute to see which object has more of/less of the attribute and describe the difference.

**2.b.8**Data analysis. The student applies mathematical process standards to collect and organize data to make it useful for interpreting information. The student is expected to:

**2.b.8.A**collect, sort, and organize data into two or three categories;

**2.b.8.B**use data to create real-object and picture graphs; and

**2.b.8.C**draw conclusions from real-object and picture graphs.

**2.b.9**Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security. The student is expected to:

**2.b.9.A**identify ways to earn income;

**2.b.9.B**differentiate between money received as income and money received as gifts;

**2.b.9.C**list simple skills required for jobs; and

**2.b.9.D**distinguish between wants and needs and identify income as a source to meet one's wants and needs.

**2c.**Represent proportional relationships by equations.

*For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.***2.CA.1**Add and subtract fluently within 100.

**2.CA.2**Solve real-world problems involving addition and subtraction within 100 in situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all parts of the addition or subtraction problem (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). Use estimation to decide whether answers are reasonable in addition problems.

**2.CA.3**Solve real-world problems involving addition and subtraction within 100 in situations involving lengths that are given in the same units (e.g., by using drawings, such as drawings of rulers, and equations with a symbol for the unknown number to represent the problem).

**2.CA.4**Add and subtract within 1000, using models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; describe the strategy and explain the reasoning used. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones, and that sometimes it is necessary to compose or decompose tens or hundreds.

**2.CA.5**Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal groups.

**2.CA.6**Show that the order in which two numbers are added (commutative property) and how the numbers are grouped in addition (associative property) will not change the sum. These properties can be used to show that numbers can be added in any order.

**2.CA.7**Create, extend, and give an appropriate rule for number patterns using addition and subtraction within 1000.

**2: Cluster: (ID 1097)**Reason with shapes and their attributes.

**2: Cluster: (ID 1101)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 1106)**Relate addition and subtraction to length.

**2: Cluster: (ID 1109)**Work with time and money.

**2: Cluster: (ID 1112)**Represent and interpret data.

**2: Cluster: (ID 1115)**Understand place value.

**2: Cluster: (ID 1122)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 1128)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 1130)**Add and subtract using numbers up to 20.

**2: Cluster: (ID 1132)**Work with equal groups of objects to gain foundations for multiplication.

**2: Cluster: (ID 11545)**Reason with shapes and their attributes

**2: Cluster: (ID 11549)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 11554)**Relate addition and subtraction to length.

**2: Cluster: (ID 11557)**Work with time and money.

**2: Cluster: (ID 11560)**Represent and interpret data.

**2: Cluster: (ID 11563)**Understand place value.

**2: Cluster: (ID 11570)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 11576)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 11578)**Add and subtract within 20.

**2: Cluster: (ID 11580)**Work with equal groups of objects to gain foundations for multiplication.

**2: Cluster: (ID 17029)**Reason with shapes and their attributes

**2: Cluster: (ID 17033)**Measure and estimate lengths in standard units

**2: Cluster: (ID 17038)**Relate addition and subtraction to length

**2: Cluster: (ID 17041)**Work with time with respect to a clock and a calendar, and work with money

**2: Cluster: (ID 17044)**Represent and interpret data

**2: Cluster: (ID 17047)**Understand place value

**2: Cluster: (ID 17054)**Use place value understanding and properties of operations to add and subtract

**2: Cluster: (ID 17060)**Represent and solve problems involving addition and subtraction

**2: Cluster: (ID 17062)**Add and subtract within 20

**2: Cluster: (ID 17064)**Work with equal groups of objects to gain foundations for multiplication

**2: Cluster: (ID 22582)**Reason with shapes and their attributes.

**2: Cluster: (ID 22586)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 22591)**Relate addition and subtraction to length.

**2: Cluster: (ID 22594)**Work with time and money.

**2: Cluster: (ID 22597)**Represent and interpret data.

**2: Cluster: (ID 22600)**Understand place value.

**2: Cluster: (ID 22607)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 22614)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 22617)**Add and subtract within 20.

**2: Cluster: (ID 22620)**Work with equal groups of objects to gain foundations for multiplication.

**2: Cluster: (ID 23298)**Reason with shapes and their attributes.

**2: Cluster: (ID 23301)**Measure and estimate lengths.

**2: Cluster: (ID 23306)**Relate addition and subtraction to length.

**2: Cluster: (ID 23309)**Build understanding of time and money.

**2: Cluster: (ID 23312)**Represent and interpret data.

**2: Cluster: (ID 23314)**Understand place value.

**2: Cluster: (ID 23319)**Use place value understanding and properties of operations.

**2: Cluster: (ID 23324)**Represent and solve problems.

**2: Cluster: (ID 23326)**Add and subtract within 20.

**2: Cluster: (ID 23328)**Work with equal groups.

**2: Cluster: (ID 23843)**Reason with shapes and their attributes (squares, circles, triangles, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons, parallelograms, quadrilaterals, cubes, spheres, cylinders, cones, triangular prisms, and rectangular prisms).

**2: Cluster: (ID 23847)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 23852)**Relate addition and subtraction to equal intervals on a number line.

**2: Cluster: (ID 23854)**Work with time and money.

**2: Cluster: (ID 23857)**Represent and interpret data.

**2: Cluster: (ID 23860)**Understand place value.

**2: Cluster: (ID 23867)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 23872)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 23874)**Add and subtract within 20.

**2: Cluster: (ID 23876)**Work with equal groups of objects to gain foundations for multiplication.

**2: Cluster: (ID 24491)**Reason with shapes and their attributes.

**2: Cluster: (ID 24495)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 24500)**Relate addition and subtraction to length.

**2: Cluster: (ID 24503)**Work with time and money.

**2: Cluster: (ID 24506)**Represent and interpret data.

**2: Cluster: (ID 24509)**Understand place value.

**2: Cluster: (ID 24516)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 24522)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 24524)**Add and subtract within 20.

**2: Cluster: (ID 24526)**Work with equal groups of objects to gain foundations for multiplication.

**2: Cluster: (ID 2479)**Reason with shapes and their attributes

**2: Cluster: (ID 2484)**Measure and estimate lengths in standard units

**2: Cluster: (ID 2489)**Relate addition and subtraction to length

**2: Cluster: (ID 2492)**Work with time and money

**2: Cluster: (ID 2495)**Represent and interpret data

**2: Cluster: (ID 2498)**Understand place value.

**2: Cluster: (ID 2503)**Use place value understanding and properties of operations to add and subtract

**2: Cluster: (ID 2509)**Represent and solve problems involving addition and subtraction

**2: Cluster: (ID 2511)**Add and subtract within 20

**2: Cluster: (ID 2513)**Work with equal groups of objects to gain foundations for multiplication

**2: Cluster: (ID 30081)**Reason with shapes and their attributes.

**2: Cluster: (ID 30085)**Measure and estimate lengths in standard units

**2: Cluster: (ID 30090)**Relate addition and subtraction to length

**2: Cluster: (ID 30093)**Work with time and money

**2: Cluster: (ID 30096)**Represent and interpret data

**2: Cluster: (ID 30099)**Understand place value

**2: Cluster: (ID 30106)**Use place value understanding and properties of operations to add and subtract

**2: Cluster: (ID 30112)**Represent and solve problems involving addition and subtraction

**2: Cluster: (ID 30114)**Fluently add and subtract within 20

**2: Cluster: (ID 30118)**Work with equal groups of objects to gain foundations for multiplication

**2: Cluster: (ID 32414)**Reason with shapes and their attributes.

**2: Cluster: (ID 32418)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 32423)**Relate addition and subtraction to length.

**2: Cluster: (ID 32426)**Work with time and money.

**2: Cluster: (ID 32429)**Represent and interpret data.

**2: Cluster: (ID 32432)**Understand place value.

**2: Cluster: (ID 32439)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 32445)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 32447)**Add and subtract within 20.

**2: Cluster: (ID 32449)**Work with equal groups of objects to gain foundations for multiplication.

**2: Cluster: (ID 35674)**Identify and continue patterns.

**2: Cluster: (ID 36)**Reason with shapes and their attributes.

**2: Cluster: (ID 40)**Measure and estimate lengths in standard units.

**2: Cluster: (ID 45)**Relate addition and subtraction to length.

**2: Cluster: (ID 48)**Work with time and money.

**2: Cluster: (ID 51)**Represent and interpret data.

**2: Cluster: (ID 54)**Understand place value.

**2: Cluster: (ID 61)**Use place value understanding and properties of operations to add and subtract.

**2: Cluster: (ID 67)**Represent and solve problems involving addition and subtraction.

**2: Cluster: (ID 69)**Add and subtract within 20.

**2: Cluster: (ID 71)**Work with equal groups of objects to gain foundations for multiplication.

**2: Curricular Indicator: (ID 40395)**No additional indicator(s) at this level. Mastery is expected at previous grade levels.

**2: Curricular Indicator: (ID 40400)**No additional indicator(s) at this level. Mastery is expected at previous grade levels.

**2: Curricular Indicator: (ID 40406)**No additional indicator(s) at this level.

**2d.**Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

**2.D.1**Collect, organize, and interpret data.

**2.D.1.1**Explain that the length of a bar in a bar graph or the number of objects in a picture graph represents the number of data points for a given category.

**2.D.1.2**Organize a collection of data with up to four categories using pictographs and bar graphs with intervals of 1s, 2s, 5s or 10s.

**2.D.1.3**Write and solve one-step word problems involving addition or subtraction using data represented within pictographs and bar graphs with intervals of one.

**2.D.1.4**Draw conclusions and make predictions from information in a graph.

**2.DA.1**Draw a picture graph (with single-unit scale) and a bar graph (with single-unit scale) to represent a data set with up to four choices (What is your favorite color? red, blue, yellow, green). Solve simple put-together, take-apart, and compare problems using information presented in the graphs.

**2: Domain: (ID 35103)**Operations & Algebraic Thinking

**2: Domain: (ID 35117)**Number & Operations in Base Ten

**2: Domain: (ID 35129)**Measurement & Data

**2: Domain: (ID 35141)**Geometry

**2: Domain: (ID 35393)**Operations and Algebraic Thinking

**2: Domain: (ID 35407)**Number and Operations in Base Ten

**2: Domain: (ID 35419)**Measurement and Data

**2: Domain: (ID 35431)**Geometry

**2: Domain: (ID 35791)**Operations & Algebraic Thinking

**2: Domain: (ID 35805)**Number & Operations in Base Ten

**2: Domain: (ID 35816)**Measurement & Data

**2: Domain: (ID 35828)**Geometry

**2: Domain: (ID 36016)**Operations and Algebraic Thinking

**2: Domain: (ID 36017)**Number and Operations in Base Ten

**2: Domain: (ID 36018)**Measurement and Data

**2: Domain: (ID 36019)**Geometry

**2: Domain: (ID 36505)**Operations & Algebraic Thinking

**2: Domain: (ID 36519)**Number & Operations in Base Ten

**2: Domain: (ID 36531)**Measurement & Data

**2: Domain: (ID 36543)**Geometry

**2: Domain: (ID 36595)**Operations & Algebraic Thinking

**2: Domain: (ID 36609)**Number & Operations in Base Ten

**2: Domain: (ID 36621)**Measurement & Data

**2: Domain: (ID 36633)**Geometry

**2: Domain: (ID 36779)**Operations and Algebraic Thinking

**2: Domain: (ID 36780)**Number and Operations in Base Ten

**2: Domain: (ID 36781)**Measurement and Data

**2: Domain: (ID 36782)**Geometry

**2: Domain: (ID 37899)**Operations & Algebraic Thinking

**2: Domain: (ID 37901)**Number & Operations in Base Ten

**2: Domain: (ID 37902)**Measurement & Data

**2: Domain: (ID 37903)**Geometry

**2: Domain: (ID 37995)**Operations & Algebraic Thinking

**2: Domain: (ID 38009)**Number & Operations in Base Ten

**2: Domain: (ID 38021)**Measurement & Data

**2: Domain: (ID 38033)**Geometry

**2: Domain: (ID 38085)**Operations & Algebraic Thinking

**2: Domain: (ID 38099)**Number & Operations in Base Ten

**2: Domain: (ID 38111)**Measurement & Data

**2: Domain: (ID 38123)**Geometry

**2: Domain: (ID 38572)**Operations & Algebraic Thinking

**2: Domain: (ID 38586)**Number & Operations in Base Ten

**2: Domain: (ID 38598)**Measurement & Data

**2: Domain: (ID 38610)**Geometry

**2: Domain: (ID 38780)**Operations & Algebraic Thinking

**2: Domain: (ID 38794)**Number & Operations in Base Ten

**2: Domain: (ID 38806)**Measurement & Data

**2: Domain: (ID 38818)**Geometry

**2: Domain: (ID 38847)**Operations and Algebraic Thinking

**2: Domain: (ID 38860)**Number and Operations in Base Ten

**2: Domain: (ID 38871)**Measurement and Data

**2: Domain: (ID 38882)**Geometry

**2: Domain: (ID 38985)**Operations & Algebraic Thinking

**2: Domain: (ID 38999)**Number & Operations in Base Ten

**2: Domain: (ID 39011)**Measurement & Data

**2: Domain: (ID 39023)**Geometry

**2: Domain: (ID 39052)**Operations & Algebraic Thinking

**2: Domain: (ID 39065)**Number & Operations in Base Ten

**2: Domain: (ID 39076)**Measurement & Data

**2: Domain: (ID 39087)**Geometry

**2: Domain: (ID 39406)**Operations and Algebraic Thinking

**2: Domain: (ID 39412)**Number and Operations in Base Ten

**2: Domain: (ID 39430)**Measurement and Data

**2: Domain: (ID 39437)**Geometry

**2: Domain: (ID 39483)**Operations & Algebraic Thinking

**2: Domain: (ID 39497)**Number & Operations in Base Ten

**2: Domain: (ID 39509)**Measurement & Data

**2: Domain: (ID 39521)**Geometry

**2: Domain: (ID 40014)**Operations and Algebraic Thinking

**2: Domain: (ID 40015)**Number and Operations in Base Ten

**2: Domain: (ID 40016)**Measurement and Data

**2: Domain: (ID 40017)**Geometry

**2: Domain: (ID 40222)**Number Sense and Operations in Base Ten

**2: Domain: (ID 40223)**Relationships and Algebraic Thinking

**2: Domain: (ID 40224)**Geometry and Measurement

**2: Domain: (ID 40225)**Data and Statistics

**2: Domain: (ID 40290)**Operations & Algebraic Thinking

**2: Domain: (ID 40304)**Number & Operations in Base Ten

**2: Domain: (ID 40316)**Measurement & Data

**2: Domain: (ID 40328)**Geometry

**2: Domain: (ID 40611)**Operations & Algebraic Thinking

**2: Domain: (ID 40625)**Number & Operations in Base Ten

**2: Domain: (ID 40637)**Measurement & Data

**2: Domain: (ID 40649)**Geometry

**2: Domain: (ID 40701)**Operations & Algebraic Thinking

**2: Domain: (ID 40715)**Number & Operations in Base Ten

**2: Domain: (ID 40727)**Measurement & Data

**2: Domain: (ID 40739)**Geometry

**2: Domain: (ID 40785)**Operations & Algebraic Thinking

**2: Domain: (ID 40787)**Number & Operations in Base Ten

**2: Domain: (ID 40788)**Measurement & Data

**2: Domain: (ID 40789)**Geometry

**2: Domain: (ID 40881)**Operations & Algebraic Thinking

**2: Domain: (ID 40895)**Number & Operations in Base Ten

**2: Domain: (ID 40907)**Measurement & Data

**2: Domain: (ID 40919)**Geometry

**2: Domain: (ID 41149)**Measurement and Data

**2: Domain: (ID 41150)**Geometry

**2: Domain: (ID 41151)**Operations and Algebraic Thinking

**2: Domain: (ID 41152)**Number and Operations in Base Ten

**2: Domain: (ID 41317)**Operations and Algebraic Thinking

**2: Domain: (ID 41318)**Number and Operations in Base Ten

**2: Domain: (ID 41319)**Measurement and Data

**2: Domain: (ID 41320)**Geometry

**2: Domain: (ID 41461)**Operations and Algebraic Thinking

**2: Domain: (ID 41462)**Number and Operations in Base Ten

**2: Domain: (ID 41463)**Measurement and Data

**2: Domain: (ID 41464)**Geometry

**2: Domain: (ID 41770)**Operations & Algebraic Thinking

**2: Domain: (ID 41784)**Number & Operations in Base Ten

**2: Domain: (ID 41796)**Measurement & Data

**2: Domain: (ID 41808)**Geometry

**2: Domain: (ID 41892)**Numbers & Operations in Base Ten

**2: Domain: (ID 41893)**Operations and Algebraic Thinking

**2: Domain: (ID 41895)**Geometry

**2: Domain: (ID 41897)**Measurement and Data

**2: Domain: (ID 41987)**Operations & Algebraic Thinking

**2: Domain: (ID 42001)**Number & Operations in Base Ten

**2: Domain: (ID 42013)**Measurement & Data

**2: Domain: (ID 42025)**Geometry

**2: Domain: (ID 42351)**Operations & Algebraic Thinking

**2: Domain: (ID 42363)**Number & Operations in Base Ten

**2: Domain: (ID 42374)**Measurement & Data

**2: Domain: (ID 42385)**Geometry

**2: Domain: (ID 42965)**Operations and Algebraic Thinking

**2: Domain: (ID 42966)**Number and Operations in Base Ten

**2: Domain: (ID 42967)**Measurement and Data

**2: Domain: (ID 42968)**Geometry

**2: Domain: (ID 44045)**Operations & Algebraic Thinking

**2: Domain: (ID 44059)**Number & Operations in Base Ten

**2: Domain: (ID 44071)**Measurement & Data

**2: Domain: (ID 44083)**Geometry

**2: Domain: (ID 44301)**Operations & Algebraic Thinking

**2: Domain: (ID 44315)**Number & Operations in Base Ten

**2: Domain: (ID 44327)**Measurement & Data

**2: Domain: (ID 44339)**Geometry

**2: Domain: (ID 44536)**Operations and Algebraic Thinking

**2: Domain: (ID 44537)**Number and Operations in Base Ten

**2: Domain: (ID 44539)**Geometry

**2: Domain: (ID 44662)**Operations & Algebraic Thinking

**2: Domain: (ID 44676)**Number & Operations in Base Ten

**2: Domain: (ID 44688)**Measurement & Data

**2: Domain: (ID 44700)**Geometry

**2: Domain: (ID 44752)**Operations & Algebraic Thinking

**2: Domain: (ID 44766)**Number & Operations in Base Ten

**2: Domain: (ID 44778)**Measurement & Data

**2: Domain: (ID 44790)**Geometry

**2.DS.A**Represent and interpret data.

**2.DS.A.1**Create a line plot to represent a set of numeric data, given a horizontal scale marked in whole numbers.

**2.DS.A.2**Generate measurement data to the nearest whole unit, and display the data in a line plot.

**2.DS.A.3**Draw a picture graph or a bar graph to represent a data set with up to four categories.

**2.DS.A.4**Solve problems using information presented in line plots, picture graphs and bar graphs.

**2.DS.A.5**Draw conclusions from line plots, picture graphs and bar graphs.

**2.G**Grade 2 - Geometry

**2.G**Geometry

**2.G**Geometry

**2.G**Geometry

**2.G**Geometry

**2.G.1**Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

**2.G.1**Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

**2.G.1**Identify, describe, and classify two- and three-dimensional shapes (triangle, square, rectangle, cube, right rectangular prism) according to the number and shape of faces and the number of sides and/or vertices. Draw two-dimensional shapes.

**2.G.1**Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Sizes are compared directly or visually, not compared by measuring. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

**2.G.1**Identify triangles, quadrilaterals, hexagons, and cubes. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.

**2.G.1**Identify and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces compared visually, not by measuring. Identify triangles, quadrilaterals, pentagons, hexagons and cubes.

**2.G.1**Recognize and identify triangles, quadrilaterals, pentagons, and hexagons based on the number of sides or vertices. Recognize and identify cubes, rectangular prisms, cones, and cylinders.

**2.G.1**Identify trapezoids, rhombuses, pentagons, hexagons, octagons, parallelograms, quadrilaterals, cubes, spheres, cylinders, cones, triangular prisms, rectangular prisms.
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.

**2.G.2**Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

**2.G.2**Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

**2.G.2**Create squares, rectangles, triangles, cubes, and right rectangular prisms using appropriate materials.

**2.G.2**Partition a rectangle into rows and columns of same-size squares to form an array and count to find the total number of parts.

**2.G.2****2.G.2**Partition a rectangle into rows and columns of same-size squares and count to find the total number.

**2.G.2****2.G.2**Partition a rectangle into rows and columns of same-size squares and count to find the total number of squares.

**2.G.3**Partition circles and rectangles into two, three, or four equal shares; describe the shares using the words *halves, thirds, half of, a third of*, etc.; and describe the whole as two halves, three thirds, or four fourths. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.3**Partition squares, rectangles and circles into two or four equal parts, and describe the parts using the words *halves, fourths, a half of, and a fourth of*. Understand that when partitioning a square, rectangle or circle into two or four equal parts, the parts become smaller as the number of parts increases.

**2.G.3**Partition circles and rectangles into two, three, or four equal shares; describe the shares using the words *halves*, *thirds*, or *fourths* and *quarters*, and use the phrases *half of*, *third of*, or *fourth of* and *quarter of*. Describe the whole as two halves, three thirds, or four fourths in real-world contexts. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.3**Partition circles and rectangles into shares, describe the shares using the words *halves*, *thirds*, *half of*, *a third of*, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.3**Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words *halves, thirds, half of, a third of*, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.3**Partition circles and rectangles into two, three, or four equal shares.
Describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that identical wholes can be equally divided in different ways.
Demonstrate understanding that partitioning shapes into more equal shares creates smaller shares.

**2.G.3**Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. *Note: fraction notation 1/2, 1/3, 1/4 is not expected at this grade level. Recognize that equal shares of identical wholes need not have the same shape.*

**2.G.3**Investigate and predict the result of composing and decomposing two- and three-dimensional shapes.

**2.G.4**Partition a rectangle into rows and columns of same-size (unit) squares and count to find the total number of same-size squares.

**2.G.5**Partition circles and rectangles into two, three, or four equal parts; describe the shares using the words halves, thirds, half of, a third of, etc.; and describe the whole as two halves, three thirds, four fourths. Recognize that equal parts of identical wholes need not have the same shape.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason about shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A**Reason with shapes and their attributes.

**2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1**Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.Sizes are compared directly or visually, not compared by measuring. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

**2.G.A.1****2.G.A.1**Recognize, identify, and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces; to include triangles, quadrilaterals, pentagons, hexagons, and cubes. (Sizes are compared directly or visually, not compared by measuring.)

**2.G.A.1****2.G.A.1****2.G.A.1**Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Draw two-dimensional shapes having specified attributes (as determined directly or visually, not by measuring), such as a given number of angles or a given number of sides of equal length.

**2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1**Identify and describe specified attributes of two-dimensional and three-dimensional shapes, according to the number and shape of faces, number of angles, and the number of sides and/or vertices. Draw two-dimensional shapes based on the specified attributes (e.g. triangles, quadrilaterals, pentagons, and hexagons).

**2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1****2.G.A.1**Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, squares, rectangles, rhombuses, trapezoids, pentagons, hexagons, and cubes.

**2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2**Partition a rectangle into rows and columns of same-sized squares and find the total number of squares.

**2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.2****2.G.A.3**Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.A.3**Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3**Partition circles and rectangles into two, three, and four equal shares, describe the shares using the words *halves, thirds, fourths, half of, a third of, and a fourth of*, and describe the whole as *two halves, three thirds, four fourths*. Recognize that equal shares of identical wholes need not have the same shape.

**2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3****2.G.A.3**Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words

*halves*,*thirds*,*half of*,*a third of*, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.**2.G.A.3****2.GM.1**Analyze attributes of two-dimensional figures and develop generalizations about their properties.

**2.GM.1.1**Recognize trapezoids and hexagons.

**2.GM.1.2**Describe, compare, and classify two-dimensional figures according to their geometric attributes.

**2.GM.1.3**Compose two-dimensional shapes using triangles, squares, hexagons, trapezoids, and rhombi.

**2.GM.1.4**Recognize right angles and classify angles as smaller or larger than a right angle.

**2.GM.2**Understand length as a measurable attribute and explore capacity.

**2.GM.2.1**Explain the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.

**2.GM.2.2**Explain the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest whole unit.

**2.GM.2.3**Explore how varying shapes and styles of containers can have the same capacity.

**2.GM.3**Tell time to the quarter hour.

**2.GM.3.1**Read and write time to the quarter-hour on an analog and digital clock. Distinguish between a.m. and p.m.

**2.GM.A**Reason with shapes and their attributes.

**2.GM.A.1**Recognize and draw shapes having specified attributes, such as a given number of angles or sides.

**2.GM.A.1.a**Identify triangles, quadrilaterals, pentagons, hexagons, circles and cubes.

**2.GM.A.1.b**Identify the faces of three-dimensional objects.

**2.GM.A.2**Partition a rectangle into rows and columns of same-size squares and count to find the total number of squares.

**2.GM.A.3**Partition circles and rectangles into two, three or four equal shares, and describe the shares and the whole.

**2.GM.A.3.a**Demonstrate that equal shares of identical wholes need not have the same shape.

**2.GM.B**Measure and estimate lengths in standard units.

**2.GM.B.4**Measure the length of an object by selecting and using appropriate tools.

**2.GM.B.5**Analyze the results of measuring the same object with different units.

**2.GM.B.6**Estimate lengths using units of inches, feet, yards, centimeters and meters.

**2.GM.B.7**Measure to determine how much longer one object is than another.

**2.GM.C**Relate addition and subtraction to length.

**2.GM.C.8**Use addition and subtraction within 100 to solve problems involving lengths that are given in the same units.

**2.GM.C.9**Represent whole numbers as lengths on a number line, and represent whole-number sums and differences within 100 on a number line.

**2.GM.D**Work with time and money.

**2.GM.D.10**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.GM.D.11**Describe a time shown on a digital clock as representing hours and minutes, and relate a time shown on a digital clock to the same time on an analog clock.

**2.GM.D.12**Find the value of combinations of dollar bills, quarters, dimes, nickels and pennies, using $ and ¢ appropriately.

**2.GM.D.13**Find combinations of coins that equal a given amount.

**2: Grade Level: (ID 35127)**Grade 2

**2: Grade Level: (ID 35417)**Grade 2

**2: Grade Level: (ID 35736)**Grade 2

**2: Grade Level: (ID 35814)**Grade 2

**2: Grade Level: (ID 36015)**Grade 2

**2: Grade Level: (ID 36529)**Grade 2

**2: Grade Level: (ID 36619)**Grade 2

**2: Grade Level: (ID 36778)**Grade 2

**2: Grade Level: (ID 37900)**Grade 2

**2: Grade Level: (ID 38019)**Grade 2

**2: Grade Level: (ID 38109)**Grade 2

**2: Grade Level: (ID 38150)**Grade 2

**2: Grade Level: (ID 38596)**Grade 2

**2: Grade Level: (ID 38711)**Grade 2

**2: Grade Level: (ID 38804)**Grade 2

**2: Grade Level: (ID 38869)**Grade 2

**2: Grade Level: (ID 39009)**Grade 2

**2: Grade Level: (ID 39074)**Grade 2

**2: Grade Level: (ID 39392)**Grade 2

**2: Grade Level: (ID 39507)**Grade 2

**2: Grade Level: (ID 39711)**Grade 2

**2: Grade Level: (ID 40013)**Grade 2

**2: Grade Level: (ID 40221)**Grade 2

**2: Grade Level: (ID 40314)**Grade 2

**2: Grade Level: (ID 40568)**Grade 2

**2: Grade Level: (ID 40635)**Grade 2

**2: Grade Level: (ID 40725)**Grade 2

**2: Grade Level: (ID 40786)**Grade 2

**2: Grade Level: (ID 40905)**Grade 2

**2: Grade Level: (ID 40956)**Grade 2

**2: Grade Level: (ID 41148)**Grade 2

**2: Grade Level: (ID 41316)**Grade 2

**2: Grade Level: (ID 41460)**Grade 2

**2: Grade Level: (ID 41713)**Grade 2

**2: Grade Level: (ID 41794)**Grade 2

**2: Grade Level: (ID 41890)**Grade 2

**2: Grade Level: (ID 42011)**Grade 2

**2: Grade Level: (ID 42372)**Grade 2

**2: Grade Level: (ID 42964)**Grade 2

**2: Grade Level: (ID 44005)**Grade 2

**2: Grade Level: (ID 44069)**Grade 2

**2: Grade Level: (ID 44116)**Grade 2

**2: Grade Level: (ID 44325)**Grade 2

**2: Grade Level: (ID 44535)**Grade 2

**2: Grade Level: (ID 44686)**Grade 2

**2: Grade Level: (ID 44776)**Grade 2

**2: : (ID 36962)**Grade 2

**2: : (ID 42117)**Grade 2

**2: : (ID 42118)**Number Sense and Base Ten

**2: : (ID 42119)**Algebraic Thinking and Operations

**2: : (ID 42120)**Geometry

**2: : (ID 42121)**Measurement and Data Analysis

**2: : (ID 7053)**Reason with shapes and their attributes.

**2: : (ID 7057)**Measure and estimate lengths in standard units.

**2: : (ID 7062)**Relate addition and subtraction to length.

**2: : (ID 7065)**Work with time and money.

**2: : (ID 7068)**Represent and interpret data.

**2: : (ID 7071)**Understand place value.

**2: : (ID 7078)**Use place value understanding and properties of operations to add and subtract.

**2: : (ID 7084)**Represent and solve problems involving addition and subtraction.

**2: : (ID 7086)**Add and subtract within 20.

**2: : (ID 7088)**Work with equal groups of objects to gain foundations for multiplication.

**2.M.1**Describe the relationships among inch, foot, and yard. Describe the relationship between centimeter and meter.

**2.M.2**Estimate and measure the length of an object by selecting and using appropriate tools, such as rulers, yardsticks, meter sticks, and measuring tapes to the nearest inch, foot, yard, centimeter and meter.

**2.M.3**Understand that the length of an object does not change regardless of the units used. Measure the length of an object twice using length units of different lengths for the two measurements. Describe how the two measurements relate to the size of the unit chosen.

**2.M.4**Estimate and measure volume (capacity) using cups and pints.

**2.M.5**Tell and write time to the nearest five minutes from analog clocks, using a.m. and p.m. Solve real-world problems involving addition and subtraction of time intervals on the hour or half hour.

**2.M.6**Describe relationships of time, including: seconds in a minute; minutes in an hour; hours in a day; days in a week; and days, weeks, and months in a year.

**2.M.7**Find the value of a collection of pennies, nickels, dimes, quarters and dollars.

**2.MD**Measurement and Data

**2.MD**Measurement and Data

**2.MD**Grade 2 - Measurement and Data

**2.MD**Measurement and Data

**2.MD**Measurement and Data

**2.MD.1**Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

**2.MD.1**Select and use appropriate tools to measure the length of an object.

**2.MD.1**Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

**2.MD.1**Measure the length of an object by selecting and using standard tools such as rulers, yardsticks, meter sticks, and measuring tapes.

**2.MD.1****2.MD.1****2.MD.10**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put together, take-apart and compare problems using information presented in a bar graph.

**2.MD.10**Draw picture graphs and bar graphs with single-unit scales to represent data sets with up to four categories.
Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

**2.MD.10**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

**2.MD.10**Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object using different units. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

**2.MD.10**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and comparison problems using information presented in a bar graph.

**2.MD.10**Organize, represent, and interpret data with up to four categories; complete picture graphs when single-unit scales are provided; complete bar graphs when single-unit scales are provided; solve simple put-together, take-apart, and compare problems in a graph.

**2.MD.11**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

**2.MD.2**Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

**2.MD.2**Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

**2.MD.2**Measure the length of an object twice using different length units for the two measurements. Describe how the two measurements relate to the size of the unit chosen.

**2.MD.2****2.MD.2**Measure the length of an object using two different standard units of measurement. Describe how the two measurements relate to the size of the units chosen.

**2.MD.2****2.MD.3**Estimate lengths using whole units of inches, feet, centimeters, and meters.

**2.MD.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.3**Estimate, measure and draw lengths using whole units of inches, feet, yards, centimeters and meters.

**2.MD.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.4**Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. *For example, after measuring a pencil and a crayon, a student uses the measurements to determine that the pencil is two inches longer than the crayon.*

**2.MD.4**Measure to compare lengths of two objects, expressing the difference in terms of a standard length unit.

**2.MD.4**Measure to determine how much longer one object is than another, expressing the difference with a standard unit of measurement.

**2.MD.4**Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

**2.MD.4**Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

**2.MD.4**Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit (inches, feet, centimeters, and meters).

**2.MD.5**Solve addition and subtraction word problems using numbers up to 100 involving length that are given in the same units (e.g., by using drawings of rulers). Write an equation with a symbol for the unknown to represent the problem.

**2.MD.5**Use addition and subtraction within 100 to solve one- and two-step word problems involving lengths that are given in the same units, *e.g. by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem*.

**2.MD.5**Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units. *For example, use drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.*

**2.MD.5**Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same whole number units, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Drawings need not show details, but should show the mathematics in the problem.

**2.MD.5**Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

**2.MD.6**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,..., and represent whole number sums and differences within 100 on a number line diagram.

**2.MD.6**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

**2.MD.6**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1,2, …, and represent whole-number sums and differences within 100 on a number line diagram.

**2.MD.6**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2… Represent whole number sums and differences within 100 on a number line diagram.

**2.MD.6**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

**2.MD.6**Represent whole numbers on a number line diagram with equally spaced points.
Represent whole-number sums and differences within 100 on a number line diagram.

**2.MD.7**Tell and write time to the nearest five minutes (including quarter after and quarter to) with a.m. and p.m. using analog and digital clocks.

**2.MD.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.7**Tell and write time from analog and digital clocks to the nearest five minutes.

**2.MD.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.7**Tell and write time to the nearest five minutes using a.m. and p.m. from analog and digital clocks.

**2.MD.8**Solve problems with money.

**2.MD.8**Solve word problems involving dollar bills and coins using the $ and ¢ symbols appropriately.

**2.MD.8**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately (Do not use decimal point, if showing 25 cents, use the word cents or ¢). *For example: If you have 2 dimes and 3 pennies, how many cents do you have? *

**2.MD.8**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

**2.MD.8**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. *For example, if you have 2 dimes and 3 pennies, how many cents do you have?*

**2.MD.8.a**Identify nickels and quarters by name and value.

**2.MD.8.a**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. *Example: If you have 2 dimes and 3 pennies, how many cents do you have?*

**2.MD.8.b**Find the value of a collection of quarters, dimes, nickels, and pennies.

**2.MD.8.b**Fluently use a calendar to answer simple real world problems such as “How many weeks are in a year?” or “James gets a $5 allowance every 2 months, how much money will he have at the end of each year?”

**2.MD.8.c**Solve word problems by adding and subtracting within 100, dollars with dollars and cents with cents (not using dollars and cents simultaneously) using the $ and ₵ symbols appropriately (not including decimal notation).

**2.MD.9**Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

**2.MD.9**Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

**2.MD.9**Generate data by measuring lengths of objects to the nearest whole standard unit.
Show the measurements by making a line plot, using a horizontal scale marked off in whole-number units.

**2.MD.9**Generate measurement data by measuring lengths of several objects to the nearest whole unit or by making repeated measurements of the same object. Show the measurements by creating a line plot, where the horizontal scale is marked off in whole number units.

**2.MD.9**Identify coins and bills and their values.

**2.MD.9**Collect, record, interpret, represent, and describe data in a table, graph or line plot.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A**Measure and estimate lengths in standard units.

**2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MD.A.1****2.MDA.1**Select and use appropriate tools (e.g., rulers, yardsticks, meter sticks, measuring tapes) to measure the length of an object.

**2.MDA.10**Draw conclusions from t-charts, object graphs, picture graphs, and bar graphs.

**2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2**Measure the length of an object using two different units of measure and describe how the two measurements relate to the size of the unit chosen.

**2.MD.A.2**Measure the length of an object twice, using different standard length units for the two measurements; describe how the two measurements relate to the size of the unit chosen. Understand that depending on the size of the unit, the number of units for the same length varies.

**2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MD.A.2****2.MDA.2**Measure the same object or distance using a standard unit of one length and then a standard unit of a different length and explain verbally and in writing how and why the measurements differ.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, yards, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.3**Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MDA.3**Estimate and measure length/distance in customary units (i.e., inch, foot, yard) and metric units (i.e., centimeter, meter).

**2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MD.A.4**Measure to determine how much longer one object is than another and express the difference in terms of a standard unit of length.

**2.MD.A.4****2.MD.A.4****2.MD.A.4****2.MDA.4**Measure to determine how much longer one object is than another, using standard length units.

**2.MDA.5**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences through 99 on a number line diagram.

**2.MDA.6**Use analog and digital clocks to tell and record time to the nearest five-minute interval using *a.m.* and *p.m.*

**2.MDA.7**Solve real-world/story problems involving dollar bills using the $ symbol or involving quarters, dimes, nickels, and pennies using the ¢ symbol.

**2.MDA.8**Generate data by measuring objects in whole unit lengths and organize the data in a line plot using a horizontal scale marked in whole number units.

**2.MDA.9**Collect, organize, and represent data with up to four categories using picture graphs and bar graphs with a single-unit scale.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B**Relate addition and subtraction to length.

**2.MD.B.5**Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

**2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5**Add and subtract within 100 to solve contextual problems involving lengths that are given in the same units by using drawings and equations with a symbol for the unknown to represent the problem.

**2.MD.B.5****2.MD.B.5**Use addition and subtraction within 100 to solve word problems within a cultural context, including those of Montana American Indians, involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. .

**2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5****2.MD.B.5**Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same unit. See Table 1.

**2.MD.B.5****2.MD.B.6**Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

**2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6****2.MD.B.6**Represent whole numbers as lengths from 0 on a number line and know that the points corresponding to the numbers on the number line are equally spaced. Use a number line to represent whole number sums and differences of lengths within 100.

**2.MD.B.6****2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C**Work with time and money.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year).

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time in quarter hours and to the nearest five minutes (in a.m. and p.m.) using analog and digital clocks.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7**Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

**2.MD.C.7.a**Know the relationships of time, including seconds in a minute, minutes in an hour, hours in a day, days in a week; days in a month and a year and approximate number of weeks in a month and weeks in a year.

**2.MD.C.8**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

**2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies (up to $10), using $ and ¢ symbols appropriately and whole dollar amounts. *For example, if you have 2 dimes and 3 pennies, how many cents do you have? If you have $3 and 4 quarters, how many dollars or cents do you have? (Students are not expected to use decimal notation.)*

**2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8**Identify and count coins and bills and apply that understanding to solve word problems.

**2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8**Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and \(¢\) symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

**2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8**Solve contextual problems involving dollar bills, quarters, dimes, nickels, and pennies using ¢ and $ symbols appropriately.

**2.MD.C.8****2.MD.C.8**Solve word problems involving collections of money, including dollar bills, quarters, dimes, nickels, and pennies. Record the total using $ and ¢ appropriately. See Table 1.

**2.MD.C.8****2.MD.C.8****2.MD.C.8****2.MD.C.8.a**Recognize and know the value of coins up to one dollar.

**2.MD.C.8.b****2.MD.C.IA.1**Describe the relationship among standard units of time: minutes, hours, days, weeks, months and years.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D**Represent and interpret data.

**2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10**Draw a pictograph and a bar graph (with intervals of one) to represent a data set with up to four categories. Solve addition and subtraction problems related to the data in a graph.

**2.MD.D.10****2.MD.D.10****2.MD.D.10**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems, using information presented in a bar graph.

**2.MD.D.10****2.MD.D.10****2.MD.D.10**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problemsSee Glossary, Table 1. using information presented in a bar graph.

**2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10**Draw a picture graph and a bar graph (with single-unit scale) to represent a data set from a variety of cultural contexts, including those of Montana American Indians, with up to four categories and solve simple put together, take-apart and compare problems using information presented in a bar graph.

**2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.10****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9**Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Organize and record the data on a line plot (dot plot) where the horizontal scale is marked off in whole-number units.

**2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9****2.MD.D.9**Generate measurement data by measuring lengths of several objects to the nearest whole unit. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

**2.MD.D.9****2.MD.D.IA.2**Use interviews, surveys, and observations to collect data that answer questions about students' interests and/or their environment.

**2.MP**Mathematical Practices

**2.MP.1**Make sense of problems and persevere in solving them.

**2.MP.2**Reason abstractly and quantitatively.

**2.MP.3**Construct viable arguments and critique the reasoning of others.

**2.MP.4**Model with mathematics.

**2.MP.5**Use appropriate tools strategically.

**2.MP.6**Attend to precision.

**2.MP.7**Look for and make use of structure.

**2.MP.8**Look for and express regularity in repeated reasoning.

**2.N.1**Compare and represent whole numbers up to 1,000 with an emphasis on place value and equality.

**2.N.1.1**Read, write, discuss, and represent whole numbers up to 1,000. Representations may include numerals, words, pictures, tally marks, number lines and manipulatives.

**2.N.1.2**Use knowledge of number relationships to locate the position of a given whole number on an open number line up to 100.

**2.N.1.3**Use place value to describe whole numbers between 10 and 1,000 in terms of hundreds, tens and ones. Know that 100 is 10 tens, and 1,000 is 10 hundreds.

**2.N.1.4**Find 10 more or 10 less than a given three-digit number. Find 100 more or 100 less than a given three-digit number.

**2.N.1.5**Recognize when to round numbers to the nearest 10 and 100.

**2.N.1.6**Use place value to compare and order whole numbers up to 1,000 using comparative language, numbers, and symbols (e.g., 425 > 276, 73 < 107, page 351 comes after page 350, 753 is between 700 and 800).

**2.N.2**Add and subtract one-and two-digit numbers in real-world and mathematical problems.

**2.N.2.1**Use the relationship between addition and subtraction to generate basic facts up to 20.

**2.N.2.2**Demonstrate fluency with basic addition facts and related subtraction facts up to 20.

**2.N.2.3**Estimate sums and differences up to 100.

**2.N.2.4**Use strategies and algorithms based on knowledge of place value and equality to add and subtract two-digit numbers.

**2.N.2.5**Solve real-world and mathematical addition and subtraction problems involving whole numbers up to 2 digits.

**2.N.2.6**Use concrete models and structured arrangements, such as repeated addition, arrays and ten frames to develop understanding of multiplication.

**2.N.3**Explore the foundational ideas of fractions.

**2.N.3.1**Identify the parts of a set and area that represent fractions for halves, thirds, and fourths.

**2.N.3.2**Construct equal-sized portions through fair sharing including length, set, and area models for halves, thirds, and fourths.

**2.N.4**Determine the value of a set of coins.

**2.N.4.1**Determine the value of a collection(s) of coins up to one dollar using the cent symbol.

**2.N.4.2**Use a combination of coins to represent a given amount of money up to one dollar.

**2.NBT**Number and Operations in Base Ten

**2.NBT**Number and Operations in Base Ten

**2.NBT**Number and Operations in Base Ten

**2.NBT**Numbers and Operations in Base Ten

**2.NBT**Grade 2 - Number and Operations in Base Ten

**2.NBT.1**Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; (*e.g. 706 equals 7 hundreds, 0 tens, and 6 ones.*) Understand the following as special cases:

**2.NBT.1**Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

**2.NBT.1**Demonstrate understanding that the three digits of a three-digit number represent amounts of hundreds, tens, and ones, including:

**2.NBT.1**Model and identify place value positions of three digit numbers. Include:

**2.NBT.1**Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; *for example, 706 equals 7 hundreds, 0 tens, and 6 ones.* Understand the following as special cases:

**2.NBT.1**Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

**2.NBT.1.a**100 can be thought of as a bundle of ten tens - called a “hundred.”

**2.NBT.1.a**100 can be thought of as a bundle of ten tens called a "hundred".

**2.NBT.1.a**100 can be thought of as a bundle of ten tens called a "hundred."

**2.NBT.1.a**100 can be thought of as a bundle of ten tens --called a "hundred".

**2.NBT.1.a**100 can be thought of as a bundle of ten tens — called a “hundred.”

**2.NBT.1.a**100 can be thought of as a bundle of ten tens—called a “hundred.”

**2.NBT.1.b**The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

**2.NBT.1.b**The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

**2.NBT.1.b**Multiples of 100 represent a number of hundreds, 0 tens, and 0 ones.

**2.NBT.1.b**The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

**2.NBT.1.b****2.NBT.1.b**The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds

**2.NBT.1.c**Show flexibility in composing and decomposing hundreds, tens and ones *(e.g. 207 can be composed from 2 hundreds 7 ones OR 20 tens 7 ones OR 207 ones OR 1 hundred 10 tens 7 ones OR 1 hundred 9 tens 17 ones, etc.*)

**2.NBT.2**Count up to 1000, skip-count by 5s, 10s and 100s.

**2.NBT.2**Count within 1000; skip-count by 5s starting at any number ending in 5 or 0. Skip-count by 10s and 100s starting at any number.

**2.NBT.2**Count within 1000; skip-count by 2s, 5s, 10s, and 100s; explain and generalize the patterns.

**2.NBT.2**Count forward and backward from any given number within 1000.
Skip-count by 5s, 10s, and 100s.

**2.NBT.2**Count within 1,000; skip-count by fives, tens, and hundreds.

**2.NBT.2**Count forward and backward within 1,000 by ones, tens, and hundreds starting at any number; skip-count by 5s starting at any multiple of 5.

**2.NBT.3**Read and write numbers to 1,000 using base-ten numerals, number names, and expanded form.

**2.NBT.3**Read and write numbers to 1,000 using base-ten numerals, number names, expanded form, and equivalent representations, e.g., 716 is 700 + 10 + 6, or 6 + 700 + 10, or 6 ones and 71 tens, etc.

**2.NBT.3**Read and write numbers within 1000 using base-ten numerals, number names, expanded form, and unit form.

**2.NBT.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.3**Read, write, order up to 1000 using base-ten numerals, number names and expanded form.

**2.NBT.4**Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, recording the results of comparisons with the symbols >, =, and <.>

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples.

Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples.

Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

**2.NBT.4**Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

**2.NBT.4**Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using > , < ,
= , and ≠ relational symbols to record the results of comparisons.

**2.NBT.4**Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

**2.NBT.4**Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

**2.NBT.4**Compare two three-digit numbers based on the meanings of the hundreds, tens and ones digits, using >, =, < symbols to record the results.

**2.NBT.5**Fluently add and subtract using numbers up to 100. Use:

- strategies based on place value
- properties of operations
- and/or the relationship between addition and subtraction.

**2.NBT.5**Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

**2.NBT.5**Use strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to fluently add and subtract within 100.

**2.NBT.5**Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

**2.NBT.5**Fluently (efficiently, accurately, and flexibly) add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (*e.g. composing/decomposing by like base-10 units, using friendly or benchmark numbers, using related equations, compensation, number line, etc.*).

**2.NBT.5**Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

**2.NBT.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.6**Use strategies based on place value and properties of operations to add up to four two-digit numbers.

**2.NBT.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.7**Add and subtract within 1,000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; record the strategy with a written numerical method (drawings and, when appropriate, equations) and explain the reasoning used.
Understand that in adding or subtracting three-digit numbers, hundreds are added or subtracted from hundreds, tens are added or subtracted from tens, ones are added or subtracted from ones; and sometimes it is necessary to compose or decompose tens or hundreds.

**2.NBT.7**Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, like base-ten units such as hundreds and hundreds, tens and tens, ones and ones are used; and sometimes it is necessary to compose or decompose tens or hundreds.

**2.NBT.7**Demonstrate understanding of place value within 1000 when adding and subtracting three-digit numbers.
Use concrete models or drawings and strategies based on place value, properties of operation, and/or the relationship between addition and subtraction to add and subtract within 1000.
Use a written method to explain the strategy.

**2.NBT.7**Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

**2.NBT.7**Add and subtract using numbers up to 1000. Use:

- concrete models or drawings and strategies based on place value
- properties of operations
- and/or relationship between addition and subtraction.

**2.NBT.7**Add and subtract within 1,000 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, and ones and ones, and that it is sometimes necessary to compose or decompose tens or hundreds.

**2.NBT.7.1**Use estimation strategies to make reasonable estimates in problem solving.

**2.NBT.8**Mentally add 10 or 100 to a given number 100 – 900, and mentally subtract 10 or 100 from a given number 100 – 900.

**2.NBT.8**Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.

**2.NBT.8**Mentally add 10 or 100 to a given number 100-900 and mentally subtract 10 or 100 from a given number.

**2.NBT.8**Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.

**2.NBT.8**Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

**2.NBT.8**Mentally add or subtract 10 or 100 to or from a given number between 100 and 900.

**2.NBT.9**Explain why addition and subtraction strategies work, using place value and the properties of operations. Explanations may be supported by drawings or objects.

**2.NBT.9**Explain why addition and subtraction strategies work, using place value and the properties of operations.

**2.NBT.9**Explain or illustrate the processes of addition or subtraction and their relationship using place value and the properties of operations.

**2.NBT.9**Explain why addition and subtraction strategies work using place value and the properties of operations. The explanations given may be supported by drawings or objects.

**2.NBT.9**Explain why addition and subtraction strategies work, using place value and the properties of operations. Explanations may be supported by drawings or objects

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value of three digit numbers.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A**Understand place value.

**2.NBT.A.1**Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

**2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1**Understand three-digit numbers are composed of hundreds, tens and ones.

**2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1****2.NBT.A.1**Know that the three digits of a three-digit number represent amounts of hundreds, tens, and ones (e.g., 706 can be represented in multiple ways as 7 hundreds, 0 tens, and 6 ones; 706 ones; or 70 tens and 6 ones).

**2.NBT.A.1****2.NBT.A.1****2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens–-called a “hundred.”

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens—called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1a**100 can be thought of as a bundle of ten tens — called a "hundred."

**2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1.b****2.NBT.A.1b****2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s, starting from any number in its skip counting sequence.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Understand that 100 can be thought of as 10 tens – called a “hundred”.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1,000; skip-count by 5s, 10s, and 100s. Identify patterns in skip counting starting at any number.

**2.NBT.A.2**Count within 1000; skip-count by 2s, 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000. Skip-count within 1000 by 5s, 10s, and 100s, starting from any number in its skip counting sequence.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.2**Count within 1000; skip-count by 5s, 10s, and 100s.

**2.NBT.A.3**Count within 1000 by 1s, 10s and 100s starting with any number.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1,000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals (standard form), number names (word form), and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using standard form, word form, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.3**Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

**2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4**Read and write numbers to 1000 using number names, base-ten numerals and expanded form.

**2.NBT.A.4**Compare two three-digit numbers based on the meanings of the digits in each place and use the symbols >, =, and < to show the relationship.

**2.NBT.A.4****2.NBT.A.4****2.NBT.A.4**Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using \(>\), =, and \(<\) symbols to record the results of comparisons.

**2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.4****2.NBT.A.5**Compare two three-digit numbers using the symbols >, = or < .

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B**Use place value understanding and properties of operations to add and subtract.

**2.NBT.B.10**Add or subtract mentally 10 or 100 to or from a given number within 1000.

**2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.5**Fluently add and subtract within 100 using properties of operations, strategies based on place value, and/or the relationship between addition and subtraction.

**2.NBT.B.5****2.NBT.B.5****2.NBT.B.5****2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using properties of operations and strategies based on place value.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Demonstrate fluency with addition and subtraction within 100.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to three two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.6**Add up to four two-digit numbers using strategies based on place value and properties of operations.

**2.NBT.B.7**Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

**2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7**Add and subtract within 1000 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; justify the reasoning used with a written explanation. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

**2.NBT.B.7****2.NBT.B.7****2.NBT.B.7**Demonstrate understanding of addition and subtraction within 1000, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form.
See Table 1.

**2.NBT.B.7****2.NBT.B.7****2.NBT.B.7**Add and subtract within 1,000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

**2.NBT.B.7****2.NBT.B.7**Add and subtract within 1000 using concrete models, drawings, strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to explain the reasoning used.

**2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7****2.NBT.B.7**Add up to four two-digit numbers.

**2.NBT.B.7****2.NBT.B.7****2.NBT.B.8**Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

**2.NBT.B.8**Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

**2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8**Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100– 900.

**2.NBT.B.8****2.NBT.B.8****2.NBT.B.8**Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

**2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.8**Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

**2.NBT.B.8****2.NBT.B.8****2.NBT.B.8**Add or subtract within 1000, and justify the solution.

**2.NBT.B.8****2.NBT.B.8****2.NBT.B.8****2.NBT.B.9**Explain why addition and subtraction strategies work, using place value and the properties of operations.

**2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9**Explain why addition and subtraction strategies work using properties of operations and place value. (Explanations may include words, drawing, or objects.)

**2.NBT.B.9****2.NBT.B.9**Explain why addition and subtraction strategies work, using place value and the properties of operations.Explanations may be supported by drawings or objects.

**2.NBT.B.9**Use the relationship between addition and subtraction to solve problems.

**2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9**Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by words, drawings or objects.)

**2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.B.9****2.NBT.C**Represent and solve problems involving addition and subtraction.

**2.NBT.C.11**Write and solve problems involving addition and subtraction within 100.

**2.NS.1**Count by ones, twos, fives, tens, and hundreds up to at least 1,000 from any given number.

**2.NS.2**Read and write whole numbers up to 1,000. Use words, models, standard form and expanded form to represent and show equivalent forms of whole numbers up to 1,000.

**2.NS.3**Plot and compare whole numbers up to 1,000 on a number line.

**2.NS.4**Match the ordinal numbers first, second, third, etc., with an ordered set up to 30 items.

**2.NS.5**Determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by placing that number of objects in two groups of the same size and recognizing that for even numbers no object will be left over and for odd numbers one object will be left over, or by pairing objects or counting them by 2s).

**2.NS.6**Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones (e.g., 706 equals 7 hundreds, 0 tens, and 6 ones). Understand that 100 can be thought of as a group of ten tens — called a “hundred." Understand that the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

**2.NS.7**Use place value understanding to compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using > , = , and < symbols to record the results of comparisons.

**2.NSBT.1**Understand place value through 999 by demonstrating that:

**2.NSBT.1.a**100 can be thought of as a bundle (group) of 10 tens called a “hundred”;

**2.NSBT.1.b**the hundreds digit in a three-digit number represents the number of hundreds, the tens digit represents the number of tens, and the ones digit represents the number of ones;

**2.NSBT.1.c**three-digit numbers can be decomposed in multiple ways (e.g., 524 can be decomposed as 5 hundreds, 2 tens and 4 ones or 4 hundreds, 12 tens, and 4 ones, etc.).

**2.NSBT.2**Count by tens and hundreds to 1,000 starting with any number.

**2.NSBT.3**Read, write and represent numbers through 999 using concrete models, standard form, and equations in expanded form.

**2.NSBT.4**Compare two numbers with up to three digits using words and symbols (i.e., > , = , or < ).

**2.NSBT.5**Add and subtract fluently through 99 using knowledge of place value and properties of operations.

**2.NSBT.6**Add up to four two-digit numbers using strategies based on knowledge of place value and properties of operations.

**2.NSBT.7**Add and subtract through 999 using concrete models, drawings, and symbols which convey strategies connected to place value understanding.

**2.NSBT.8**Determine the number that is 10 or 100 more or less than a given number through 1,000 and explain the reasoning verbally and in writing.

**2.OA**Operations and Algebraic Thinking

**2.OA**Grade 2 - Operations and Algebraic Thinking

**2.OA**Operations and Algebraic Thinking

**2.OA**Operations and Algebraic Thinking

**2.OA**Operations and Algebraic Thinking

**2.OA.1**Use strategies to add and subtract within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

**2.OA.1**Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**2.OA.1**Use addition and subtraction within 100 to solve one- and two- step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**2.OA.1**Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, *for example, by using drawings and equations with a symbol for the unknown number to represent the problem.*

**2.OA.1**Use addition and subtraction strategies to estimate, then solve one- and two-step word problems (using numbers up to 100) involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.

**2.OA.1**Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, (*e.g. by using drawings and situation equations and/or solution equations with a symbol for the unknown number to represent the problem.*)

**2.OA.2**Fluently add and subtract using numbers up to 20 using mental strategies. Know from memory all sums of two one-digit numbers.

**2.OA.2**Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.2**Use mental strategies to fluently add and subtract within 20.

**2.OA.2**Fluently add and subtract within 20.

**2.OA.2**Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.2**Fluently (efficiently, accurately, and flexibly) add and subtract within 20 using mental strategies (counting on, making a ten, decomposing a number, creating an equivalent but easier and known sum, and using the relationship between addition and subtraction) Work with equal groups of objects to gain foundations for multiplication.

**2.OA.2.a**Add and subtract within 20 using mental strategies such as counting on; making ten (*for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14*); decomposing a number leading to a ten (*for example, 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9*); using the relationship between addition and subtraction (*for example, knowing that 8 + 4 = 12, one knows 12 – 8 = 4*); and creating equivalent but easier or known sums (*for example, adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13*).

**2.OA.2.b**By the end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.3**Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

**2.OA.3**Determine whether a group of objects (up to 20) has an odd or even number of members, (*for example, by pairing objects or counting them by twos*). Write an equation to express an even number as a sum of two equal addends.

**2.OA.3**Determine whether a group of objects (up to 20) has an odd or even number of members, (*e.g. by pairing objects or counting them by 2s*); write an equation to express an even number as a sum of two equal addends.

**2.OA.3**Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

**2.OA.3**Determine whether a given number of objects up to 20 is odd or even.
Write an equation to represent an even number using two equal addends or groups of 2.

**2.OA.3**Determine whether a group of objects (up to 20) is odd or even (e.g., by pairing objects and comparing, counting by 2s). Model an even number as two equal groups of objects and then write an equation as a sum of two equal addends.

**2.OA.4**Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Write an equation to express the total as repeated addition (e.g., array of 4 by 5 would be 5 + 5 + 5 + 5 = 20).

**2.OA.4**Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

**2.OA.4**Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns.
Write an equation to express the total as a sum of equal addends.

**2.OA.4**Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

**2.OA.4**Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

**2.OA.4****2.OA.5**Identify, continue and label number patterns (e.g., aabb, abab). Describe a rule that determines and continues a sequence or pattern.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A**Represent and solve problems involving addition and subtraction.

**2.OA.A.1**Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1**Use addition and subtraction within 100 to solve one- and two-step word problems involving situations within a cultural context, including those of Montana American Indians, of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1**Add and subtract within 100 to solve one- and two-step contextual problems, with unknowns in all positions, involving situations of *add to, take from, put together/take apart*, and *compare*. Use objects, drawings, and equations with a symbol for the unknown number to represent the problem.

**2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1**Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.See Glossary, Table 1.

**2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1****2.OA.A.1**Use addition and subtraction within 100 to solve one- and two-step word problems. Represent a word problem as an equation with a symbol for the unknown. See Table 1.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 30.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B**Add and subtract within 20.

**2.OA.B.2**Fluently add and subtract within 20. By end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2**Add and subtract within 20.

**2.OA.B.2**Fluently add and subtract within 20 using mental strategies.See standard 1.OA.6 for a list of mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2****2.OA.B.2**Fluently add and subtract within 20 using mental strategies. By end of grade 2, know from memory all sums of two single-digit numbers and related differences. *For example, the sum 6 + 5 = 11 has related differences of 11 – 5 = 6 and 11 – 6 = 5.*

**2.OA.B.2**Fluently add and subtract within 20 using mental strategies. By the end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.B.2****2.OA.B.2**Fluently add and subtract within 30 using mental strategies. By the end of 2nd grade, know from memory all sums of two one-digit numbers and related subtraction facts.

**2.OA.B.2****2.OA.B.2****2.OA.B.2.a**Fluently add and subtract within 20 using mental strategies. (See standard 1.OA.6 for a list of mental strategies.)

**2.OA.B.2.b**By end of Grade 2, know from memory all sums of two one-digit numbers.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C**Work with equal groups of objects to gain foundations for multiplication.

**2.OA.C.3**Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

**2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3**Determine whether a group of objects (up to 20) has an odd or even number of members by pairing objects or counting them by 2s. Write an equation to express an even number as a sum of two equal addends.

**2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.3**Determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by pairing objects or counting them by 2's).

**2.OA.C.3****2.OA.C.3****2.OA.C.3****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4**Use repeated addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

**2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4**Use addition to find the total number of objects arranged in rectangular arrays with up to five rows and up to five columns; write an equation to express the total as a sum of equal addends.

**2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.OA.C.4****2.RA.A**Add and subtract within 20.

**2.RA.A.1**Demonstrate fluency with addition and subtraction within 20.

**2.RA.B**Develop foundations for multiplication and division.

**2.RA.B.2**Determine if a set of objects has an odd or even number of members.

**2.RA.B.2.a**Count by 2s to 100 starting with any even number.

**2.RA.B.2.b**Express even numbers as pairings/groups of 2, and write an expression to represent the number using addends of 2.

**2.RA.B.2.c**Express even numbers as being composed of equal groups and write an expression to represent the number with 2 equal addends.

**2.RA.B.3**Find the total number of objects arranged in a rectangular array with up to 5 rows and 5 columns, and write an equation to represent the total as a sum of equal addends.

**2.SMP**Grade 2 - Standards for Mathematical Practice

**2.SMP.1**Make sense of problems and persevere in solving them.

**2.SMP.2**Reason abstractly and quantitatively.

**2.SMP.3**Construct viable arguments and critique the reasoning of others.

**2.SMP.4**Model with mathematics.

**2.SMP.5**Use appropriate tools strategically.

**2.SMP.6**Attend to precision.

**2.SMP.7**Look for and make use of structure.

**2.SMP.8**Look for and express regularity in repeated reasoning.

**2: Standard: (ID 39551)**Compare and represent whole numbers up to 1000 with an emphasis on place value and equality.

**2: Standard: (ID 39552)**Demonstrate mastery of addition and subtraction basic facts; add and subtract one- and two-digit numbers in real-world and mathematical problems.

**2: Standard: (ID 39593)**Recognize, create, describe, and use patterns and rules to solve real-world and mathematical problems.

**2: Standard: (ID 39595)**Use number sentences involving addition, subtraction and unknowns to represent and solve real-world and mathematical problems; create real-world situations corresponding to number sentences.

**2: Standard: (ID 39638)**Identify, describe and compare basic shapes according to their geometric attributes.

**2: Standard: (ID 39639)**Understand length as a measurable attribute; use tools to measure length.

**2: Standard: (ID 39640)**Use time and money in real-world and mathematical situations.

**2: Strand: (ID 38179)**Number Sense

**2: Strand: (ID 38181)**Computation and Algebraic Thinking

**2: Strand: (ID 38183)**Geometry

**2: Strand: (ID 38186)**Measurement

**2: Strand: (ID 38190)**Data Analysis

**2: Strand: (ID 39712)**Number & Operation

**2: Strand: (ID 39713)**Algebra

**2: Strand: (ID 39714)**Geometry & Measurement

**2: Strand: (ID 41714)**Number & Operations (N)

**2: Strand: (ID 41715)**Algebraic Reasoning & Algebra (A)

**2: Strand: (ID 41716)**Geometry & Measurement (GM)

**2: Strand: (ID 41717)**Data & Probability (D)

**2: Strand: (ID 44131)**Number and Number Sense

**2: Strand: (ID 44133)**Computation and Estimation

**2: Strand: (ID 44134)**Measurement and Geometry

**2: Strand: (ID 44136)**Probability and Statistics

**2: Strand: (ID 44138)**Patterns, Functions, and Algebra

**3**Grade 3

**3**Grade 1, Adopted 2012.

**3.**Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

**3.**Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

**3.**Communicates mathematical ideas effectively.

**3.**Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

**3.**Calculate limits based on convergent and divergent series.

**3.**Construct viable arguments and critique the reasoning of others.

**3.**Apply properties of operations as strategies to add and subtract.

**3.**Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

**3.**Use special numbers, including e, i, Π, p and the golden ratio, to solve application-based problems.

**3.**Use proportional relationships to solve multistep ratio and percent problems.

**3.**Use the recursive process and difference equations to create fractals, population growth models, sequences, series, and compound interest models.

**3.**Use formulas or equations of functions to calculate outcomes of exponential growth or decay. *Example:Solve problems involving compound interest, bacterial growth, carbon-14 dating, and depreciation.*

**3.**Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**3.**Know and apply the properties of integer exponents to generate equivalent numerical expressions.

*For example, 3² × 3*^{-5}= 3^{-3}= 1/3³ = 1/27.**3.**Use the structure of an expression to identify ways to rewrite it.

**3.****3.**Prove statements using mathematical induction.

**3.**(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Find the dot product and the cross product of vectors.

**3.**Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

*For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.***3.**Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

**3.**(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

**3.**(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

**3.**(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

**3.0**Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.

**3.0**Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.

**30.**Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

**30.**Use coordinates to prove simple geometric theorems algebraically. *Example: Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).*

**30.**Interpret the concept of definite integral as a limit of Riemann sums over equal subdivisions.

**30.**(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

**30.**Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.

**30.**Understand the relationship between the constants of a quadratic equation and the attributes of the graph. Recognize the relationship between the value of the discriminant and the type and number of solutions (i.e., *predict the characteristics of a graph given the equation*).

**30.**Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

**30.**Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

**30a.**Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

**30a.**Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

**30b.**Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

**30b.**Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

**30c.**Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

**30c.**Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

**3.1**The student will

- read, write, and identify the place and value of each digit in a six-digit whole number, with and without models;
- round whole numbers, 9,999 or less, to the nearest ten, hundred, and thousand; and
- compare and order whole numbers, each 9,999 or less.

**31.**Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

**31.**Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

**31.**Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

**31.**Describe and identify a polynomial of degree one, two, three and four by examining a polynomial expression or a graph.

**31.**Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

**31.**(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

**31.**Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

**3.10**The student will read temperature to the nearest degree.

**3.11**The student will identify and draw representations of points, lines, line segments, rays, and angles.

**3.1.1.1**Read, write and represent whole numbers up to 100,000. Representations may include numerals, expressions with operations, words, pictures, number lines, and manipulatives such as bundles of sticks and base 10 blocks.

**3.1.1.2**Use place value to describe whole numbers between 1000 and 100,000 in terms of ten thousands, thousands, hundreds, tens and ones.
*For example*: Writing 54,873 is a shorter way of writing the following sums:
5 ten thousands + 4 thousands + 8 hundreds + 7 tens + 3 ones
54 thousands + 8 hundreds + 7 tens + 3 ones.

**3.1.1.3**Find 10,000 more or 10,000 less than a given five-digit number. Find 1000 more or 1000 less than a given four- or five-digit. Find 100 more or 100 less than a given four- or five-digit number.

**3.1.1.4**Round numbers to the nearest 10,000, 1000, 100 and 10. Round up and round down to estimate sums and differences.
*For example*: 8726 rounded to the nearest 1000 is 9000, rounded to the nearest 100 is 8700, and rounded to the nearest 10 is 8730.
*Another example*: 473 – 291 is between 400 – 300 and 500 – 200, or between 100 and 300.

**3.1.1.5**Compare and order whole numbers up to 100,000.

**3.12**The student will

- define polygon;
- identify and name polygons with 10 or fewer sides; and
- combine and subdivide polygons with three or four sides and name the resulting polygon(s).

**3.1.2.1**Add and subtract multi-digit numbers, using efficient and generalizable procedures based on knowledge of place value, including standard algorithms.

**3.1.2.2**Use addition and subtraction to solve real-world and mathematical problems involving whole numbers. Use various strategies, including the relationship between addition and subtraction, the use of technology, and the context of the problem to assess the reasonableness of results.
*For example*: The calculation 117 – 83 = 34 can be checked by adding 83 and 34.

**3.1.2.3**Represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.

**3.1.2.4**Solve real-world and mathematical problems involving multiplication and division, including both "how many in each group" and "how many groups" division problems.
*For example*: You have 27 people and 9 tables. If each table seats the same number of people, how many people will you put at each table?
*Another example*: If you have 27 people and tables that will hold 9 people, how many tables will you need?

**3.1.2.5**Use strategies and algorithms based on knowledge of place value, equality and properties of addition and multiplication to multiply a two- or three-digit number by a one-digit number. Strategies may include mental strategies, partial products, the standard algorithm, and the commutative, associative, and distributive properties.
*For example*: 9 × 26 = 9 × (20 + 6) = 9 × 20 + 9 × 6 = 180 + 54 = 234.

**3.13**The student will identify and describe congruent and noncongruent figures.

**3.1.3.1**Read and write fractions with words and symbols. Recognize that fractions can be used to represent parts of a whole, parts of a set, points on a number line, or distances on a number line.
*For example*: Parts of a shape (3/4 of a pie), parts of a set (3 out of 4 people), and measurements (3/4 of an inch).

**3.1.3.2**Understand that the size of a fractional part is relative to the size of the whole.
*For example*: One-half of a small pizza is smaller than one-half of a large pizza, but both represent one-half.

**3.1.3.3**Order and compare unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator.

**3.14**The student will investigate and describe the concept of probability as a measurement of chance and list possible outcomes for a single event.

**3.15**The student will

- collect, organize, and represent data in pictographs or bar graphs; and
- read and interpret data represented in pictographs and bar graphs.

**3.16**The student will identify, describe, create, and extend patterns found in objects, pictures, numbers and tables.

**3.17**The student will create equations to represent equivalent mathematical relationships.

**31a.**Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. *Example: f(x) - xÂ² + 6x + 5. Locate the vertex, axis of symmetry, show the zeros and extreme values.*

**31a.**Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. *Example: f(x) - xÂ² + 6x + 5. Locate the vertex, axis of symmetry, show the zeros and extreme values.*

**31a.**Graph linear and quadratic functions, and show intercepts, maxima, and minima.

**31b.**Use the properties of exponents to interpret expressions for exponential functions.

*Example: Identify percent rate of change in functions such as y = (1.02)*^{t}, y = (0.97)^{t}, y = (1.01^{12t}, y = (1.2)^{t/10}, and classify them as representing exponential growth or decay.**31b.****31b.**Use the properties of exponents to interpret expressions for exponential functions.

*Example: Identify percent rate of change in functions such as y = (1.02)*^{t}, y = (0.97)^{t}, y = (1.01^{12t}, y = (1.2)^{t/10}, and classify them as representing exponential growth or decay.**3.2**The student will

- name and write fractions and mixed numbers represented by a model;
- represent fractions and mixed numbers with models and symbols; and
- compare fractions having like and unlike denominators, using words and symbols (> , < , = , or ≠), with models.

**32.**Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

**32.**Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

*Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.***32.**Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

*Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.***32.**Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

**32.**(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

**32.**Add and subtract polynomials using appropriate strategies (e.g. by using Algebra Tiles).

**32.**Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

**3.2.1.1**Create, describe, and apply single-operation input-output rules involving addition, subtraction and multiplication to solve problems in various contexts.
*For example*: Describe the relationship between number of chairs and number of legs by the rule that the number of legs is four times the number of chairs.

**3.2.2.1**Understand how to interpret number sentences involving multiplication and division basic facts and unknowns. Create real-world situations to represent number sentences.

**3.2.2.2**Use multiplication and division basic facts to represent a given problem situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.
*For example*: Find values of the unknowns that make each number sentence true
6 = p ÷ 9
24 = a × b
5 × 8 = 4 × t.
*Another example*: How many math teams are competing if there is a total of 45 students with 5 students on each team? This situation can be represented by 5 × n = 45 or 45/5 = n or 45/n = 5.

**32a.**Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

**32b.**Use the properties of exponents to interpret expressions for exponential functions.

*For example, identify percent rate of change in functions such as y = (1.02)*^{t}, y = (0.97)^{t}, y = (1.01^{12t}, y = (1.2)^{t/10}, and classify them as representing exponential growth or decay.**3.3**The student will

- estimate and determine the sum or difference of two whole numbers; and
- create and solve single-step and multistep practical problems involving sums or differences of two whole numbers, each 9,999 or less.

**33.**Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

**33.**Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

**33.**Factor polynomials using the greatest common factor and factor quadratics that have only rational zeros.

**33.**Write a function that describes a relationship between two quantities.

**33.**Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

**33.**Write a function that describes a relationship between two quantities.

**33.**Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). *Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*

**3.3.1.1**Identify parallel and perpendicular lines in various contexts, and use them to describe and create geometric shapes, such as right triangles, rectangles, parallelograms and trapezoids.

**3.3.1.2**Sketch polygons with a given number of sides or vertices (corners), such as pentagons, hexagons and octagons.

**3.3.2.1**Use half units when measuring distances.
*For example*: Measure a person's height to the nearest half inch.

**3.3.2.2**Find the perimeter of a polygon by adding the lengths of the sides.

**3.3.2.3**Measure distances around objects.
*For example*: Measure the distance around a classroom, or measure a person's wrist size.

**3.3.3.1**Tell time to the minute, using digital and analog clocks. Determine elapsed time to the minute.
*For example*: Your trip began at 9:50 a.m. and ended at 3:10 p.m. How long were you traveling?

**3.3.3.2**Know relationships among units of time.
*For example*: Know the number of minutes in an hour, days in a week and months in a year.

**3.3.3.3**Make change up to one dollar in several different ways, including with as few coins as possible.
*For example*: A chocolate bar costs $1.84. You pay for it with $2. Give two possible ways to make change.

**3.3.3.4**Use an analog thermometer to determine temperature to the nearest degree in Fahrenheit and Celsius.
*For example*: Read the temperature in a room with a thermometer that has both Fahrenheit and Celsius scales. Use the thermometer to compare Celsius and Fahrenheit readings.

**33a.**Combine standard function types using arithmetic operations.

*Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.***33a.**Combine standard function types using arithmetic operations.

*Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.***3.4**The student will

- represent multiplication and division through 10 × 10, using a variety of approaches and models;
- create and solve single-step practical problems that involve multiplication and division through 10 x 10; and
- demonstrate fluency with multiplication facts of 0, 1, 2, 5, and 10; and
- solve single-step practical problems involving multiplication of whole numbers, where one factor is 99 or less and the second factor is 5 or less.

**34.**Identify the effect on the graph of replacing *f(x)* by *f(x) + k*, *k f(x)*, *f(kx)*, and* f(x + k)* for specific values of *k* (both positive and negative); find the value of *k* given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

**34.**Identify the effect on the graph of replacing *f(x)* by *f(x) + k*, *k f(x)*, *f(kx)*, and* f(x + k)* for specific values of *k* (both positive and negative); find the value of *k* given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

**34.**Write a function that describes a relationship between two quantities.

**34.**Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.

**34.**Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

**34.**Justify why some polynomials are prime over the rational number system.

**34.**(+) Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

**3.4.1.1**Collect, display and interpret data using frequency tables, bar graphs, picture graphs and number line plots having a variety of scales. Use appropriate titles, labels and units.

**34a.**Determine an explicit expression, a recursive process, or steps for calculation from a context.

**34b.**Combine standard function types using arithmetic operations. *Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.*

**3.5**The student will solve practical problems that involve addition and subtraction with proper fractions having like denominators of 12 or less.

**35.**(+) Derive the formula *A = (½)ab sin(C)* for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (*Apply formulas previously derived in Geometry*.)

**35.**Use the zeros of a polynomial to construct a rough graph of the function.

**35.**Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

**35.**Find inverse functions.

**35.**Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. *Use dissection arguments, Cavalieri’s principle, and informal limit arguments. *

**35.**Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

**35.**Find inverse functions.

**35a.**Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. *Example: f(x) =2x³ or f(x) = (x+1)/(x-1) for x ≠ 1.*

**35a.**Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. *Example: f(x) =2x³ or f(x) = (x+1)/(x-1) for x ≠ 1.*

**3.6**The student will

- determine the value of a collection of bills and coins whose total value is $5.00 or less;
- compare the value of two sets of coins or two sets of coins and bills; and
- make change from $5.00 or less.

**36.**Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

**36.**For exponential models, express as a logarithm the solution to *ab

^{ct}= d* where *a*, *c*, and *d* are numbers, and the base *b* is 2, 10, or *e*; evaluate the logarithm using technology.**36.**Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

**36.**Explain and apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

**36.**(+) Derive the equations of a parabola given a focus and directrix.

**36.**For exponential models, express as a logarithm the solution to *ab

^{ct}= d* where *a*, *c*, and *d* are numbers, and the base *b* is 2, 10, or *e*; evaluate the logarithm using technology.**36.**Identify the effect on the graph of replacing *f(x)* by *f(x) + k*, *k f(x)*, *f(kx)*, and *f(x + k)* for specific values of *k* (both positive and negative); find the value of *k* given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

**3.7**The student will estimate and use U.S. Customary and metric units to measure

- length to the nearest 1/2 inch, inch, foot, yard, centimeter, and meter; and
- liquid volume in cups, pints, quarts, gallons, and liters.

**37.**Graph piecewise defined functions and determine continuity or discontinuities.

**37.**Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

**37.**Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

**37.**Distinguish between situations that can be modeled with linear functions and with exponential functions.

**37.**(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

**37.**(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

**37.**Determine the relationship between surface areas of similar figures and volumes of similar figures.

**37a.**Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

**37b.**Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

**37c.**Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

**3.8**The student will estimate and

- measure the distance around a polygon in order to determine its perimeter using U.S. Customary and metric units; and
- count the number of square units needed to cover a given surface in order to determine its area.

**38**Implementation of Texas Essential Knowledge and Skills for Mathematics, High School, Adopted 2012.

**38.**Describe the attributes of graphs and the general equations of parent functions (linear, quadratic, cubic, absolute value, rational, exponential, logarithmic, square root, cube root, and greatest integer).

**38.**Fluently use formulas and/or appropriate measuring tools to find length and angle measures, perimeter, area, volume, and surface area of polygons, circles, spheres, cones, cylinders, pyramids, and composite or irregular figures. Use them to solve real-world and mathematical problems.

**38.**Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

**38.**Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

**38.**(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

**38.**Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

**38.**(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

**38.a**The provisions of §§111.39-111.45 of this subchapter shall be implemented by school districts.

**38.b**No later than June 30, 2015, the commissioner of education shall determine whether instructional materials funding has been made available to Texas public schools for materials that cover the essential knowledge and skills for mathematics as adopted in §§111.39-111.45 of this subchapter.

**38.c**If the commissioner makes the determination that instructional materials funding has been made available under subsection (b) of this section, §§111.39-111.45 of this subchapter shall be implemented beginning with the 2015-2016 school year and apply to the 2015-2016 and subsequent school years.

**38.d**If the commissioner does not make the determination that instructional materials funding has been made available under subsection (b) of this section, the commissioner shall determine no later than June 30 of each subsequent school year whether instructional materials funding has been made available. If the commissioner determines that instructional materials funding has been made available, the commissioner shall notify the State Board of Education and school districts that §§111.39-111.45 of this subchapter shall be implemented for the following school year.

**38.e**Sections 111.31-111.37 of this subchapter shall be superseded by the implementation of §§111.38-111.45 under this section.

**3.9**The student will

- tell time to the nearest minute, using analog and digital clocks;
- solve practical problems related to elapsed time in one-hour increments within a 12-hour period; and
- identify equivalent periods of time and solve practical problems related to equivalent periods of time.

**39**Algebra I, Adopted 2012 (One Credit).

**39.**Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

**39.**Explain the effects of changing the parameters in transformations of functions.

**39.**Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. *(Focus on increasing rigor using standard deviation.) *

**39.**Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

**39.**Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions.

**39.**Solve real-world and mathematical problems involving two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

**39.**Describe events as subsets of a sample space (the set of outcomes), using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

**39.a**General requirements. Students shall be awarded one credit for successful completion of this course. This course is recommended for students in Grade 8 or 9. Prerequisite: Mathematics, Grade 8 or its equivalent.

**39.b**Introduction.

**39.b.1**The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on fluency and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.

**39.b.2**The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, paper and pencil, and technology and techniques such as mental math, estimation, and number sense to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

**39.b.3**In Algebra I, students will build on the knowledge and skills for mathematics in Grades 6-8, which provide a foundation in linear relationships, number and operations, and proportionality. Students will study linear, quadratic, and exponential functions and their related transformations, equations, and associated solutions. Students will connect functions and their associated solutions in both mathematical and real-world situations. Students will use technology to collect and explore data and analyze statistical relationships. In addition, students will study polynomials of degree one and two, radical expressions, sequences, and laws of exponents. Students will generate and solve linear systems with two equations and two variables and will create new functions through transformations.

**39.b.4****39.c**Knowledge and skills.

**39.c.1****39.c.10**Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to:

**39.c.10.A**add and subtract polynomials of degree one and degree two;

**39.c.10.B**multiply polynomials of degree one and degree two;

**39.c.10.C**determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend;

**39.c.10.D**rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property;

**39.c.10.E**factor, if possible, trinomials with real factors in the form $ax^2 + bx + c$, including perfect square trinomials of degree two; and

**39.c.10.F**decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.

**39.c.11**Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to:

**39.c.11.A**simplify numerical radical expressions involving square roots; and

**39.c.11.B**simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

**39.c.12**Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:

**39.c.12.A**decide whether relations represented verbally, tabularly, graphically, and symbolically define a function;

**39.c.12.B**evaluate functions, expressed in function notation, given one or more elements in their domains;

**39.c.12.C**identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes;

**39.c.12.D**write a formula for the $n^{th}$ term of arithmetic and geometric sequences, given the value of several of their terms; and

**39.c.12.E**solve mathematic and scientific formulas, and other literal equations, for a specified variable.

**39.c.1.A**apply mathematics to problems arising in everyday life, society, and the workplace;

**39.c.1.B****39.c.1.C****39.c.1.D****39.c.1.E**create and use representations to organize, record, and communicate mathematical ideas;

**39.c.1.F**analyze mathematical relationships to connect and communicate mathematical ideas; and

**39.c.1.G****39.c.2**Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:

**39.c.2.A**determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities;

**39.c.2.B**write linear equations in two variables in various forms, including $y = mx + b$, $Ax + By = C$, and $y - y_1= m(x - x_1)$, given one point and the slope and given two points;

**39.c.2.C**write linear equations in two variables given a table of values, a graph, and a verbal description;

**39.c.2.D**write and solve equations involving direct variation;

**39.c.2.E**write the equation of a line that contains a given point and is parallel to a given line;

**39.c.2.F**write the equation of a line that contains a given point and is perpendicular to a given line;

**39.c.2.G**write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined;

**39.c.2.H**write linear inequalities in two variables given a table of values, a graph, and a verbal description; and

**39.c.2.I**write systems of two linear equations given a table of values, a graph, and a verbal description.

**39.c.3**Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:

**39.c.3.A**determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including $y = mx + b$, $Ax + By = C$, and $y - y_1= m(x - x_1)$;

**39.c.3.B**calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems;

**39.c.3.C**graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems;

**39.c.3.D**graph the solution set of linear inequalities in two variables on the coordinate plane;

**39.c.3.E**determine the effects on the graph of the parent function $f(x) = x$ when $f(x)$ is replaced by $af(x$), $f(x) + d$, $f(x - c)$, $f(bx)$ for specific values of $a$, $b$, $c$, and $d$;

**39.c.3.F**graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist;

**39.c.3.G**estimate graphically the solutions to systems of two linear equations with two variables in real-world problems; and

**39.c.3.H**graph the solution set of systems of two linear inequalities in two variables on the coordinate plane.

**39.c.4**Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. The student is expected to:

**39.c.4.A**calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association;

**39.c.4.B**compare and contrast association and causation in real-world problems; and

**39.c.4.C**write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.

**39.c.5**Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:

**39.c.5.A**solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides;

**39.c.5.B**solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides; and

**39.c.5.C**solve systems of two linear equations with two variables for mathematical and real-world problems.

**39.c.6**Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to:

**39.c.6.A**determine the domain and range of quadratic functions and represent the domain and range using inequalities;

**39.c.6.B**write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form $(f(x) = a(x - h)^2+ k)$, and rewrite the equation from vertex form to standard form $(f(x) = ax^2+ bx + c)$; and

**39.c.6.C**write quadratic functions when given real solutions and graphs of their related equations.

**39.c.7**Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations. The student is expected to:

**39.c.7.A**graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry;

**39.c.7.B**describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions; and

**39.c.7.C**determine the effects on the graph of the parent function $f(x) = x^2$ when $f(x)$ is replaced by $af(x)$, $f(x) + d$, $f(x - c)$, $f(bx)$ for specific values of $a$, $b$, $c$, and $d$.

**39.c.8**Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:

**39.c.8.A**solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula; and

**39.c.8.B**write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.

**39.c.9**Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:

**39.c.9.A**determine the domain and range of exponential functions of the form $f(x) = ab^x$ and represent the domain and range using inequalities;

**39.c.9.B**interpret the meaning of the values of $a$ and $b$ in exponential functions of the form $f(x) = ab^x$ in real-world problems;

**39.c.9.C**write exponential functions in the form $f(x) = ab^x$ (where $b$ is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay;

**39.c.9.D**graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems; and

**39.c.9.E**write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems.

**3.a**Introduction.

**3a.**Identify transcendental numbers.

**3a.**Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

**3.a.1****3.A.1**Describe and create representations of numerical and geometric patterns.

**3.A.1.1**Create, describe, and extend patterns involving addition, subtraction, or multiplication to solve problems in a variety of contexts.

**3.A.1.2**Describe the rule (single operation) for a pattern from an input/output table or function machine involving addition, subtraction, or multiplication.

**3.A.1.3**Explore and develop visual representations of growing geometric patterns and construct the next steps.

**3.a.2****3.A.2**Use number sentences involving multiplication and unknowns to represent and solve real-world and mathematical problems.

**3.A.2.1**Find unknowns represented by symbols in arithmetic problems by solving one-step open sentences (equations) and other problems involving addition, subtraction, and multiplication. Generate real-world situations to represent number sentences.

**3.A.2.2**Recognize, represent and apply the number properties (commutative, identity, and associative properties of addition and multiplication) using models and manipulatives to solve problems.

**3.a.3**For students to become fluent in mathematics, students must develop a robust sense of number. The National Research Council's report, "Adding It Up," defines procedural fluency as "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately." As students develop procedural fluency, they must also realize that true problem solving may take time, effort, and perseverance. Students in Grade 1 are expected to perform their work without the use of calculators.

**3.a.4**The primary focal areas in Grade 1 are understanding and applying place value, solving problems involving addition and subtraction, and composing and decomposing two-dimensional shapes and three-dimensional solids.

**3.a.4.A**Students use relationships within the numeration system to understand the sequential order of the counting numbers and their relative magnitude.

**3.a.4.B**Students extend their use of addition and subtraction beyond the actions of joining and separating to include comparing and combining. Students use properties of operations and the relationship between addition and subtraction to solve problems. By comparing a variety of solution strategies, students use efficient, accurate, and generalizable methods to perform operations.

**3.a.4.C**Students use basic shapes and spatial reasoning to model objects in their environment and construct more complex shapes. Students are able to identify, name, and describe basic two-dimensional shapes and three-dimensional solids.

**3.a.5****3.AT.1**Solve real-world problems involving addition and subtraction of whole numbers within 1000 (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem).

**3.AT.2**Solve real-world problems involving whole number multiplication and division within 100 in situations involving equal groups, arrays, and measurement quantities (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem).

**3.AT.3**Solve two-step real-world problems using the four operations of addition, subtraction, multiplication and division (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem).

**3.AT.4**Interpret a multiplication equation as equal groups (e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each). Represent verbal statements of equal groups as multiplication equations.

**3.AT.5**Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

**3.AT.6**Create, extend, and give an appropriate rule for number patterns using multiplication within 100.

**3.ATO.1**Use concrete objects, drawings and symbols to represent multiplication facts of two single-digit whole numbers and explain the relationship between the factors (i.e., 0 – 10) and the product.

**3.ATO.2**Use concrete objects, drawings and symbols to represent division without remainders and explain the relationship among the whole number quotient (i.e., 0 – 10), divisor (i.e., 0 – 10), and dividend.

**3.ATO.3**Solve real-world problems involving equal groups, area/array, and number line models using basic multiplication and related division facts. Represent the problem situation using an equation with a symbol for the unknown.

**3.ATO.4**Determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is a missing factor, product, dividend, divisor, or quotient.

**3.ATO.5**Apply properties of operations (i.e., Commutative Property of Multiplication, Associative Property of Multiplication, Distributive Property) as strategies to multiply and divide and explain the reasoning.

**3.ATO.6**Understand division as a missing factor problem.

**3.ATO.7**Demonstrate fluency with basic multiplication and related division facts of products and dividends through 100.

**3.ATO.8**Solve two-step real-world problems using addition, subtraction, multiplication and division of whole numbers and having whole number answers. Represent these problems using equations with a letter for the unknown quantity.

**3.ATO.9**Identify a rule for an arithmetic pattern (e.g., patterns in the addition table or multiplication table).

**3.b**Knowledge and skills.

**3b.**Solve unit rate problems including those involving unit pricing and constant speed.

*For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?***3.b.1****3.b.1.A**apply mathematics to problems arising in everyday life, society, and the workplace;

**3.b.1.B****3.b.1.C****3.b.1.D****3.b.1.E**create and use representations to organize, record, and communicate mathematical ideas;

**3.b.1.F**analyze mathematical relationships to connect and communicate mathematical ideas; and

**3.b.1.G****3.b.2**Number and operations. The student applies mathematical process standards to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system related to place value. The student is expected to:

**3.b.2.A**recognize instantly the quantity of structured arrangements;

**3.b.2.B**use concrete and pictorial models to compose and decompose numbers up to 120 in more than one way as so many hundreds, so many tens, and so many ones;

**3.b.2.C**use objects, pictures, and expanded and standard forms to represent numbers up to 120;

**3.b.2.D**generate a number that is greater than or less than a given whole number up to 120;

**3.b.2.E**use place value to compare whole numbers up to 120 using comparative language;

**3.b.2.F**order whole numbers up to 120 using place value and open number lines; and

**3.b.2.G**represent the comparison of two numbers to 100 using the symbols > , < , or =.

**3.b.3**Number and operations. The student applies mathematical process standards to develop and use strategies for whole number addition and subtraction computations in order to solve problems. The student is expected to:

**3.b.3.A**use concrete and pictorial models to determine the sum of a multiple of 10 and a one-digit number in problems up to 99;

**3.b.3.B**use objects and pictorial models to solve word problems involving joining, separating, and comparing sets within 20 and unknowns as any one of the terms in the problem such as 2 + 4 = [ ]; 3 + [ ] = 7; and 5 = [ ] - 3;

**3.b.3.C**compose 10 with two or more addends with and without concrete objects;

**3.b.3.D**apply basic fact strategies to add and subtract within 20, including making 10 and decomposing a number leading to a 10;

**3.b.3.E**explain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences; and

**3.b.3.F**generate and solve problem situations when given a number sentence involving addition or subtraction of numbers within 20.

**3.b.4**Number and operations. The student applies mathematical process standards to identify coins, their values, and the relationships among them in order to recognize the need for monetary transactions. The student is expected to:

**3.b.4.A**identify U.S. coins, including pennies, nickels, dimes, and quarters, by value and describe the relationships among them;

**3.b.4.B**write a number with the cent symbol to describe the value of a coin; and

**3.b.4.C**use relationships to count by twos, fives, and tens to determine the value of a collection of pennies, nickels, and/or dimes.

**3.b.5**Algebraic reasoning. The student applies mathematical process standards to identify and apply number patterns within properties of numbers and operations in order to describe relationships. The student is expected to:

**3.b.5.A**recite numbers forward and backward from any given number between 1 and 120;

**3.b.5.B**skip count by twos, fives, and tens to determine the total number of objects up to 120 in a set;

**3.b.5.C**use relationships to determine the number that is 10 more and 10 less than a given number up to 120;

**3.b.5.D**represent word problems involving addition and subtraction of whole numbers up to 20 using concrete and pictorial models and number sentences;

**3.b.5.E**understand that the equal sign represents a relationship where expressions on each side of the equal sign represent the same value(s);

**3.b.5.F**determine the unknown whole number in an addition or subtraction equation when the unknown may be any one of the three or four terms in the equation; and

**3.b.5.G**apply properties of operations to add and subtract two or three numbers.

**3.b.6**Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties. The student is expected to:

**3.b.6.A**classify and sort regular and irregular two-dimensional shapes based on attributes using informal geometric language;

**3.b.6.B**distinguish between attributes that define a two-dimensional or three-dimensional figure and attributes that do not define the shape;

**3.b.6.C**create two-dimensional figures, including circles, triangles, rectangles, and squares, as special rectangles, rhombuses, and hexagons;

**3.b.6.D**identify two-dimensional shapes, including circles, triangles, rectangles, and squares, as special rectangles, rhombuses, and hexagons and describe their attributes using formal geometric language;

**3.b.6.E**identify three-dimensional solids, including spheres, cones, cylinders, rectangular prisms (including cubes), and triangular prisms, and describe their attributes using formal geometric language;

**3.b.6.F**compose two-dimensional shapes by joining two, three, or four figures to produce a target shape in more than one way if possible;

**3.b.6.G**partition two-dimensional figures into two and four fair shares or equal parts and describe the parts using words; and

**3.b.6.H**identify examples and non-examples of halves and fourths.

**3.b.7**Geometry and measurement. The student applies mathematical process standards to select and use units to describe length and time. The student is expected to:

**3.b.7.A**use measuring tools to measure the length of objects to reinforce the continuous nature of linear measurement;

**3.b.7.B**illustrate that the length of an object is the number of same-size units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other;

**3.b.7.C**measure the same object/distance with units of two different lengths and describe how and why the measurements differ;

**3.b.7.D**describe a length to the nearest whole unit using a number and a unit; and

**3.b.7.E**tell time to the hour and half hour using analog and digital clocks.

**3.b.8**Data analysis. The student applies mathematical process standards to organize data to make it useful for interpreting information and solving problems. The student is expected to:

**3.b.8.A**collect, sort, and organize data in up to three categories using models/representations such as tally marks or T-charts;

**3.b.8.B**use data to create picture and bar-type graphs; and

**3.b.8.C**draw conclusions and generate and answer questions using information from picture and bar-type graphs.

**3.b.9**Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security. The student is expected to:

**3.b.9.A**define money earned as income;

**3.b.9.B**identify income as a means of obtaining goods and services, oftentimes making choices between wants and needs;

**3.b.9.C**distinguish between spending and saving; and

**3.b.9.D**consider charitable giving.

**3c.**Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

**3.C.1**Add and subtract whole numbers fluently within 1000.

**3.C.2**Represent the concept of multiplication of whole numbers with the following models: equal-sized groups, arrays, area models, and equal "jumps" on a number line. Understand the properties of 0 and 1 in multiplication.

**3.C.3**Represent the concept of division of whole numbers with the following models: partitioning, sharing, and an inverse of multiplication. Understand the properties of 0 and 1 in division.

**3.C.4**Interpret whole-number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each).

**3.C.5**Multiply and divide within 100 using strategies, such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8), or properties of operations.

**3.C.6**Demonstrate fluency with multiplication facts and corresponding division facts of 0 to 10.

**3: Cluster: (ID 109)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 1135)**Reason with shapes and their attributes.

**3: Cluster: (ID 1138)**Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

**3: Cluster: (ID 114)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 1141)**Represent and interpret data.

**3: Cluster: (ID 1144)**Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

**3: Cluster: (ID 1154)**Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

**3: Cluster: (ID 1156)**Use place value understanding and properties of operations to perform multi-digit arithmetic.

**3: Cluster: (ID 11583)**Reason with shapes and their attributes.

**3: Cluster: (ID 11586)**Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

**3: Cluster: (ID 11590)**Represent and interpret data.

**3: Cluster: (ID 11593)**Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

**3: Cluster: (ID 1160)**Develop understanding of fractions as numbers.

**3: Cluster: (ID 11603)**Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

**3: Cluster: (ID 11605)**Use place value understanding and properties of operations to perform multi-digit arithmetic.

**3: Cluster: (ID 11609)**Develop understanding of fractions as numbers.

**3: Cluster: (ID 11619)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 11624)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 11627)**Multiply and divide within 100 (basic facts up to 10 x 10).

**3: Cluster: (ID 11629)**Solve problems involving the four operations, and identify and explain patterns in arithmetic.

**3: Cluster: (ID 117)**Multiply and divide within 100.

**3: Cluster: (ID 1170)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 1175)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 1178)**Multiply and divide up to 100.

**3: Cluster: (ID 1180)**Solve problems involving the four operations, and identify and explain patterns in arithmetic.

**3: Cluster: (ID 119)**Solve problems involving the four operations, and identify and explain patterns in arithmetic.

**3: Cluster: (ID 17067)**Reason with shapes and their attributes

**3: Cluster: (ID 17070)**Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects

**3: Cluster: (ID 17073)**Represent and interpret data

**3: Cluster: (ID 17076)**Geometric measurement: understand concepts of area and relate area to multiplication and to addition

**3: Cluster: (ID 17086)**Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures

**3: Cluster: (ID 17088)**Use place value understanding and properties of operations to perform multi-digit arithmetic

**3: Cluster: (ID 17092)**Develop understanding of fractions as numbers

**3: Cluster: (ID 17102)**Represent and solve problems involving multiplication and division

**3: Cluster: (ID 17107)**Understand properties of multiplication and the relationship between multiplication and division

**3: Cluster: (ID 17110)**Multiply and divide within 100

**3: Cluster: (ID 17112)**Solve problems involving the four operations, and identify and explain patterns in arithmetic

**3: Cluster: (ID 22624)**Reason with shapes and their attributes.

**3: Cluster: (ID 22627)**Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

**3: Cluster: (ID 22631)**Represent and interpret data.

**3: Cluster: (ID 22634)**Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

**3: Cluster: (ID 22644)**Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

**3: Cluster: (ID 22647)**Use place value understanding and properties of operations to perform multi-digit arithmetic.

**3: Cluster: (ID 22651)**Develop understanding of fractions as numbers.

**3: Cluster: (ID 22661)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 22666)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 22669)**Multiply and divide within 100.

**3: Cluster: (ID 22672)**Solve problems involving the four operations, and identify and extend patterns in arithmetic.

**3: Cluster: (ID 23331)**Reason with shapes and their attributes.

**3: Cluster: (ID 23333)**Solve problems involving measurement.

**3: Cluster: (ID 23336)**Represent and interpret data.

**3: Cluster: (ID 23338)**Understand the concept of area.

**3: Cluster: (ID 23341)**Understand the concept of perimeter.

**3: Cluster: (ID 23343)**Use place value to add and subtract.

**3: Cluster: (ID 23346)**Understand fractions as numbers.

**3: Cluster: (ID 23351)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 23357)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 23359)**Multiply and divide within 100.

**3: Cluster: (ID 23360)**Solve two-step problems.

**3: Cluster: (ID 23879)**Reason with shapes and their attributes.

**3: Cluster: (ID 23882)****3: Cluster: (ID 23885)**Represent and interpret data

**3: Cluster: (ID 23888)****3: Cluster: (ID 23898)****3: Cluster: (ID 23900)**Use place value understanding and properties of operations to perform multi-digit arithmetic. Note: A range of algorithms may be used.

**3: Cluster: (ID 23904)**Develop understanding of fractions as numbers.

**3: Cluster: (ID 23916)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 23921)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 23924)**Multiply and divide within 10

**3: Cluster: (ID 23926)**Solve problems involving the four operations, and identify and explain patterns in arithmetic.

**3: Cluster: (ID 24529)**Reason with shapes and their attributes.

**3: Cluster: (ID 24532)**Solve problems involving money, measurement, and estimation of intervals of time, liquid volumes, and masses of objects.

**3: Cluster: (ID 24536)**Represent and interpret data.

**3: Cluster: (ID 24539)****3: Cluster: (ID 24549)****3: Cluster: (ID 24551)**Use place value understanding and properties of operations to perform multi-digit arithmetic. A range of strategies and algorithms may be used.

**3: Cluster: (ID 24555)**Develop understanding of fractions as numbers. Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

**3: Cluster: (ID 24565)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 24570)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 24573)**Multiply and divide within 100.

**3: Cluster: (ID 24575)**Solve problems involving the four operations, and identify and explain patterns in arithmetic.

**3: Cluster: (ID 2516)**Reason with shapes and their attributes

**3: Cluster: (ID 2519)**Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects

**3: Cluster: (ID 2522)**Represent and interpret data

**3: Cluster: (ID 2525)**Geometric measurement: understand concepts of area and relate area to multiplication and to addition

**3: Cluster: (ID 2529)**Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures

**3: Cluster: (ID 2531)**Use place value understanding and properties of operations to preform multi-digit arithmetic

**3: Cluster: (ID 2535)**Develop understanding of fractions as numbers

**3: Cluster: (ID 2539)**Represent and solve problems involving multiplication and division

**3: Cluster: (ID 2544)**Understand properties of multiplication and the relationship between multiplication and division

**3: Cluster: (ID 2547)**Multiply and divide within 100

**3: Cluster: (ID 2549)**Solve problems involving the four operations, and identify and explain patterns in arithmetic

**3: Cluster: (ID 30121)**Reason with shapes and their attributes

**3: Cluster: (ID 30124)****3: Cluster: (ID 30127)**Represent and interpret data

**3: Cluster: (ID 30130)**Understand concepts of area and relate area to multiplication and addition

**3: Cluster: (ID 30140)**Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures

**3: Cluster: (ID 30142)**Use place value understanding and properties of operations to perform multi-digit arithmetic.

**3: Cluster: (ID 30146)**Develop understanding of fractions as numbers.

**3: Cluster: (ID 30158)**Represent and solve problems involving multiplication and division within 100

**3: Cluster: (ID 30163)**Demonstrate understanding of the properties of multiplication and the relationship between multiplication and division

**3: Cluster: (ID 30169)**Use the four operations to identify and explain patterns in arithmetic

**3: Cluster: (ID 32452)**Reason with shapes and their attributes.

**3: Cluster: (ID 32455)****3: Cluster: (ID 32458)**Represent and interpret data.

**3: Cluster: (ID 32461)****3: Cluster: (ID 32471)****3: Cluster: (ID 32473)**Use place value understanding and properties of operations to perform multi-digit arithmetic.

**3: Cluster: (ID 32477)**Develop understanding of fractions as numbers.

**3: Cluster: (ID 32487)**Represent and solve problems involving multiplication and division.

**3: Cluster: (ID 32492)**Understand properties of multiplication and the relationship between multiplication and division.

**3: Cluster: (ID 32495)**Multiply and divide within 100.

**3: Cluster: (ID 32497)**Solve problems involving the four operations, and identify and explain patterns in arithmetic.

**3: Cluster: (ID 41126)**Explore patterns of numbers.

**3: Cluster: (ID 41127)**Generalize place value understanding for multi-digit numbers.

**3: Cluster: (ID 74)**Reason with shapes and their attributes.

**3: Cluster: (ID 77)****3: Cluster: (ID 80)**Represent and interpret data.

**3: Cluster: (ID 83)****3: Cluster: (ID 93)****3: Cluster: (ID 95)**Use place value understanding and properties of operations to perform multi-digit arithmetic.

**3: Cluster: (ID 99)**Develop understanding of fractions as numbers.

**3: Curricular Indicator: (ID 40418)**No additional indicator(s) at this level. Mastery is expected at previous grade levels.

**3: Curricular Indicator: (ID 40424)**No additional indicator(s) at this level.

**3d.**Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

**3.D.1**Summarize, construct, and analyze data.

**3.D.1.1**Summarize and construct a data set with multiple categories using a frequency table, line plot, pictograph, and/or bar graph with scaled intervals.

**3.D.1.2**Solve one- and two-step problems using categorical data represented with a frequency table, pictograph, or bar graph with scaled intervals.

**3.DA.1**Create scaled picture graphs, scaled bar graphs, and frequency tables to represent a data set—including data collected through observations, surveys, and experiments—with several categories. Solve one- and two-step “how many more” and “how many less” problems regarding the data and make predictions based on the data.

**3.DA.2**Generate measurement data by measuring lengths with rulers to the nearest quarter of an inch. Display the data by making a line plot, where the horizontal scale is marked off in appropriate units, such as whole numbers, halves, or quarters.

**3: Domain: (ID 35104)**Operations & Algebraic Thinking

**3: Domain: (ID 35118)**Number & Operations in Base Ten

**3: Domain: (ID 35130)**Number & Operations—Fractions

**3: Domain: (ID 35142)**Measurement & Data

**3: Domain: (ID 35152)**Geometry

**3: Domain: (ID 35394)**Operations and Algebraic Thinking

**3: Domain: (ID 35408)**Number and Operations in Base Ten

**3: Domain: (ID 35420)**Number and Operations—Fractions

**3: Domain: (ID 35432)**Measurement and Data

**3: Domain: (ID 35442)**Geometry

**3: Domain: (ID 35792)**Operations & Algebraic Thinking

**3: Domain: (ID 35806)**Number & Operations in Base Ten

**3: Domain: (ID 35817)**Number & Operations—Fractions

**3: Domain: (ID 35829)**Measurement & Data

**3: Domain: (ID 35838)**Geometry

**3: Domain: (ID 36021)**Operations and Algebraic Thinking

**3: Domain: (ID 36022)**Number and Operations in Base Ten

**3: Domain: (ID 36023)**Number and Operations - Fractions

**3: Domain: (ID 36024)**Measurement and Data

**3: Domain: (ID 36025)**Geometry

**3: Domain: (ID 36506)**Operations & Algebraic Thinking

**3: Domain: (ID 36520)**Number & Operations in Base Ten

**3: Domain: (ID 36532)**Number & Operations—Fractions

**3: Domain: (ID 36544)**Measurement & Data

**3: Domain: (ID 36554)**Geometry

**3: Domain: (ID 36596)**Operations & Algebraic Thinking

**3: Domain: (ID 36610)**Number & Operations in Base Ten

**3: Domain: (ID 36622)**Number & Operations—Fractions

**3: Domain: (ID 36634)**Measurement & Data

**3: Domain: (ID 36644)**Geometry

**3: Domain: (ID 36784)**Operations and Algebraic Thinking

**3: Domain: (ID 36785)**Number and Operations in Base Ten

**3: Domain: (ID 36786)**Number and Operations – Fractions

**3: Domain: (ID 36787)**Measurement and Data

**3: Domain: (ID 36788)**Geometry

**3: Domain: (ID 37904)**Operations & Algebraic Thinking

**3: Domain: (ID 37906)**Number & Operations in Base Ten

**3: Domain: (ID 37907)**Measurement & Data

**3: Domain: (ID 37908)**Geometry

**3: Domain: (ID 37909)**Number & Operations—Fractions

**3: Domain: (ID 37996)**Operations & Algebraic Thinking

**3: Domain: (ID 38010)**Number & Operations in Base Ten

**3: Domain: (ID 38022)**Number & Operations—Fractions

**3: Domain: (ID 38034)**Measurement & Data

**3: Domain: (ID 38044)**Geometry

**3: Domain: (ID 38086)**Operations & Algebraic Thinking

**3: Domain: (ID 38100)**Number & Operations in Base Ten

**3: Domain: (ID 38112)**Number & Operations—Fractions

**3: Domain: (ID 38124)**Measurement & Data

**3: Domain: (ID 38134)**Geometry

**3: Domain: (ID 38573)**Operations & Algebraic Thinking

**3: Domain: (ID 38587)**Number & Operations in Base Ten

**3: Domain: (ID 38599)**Number & Operations—Fractions

**3: Domain: (ID 38611)**Measurement & Data

**3: Domain: (ID 38621)**Geometry

**3: Domain: (ID 38781)**Operations & Algebraic Thinking

**3: Domain: (ID 38795)**Number & Operations in Base Ten

**3: Domain: (ID 38807)**Number & Operations—Fractions

**3: Domain: (ID 38819)**Measurement & Data

**3: Domain: (ID 38829)**Geometry

**3: Domain: (ID 38848)**Operations and Algebraic Thinking

**3: Domain: (ID 38861)**Number and Operations in Base Ten

**3: Domain: (ID 38872)**Number and Operations—Fractions

**3: Domain: (ID 38883)**Measurement and Data

**3: Domain: (ID 38892)**Geometry

**3: Domain: (ID 38986)**Operations & Algebraic Thinking

**3: Domain: (ID 39000)**Number & Operations in Base Ten

**3: Domain: (ID 39012)**Number & Operations—Fractions

**3: Domain: (ID 39024)**Measurement & Data

**3: Domain: (ID 39034)**Geometry

**3: Domain: (ID 39053)**Operations & Algebraic Thinking

**3: Domain: (ID 39066)**Number & Operations in Base Ten

**3: Domain: (ID 39077)**Number & Operations—Fractions

**3: Domain: (ID 39088)**Measurement & Data

**3: Domain: (ID 39097)**Geometry

**3: Domain: (ID 39407)**Operations and Algebraic Thinking

**3: Domain: (ID 39413)**Number and Operations in Base Ten

**3: Domain: (ID 39416)**Number and Operations – Fractions

**3: Domain: (ID 39431)**Measurement and Data

**3: Domain: (ID 39438)**Geometry

**3: Domain: (ID 39484)**Operations & Algebraic Thinking

**3: Domain: (ID 39498)**Number & Operations in Base Ten

**3: Domain: (ID 39510)**Number & Operations—Fractions

**3: Domain: (ID 39522)**Measurement & Data

**3: Domain: (ID 39532)**Geometry

**3: Domain: (ID 40019)**Operations and Algebraic Thinking

**3: Domain: (ID 40020)**Number and Operations in Base Ten

**3: Domain: (ID 40021)**Number and Operations—Fractions

**3: Domain: (ID 40022)**Measurement and Data

**3: Domain: (ID 40023)**Geometry

**3: Domain: (ID 40227)**Number Sense and Operations in Base Ten

**3: Domain: (ID 40228)**Number Sense and Operations in Fractions

**3: Domain: (ID 40229)**Relationships and Algebraic Thinking

**3: Domain: (ID 40230)**Geometry and Measurement

**3: Domain: (ID 40231)**Data and Statistics

**3: Domain: (ID 40291)**Operations & Algebraic Thinking

**3: Domain: (ID 40305)**Number & Operations in Base Ten

**3: Domain: (ID 40317)**Number & Operations—Fractions

**3: Domain: (ID 40329)**Measurement & Data

**3: Domain: (ID 40339)**Geometry

**3: Domain: (ID 40612)**Operations & Algebraic Thinking

**3: Domain: (ID 40626)**Number & Operations in Base Ten

**3: Domain: (ID 40638)**Number & Operations—Fractions

**3: Domain: (ID 40650)**Measurement & Data

**3: Domain: (ID 40660)**Geometry

**3: Domain: (ID 40702)**Operations & Algebraic Thinking

**3: Domain: (ID 40716)**Number & Operations in Base Ten

**3: Domain: (ID 40728)**Number & Operations—Fractions

**3: Domain: (ID 40740)**Measurement & Data

**3: Domain: (ID 40750)**Geometry

**3: Domain: (ID 40790)**Operations & Algebraic Thinking

**3: Domain: (ID 40792)**Number & Operations in Base Ten

**3: Domain: (ID 40793)**Measurement & Data

**3: Domain: (ID 40794)**Geometry

**3: Domain: (ID 40795)**Number & Operations—Fractions

**3: Domain: (ID 40882)**Operations & Algebraic Thinking

**3: Domain: (ID 40896)**Number & Operations in Base Ten

**3: Domain: (ID 40908)**Number & Operations—Fractions

**3: Domain: (ID 40920)**Measurement & Data

**3: Domain: (ID 40930)**Geometry

**3: Domain: (ID 41154)**Measurement and Data

**3: Domain: (ID 41155)**Geometry

**3: Domain: (ID 41156)**Number and Operations – Fractions

**3: Domain: (ID 41157)**Operations and Algebraic Thinking

**3: Domain: (ID 41158)**Number and Operations in Base Ten

**3: Domain: (ID 41322)**Operations and Algebraic Thinking

**3: Domain: (ID 41323)**Number and Operations in Base Ten

**3: Domain: (ID 41324)**Measurement and Data

**3: Domain: (ID 41325)**Geometry

**3: Domain: (ID 41326)**Number and Operations - Fractions

**3: Domain: (ID 41466)**Operations and Algebraic Thinking

**3: Domain: (ID 41467)**Number and Operations in Base Ten

**3: Domain: (ID 41468)**Measurement and Data

**3: Domain: (ID 41469)**Geometry

**3: Domain: (ID 41470)**Number and Operations - Fractions

**3: Domain: (ID 41771)**Operations & Algebraic Thinking

**3: Domain: (ID 41785)**Number & Operations in Base Ten

**3: Domain: (ID 41797)**Number & Operations—Fractions

**3: Domain: (ID 41809)**Measurement & Data

**3: Domain: (ID 41819)**Geometry

**3: Domain: (ID 41901)**Numbers & Operations in Base Ten

**3: Domain: (ID 41902)**Numbers & Operations - Fractions

**3: Domain: (ID 41903)**Operations and Algebraic Thinking

**3: Domain: (ID 41905)**Geometry

**3: Domain: (ID 41907)**Measurement and Data

**3: Domain: (ID 41988)**Operations & Algebraic Thinking

**3: Domain: (ID 42002)**Number & Operations in Base Ten

**3: Domain: (ID 42014)**Number & Operations—Fractions

**3: Domain: (ID 42026)**Measurement & Data

**3: Domain: (ID 42036)**Geometry

**3: Domain: (ID 42352)**Operations & Algebraic Thinking

**3: Domain: (ID 42364)**Number & Operations in Base Ten

**3: Domain: (ID 42375)**Number & Operations—Fractions

**3: Domain: (ID 42386)**Measurement & Data

**3: Domain: (ID 42395)**Geometry

**3: Domain: (ID 42970)**Operations and Algebraic Thinking

**3: Domain: (ID 42971)**Number and Operations in Base Ten

**3: Domain: (ID 42972)**Number and Operations - Fractions

**3: Domain: (ID 42973)**Measurement and Data

**3: Domain: (ID 42974)**Geometry

**3: Domain: (ID 44046)**Operations & Algebraic Thinking

**3: Domain: (ID 44060)**Number & Operations in Base Ten

**3: Domain: (ID 44072)**Number & Operations—Fractions

**3: Domain: (ID 44084)**Measurement & Data

**3: Domain: (ID 44094)**Geometry

**3: Domain: (ID 44302)**Operations & Algebraic Thinking

**3: Domain: (ID 44316)**Number & Operations in Base Ten

**3: Domain: (ID 44328)**Number & Operations—Fractions

**3: Domain: (ID 44340)**Measurement & Data

**3: Domain: (ID 44350)**Geometry

**3: Domain: (ID 44541)**Operations and Algebraic Thinking

**3: Domain: (ID 44542)**Number and Operations in Base Ten

**3: Domain: (ID 44543)**Number and Operations - Fractions

**3: Domain: (ID 44544)**Measurement and Data

**3: Domain: (ID 44545)**Geometry

**3: Domain: (ID 44663)**Operations & Algebraic Thinking

**3: Domain: (ID 44677)**Number & Operations in Base Ten

**3: Domain: (ID 44689)**Number & Operations—Fractions

**3: Domain: (ID 44701)**Measurement & Data

**3: Domain: (ID 44711)**Geometry

**3: Domain: (ID 44753)**Operations & Algebraic Thinking

**3: Domain: (ID 44767)**Number & Operations in Base Ten

**3: Domain: (ID 44779)**Number & Operations—Fractions

**3: Domain: (ID 44791)**Measurement & Data

**3: Domain: (ID 44801)**Geometry

**3.DS.A**Represent and analyze data.

**3.DS.A.1**Create frequency tables, scaled picture graphs and bar graphs to represent a data set with several categories.

**3.DS.A.2**Solve one- and two-step problems using information presented in bar and/or picture graphs.

**3.DS.A.3**Create a line plot to represent data.

**3.DS.A.4**Use data shown in a line plot to answer questions.

**3.G**Geometry

**3.G**Geometry

**3.G**Geometry

**3.G**Geometry

**3.G**Grade 3 - Geometry

**3.G.1**Understand that shapes in different categories (e.g., rhombuses, rectangles, circles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

**3.G.1**Understand that shapes in different categories (*e.g. rhombuses, rectangles, trapezoids, kites and others*) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (*e.g. quadrilaterals*). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Refer to inclusive definitions noted in the glossary.

**3.G.1**Categorize shapes by different attribute classifications and recognize that shared attributes can define a larger category. Generalize to create examples or non-examples.

**3.G.1**Draw and describe triangles, quadrilaterals (rhombuses, rectangles, and squares), and polygons (up to 8 sides) based on the number of sides and the presence or absence of square corners (right angles).

**3.G.1**Understand that shapes in different categories (*for example, rhombuses, rectangles, and others*) may share attributes (*for example, having four sides*), and that the shared attributes can define a larger category (*for example, quadrilaterals*). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

**3.G.1**Identify and describe the following: cube, sphere, prism, pyramid, cone, and cylinder.

**3.G.1**Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).
Recognize rhombuses, rectangles, and squares as examples of quadrilaterals.
Draw examples of quadrilaterals that do not belong to any of these subcategories.

**3.G.1**Understand that shapes in different categories (e.g., rhombus, rectangle, square, and other 4-sided shapes) may share attributes (e.g., 4-sided figures) and the shared attributes can define a larger category (e.g., quadrilateral). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

**3.G.2**Understand that shapes (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize and draw rhombuses, rectangles, and squares as examples of quadrilaterals. Recognize and draw examples of quadrilaterals that do not belong to any of these subcategories.

**3.G.2**Partition shapes into parts with equal areas.
Express the area of each part as a unit fraction of the whole

**3.G.2**Partition two-dimensional shapes into 2, 3, 4, 6, or 8 parts with equal areas and express the area of each part using the same unit fraction. Recognize that equal parts of identical wholes need not have the same shape.