## Scope and Sequence

### Narrative

The big ideas in grade 3 include: developing understanding of multiplication and division and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions (fractions with numerator 1); developing understanding of the structure of rectangular arrays and of area; and describing and analyzing two-dimensional shapes.

The mathematical work for grade 3 is partitioned into 8 units:

1. Introducing Multiplication
2. Area and Multiplication
3. Wrapping Up Addition and Subtraction within 1,000
4. Relating Multiplication to Division
5. Fractions as Numbers
6. Measuring Length, Time, Liquid Volume, and Weight
7. Two-dimensional Shapes and Perimeter
8. Putting it All Together

### Unit 1: Introducing Multiplication

Unit Learning Goals
• Students represent and solve multiplication problems through the context of picture and bar graphs that represent categorical data.

In this unit, students interpret and represent data on scaled picture graphs and scaled bar graphs. Then, they learn the concept of multiplication.

This is the first of four units that focus on multiplication. In this unit, students explore scaled picture graphs and bar graphs as an entry point for learning about equal-size groups and multiplication.

In grade 2, students analyzed picture graphs in which one picture represented one object and bar graphs that were scaled by single units. Here, students encounter picture graphs in which each picture represents more than one object and bar graphs that were scaled by 2 or 5 units. The idea that one picture can represent multiple objects helps to introduce the idea of equal-size groups.

Students learn that multiplication can mean finding the total number of objects in $$a$$ groups of $$b$$ objects each, and can be represented by $$a \times b$$. They then relate the idea of equal groups and the expression $$a \times b$$ to the rows and columns of an array. In working with arrays, students begin to notice the commutative property of multiplication.

In all cases, students make sense of the meaning of multiplication expressions before finding their value, and before writing equations that relate two factors and a product.

Later in the unit, students see situations in which the total number of objects is known but either the number of groups or the size of each group is not known. Problems with a missing factor offer students a preview to division.

Throughout the unit, provide access to connecting cubes or counters, as students may choose to use them to represent and solve problems.

#### Section A: Interpret and Represent Data on Scaled Graphs

Standards Alignments
Section Learning Goals
• Interpret scaled picture and bar graphs.
• Represent data using scaled picture and bar graphs.
• Solve one- and two-step story problems using addition and subtraction.

In this section, students interpret and draw picture graphs and bar graphs to represent data, building on their experience with data representation and with skip-counting by 2, 5, and 10 in grade 2.

Students see that each picture in a picture graph and each line or increment in a bar graph can represent more than one object. They work with familiar number scales of 2, 5, and 10.

Students use the information in scaled bar graphs to solve one- and two-step “how many more” and “how many fewer” problems within 100. This work allows teachers to formatively assess students’ fluency with addition and subtraction within 100, a grade 2 expectation.

PLC: Lesson 3, Activity 1, So Many Responses

#### Section B: From Graphs to Multiplication

Standards Alignments
 Addressing 3.OA.A, 3.OA.A.1, 3.OA.A.3, 3.OA.A.4, 3.OA.D.9
Section Learning Goals
• Represent and solve multiplication problems involving equal groups.
• Understand multiplication in terms of equal groups.

In this section, students make sense of multiplication in terms of equal groups of objects. They use discrete drawings and tape diagrams that show equal groups to represent multiplication, and then relate these representations to expressions such as $$3 \times 2$$, interpreting them to mean “3 groups of 2.”

Note that expressions of the form $$a \times b$$ could be interpreted to mean $$a$$ groups of $$b$$ or $$b$$ groups of $$a$$. Because we tend to say “___ groups of ___” when referring to equal groups, however, in these materials we write multiplication expressions in that order:

$$\text{number of groups} \ \times \ \text{size of each group}$$

It is not necessary for students to use this convention as long as they can explain what each number in their expression represents.

Later, students write equations to represent multiplication situations and find unknown products or factors. In reasoning about the latter, they begin to make sense of the relationship between multiplication and division, without formally using the language of division.

PLC: Lesson 10, Activity 2, Card Sort: Equal Groups

#### Section C: Represent Multiplication with Arrays and the Commutative Property

Standards Alignments
 Addressing 3.MD.B.3, 3.OA.A, 3.OA.A.1, 3.OA.A.3, 3.OA.B.5, 3.OA.C.7, 3.OA.D.9
Section Learning Goals
• Represent and solve multiplication problems involving arrays.

In this section, students relate the idea of equal groups to the structure of an array, a representation introduced in grade 2.

Students see that the rows and columns of an array represent equal groups. The number of rows (or columns), the number of items in each row (or column), and the total number of objects in an array can therefore be represented with a multiplication equation. The equations may involve an unknown value, be it one of the factors or the product. As students reason about arrays, they also notice that multiplication is commutative.

Estimated Days: 20 - 21

### Unit 2: Area and Multiplication

Unit Learning Goals
• Students learn about area concepts and relate area to multiplication and to addition.

In this unit, students encounter the concept of area, relate the area of rectangles to multiplication, and solve problems involving area.

In grade 2, students explored attributes of shapes, such as number of sides, number of vertices, and length of sides. They measured and compared lengths (including side lengths of shapes).

In this unit, students make sense of another attribute of shapes: a measure of how much a shape covers. They begin informally, by comparing two shapes and deciding which one covers more space. Later, they compare more precisely by tiling shapes with pattern blocks and square tiles. Students learn that the area of a flat figure is the number of square units that cover it without gaps or overlaps.

Students then focus on the area of rectangles. They notice that a rectangle tiled with squares forms an array, with the rows and columns as equal-size groups. This observation allows them to connect the area of rectangles to multiplication—as a product of the number of rows and number of squares per row.

To transition from counting to multiplying side lengths, students reason about area using increasingly more abstract representations. They begin with tiled or gridded rectangles, move to partially gridded rectangles or those with marked sides, and end with rectangles labeled with their side lengths.

$$6\times 3=18$$

Students also learn some standard units of area—square inches, square centimeters, square feet, and square meters—and solve real-world problems involving area of rectangles.

Later in the unit, students find the area and missing side lengths of figures composed of non-overlapping rectangles. This work includes cases with two non-overlapping rectangles sharing one side of the same length, which lays the groundwork for understanding the distributive property of multiplication in a later unit.

#### Section A: Concepts of Area Measurement

Standards Alignments
 Addressing 3.MD.C.5, 3.MD.C.5.a, 3.MD.C.5.b, 3.MD.C.6, 3.OA.A.1
Section Learning Goals
• Describe area as the number of unit squares that cover a plane figure without gaps and overlaps.
• Measure the area of rectangles by counting unit squares.

In this section, students reason about area as an attribute of two-dimensional shapes and develop a sense of area as the amount of space covered by a shape.

They begin by considering how to show or explain a shape as being larger or smaller than another. Next, they see that they can quantify the size of shapes more precisely by covering them with units of the same size, such as pattern blocks or square tiles.

Students then learn that the area of a shape is the number of squares that covers it with no gaps or overlaps. To find the number of square tiles used to cover a space, students may skip-count or use multiplication.

PLC: Lesson 1, Activity 1, Compare Shapes

#### Section B: Relate Area to Multiplication

Standards Alignments
Section Learning Goals
• Explain why the area of a rectangle can be determined by multiplying the side lengths.
• Solve problems involving the area of rectangles.

In this section, students relate the area of rectangles to multiplication expressions.

Students see equal-size groups in rectangles that are tiled with squares. They learn to express the area of rectangles as a product of two numbers that describe the equal groups. For example, in a rectangle that is 8 units by 4 units, students see 8 groups of 4 or 4 groups of 8. The product of the two numbers, $$8 \times 4$$ or $$4 \times 8$$, gives the number of squares that covers a rectangle completely with no gaps or overlaps.

Use of the structure of a rectangle enables students to transition from gridded rectangles to rectangles showing only side lengths (MP7). The progression in visual representations matches the progression in strategies for reasoning about area: moving from concrete (counting) to abstract (finding products of two numbers).

In this section, students also learn about standard units of area in inches, feet, centimeters, and meters. They explore these units in the context of real-world and mathematical problems.

PLC: Lesson 5, Activity 1, Match Expressions and Areas

#### Section C: Find Area of Figures Composed of Rectangles

Standards Alignments
 Addressing 3.MD.C.5, 3.MD.C.6, 3.MD.C.7.b, 3.MD.C.7.d, 3.NBT.A.2
Section Learning Goals
• Find the area of figures composed of rectangles.

In this section, students encounter figures composed of non-overlapping rectangles and find their area.

As with the rectangles in earlier lessons, students see increasingly abstract diagrams, starting with figures that are fully gridded, moving to those with a partial grid, and ending with figures showing only side lengths and no grid. The progression encourages students to decompose the figures and use multiplication to reason about area. The work here highlights the additive nature of area.

Students also use their understanding of rectangles (that opposite sides are equal) to find missing side lengths in figures composed of rectangles.

PLC: Lesson 12, Activity 2, Find the Rectangles

Estimated Days: 13 - 15

### Unit 3: Wrapping Up Addition and Subtraction Within 1,000

Unit Learning Goals
• Students use place value understanding to round whole numbers and add and subtract within 1,000. They also represent and solve two-step word problems using addition, subtraction, and multiplication and assess the reasonableness of answers.

In this unit, students work toward the goal of fluently adding and subtracting within 1,000. They use mental math strategies developed in grade 2 and learn algorithms based on place value.

In grade 2, students added and subtracted within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction. When students combine hundreds, tens, and ones, they use place value understanding. When they decompose numbers to add or subtract, they rely on the commutative and associative properties. When students count up to subtract, they use the relationship between addition and subtraction.

To move toward fluency, students learn a few different algorithms that work with any numbers and are generalizable to larger numbers and decimals. Students work with a variety of algorithms, starting with those that show expanded form, and moving toward algorithms that are more streamlined and closer to the standard algorithm.

Students explore various algorithms but are not required to use a specific one. They should, however, move from strategy-based work of grade 2 to algorithm-based work to set the stage for using the standard algorithm in grade 4. If students begin the unit with knowledge of the standard algorithm, it is still important for them to make sense of the place-value basis of the algorithm.

Understanding of place value also comes into play as students round numbers to the nearest multiple of 10 and 100. Students do not need to know a formal definition of “multiples” until grade 4. At this point, it is enough to recognize that a multiple of 10 is a number called out when counting by 10, or the total in a whole-number of tens (such as 8 tens). Likewise, a multiple of 100 is a number called out when counting by 100, or the total in a whole-number of hundreds (such as 6 hundreds). Students use rounding to estimate answers to two-step problems and determine if answers are reasonable.

#### Section A: Add Within 1,000

Standards Alignments
Section Learning Goals
• Fluently add within 1,000 using algorithms based on place value and properties of operations.
• Use place value understanding to compose and decompose numbers.

Students begin this section by revisiting the idea of place value, reasoning about different ways to decompose numbers within 1,000, and using familiar strategies from grade 2 to add and subtract within 1,000.

From there, they progress toward more abstract addition strategies, but ones that are still based on place value. To support this progression toward algorithms, students use base-ten blocks or diagrams, express numbers in expanded form, and rely on their understanding of properties of operations. For example, here are three ways to add $$362+354$$:

Students look for and make use of structure as they relate the compositions of numbers, expressions, and base-ten blocks or diagrams to find sums and differences (MP7).

PLC: Lesson 4, Activity 1, What is an Algorithm?

#### Section B: Subtract Within 1,000

Standards Alignments
Section Learning Goals
• Fluently subtract within 1,000 using algorithms based on place value, properties of operations, and the relationship between addition and subtraction.

In this section, students analyze and use subtraction algorithms. They begin by using base-ten blocks and diagrams to subtract numbers. Because it is difficult to record regrouping using drawings, however, they see algorithms as a helpful way to find differences.

As is the case with addition, students first make sense of a subtraction algorithm that uses expanded form, which allows them to see how the hundreds and tens are decomposed into smaller units.

This non-conventional notation allows students to see the meaning behind the digits used above the numbers in the standard algorithm.

PLC: Lesson 10, Activity 1, A New Subtraction Algorithm

#### Section C: Round Within 1,000

Standards Alignments
Section Learning Goals
• Round whole numbers to the nearest multiple of 10 and 100.

In this section, students learn the conventions of rounding whole numbers to the nearest multiple of 10 or 100. This work relies on and reinforces their understanding of place value. Number line diagrams are used to help students think about the multiple of 10 or 100 to which a given number is closest.

Students learn that when we find the nearest multiple of 10 or 100, we are rounding “to the nearest ten” or rounding “to the nearest hundred.” They also see that rounding a number to the nearest ten and nearest hundred can produce the same result.

Students explore how rounding to the nearest ten or hundred can change the estimate of a sum. This prepares them to use rounding to see if solutions to problems are reasonable in the next section.

PLC: Lesson 13, Activity 2, Close to Multiples of 100

#### Section D: Solve Two-Step Problems

Standards Alignments
Section Learning Goals
• Assess the reasonableness of answers.
• Solve two-step word problems using addition, subtraction, and multiplication.

In this section, students encounter more complex problems, think about the reasonableness of their answers, and use rounding to make estimates.

Students analyze tape diagrams that could represent the relationships in given situations and write corresponding equations to represent them. Previously, they worked with diagrams and equations with a ? or ___ to represent an unknown. Now, students interpret and write letters to stand for an unknown number.

$$124-(2\times10) = {n}$$

Finally, students apply what they’ve learned about adding and subtracting within 1,000 to solve two-step word problems that involve multiplication, addition, and subtraction.

PLC: Lesson 17, Activity 2, Solve and Reason

Estimated Days: 20 - 21

### Unit 4: Relating Multiplication to Division

Unit Learning Goals
• Students learn about and use the relationship between multiplication and division, place value understanding, and the properties of operations to multiply and divide whole numbers within 100. They also represent and solve two-step word problems using the four operations.

This unit introduces students to the concept of division and its relationship to multiplication.

Previously, students learned that multiplication can be understood in terms of equal-size groups. The expression $$5 \times 2$$ can represent the total number of objects when there are 5 groups of 2 objects, or when there are 2 groups of 5 objects.

Here, students make sense of division also in terms of equal-size groups. For instance, the expression $$30 \div 5$$ can represent putting 30 objects into 5 equal groups, or putting 30 objects into groups of 5. They see that, in general, dividing can mean finding the size of each group, or finding the number of equal groups.

30 objects put into 5 equal groups

30 objects put into groups of 5

Students use the relationship between multiplication and division to develop fluency with single-digit multiplication and division facts. They continue to reason about products of two numbers in terms of the area of rectangles whose side lengths represent the factors, decomposing side lengths and applying properties of operations along the way.

As they multiply numbers greater than 10, students see that it is helpful to decompose the two-digit factor into tens and ones and distribute the multiplication. For instance, to find the value of $$26 \times 3$$, they can decompose the 26 into 20 and 6, and then multiply each by 3.

Toward the end of the unit, students solve two-step problems that involve all four operations. In some situations, they work with expressions that use parentheses to indicate which operation is completed first (for example: $$276 + (45 \div 5) = {?}$$).

#### Section A: What is Division?

Standards Alignments
Section Learning Goals
• Represent and solve “how many groups?” and “how many in each group?” problems.

In this section, students encounter situations involving the questions “how many in each group?” and “how many groups?” They make sense of division in terms of finding the answers to these questions.

The focus here is on interpreting descriptions, diagrams, and expressions that represent division situations. Students see that the same diagram or expression can represent different questions. For example, the expression $$6 \div 2$$ can represent two different questions about 6 blocks being put into stacks of 2 or into 2 equal stacks.

Later, students generalize their observations about division situations and interpret division expressions without a context.

PLC: Lesson 4, Activity 3, Stacks of Blocks

#### Section B: Relate Multiplication and Division

Standards Alignments
 Addressing 3.MD.C.7.c, 3.NBT.A.3, 3.OA.A.2, 3.OA.A.3, 3.OA.B.6, 3.OA.C.7, 3.OA.D.9
Section Learning Goals
• Understand division as a missing-factor problem.
• Use properties of operations to develop fluency with single-digit multiplication facts, and their related division facts.

In this section, students explicitly relate division to the missing factor in a multiplication equation. For example, the quotient in $$30 \div 6 = \underline{\hspace{1 cm}}$$ is the missing factor in $$\underline{\hspace{1 cm}} \times 6 = 30$$. They use this insight and their knowledge of multiplication facts to identify division facts.

To develop fluency, students reason about patterns in a multiplication table and notice that multiplication is commutative. For instance, if they know the value of $$4 \times 7$$, they also know that of $$7 \times 4$$.

Students also reason about the product of two factors by decomposing one of the factors. For instance, to find the value of $$7 \times 3$$, they can decompose the 7 into 5 and 2 and find the value of $$(5 \times 3) + (2 \times 3)$$. Visually, the product can be represented by the area of a 7-by-3 rectangle that has been decomposed into two rectangles that are 5 by 3 and 2 by 3.

This line of reasoning develops students' intuition for the distributive property of multiplication. (Note that students are not expected to know the names of the properties of operations.)

PLC: Lesson 7, Activity 1, Division Round Table

#### Section C: Multiplying Larger Numbers

Standards Alignments
 Addressing 3.MD.C.7.c, 3.NBT.A.3, 3.OA.A.3, 3.OA.B.5, 3.OA.D.8
Section Learning Goals
• Use properties of operations and place value understanding to develop strategies to multiply within 100 and to multiply one-digit numbers by a multiple of 10.

In this section, students use various strategies based on place value and properties of operations to multiply larger numbers.

Students first multiply one-digit numbers and multiples of 10 and observe the associative property of multiplication. They interpret $$3 \times 20$$ to mean 3 groups of 2 tens, which is 6 tens. This means $$3 \times 20$$ can be evaluated by finding $$3 \times 2 \times 10$$ or $$6 \times 10$$.

These insights enable students to then multiply other one- and two-digit factors (not limited to multiples of 10) and find products within 100.

The representations used here (base-ten blocks, gridded rectangles, and ungridded diagrams) encourage students to also use their understanding of place value and to decompose two-digit factors into tens and ones as they multiply.

PLC: Lesson 14, Activity 1, A Factor Greater than Ten

#### Section D: Dividing Larger Numbers

Standards Alignments
 Addressing 3.MD.C.7, 3.NBT.A.3, 3.OA.A.2, 3.OA.A.3, 3.OA.A.4, 3.OA.B.5, 3.OA.C.7, 3.OA.D.8
Section Learning Goals
• Use properties of operations, place value understanding, and the relationship between multiplication and division to divide within 100.

In this section, students perform division in which the quotient or divisor is larger than 10. They apply what they know about place value, the two interpretations of division, and the relationship between multiplication and division to divide larger numbers.

The numbers in the division expressions encourage students to see the divisor as either the number of groups or the number in each group. For example, they may interpret $$57 \div 3$$ to mean dividing 57 into 3 equal groups. However, given $$90 \div 15$$, students may make groups of 15 and see how many are needed to make 90. This flexibility helps students choose methods that are most efficient for them for any given problem.

Students also use the relationship between multiplication and division and place value understanding to find quotients. For instance, to find the value of $$78 \div 3$$, students may reason as follows:

$$3 \times 10 = 30$$
$$3 \times 10 = 30$$
$$3 \times 6 = 18$$
$$10 + 10 + 6 = 26$$

$$3 \times 20 = 60$$
$$3 \times 6 = 18$$
$$20 + 6 = 26$$

In both cases, students see that there are 3 groups of 26 in 78.

PLC: Lesson 21, Activity 1, Apple Adventure

Estimated Days: 21 - 22

### Unit 5: Fractions as Numbers

Unit Learning Goals
• Students develop an understanding of fractions as numbers and of fraction equivalence by representing fractions on diagrams and number lines, generating equivalent fractions, and comparing fractions.

In this unit, students make sense of fractions as numbers, using various diagrams to represent and reason about fractions, compare their size, and relate them to whole numbers. The denominators of the fractions explored here are limited to 2, 3, 4, 6, and 8.

In grade 2, students partitioned circles and rectangles into equal parts and used the language “halves,” “thirds,” and “fourths.” Students begin this unit in a similar way, by reasoning about the size of shaded parts in shapes. Next, they create fraction strips by folding strips of paper into equal parts and later represent the strips as tape diagrams.

Using fraction strips and tape diagrams to represent fractions prepare students to think about fractions more abstractly: as lengths and locations on the number line. This work builds on students’ prior experience with representing whole numbers on the number line.

In each representation, students take care to identify 1 whole. This helps them reason about the size of the parts and whether a fraction is less or greater than 1. (Fractions greater than 1 are not treated as special cases.)

Students then use these representations to learn about equivalent fractions and to compare fractions.

They see that fractions are equivalent if they are the same size or at the same location on the number line, and that some fractions are the same size as whole numbers.

$$3 = \frac{12}{4}$$

Later in the unit, students compare fractions with the same denominator and those with the same numerator. They recognize that as the numerator gets larger, more parts are being counted, and as the denominator gets larger, the size of each part in a whole gets smaller.

#### Section A: Introduction to Fractions

Standards Alignments
Section Learning Goals
• Understand that fractions are built from unit fractions such that a fraction $\frac{a}{b}$ is the quantity formed by $a$ parts of size $\frac{1}{b}$.
• Understand that unit fractions are formed by partitioning shapes into equal parts.

In this section, students use shaded diagrams and fraction strips to learn about fractions, building on their prior knowledge of halves, thirds, and fourths.

Students partition rectangles into 6 or 8 equal parts and describe each part as “a sixth” or “an eighth” and write the notation $$\frac{1}{6}$$ or $$\frac{1}{8}$$.

They learn that the notation $$\frac{1}{b}$$ refers to a unit fraction, or the size of each part if the whole is partitioned into $$b$$ parts. Working with fraction strips allows students to see non-unit fractions as being composed of unit fractions, so $$a$$ parts of unit fractions of size $$\frac{1}{b}$$ gives a non-unit fraction $$\frac{a}{b}$$.

For example, putting together 3 pieces of fourths or 3 parts of the unit fraction $$\frac{1}{4}$$ gives $$\frac{3}{4}$$.

$$\frac{3}{4}$$

As students develop their understanding, they make connections between the meaning, language, and notation of fractions—between what fractions represent and how they are expressed in words and in numbers. (The terminology “numerator” and “denominator” are not introduced until later so students can focus on meaning making.)

PLC: Lesson 3, Activity 1, Write and Read Fractions

#### Section B: Fractions on the Number Line

Standards Alignments
 Addressing 3.NF.A.2, 3.NF.A.2.a, 3.NF.A.2.b, 3.NF.A.3.c, 3.OA.C.7
Section Learning Goals
• Understand a fraction as a number and represent fractions on the number line.

In this section, students reason about fractions on the number line. This work relies on two prior experiences: locating whole numbers on the number line, and partitioning a whole into equal parts.

Students have previously learned that numbers can be represented as distances from 0 on the number line. Here, students learn that the same is true about fractions. Students begin by partitioning the interval between 0 and 1 into equal parts, just as they had done with fraction strips and tape diagrams.

They then mark the first tick mark with a unit fraction $$\frac{1}{b}$$ and locate non-unit fractions by counting lengths the size of $$\frac{1}{b}$$. They reason that a tick mark that is $$a$$ intervals away represents a fraction $$\frac{a}{b}$$. The terms “numerator” and “denominator” are introduced here.

Students also notice that certain fractions are in the same location as whole numbers on the number line. For example, $$\frac{4}{4}$$ and $$\frac{8}{4}$$ are at the same location as 1 and 2, respectively. This observation helps students understand that whole numbers can be represented as fractions.

PLC: Lesson 7, Activity 2, Fractions on the Number Line

#### Section C: Equivalent Fractions

Standards Alignments
Section Learning Goals
• Explain equivalence of fractions in special cases and express whole numbers as fractions and fractions as whole numbers.

In this section, students learn that equivalent fractions are fractions that are the same size.

They first identify equivalent fractions by noticing parts that are of equal length on fraction strips and tape diagrams.

For example, the shaded third in the first diagram is the same size as the two shaded sixths in the second diagram, so $$\frac{1}{3}$$ and $$\frac{2}{6}$$ are equivalent.

Students see that they can show equivalence by decomposing each fractional part into smaller parts, or by grouping fractional parts to make larger parts.

Suppose we want to show that the shaded parts of this diagram represent both $$\frac{6}{8}$$ and $$\frac{3}{4}$$.

If we group 2 eighths together, we have 4 equal groups, each being a fourth. We can see that the 6 shaded eighths and 3 shaded fourths are the same size.

Later, students learn that equivalent fractions are the same distance away from 0 and are therefore located at the same point on the number line. They write equations to express equivalence, including for fractions that are equivalent to whole numbers.

$$\frac{6}{8} = \frac{3}{4}$$

PLC: Lesson 12, Activity 2, Locate and Pair

#### Section D: Fraction Comparisons

Standards Alignments
Section Learning Goals
• Compare two fractions with the same numerator or denominator, record the results with the symbols >, =, or

In this section, students compare fractions using any representation or reasoning strategies that make sense to them. They learn that comparisons are only valid if the fractions being compared refer to the same whole.

Students begin by deciding if two fractions are equivalent. They use diagrams, number lines, and the meaning of fractions to support their reasoning.

Next, students compare fractions with the same denominator. They see that these fractions are composed of parts of the same size, so to compare them involves looking at the numerators to see which fraction has more parts.

For example, there are 4 sixths in $$\frac{4}{6}$$ and 5 sixths in $$\frac{5}{6}$$, so $$\frac{4}{6}$$ is less than $$\frac{5}{6}$$. On the number line, $$\frac{4}{6}$$ would to the left of $$\frac{5}{6}$$, closer to 0.

$$\frac{4}{6} < \frac{5}{6}$$

In contrast, fractions with the same numerator have the same number of parts, so to compare them involves looking at the denominators to see which fraction is made up of larger parts.

For instance, 5 sixths is greater than 5 eighths because a sixth is larger than an eighth.

$$\frac{5}{6} > \frac{5}{8}$$

The work here reinforces the idea that as the denominator increases, the size of each part gets smaller.

PLC: Lesson 17, Activity 1, Comparison Problems

Estimated Days: 17 - 18

### Unit 6: Measuring Length, Time, Liquid Volume, and Weight

Unit Learning Goals
• Students generate and represent length measurement data in halves and fourths of an inch on line plots. They learn about and estimate relative units of measure including weight, liquid volume, and time, and use the four operations to solve problems involving measurement.

In this unit, students measure length, weight, liquid volume, and time. They begin with a study of length measurement, building on their recent work with fractions.

In grade 2, students measured lengths using informal and formal units to the nearest whole number. They plotted length data on line plots. Here, students explore length measurements in halves and fourths of an inch. They use a ruler to collect measurements and then display the data on line plots, learning about mixed numbers and revisiting equivalent fractions along the way.

Kiran says that the worm is $$4\frac{2}{4}$$ inches long.
Jada says that the worm is $$4\frac{1}{2}$$ inches long.
Use the ruler to explain how both of their measurements are correct.

Next, students learn about standard units for measuring weight (kilograms and grams) and liquid volume (liters). To build a sense of weights such as 1 gram or 1 kilogram, students hold common objects such as paper clips and bottles of water.

To gain familiarity with liters, they fill a container with water by the liter and estimate the volume of everyday containers such as pots, tubs, and buckets. They then use the scale on measurement tools to measure and represent liquid volume.

From there, students move on to measure time. In grade 2, they told and wrote time to the nearest 5 minutes. Now, they tell time to the minute, using the relationship between the hour hand and the minute hand to make sense of times such as 3:57 p.m.

In the final section of the unit, students make sense of and solve problems related to all three measurements. The work here allows students to continue to develop their fluency with addition and subtraction within 1,000 and understanding of properties of operations. It also prompts them to use the relationship between multiplication and division to solve problems.

#### Section A: Measurement Data on Line Plots

Standards Alignments
Section Learning Goals
• Measure lengths using rulers marked with halves and fourths of an inch to generate data for making a line plot.

In this section, students learn to measure lengths in fractions of an inch—first in halves of an inch, and then fourths of an inch. They partition rulers with whole-number inch marks into equal intervals and then use them to measure lengths of objects in the classroom.

Students learn that measurements that are greater than 1 can be expressed with mixed numbers, which combine a whole number and a fraction less than 1.

As they measure with greater levels of precision, students revisit the idea of equivalent fractions. They see that the half-inch marks are also two-fourths of an inch, and that each whole number of inches can also be expressed as some number of halves or fourths.

Students then use their understanding of the number line and rulers to interpret and create line plots that represent lengths measured in half inches and quarter inches. They see that all three representations—number lines, rulers, and line plots—have the same structure, which shows whole-number intervals being partitioned into equal parts.

PLC: Lesson 3, Activity 1, Halves and Quarters

#### Section B: Weight and Liquid Volume

Standards Alignments
Section Learning Goals
• Measure and estimate weights and liquid volumes of objects.

In earlier grades, students learned that weight is a measurable attribute and directly compared the weights of two objects. In this section, they learn that weight is a measure of how heavy something is and that grams and kilograms are units for measuring weight.

To establish some benchmarks for weights, they hold objects of different numbers of grams and kilograms. Then, they estimate the weight of other objects relative to those benchmarks.

Next, students learn that liquid volume is the amount of space that a liquid takes up. They first use informal units (such as plastic cups, spoons, and so on) to compare the liquid volume that two containers can hold before learning about liters as a unit for measurement.

Students gain concrete experience with the new unit by filling a large container in 1-liter increments. They also estimate the liquid volume of everyday objects such a sink, a bucket, and a bathtub.

Later, students make sense of fractional units of liquid volume, learn to read the scale on liquid measurement tools (such as beakers), and compare the scales to the marks on rulers.

PLC: Lesson 7, Activity 2, Liquid Volume in Liters

#### Section C: Problems Involving Time

Standards Alignments
Section Learning Goals
• Solve problems involving addition and subtraction of time intervals in minutes.
• Tell time to the minute.

In this section, students learn to tell and write time to the nearest minute and to show given time on an analog clock. They also solve elapsed time problems with an unknown start time, unknown duration, or unknown end time.

Han got on the bus:

Han got off the bus:

For how many minutes was Han on the bus?

To reason about the problems, students can use any representation that makes sense to them, such as tables, words, equations, or marks on a clock. They also examine a variety of reasoning strategies and adjust their approach depending on the problem at hand.

Elena arrived at the bus stop at 3:45 p.m.
She waited 24 minutes for her bus to arrive.
What time did the bus arrive?
Show your thinking. Organize it so it can be followed by others.

As they solve problems, students continue to build their fluency with multiplication (especially multiples of 5, 10, and 15), addition, and subtraction.

PLC: Lesson 10, Activity 1, Time at the Bus Stop

#### Section D: Measurement Problems in Context

Standards Alignments
 Addressing 3.MD.A.1, 3.MD.A.2, 3.NBT.A.2, 3.OA.A.3, 3.OA.C.7
Section Learning Goals
• Solve problems involving the four operations and measurement contexts.

In this section, students solve problems that involve measurements of weight, liquid volume, and time in the context of a state or county fair. The problems prompt students to use all four operations: addition and subtraction within 1,000, and multiplication and division within 100.

The problems prompt students to make sense of the situations and the questions being asked, consider information that might be needed to answer questions. They explain why they need that information and may need to ask different questions if their partner does not have the information requested (MP1). In each situation, students make sense of quantities and their relationships (MP2).

At one point during the growing season, a giant pumpkin gained 12 kilograms per day for 7 days.
How much weight did the pumpkin gain during that week?

An optional lesson at the end of the section gives students a chance to examine carnival games and design a game that incorporates concepts of measurement and operations.

PLC: Lesson 12, Activity 1, Giant Pumpkin Event

Estimated Days: 15 - 16

### Unit 7: Two-dimensional Shapes and Perimeter

Unit Learning Goals
• Students reason about shapes and their attributes, with a focus on quadrilaterals. They solve problems involving the perimeter and area of shapes.

In this unit, students reason about attributes of two-dimensional shapes and learn about perimeter.

Students began to describe, compare, and sort two-dimensional shapes in earlier grades. Here, they continue to do so and to develop language that is increasingly more precise to describe and categorize shapes. Students learn to classify broader categories of shapes (quadrilaterals and triangles) into more specific sub-categories based on their attributes. For instance, they study examples and non-examples of rhombuses, rectangles, and squares, and come to recognize their specific attributes.

These are rectangles.

These are not rectangles.

Students also expand their knowledge about attributes that can be measured.

Previously, they learned the meaning of area and found the area of rectangles and figures composed of rectangles. In this unit, students learn the meaning of perimeter and find the perimeter of shapes. They consider geometric attributes of shapes (such as opposite sides having the same length) that can help them find perimeter.

Find the perimeter of this rectangle.

As the lessons progress, they consider situations that involve perimeter, and then those that involve both perimeter and area. These lessons aim to distinguish the two attributes (which are commonly confused) and reinforce that perimeter measures length or distance (in length units) and area measures the amount of space covered by a shape (in square units).

At the end of the unit, students solve problems in a variety of contexts. They apply what they learn about geometric attributes of shapes, perimeter, and area, to design a park, a West African wax print pattern, and a robot. They then solve problems within the context of their design.

#### Section A: Reason with Shapes

Standards Alignments
Section Learning Goals
• Reason about shapes and their attributes.

In this section, students describe, compare, and sort a variety of shapes. They have previously used terms such as square, rectangle, triangle, quadrilateral, pentagon, and hexagon to name shapes. Here, students think about ways to further categorize triangles and quadrilaterals. They see that triangles and quadrilaterals can be classified based on their sides (whether some are of equal length) and their angles (whether one or more right angles are present).

Although students will not learn the formal definition of an angle until grade 4, they are introduced to the terms “angle in a shape” and “right angle in a shape” to describe the corners of shapes. This allows students to distinguish right triangles and to describe defining attributes of squares and rectangles.

These are right triangles.

These are not right triangles.

What makes a shape a right triangle?

Students come to understand that a shape can have more than one name if it has the attributes that define different types of shapes. They also see that some quadrilaterals aren’t squares, rhombuses, or rectangles because they don’t have the defining attributes of these shapes.

For example, here are three quadrilaterals. The first one is a rectangle, a rhombus, and a square. The other two are not squares, rhombuses, or rectangles.

PLC: Lesson 4, Activity 1, What Makes These Shapes So?

#### Section B: What is Perimeter?

Standards Alignments
Section Learning Goals
• Find the perimeter of two-dimensional shapes, including when all or some side lengths are given.

In this section, students are introduced to the idea of perimeter. Students begin to conceptualize perimeter as a measurable geometric attribute with a concrete experience: using paper clips to build the boundary of shapes and using the length of a paper clip as the unit for measuring the distance around each shape.

From there, they transition to analyzing shapes with equal-size intervals marked on their sides or shapes drawn on dot paper or grid paper. They quantify the distance around the shape by counting the intervals or adding the number of units on each side.

Later, students find the perimeter of shapes labeled with their side lengths. They learn to leverage the geometric attributes of shapes to find perimeter more efficiently (for instance, by recognizing sides that are the same length and using multiplication).

As they find the perimeter of shapes, students see that different shapes can have the same perimeter and draw shapes with a specified perimeter. Finally, students find missing side lengths of shapes given the perimeter and solve perimeter problems in context.

This pentagon has a perimeter of 32 cm. What is the length of the missing side?

​​​​​​

PLC: Lesson 8, Activity 1, Ways to Find Perimeter

#### Section C: Expanding on Perimeter

Standards Alignments
Section Learning Goals
• Solve problems involving perimeter and area, in and out of context.

In this section, students analyze the area and perimeter of shapes. They begin by solving contextual problems that require considerations of both measurements. They then draw rectangles with the same perimeter and different areas, and rectangles with the same area and different perimeters.

Students come to see that, given the perimeter of a rectangle, they can find rectangles with different whole-number areas. Likewise, given the area, they can find rectangles with different perimeters.

rectangles with a perimeter of 16 units

rectangles with an area of 24 square units

PLC: Lesson 11, Activity 2, Same Perimeter, Different Area

#### Section D: Design with Perimeter and Area

Standards Alignments
Section Learning Goals
• Apply geometric understanding to solve problems.

In this section, students apply what they’ve learned about shapes, geometric attributes, perimeter, and area to solve problems and create designs in different contexts.

Students begin by designing a small park with certain features and then finding the area and perimeter of the park. Next, they examine geometric features in West African wax print patterns and then design their own pattern. Finally, students use their knowledge of area and perimeter to create a drawing of a robot whose parts are rectangles with a certain area or perimeter.

Throughout these activities, students draw on dot paper and use the intervals between dots as a unit of measurement.

PLC: Lesson 14, Activity 1, Create a Wax Print Pattern

Estimated Days: 15

### Unit 8: Putting It All Together

Unit Learning Goals
• Students consolidate and solidify their understanding of various concepts and skills related to major work of the grade. They also continue to work toward fluency goals of the grade.

In this unit, students revisit major work and fluency goals of the grade, applying their learning from the year.

In section A, students reinforce what they learned about fractions, their size, and their location on the number line. In section B, students deepen their understanding of perimeter, area, and scaled graphs by solving problems about measurement and data. Two of the lessons invite students to design a tiny house that meet certain conditions and calculate the cost for furnishing it.

Section C enables students to work toward multiplication and division fluency goals through games. In the final section, students review major work of the grade as they create activities in the format of the warm-up routines they have encountered throughout the year (Notice and Wonder, Estimation Exploration, Number Talk, and How Many Do You See?).

How many do you see? How do you see them?

The concepts and skills strengthened in this unit prepare students for major work in grade 4: comparing, adding, and subtracting fractions, multiplying and dividing within 1,000, and using the standard algorithm to add and subtract multi-digit numbers within 1 million.

The sections in this unit are standalone sections, not required to be completed in order. Within each section, many lessons can also be completed independently of the ones preceding them. The goal is to offer ample opportunities for students to integrate the knowledge they have gained and to practice skills related to the expected fluencies of the grade.

#### Section A: Fraction Fun

Standards Alignments
Section Learning Goals
• Understand a fraction as a number and represent fractions on the number line.

In this section, students revisit and build on important fraction ideas that they have learned in the course. They think about different ways to represent fractions and to estimate the size of fractions presented in different forms: as an area diagram, a shaded strip, and a number line.

What fraction of the square is shaded?

What number is represented by the point on the number line?

Students also practice identifying and locating fractions on the number line, using tape to create a number line that shows a large number of fractions.

Later in the section, students consolidate their understanding by analyzing general statements about fractions (for instance, “a fraction is a number less than 1,” or “whole numbers are fractions”). They express their agreement or disagreement with the statements and have the chance to revise their thinking upon discussions with others.

PLC: Lesson 3, Activity 1, Fractions Round Table

#### Section B: Measurement and Data

Standards Alignments
 Addressing 3.MD.B.3, 3.MD.C.7.b, 3.MD.C.7.d, 3.MD.D.8, 3.NBT.A.2, 3.OA.D.8
Section Learning Goals
• Apply concepts of measurement and data to solve problems.

In this section, students further investigate ideas on measurement (area and perimeter) and data (scaled graphs).

Students begin by analyzing features of tiny houses. They then use their knowledge of shapes, perimeter, and area to design their own tiny house, and then write questions about the area and perimeter of shapes in their design.

Later, students apply their knowledge of addition and subtraction to calculate the cost of finishing a room in their tiny house.

In the second half of the section, students focus on data collection and representation. They think about survey questions to ask others in the class and in the school, how to present the answer choices, and how to collect and record a large set of data.

Students then conduct their survey in the school community, organize their data, and represent the data with a scaled graph. They also ask and answer questions about the data.

Write questions that could be answered with your bar graph by completing these sentences.

How many more students liked _________ than _________ ?

How many fewer students liked _________ than _________ ?

PLC: Lesson 4, Activity 1, Design a Tiny House

#### Section C: Multiplication and Division Games

Standards Alignments
Section Learning Goals
• Develop fluency with single-digit multiplication facts and their related division facts.

In this section, students continue their work of building fluency with multiplication and division. They begin by reflecting on the products within 100 they know from memory or can find quickly and the ones they don’t know yet. Students then practice multiplication facts (focusing on the ones that are least familiar to them) through games.

Next, students reinforce their understanding of the connections between multiplication and division by matching equations and diagrams that represent the same quantities and relationships. For instance, the equations $$56 \div 7 = {?}$$ and $${?} \times 7 = 56$$, and a diagram of a rectangle with an area of 56 and a side length of 7 can all describe the same situation.

Here is another example that shows different representations of multiplication and division:

Which one doesn’t belong?

Students then play games to improve their facility with multiplication and division. They revisit familiar center activities and learn new ones. Compare, Rectangle Rumble, and How Close? are the centers used in this section.

PLC: Lesson 10, Activity 1, Card Sort: Find the Match

#### Section D: Create and Design

Standards Alignments
Section Learning Goals
• Review the major work of the grade by creating and designing instructional routines.

Throughout the course, students have engaged in warm-up routines such as How Many Do You See, Exploration Estimation, Which One Doesn’t Belong, True or False, and Number Talk. This section enables them to apply the mathematics they have learned (the four operations, fractions, and measurement, in particular) to design warm-ups that incorporate some of these routines.

Each lesson is devoted to a particular routine. Students begin by completing partially created tasks. They practice anticipating responses that others might give to the prompts they pose.

What do you notice? What do you wonder?

Along the way, students gain the skills and insights needed to create an activity from scratch or with minimal scaffolding. In each lesson, students have the option to facilitate their activity with another group in the class.

PLC: Lesson 13, Activity 1, Design Your How Many Do You See

Estimated Days: 15