Narrative

Students begin the course with a study of sequences, which is also an opportunity to revisit linear and exponential functions. Students represent functions in a variety of ways while addressing some aspects of mathematical modeling. This work leads to looking at situations that are well modeled by polynomials before pivoting to a study of the structure of polynomial graphs and expressions. Students do arithmetic on polynomials and rational functions and use different forms to identify asymptotes and end behavior. Students also study polynomial identities and use some key identities to establish the formula for the sum of the first $$n$$ terms of a geometric sequence.

Next, students extend exponent rules to include rational exponents. They solve equations involving square and cube roots before developing the idea of $$i$$, a number whose square is $$\text-1$$, expanding the number system to include complex numbers. This allows them to solve quadratic equations with non-real solutions.

Building on rational exponents, students return to their study of exponential functions and establish that the property of growth by equal factors over equal intervals holds even when the interval has non-integer length. They use logarithms to solve for unknown exponents, and are introduced to the number $$e$$ and its use in modeling continuous growth. Logarithm functions and some situations they model well are also briefly addressed.

Students learn to transform functions graphically and algebraically. In previous courses and units, students adjusted the parameters of particular types of models to fit data. Here, they consolidate and generalize this understanding. This work is useful in the study of periodic functions that comes next. Students work with the unit circle to make sense of trigonometric functions and use those functions to model periodic relationships.

The last unit, on statistical inference, focuses on analyzing data from experiments using normal distributions. Students learn to account for variability in data and estimate population mean, margin of error, and proportions using sampling and simulations. They develop skepticism about news stories that summarize data inappropriately.

Within the classroom activities, students have opportunities to engage in aspects of mathematical modeling. Additionally, modeling prompts are provided for use throughout the course. Modeling prompts offer opportunities for students to engage in the full modeling cycle. These can be implemented in a variety of ways. Please see the course guide for a more detailed explanation of modeling prompts.

Unit 1: Sequences and Functions

This unit provides an opportunity to revisit representations of functions (including graphs, tables, and expressions) at the beginning of the Algebra 2 course, using the example of a sequence as a particular type of function. Through many concrete examples, students learn to identify geometric and arithmetic sequences. They see them as examples of the exponential and linear functions they learned about in previous courses, defined by a subset of the integers.

Students begin with an invitation to describe sequences with informal language. They write out the terms of sequences arising from mathematical situations, in addition to interpreting and creating tables and graphs about the given relationship. Students learn that sequences are a type of function in which the input variable is the position and the output variable is the term at that position. They learn to interpret and then write their own definitions for sequences recursively using function notation.

Expressions for the $$n^{\text{th}}$$ term of a sequence are built up through expressing regularity in repeated reasoning (MP8), building on students’ prior experiences studying linear and exponential functions. For example, the geometric sequence 6, 18, 54, 162, . . . could be written as $$6, 6\boldcdot3, 6\boldcdot3\boldcdot3, 6\boldcdot3\boldcdot3\boldcdot3, . . .$$ which makes it clearer to see that the $$n$$th term can be defined by $$f(n)=6 \boldcdot 3^{n-1}$$, assuming we start at $$f(1)=6.$$

In the last part of the unit, students use sequences to model several situations represented in different ways (MP4). This isn’t meant to be full-blown modeling, but to touch on some practices that must be attended to while modeling, such as choosing a good model, identifying an appropriate domain, or expressing numbers with an appropriate level of precision given the situation. Students also recognize that a sequence is an appropriate type of function to use as a model for these situations since the domain of each is a subset of the integers. Finally, students encounter some situations in which it makes sense to compute the sum of a finite sequence. Developing a formula for such a sum occurs in a future unit.

Unit 2: Polynomials and Rational Functions

In previous courses, students learned about linear and quadratic functions. They rewrote expressions for these functions in different forms to reveal structure and identified key features of their graphs, such as intercepts. In this unit, students will expand their earlier work as they investigate polynomials of higher degree and the features that all polynomial functions have in common.

The unit begins with an introduction to some situations polynomial functions are good at modeling. Students learn to identify the degree of a polynomial and gain exposure to what graphs of polynomials can look like as they build intuition for what features these graphs can and cannot have. Some aspects of graphs brought out in this unit, such as symmetry and end behavior, will continue to be important in a later unit when students compare square and cube functions in order to understand why there can be two or zero square roots of a number but only one cube root.

Focusing on functions expressed in factored form and their graphs, students make connections between linear factors and horizontal intercepts, identifying that a factor of $$(x-a)$$ means $$a$$ is a zero of the function and $$(a,0)$$ is a horizontal intercept. The effect of the degree and leading coefficient on end behavior is established along with the effect of multiplicity on the shape of the graph near zeros of the function. Taking in all of these features, students learn to make rough sketches of polynomial functions expressed as a product of linear factors.

Embedded throughout the first half of the unit are opportunities for students to practice multiplying polynomials and some optional review of factoring. This practice is meant to help pave the way for understanding division, which in this unit focuses on dividing a polynomial written in standard form by a suspected factor for the purposes of rewriting the equation in factored form. From there, the connection between division and multiplication equations is used to establish the Remainder Theorem. This allows the conclusion that not only does a known factor of the form $$(x-a)$$ mean a polynomial has a zero at $$x=a$$, but that if a polynomial has a zero at $$x=a$$, then it must also have $$(x-a)$$ as a factor.

Students transition to working with rational functions by considering situations they model, such as average cost. The asymptotic behavior of their graphs is examined as it relates to the structure of the equation. Also building on structure, students use polynomial division to rewrite rational expressions for the purpose of identifying the end behavior of the function. Students then focus on solving rational equations and making sense of how the process can lead to possible solutions that are in fact not solutions (so-called extraneous solutions).

In the final section students study polynomial identities. They hone skills manipulating polynomial expressions while proving, or disproving, that two expressions are equivalent. The unit concludes with a return to geometric sequences first examined in the previous unit and, using polynomial identities, students derive the formula for the sum of the first $$n$$ terms in a geometric sequence before using the formula to solve problems.

Unit 3: Complex Numbers and Rational Exponents

In an earlier grade, students learned various techniques for solving quadratic equations including solving by inspection (for example, solving $$x^2=49$$ by knowing that -7 and 7 are the two numbers that square to make 49), finding square roots, graphing, completing the square, the quadratic formula, and factoring. Students have also worked with radicals including $$\sqrt{}$$ and $$\sqrt[3]{}$$ in various geometric contexts and worked with expressions involving integer exponents to establish exponent rules.

In this unit, students use what they know about exponents and radicals to extend exponent rules to include rational exponents (for example, $$5^{\frac{1}{3}}=\sqrt[3]{5}$$), solve various equations involving squares and square roots, develop the concept of complex numbers by defining a new number $$i$$ whose square is -1, and use complex numbers to find solutions to quadratic equations.

The first set of lessons in the unit provides an opportunity for students to review what they know about exponent rules and radicals, and extend those rules to make sense of expressions with rational exponents. Students eventually add the rule $$x^{\frac{a}{b}}=\sqrt[b]{x^a}$$, where $$a$$ and $$b$$ are whole numbers, to all the other exponent rules they know. The first two lessons are optional and are designed to remind students of the properties of positive and negative integer exponents, and the geometric meaning of square and cube roots.

In the next set of lessons, students connect the $$\sqrt{}$$ and $$\sqrt[3]{}$$ symbols with solutions to quadratic and cubic equations. Students learn that a number is a square root of $$c$$ if it squares to make $$c$$. In other words, square roots of $$c$$ are solutions to the equation $$x^2=c$$. Students use the graph of $$y=x^2$$ to see that all positive numbers have two square roots, one positive and one negative. They learn the convention that the positive square root is given the symbol $$\sqrt{}$$, so the positive square root of $$c$$ is written $$\sqrt{c}$$ and the negative square root is written $$\text-\sqrt{c}$$.

Similarly, students use the graph of $$y=x^3$$ to see that all numbers, positive or negative, have a single cube root, and so the solution to the equation $$x^3=c$$ is written as $$\sqrt[3]{c}$$. Students solve equations like $$\sqrt{x}=c$$ to find they have one solution if $$c$$ is positive and no solutions if $$c$$ is negative because of the definition of the $$\sqrt{}$$ symbol (a positive number cannot be equal to a negative number).

Critically, students use this connection between square roots and solutions to $$x^2=c$$ to understand that squaring each side of an equation can sometimes introduce new solutions that aren't solutions to the original equation, and that applying the $$\sqrt{}$$ symbol on each side of an equation ignores the existence of negative square roots. This is important in later lessons when students must take care to account for both positive and negative square roots in the process of solving quadratic equations. The last lesson of this section is optional, and it gives students an opportunity for extra practice solving these kinds of equations, as well as constructing their own examples that have different numbers of solutions. Note that there are claims in these lessons like, “All numbers have exactly one cube root,” which would be more precisely stated as “All real numbers have exactly one real cube root.” However, students don’t know about any numbers other than real numbers, so it doesn't make sense to make this distinction until they expand their concept of number to include imaginary and complex numbers in subsequent lessons.

The next few lessons introduce imaginary and complex numbers. The number line is renamed the real number line, and students observe that there are no real numbers that square to make negative numbers. The symbol $$\sqrt{\text-1}$$ is used to define a number that squares to make -1, that is, $$\sqrt{\text-1}$$ is defined as a solution to the equation $$x^2=\text-1$$. Since no real numbers are solutions to this equation, this number is represented as a point off of the real number line. This leads to the construction of the imaginary axis and the complex plane.

The symbol $$\sqrt{\text-1}$$ is quickly renamed $$i$$, and students find that negative real numbers also have two square roots, one on the positive imaginary axis, and one on the negative imaginary axis. In other words, if $$c$$ is a negative number, then the equation $$x^2=c$$ has two solutions, $$i\sqrt{|c|}$$ and $$\text-i\sqrt{|c|}$$. Just as the $$\sqrt{}$$ symbol is defined to describe the positive square root of a real number, students learn the convention that if $$c$$ is a negative number, the notation $$\sqrt{c}$$ is taken to mean the square root of $$c$$ on the positive imaginary axis, $$i\sqrt{|c|}$$. Students then use the fact that $$i^2 = \text- 1$$ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers to express them in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. The section ends with an optional practice lesson on complex arithmetic and powers of $$i$$, and then a lesson that wraps up what students have learned about multiplying complex numbers.

In the final few lessons, students use what they have learned about square roots and complex numbers to solve quadratic equations that have complex solutions. Students practice completing the square and using the quadratic formula. The first lesson of this section is optional review of these techniques. Sometimes these methods appear to produce different solutions, for example one method may give a solution that includes $$\frac{\sqrt{32}}{2}$$ while another method produces $$\sqrt{8}$$ instead. Students use what they know about square roots to determine whether solutions are equivalent.

Unit 4: Exponential Functions and Equations

Students were introduced to exponential functions in an earlier course. This unit begins by activating students’ prior knowledge. Students recall that an exponential function involves a change by equal factors over equal intervals and can be expressed as $$f(x)=a \boldcdot b^x$$, where $$a$$ is the initial value of the function (the value when $$x$$ is 0), and $$b$$ is the growth factor. They review the use of verbal descriptions, tables, and graphs to represent exponential functions.

The next few lessons extend students’ ability to write, interpret, and compare exponential functions. Previously, students saw exponential functions defined for integer inputs. In this unit, they extend that work to include exponential functions defined for all real number inputs. Students first studied the meaning of rational exponents in a previous unit. Here, they write and interpret exponential functions evaluated at rational number inputs in context. For example, if $$f(w)=200 \boldcdot \left(\frac12\right)^w$$ represents the area in square meters of a pond covered by algae $$w$$ weeks after an algae-control treatment is applied, then $$f(\frac17)=200 \boldcdot \left(\frac12\right)^\frac17$$ represents the area covered after 1 day ($$\frac17$$ of a week). They see that, aside from using graphs, they can use properties of exponents to estimate or find the value of a function when the input is a rational number.

This work allows students to identify growth factors over fractional intervals of input, for example, to find the annual growth factor of a population given its growth factor every decade. It also enables them to write expressions to highlight different aspects of the same situation. For instance, if $$d$$ is time in days, $$f(d)=7\boldcdot (3)^{\frac15 d}$$ shows a quantity that starts at 7 and triples every 5 days. The growth factor per day is $$3^\frac15$$, which is approximately 1.25, or $$1 + 0.25$$. Since $$7 \boldcdot (3)^{\frac{1}{5} d}$$ is approximately equivalent to $$7\boldcdot (1+0.25)^d$$, we can conclude that the population is growing by roughly 25% each day.

In the second half of the unit, students learn that logarithms are a way to express the exponent that makes an exponential equation true. For example, if the expression $$2^y$$ has a value of 32, we can reason that the exponent $$y$$ is 5, but we can also write $$\log_2 32$$ to express the value of $$y$$. They see that value of $$y$$ that makes the equation $$2^y=32$$ true can be written as $$y=\log_2 32$$, and that these two equations are equivalent. Students then learn to solve exponential equations using logarithms, working at first mainly with base 2 and 10.

Next, students encounter the constant $$e$$. They learn that it is irrational, its value is approximately 2.7, and it is used in many exponential functions that model real-life situations with a continuous growth rate. Students make sense of exponential functions of the form $$f(t)=P \boldcdot e^{rt}$$, interpret them in context, and graph them. (Students are not expected to build exponential functions with base $$e$$ in this course.) They also learn that they can express the solution to exponential equations in base $$e$$ using the natural logarithm, connecting to earlier work and reinforcing the idea that $$e$$ is just a number. Students then explore several situations that are modeled with an exponential equation using base $$e$$. They solve problems about the situations both algebraically, using logarithms to rewrite equations, and graphically.

The last few lessons expose students to logarithmic functions in base 2 and base 10. Students analyze the graphs, interpret them in context, and use them to answer questions about real-life situations such as population growth, acidity of substances, and intensity of earthquakes. Logarithmic functions are not studied in depth in this course.

Unit 5: Transformations of Functions

Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. The purpose of this unit is for students to consider functions as a whole and understand how they can be transformed to fit the needs of a situation, which is an aspect of modeling with mathematics (MP4). An important takeaway of the unit is that we can transform functions in a predictable manner using translations, reflections, scale factors, and by combining multiple functions. Throughout the unit students analyze graphs, tables, equations, and contexts as they work to connect representations and understand the structure of different transformations (MP7).

The unit begins with students informally describing transformations of graphs, eliciting their prior knowledge and establishing language that will be refined throughout the unit. Students consider the graphs of two possible functions as fits for a data set and make an argument about why one is a better fit (MP3). Students return to this data set in future lessons as they learn more ways to transform a given equation to fit data.

The first types of transformations students consider are vertical and horizontal translations. While these types of transformations have been studied briefly for specific function types, such as quadratics, here they are studied for all function types. In parallel with their study of the effect of translations on graphs and tables, students learn to write equations for functions that are defined in terms of another to describe transformations using function notation.

Next, students investigate how transformations such as reflections across the horizontal and vertical axes are defined using function notation and make connections to the same topic from geometry. These ideas are expanded to consider the properties of even functions, odd functions, and functions that are neither even nor odd from both a graphical and algebraic perspective.

From translations and reflections, students move on to explore the effect of multiplying the output or input of a function by a scale factor. They fit quadratic functions to parabolic arches in photos in order to better understand how to “squash” or “stretch” outputs. Students consider the change in height over time of a rider on different Ferris wheels as another application of scale factors as they contrast multiplying outputs against multiplying inputs of functions. The use of clear and precise language is emphasized as students make sense of the effects of different scale factors (MP6).

In a future unit, students use their knowledge of transformations to transform trigonometric functions to model a variety of periodic situations. By saving the introduction of trigonometric functions until after a study of transformations, students have the opportunity to revisit transformations from a new perspective which reinforces the idea that all functions, even periodic ones, behave the same way with respect to translations, reflections, and scale factors.

Unit 6: Trigonometric Functions

In this unit, students are introduced to trigonometric functions. While they have previously studied a variety of function types with different key features, this is the first time students are asked to consider periodic functions, that is, functions whose output values repeat at regular intervals. This unit also builds directly on the work of the previous unit by having students apply their knowledge of transformations to trigonometric functions and use these functions to model periodic situations.

The unit opens with a study of the motion of clock hands. Using the structure of the coordinate plane with the clock centered at the origin allows students to quantify the motion of the end of a clock hand as time passes. Focusing on quadrant 1, students first use the Pythagorean Theorem and then right triangle trigonometry to determine the coordinates of the end of clock hand. Students recall ratios for cosine, sine, and tangent, focusing on the important fact that when the hypotenuse has unit length, the length of the legs can be expressed with cosine and sine.

The ability to find the coordinates of a point 1 unit away from the origin using cosine, sine, and an angle leads to the development of the unit circle. Students return to radian angle measurements, which they studied in a previous course, and apply them to the geometry of the unit circle. They study specific radian angles and learn to use the symmetry of the unit circle. Students learn that the cosine and sine of an angle $$\theta$$ can be defined as the $$x$$- and $$y$$-coordinates, respectively, of the point on the unit circle corresponding to $$\theta$$. This work also leads to establishing the Pythagorean Identity and defining the tangent of an angle using the unit circle.

Once students have a working understanding of the unit circle and its relationship to trigonometry, they are ready to transition to thinking about cosine and sine as functions. Using the unit circle, students graph cosine and sine and identify common features between the two functions. They expand the domain of cosine and sine to all real numbers as they learn the meaning of radian angles greater than $$2\pi$$ and less than 0. This work naturally leads to expanding the domain of the tangent function, but, as with rational functions, students reason about the input values where the tangent function does not exist and how the output values repeat at regular intervals, leading to the tangent function’s periodic nature.

Next, students apply their previous work with transformations of graphs to trigonometric functions. They relate vertical translations to the midline of the graph and vertical stretches to the amplitude. Students apply their understanding of horizontal scale factors to sine and cosine, learning, both algebraically and geometrically, how these transformations affect the period.

Finally, students apply trigonometric functions to model different situations. Some of these involve motion in a circle, such as examining a point on a Ferris wheel or carousel. But there are also periodic or approximately periodic phenomena that are not directly associated with position on a circle, such as tides and the amount of the moon that is visible over the course of a month. These investigations bring out different aspects of the modeling cycle.

A note on the unit circle and technology as a reference tool: early in this unit students create their own unit circles showing the angles and coordinates of 24 points evenly spaced around the circle. This display is meant to be a reference tool for students throughout the unit as they transition from a right triangle-focused understanding of trigonometry to seeing cosine, sine, and tangent as functions with their own inputs and outputs. Students should have access to a unit circle display throughout the unit unless otherwise noted.

Unit 7: Statistical Inferences

All statistics are based on a question about a population of interest, but conducting a census of a population is often impossible due to the constraints of time and money. So, instead, we collect data from a sample. In order for the data to be useful, it must be representative of the population in question. To assure this is true, we use a process of randomization to reduce the amount of bias in our sample selection. We use the sample to create statistical measures, but it is important to also measure by how much we expect this value to vary if we had chosen a different random sample. We often use a margin of error to estimate how much we expect the statistic to vary. Lastly, we can make inferences using the statistics and margin of error we find about the overall population based on the sample if it is representative and normalized. The normal distribution allows us to gain additional information about the proportions that commonly occur with samples of this standardized shape. These inferences allow us to answer questions we have about the population of interest in a way that is both cost and time effective.

In grade 7, students examined processes for collecting samples from a population and using information from the samples to estimate characteristics for the population. In this unit, students expand on this idea by exploring the normal distribution and applying their understanding of the distribution to provide estimates with a margin of error. The unit also examines experimental studies, observational studies, and surveys. For experimental studies, students use a method for analyzing the data using a randomization distribution along with normal distributions.

The unit begins with an exploration of statistical questions and the type of study that is used to answer different kinds of questions. In particular, it emphasizes the importance of random selection for gathering a sample for surveys and observational studies and the importance of random assignment in experimental studies.

The unit then transitions to an analysis of the data collected by examining the shapes of distributions and focusing on the normal distribution as a common and standardized shape. Data with an approximately normal shape can be modeled by a normal distribution to gain additional information such as the proportion of data expected within certain intervals.

The unit concludes with ways to analyze the results from experimental studies. Data from surveys and observational studies using random samples are used to estimate population means and proportions with a margin of error. This analysis is based on the understanding of the normal distribution students gained in the previous section.

Students ultimately collect data from an experiment involving their heart rates and analyze the data using randomization distributions. Again, an understanding of the normal distribution is used, this time to determine whether the experimental data is likely due to the chance arrangement of the groups or the experimental treatment.