Narrative

Students begin the course with one-variable statistics, building on ideas from middle school. Starting with data collection and analysis sets a tone for the course of understanding quantities in context. It also allows students to access grade-level mathematics that isn't as dependent on prior skills as some other topics. Gathering and displaying data, measuring data distribution, and interpreting statistical results encourages students to collaborate, communicate, and explore new tools and routines.

From there, students move on to expand their understanding of linear equations, inequalities, and systems of linear equations and inequalities. They use these representations to model relationships and constraints but also reason with them abstractly. Students write, rearrange, evaluate, and solve equations and inequalities, explaining and validating their reasoning with increased precision. They then take these insights to a unit on two-variable statistics, where they extend their prior knowledge of scatter plots and lines of best fit. Students use residuals and correlation coefficients to assess linear models, interpret quantitative data, and distinguish correlation and causality. They also determine associations in categorical data, by using two-way tables and relative frequencies.

Next, students study functions, continuing the work begun in grade 8. Over the next few units, they deepen their understanding of functions and deepen their ability to represent, interpret, and communicate about them—using function notation, domain and range, average rate of change, and features of graphs. They also see categories of functions, starting with linear functions (including their inverses) and piecewise-defined functions (including absolute value functions), followed by exponential and quadratic functions. For each function type, students begin their investigation with real-world and mathematical contexts, look closely at the structural attributes of the function, and analyze how these attributes are expressed in different representations.

The course ends with a close look at quadratic equations. Students extend their ability to use equations to model relationships and solve problems. They develop their capacity to write, transform, graph, and solve equations—by reasoning, rearranging equations into useful forms, and applying the quadratic formula. In solving quadratic equations students encounter rational and irrational solutions, providing an opportunity to deepen their understanding of the real number system.

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: One-variable Statistics

In grades 6–8, students used data displays (histograms, dot plots, and box plots) as a way to summarize data. These data displays are revisited in this unit, but with a focus on interpretation and what they reveal about the data in addition to the mechanics of constructing the data displays.

Students were introduced to measures of center (mean and median) and measures of variability (mean absolute deviation and interquartile range) in grades 6–8. Understanding of the mean absolute deviation (MAD) provides a foundation for the standard deviation, which is a more widely used measure of variability introduced in this unit.

The first five lessons of the unit give students an opportunity to review ideas from middle school while taking the analysis of the data displays a little deeper. The lessons build on student understanding gained in middle school grades of statistical variability, ability to describe distributions, and informally comparing distributions. They represent and interpret data using data displays such as dot plots, histograms, and box plots. They describe distributions using the appropriate terminology such as “symmetric,” “skewed,” “uniform,” “bimodal,” and “bell-shaped.” They create data displays and calculate summary statistics using technology, then interpret the values in context. They recognize a relationship between the shape of a distribution and the mean and median. They compare data sets with different measures of variability and interpret data sets with greater MADs or interquartile ranges as having greater variability.

Lessons 6 through 9 familiarize students with spreadsheets and technology that will be used to calculate statistics such as mean, median, quartiles, and standard deviation as well as create data displays. The familiarity gained with technology will continue to help students throughout upcoming units as they use the tools to explore algebraic structures and functions throughout this course.

Geogebra’s spreadsheets were chosen for their versatility for the on-level mathematics in this course. While other spreadsheet programs have additional functionality and uses, they are limited in other ways (such as creating histograms, dot plots, or box plots and computing quartiles correctly). That said, please adapt the materials to the needs of your students.

By the end of lesson 9, students should be able to create and interpret data displays such as histograms, dot plots, and box plots as well as calculate and interpret measures of center (mean and median) and measures of variability (interquartile range and mean absolute deviation). Since most of these concepts address ideas from middle school, a mid-unit assessment isn't included with this unit. Teachers may wish to use the information gained from cool-downs and practice problems to informally assess student understanding of these concepts before proceeding to the remaining lessons of the unit.

Lessons 10 through 15 explore standard deviation, outliers, and comparing data sets using measures of center and measures of variability. They learn that standard deviation is a measure of variability, and they interpret standard deviation in context. (Note that this unit only includes population standard deviation. Standard deviation based on a sample appears in a unit in later grades.) They recognize outliers, investigate their source, make decisions about excluding them from the data set, and understand how the presence of outliers impacts measures of center and measures of variability. They compare measures of center and the standard deviation and the interquartile range for different data sets.

The last lesson gives students a chance to practice their skills by collecting data and analyzing the values. In the culminating activity, students pose and answer a statistical question by designing an experiment, collecting data, and analyzing data.

Unit 2: Linear Equations, Inequalities, and Systems

In middle school, students began building an understanding of how variables, expressions, equations, and inequalities could be used to represent quantities and relationships. Students also made connections among different kinds of representations—algebraic, verbal, tabular, and graphical. In this unit, students further develop their capacity to create, manipulate, interpret, and connect these representations and to use them for modeling.

In the first few lessons, students learn to think of equations as a way to represent constraints or limitations on quantities. (For instance, if the cost of food, $$f$$, and the cost of drinks, $$d$$, for a party add up to \$80, we can write $$f + d = 80$$ to represent this constraint.) Students understand that when we solve equations, we are looking for values that satisfy the constraints and make the equations true. (For example, $$f=53$$ and $$d=27$$ could be a pair of solutions to $$f + d = 80$$, but $$f=50$$ and $$d=35$$ could not be.) Students also see that graphs of equations can help us make sense of constraints and identify values that satisfy them. Students then investigate different ways to express the same relationship or constraint—by analyzing and writing equivalent equations. They look at moves that can transform one equation to an equivalent equation, recognizing that these are the moves we make to solve equations. The focus here is not only on identifying acceptable moves for solving, but also on explaining why these moves keep each subsequent equation true and maintain the solutions of the original equation. Along the way, students realize that some equations are more helpful than others, depending on what we want to know. In some equations, the quantity of interest is easy to pin down. In others, we may need to manipulate the equation and solve for a particular variable. Students also explore how the form and the parts of a linear equation in two variables are related to the features of its graph. They see that understanding the structure and connections across representations can give us deeper insights about the situation being studied. Next, students encounter situations that involve two or more constraints. In those cases, we often want to find values that satisfy both or all constraints simultaneously. Systems of equations are helpful for representing these constraints. (The work here is limited to systems of linear equations in two variables.) Students draw on their understanding of systems of linear equations from grade 8 to solve problems, but soon notice the limitations of solving systems by graphing and by substitution. They then learn to solve systems of equations by elimination, to explain why the steps taken to eliminate a variable are valid and productive, and to articulate how the process essentially entails writing a series of equivalent systems. Additionally, students reinforce their awareness that a system of equations could have one solution, no solutions, or infinitely many solutions. In the last third of the unit, students rely on their understanding of equations to explore inequalities in one and two variables. They see that inequalities are a handy way to express constraints that involve an upper or lower limit, and can be satisfied by a range of values rather than a single value. (For instance, if the weekly work hours, $$h$$, of an employee must be at least 40 or $$h\geq40$$, any value that is 40 or greater meets this constraint.) Students see that a solution to an inequality (in one or two variables) is a value or a pair of values that makes the inequality true, and a solution to a system of inequalities in two variables is any pair of values that make both inequalities in the system true. The solution set of a system of inequalities, they learn, can be best represented by graphing. Unit 3: Two-variable Statistics In grade 8, students informally constructed scatter plots and lines of fit, noticed linear patterns, and observed associations in categorical data using two-way tables. In this unit, students build on this previous knowledge by assessing how well a linear model matches the data using residuals as well as the correlation coefficient for best-fit lines (found using technology). The unit also revisits two-way tables to find associations in categorical data using relative frequencies. The unit begins with categorical data arranged in two-way tables that students are asked to analyze. Students then examine the relative frequencies of the combinations of those categorical variables. Students find the relative frequencies for combinations relative to the whole data set, as well as row or column relative frequencies to look at subgroups within categories. The row and column relative frequency tables are ultimately used to find evidence to determine if any associations are present in the data. The unit then transitions to bivariate numerical data, which are visualized using scatter plots and lines of best fit. Students use technology to compute the lines of best fit and observe how well the linear models match the data. Residuals and correlation coefficients are used to quantify the goodness of fit for linear models. The unit closes with an exploration of the difference between correlation and causal relationships, as well as an opportunity to apply this learning to areas of interest, like anthropology and sports. Unit 4: Functions In grade 8, students learned that a function is a rule that assigns exactly one output to each input. They represented functions in different ways—with verbal descriptions, algebraic expressions, graphs, and tables—and used functions to model relationships between quantities, linear relationships in particular. In this unit, students expand and deepen their understanding of functions. They develop new knowledge and skills for communicating about functions clearly and precisely, investigate different kinds of functions, and hone their ability to interpret functions. Students also use functions to model a wider variety of mathematical and real-world situations. The unit opens with a refresher on what functions are and what they are not. Students use descriptions, tables, and graphs to reason about the idea of “exactly one output for each input.” Then, students learn that function notation is an efficient way to communicate succinctly about functions and devote some focused time to interpret this new notation and use it. They continue this work throughout the unit, employing the notation to perform increasingly sophisticated mathematical work: to analyze and compare functions, to write rules of functions (primarily linear functions), to solve for an input, to graph functions, and more. Next, students focus their attention on graphs of functions and on how they help to tell stories about the relationships between the quantities in the functions. Students interpret features of graphs and relate them to features of situations, using terms such as “maximum,” “minimum,” and “intercepts” to describe their observations. From a graph, students can see intervals where the values of a function increase or decrease. They learn to use average rates of change to more precisely describe how quickly these values rise or fall. Students also sketch graphs to depict qualitative behavior of functions. Students then go on to take a closer at look at the input and output of a function. They think about possible and reasonable input and output values and learn to identify the domain and range of a function based on contextual and graphical information. This new awareness of input and output in turn helps students make sense of piecewise-defined functions, in which different rules apply to different intervals of the domain, producing different sets of output values. Two variations of piecewise functions are studied here: step functions and absolute value functions. The latter are introduced with the idea of absolute errors as an entry point. Thinking about “how far away from a value” primes students to regard the absolute value function as a distance function. The graph of such a function is a distinct V shape, which is convenient for noticing the graphical effects of changing an expression that defines a function. Later in the unit, students continue to mind inputs and outputs as they explore inverse functions. Message encryption and decryption, as well as currency exchange, provide helpful contexts for getting a feel for inverse functions, as both involve a two-way process—doing and undoing. Students see that, while an equation that defines a function is useful for finding output values, an equation for its inverse is useful for finding its input values. The work here focuses on finding the inverse of linear functions. Students close the unit by applying their insights about functions to model real-world situations and solve problems. In subsequent units and courses, students will use what they learned here to study exponential, quadratic, logarithmic, and periodic functions. Unit 5: Introduction to Exponential Functions Before starting this unit, students are familiar with linear functions from previous units in this course and from work in grade 8. They have been formally introduced to functions and function notation and have explored the behaviors and traits of both linear and non-linear functions. Additionally, students have spent significant time graphing, interpreting graphs, and exploring how to compare the graphs of two linear functions to each other. In this unit, students frequently use the properties of exponents, a topic developed in grade 8. They also apply their understanding of percent change from grade 7 and use an exponent to express repeated increase or decrease by the same percentage. In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. Students investigate these exponential relationships without using function notation and language so that they can focus on gaining an appreciation for critical properties and characteristics of exponential relationships. Students subsequently view these new types of relationships as functions and employ the notation and terminology of functions (for example, dependent and independent variables). They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don’t represent a particular context, observing the effect of different values of $$a$$ and $$b$$ on the graph of the function $$f$$ represented by $$f(x)=ab^x$$ The context of credit (both in terms of loans and savings) is used through several lessons to: • contextualize a percent change applied repeatedly • make a distinction between (for example) applying 10% increase followed by another 10% increase versus applying a 20% increase to the original amount • strategically write and interpret expressions and relate them back to a context • write equivalent expressions in a different way to highlight a different aspect of the situation In this unit, students learn that the output of an increasing exponential function is eventually greater than the output of an increasing linear function for the same input. In a later unit, students are introduced to quadratic functions. At that time, students will also extend their understanding of exponential functions by how they relate to quadratic functions, understanding that an exponential growth function will eventually exceed both a linear and a quadratic function. The contexts used earlier in this unit lead to functions where the domain is the integers. Later, students encounter functions where the domain includes the real numbers. Students do not yet know how to manipulate exponential expressions where the exponents are not whole numbers, but they can interpret the meaning of fractional values in the domain and use a graph to approximate corresponding values in the range. Note on materials: Students should have access to a calculator with an exponent button throughout the unit. Access to graphing technology is necessary for some activities, starting in lesson 7. Examples of graphing technology include a handheld graphing calculator, a computer with a graphing calculator application installed, or an internet-enabled device with access to a site like desmos.com/calculator or geogebra.org/graphing. Interactive applets are embedded throughout, and a graphing calculator tool is accessible in the Math Tools in the digital version. Unit 6: Introduction to Quadratic Functions Prior to this unit, students have studied what it means for a relationship to be a function, used function notation, and investigated linear and exponential functions. In this unit, they begin by looking at some patterns that grow quadratically. They contrast this growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than linear patterns but more slowly than exponential patterns. Students examine the important example of free-falling objects whose height over time can be modeled with quadratic functions. They use tables, graphs, and equations to describe the movement of these objects, eventually looking at the situation where a projectile is launched upward. This leads to the important interpretation that in a quadratic function such as $$f(t) = 5 + 30t - 16t^2$$, representing the vertical position of an object after $$t$$ seconds, 5 represents the initial height of the object, $$30t$$ represents its initial upward path, and $$\text-16t^2$$ represents the effect of gravity. Through this investigation, students also begin to appreciate how the different coefficients in a quadratic function influence the shape of the graph. In addition to projectile motion, students examine other situations represented by quadratic functions including area and revenue. Next, students examine the standard and factored forms of quadratic expressions. They investigate how each form is useful for understanding the graph of the function defined by these equivalent forms. The factored form is helpful for finding when the quadratic function takes the value 0 to obtain the $$x$$-intercept(s) of its graph, while the constant term in the standard form shows the $$y$$-intercept. Students also find that the factored form is useful for finding the vertex of the graph because its $$x$$-coordinate is halfway between the points where the graph intersects the $$x$$-axis (if it has two $$x$$-intercepts). As for the standard form, students investigate the coefficients of the quadratic and linear terms further, noticing that the coefficient of the quadratic term determines if it opens upward or downward. The effect of the coefficient of the linear term is somewhat mysterious and more complicated. Students explore how it shifts the graph both vertically and horizontally in an optional lesson. Finally, students investigate the vertex form of a quadratic function and understand how the parameters in the vertex form influence the graph. They learn how to determine the vertex of the graph from the vertex form of the function. They also begin to relate the different parameters in the vertex form to the general ideas of horizontal and vertical translation and vertical stretch, ideas which will be investigated further in a later course. Note on materials: Access to graphing technology is necessary for many activities. Examples of graphing technology are: a handheld graphing calculator, a computer with a graphing calculator application installed, and an internet-enabled device with access to a site like desmos.com/calculator or geogebra.org/graphing. For students using the digital version of these materials, a separate graphing calculator tool isn’t necessary. Interactive applets are embedded throughout, and a graphing calculator tool is accessible in the student math tools. Unit 7: Quadratic Equations Prior to this unit, students have studied quadratic functions. They analyzed and represented quadratic functions using expressions, tables, graphs, and descriptions. Students also evaluated the functions and interpreted the input and the output values in context. They encountered the terms “standard form,” “factored form,” and “vertex form” and examined the advantages of each form. They also rewrote expressions from factored form and vertex form to standard form. In this unit, students interpret, write, and solve quadratic equations. They see that writing and solving quadratic equations enables them to find input values that produce certain output values. Suppose the revenue of a theater is a function of the ticket price for a performance. At what ticket price would the theater earn \$10,000? Previously, students were only able to solve such problems by observing graphs and estimating, or by guessing and checking. Here, they learn to answer such questions algebraically.

Students begin solving quadratic equations by reasoning. For instance, to solve $$x^2+9 = 25$$, they think: Adding 9 to a squared number makes 25. That squared number must be 16, so $$x$$ must be 4 or -4. Along the way, students see that quadratic equations can have 2, 1, or 0 solutions.

Next, students learn that equations of the form $$(x-m)(x-n)=0$$ can be easily solved by applying the zero product property, which says that when two factors have a product of 0, one of the factors must be 0. When the equations are not in factored form, students rearrange them so that one side is 0, and rewrite the expressions from standard form to factored form. Students soon recognize that not all quadratic expressions in standard form can be rewritten into factored form. Even when it is possible, finding the right two numbers may be tedious, so another strategy is needed.

Students encounter perfect squares and notice that solving a quadratic equation is pretty straightforward when the equation contains a perfect square on one side and a number on the other. They learn that we can put equations into this helpful format by completing the square, that is, by rewriting the equation such that one side is a perfect square. Although this method can be used to solve any quadratic equation, it is not practical for solving all equations. This challenge motivates the quadratic formula.

Once introduced to the formula, students apply it to solve contextual and abstract problems, including those that they couldn’t previously solve. After gaining some experience using the formula, students investigate how it is derived. They find that the formula essentially encapsulates all the steps of completing the square into a single expression. Just like completing the square, the quadratic formula can be used to solve any equation, but it may not always be the quickest method. Students consider how to use the different methods strategically.

Throughout the unit, students see that solutions to quadratic equations are often irrational numbers. Sometimes they are expressed as sums or products of a rational number and an irrational number (such as $$4 \pm \sqrt {7}$$ or $$\pm \frac12 \sqrt3$$). Students reason about whether such sums and products are rational or irrational.

Toward the end of the unit, students revisit the vertex form and recall that it can be used to identify the maximum or minimum of a quadratic function. Previously students learned to rewrite expressions from vertex form to standard form. Now they can go in reverse—by completing the square. Being able to rewrite expressions in vertex form allows students to effectively solve problems about maximum and minimum values of quadratic functions.

In the final lesson, students integrate their insights and choose appropriate strategies to solve an applied problem and a mathematical problem (a system of linear and quadratic equations).