Algebra I

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra I than has generally been offered. The modules deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

High School Pacing Guide SY 2022-2023

Module 1: Relationships Between Quantities and Reasoning with Equations and Their Graphs

(28 Hours)

Focus Standards

N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N-Q.A.2 2 Define appropriate quantities for the purpose of descriptive modeling.

N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y 4 as (x2 ) 2 – (y2 ) 2 , thus recognizing it as a difference of squares that can be factored as (x2 – y 2 )(x2 + y 2 ).

A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A-CED.A.1 3 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Overview

In this module students analyze and explain precisely the process of solving an equation. Through repeated reasoning, students develop fluency in writing, interpreting, and translating between various forms of linear equations and inequalities and make conjectures about the form that a linear equation might take in a solution to a problem. They reason abstractly and quantitatively by choosing and interpreting units in the context of creating equations in two variables to represent relationships between quantities. They master the solution of linear equations and apply related solution techniques and the properties of exponents to the creation and solution of simple exponential equations. They learn the terminology specific to polynomials and understand that polynomials form a system analogous to the integers.

Unit Vocab

Piecewise-Linear Function (Given a finite number of non-overlapping intervals on the real number line, a (real) piecewise-linear function is a function from the union of the intervals to the set of real numbers such that the function is defined by (possibly different) linear functions on each interval.)

Numerical Symbol (A numerical symbol is a symbol that represents a specific number.)

Variable Symbol (A variable symbol is a symbol that is a placeholder for a number. It is possible that a question may restrict the type of number that a placeholder might permit, maybe integers only or a positive real number, for instance.)

Numerical Expression (A numerical expression is an algebraic expression that contains only numerical symbols (no variable symbols) and that evaluates to a single number.)

Algebraic Expression (An algebraic expression is either: (1) a numerical symbol or a variable symbol or (2) the result of placing previously generated algebraic expressions into the two blanks of one of the four operators ((__) (__), (__) (__), (__) (__), (__) (__)) or into the base blank of an exponentiation with an exponent that is a rational number.)

Equivalent Numerical Expressions (Two numerical expressions are equivalent if they evaluate to the same number.)

Equivalent Algebraic Expressions (Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative, associative, and distributive properties and the properties of rational exponents to components of the first expression.)

Polynomial Expression (A polynomial expression is either: (1) a numerical expression or a variable symbol or (2) the result of placing two previously generated polynomial expressions into the blanks of the addition operator (__ __) or the multiplication operator (__ __).)

Monomial (A monomial is a polynomial expression generated using only the multiplication operator (__ __). Monomials are products whose factors are numerical expressions or variable symbols.)

Degree of a Monomial (The degree of a non-zero monomial is the sum of the exponents of the variable symbols that appear in the monomial.)

Standard Form of a Polynomial Expression in One Variable (A polynomial expression with one variable symbol is in standard form if it is expressed as where is a non-negative integer, and are constant coefficients with . A polynomial expression in that is in standard form is often called a polynomial in .)

Degree of a Polynomial in Standard Form (The degree of a polynomial in standard form is the highest degree of the terms in the polynomial, namely .)

Leading Term and Leading Coefficient of a Polynomial in Standard Form (The term is called the leading term, and is called the leading coefficient.)

Constant Term of a Polynomial in Standard Form (The constant term is the value of the numerical expression found by substituting into all the variable symbols of the polynomial, namely .)

Solution (A solution to an equation with one variable is a number in the domain of the variable that, when substituted for all instances of the variable in both expressions, makes the equation a true number sentence.)

Solution Set (The set of solutions of an equation is called its solution set.)

Graph of an Equation in Two Variables (The set of all points in the coordinate plane that are solutions to an equation in two variables is called the graph of the equation.)

Zero Product Property (The Zero Product Property states that given real numbers, and if then either or or both and .)

Relationship between quantities... (All materials here)

Topic A: Introduction to Functions - Graphing Stories

Focus Standards

N-Q.A.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; and choose and interpret the scale and the origin in graphs and data displays.

N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Overview

In Topic A, students explore the main functions that they will work with in Grade 9: linear, quadratic, and exponential. The goal is to introduce students to these functions by having them make graphs of situations (usually based upon time) in which the functions naturally arise (A-CED.2). As they graph, they reason abstractly and quantitatively as they choose and interpret units to solve problems related to the graphs they create (N-Q.1, N-Q.2, N-Q.3).

Lessons

Topic B: The Structure of Expressions

Focus Standards

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 – y 4 as (x2 ) 2 – (y2 ) 2 , thus recognizing it as a difference of squares that can be factored as (x2 – y 2 )(x2 + y2 ).

A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Overview

in Topic B, students use the structure of expressions to define what it means for two algebraic expressions to be equivalent. In doing so, they discern that the commutative, associative, and distributive properties help link each of the expressions in the collection together, even if the expressions look very different themselves (A-SSE.2). They learn the definition of a polynomial expression and build fluency in identifying and generating polynomial expressions as well as adding, subtracting, and multiplying polynomial expressions (A-APR.1).

Lessons

Topic C: Solving Equations and Inequalities

Focus Standards

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law 𝑉 = 𝐼𝑅 to highlight resistance 𝑅.

A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-

REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Overview

in Topic C, instead of just solving equations, they formalize descriptions of what they learned before (variable, solution sets, etc.) and are able to explain, justify, and evaluate their reasoning as they strategize methods for solving linear and non-linear equations (A-REI.1, A-REI.3, A-CED.4). Students take their experience solving systems of linear equations further as they prove the validity of the addition method, learn a formal definition for the graph of an equation and use it to explain the reasoning of solving systems graphically, and graphically represent the solution to systems of linear inequalities (A-CED.3, A-REI.5, A-REI.6, A-REI.10, A-REI.12).

Lessons

Topic D: Creating Equations to Solve Problems

Focus Standards

N-Q.A.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

A-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients.

A-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Overview

In Topic D, students are formally introduced to the modeling cycle (see page 61 of the CCLS) through problems that can be solved by creating equations and inequalities in one variable, systems of equations, and graphing (N-Q.1, A-SSE.1, A-CED.1, A-CED.2, A-REI.3)

Lessons

Module 2: Descriptive Statistics

(20 Hours)

Focus Standards

S-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.

S-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

S-ID.C.9 Distinguish between correlation and causation.

Overview

In this module, students reconnect with and deepen their understanding of statistics and probability concepts first introduced in Grades 6, 7, and 8. There is variability in data, and this variability often makes learning from data challenging. Students develop a set of tools for understanding and interpreting variability in data and begin to make more informed decisions from data. Students work with data distributions of various shapes, centers, and spreads. Measures of center and measures of spread are developed as ways of describing distributions. The choice of appropriate measures of center and spread is tied to distribution shape. Symmetric data distributions are summarized by the mean and mean absolute deviation, or standard deviation. The median and the interquartile range summarize data distributions that are skewed. Students calculate and interpret measures of center and spread and compare data distributions using numerical measures and visual representations.

Students build on their experience with bivariate quantitative data from Grade 8; they expand their understanding of linear relationships by connecting the data distribution to a model and informally assessing the selected model using residuals and residual plots. Students explore positive and negative linear relationships and use the correlation coefficient to describe the strength and direction of linear relationships. Students also analyze bivariate categorical data using two-way frequency tables and relative frequency tables. The possible association between two categorical variables is explored by using data summarized in a table to analyze differences in conditional relative frequencies.

Unit Vocab

Skewed Data Distribution (A data distribution is said to be skewed if the distribution is not symmetric with respect to its mean. Left-skewed or skewed to the left is indicated by the data spreading out longer (like a tail) on the left side.

Right-skewed or skewed to the right is indicated by the data spreading out longer (like a tail) on the right side.)

Outlier (An outlier of a finite numerical data set is a value that is greater than 𝑄3 by a distance of 1.5 ∙ 𝐼𝑄𝑅, or a value that is less than 𝑄1 by a distance of 1.5 ∙ 𝐼𝑄𝑅. Outliers are usually identified by an “*” or a “•” in a box plot.)

Sample Standard Deviation (The sample variance for a numerical sample data set of 𝑛-values is the sum of the squared distances the values are from the mean divided by (𝑛 − 1). The sample standard deviation is the principle (positive) square root of the sample variance.)

Interquartile Range (The interquartile range (or 𝐼𝑄𝑅) is the distance between the first quartile and the second quartile: 𝐼𝑄𝑅 = 𝑄3 − 𝑄1. The 𝐼𝑄𝑅 describes variability by identifying the length of the interval that contains the middle 50% of the data values.)

Association (A statistical association is any relationship between measures of two types of quantities so that one is statistically dependent on the other.)

Conditional Relative Frequency (A conditional relative frequency compares a frequency count to the marginal total that represents the condition of interest.)

Residual (The residual of the data point (𝑥𝑖 , 𝑦𝑖) is the (actual 𝑦𝑖 -value) − (predicted 𝑦-value) for the given 𝑥𝑖 .)

Residual Plot (Given a bivariate data set and linear equation used to model the data set, a residual plot is the graph of all ordered pairs determined as follows: For each data point (𝑥𝑖 , 𝑦𝑖) in the data set, the first entry of the ordered pair is the 𝑥-value of the data point, and the second entry is the residual of the data point.)

Correlation Coefficient (The correlation coefficient, often denoted by 𝑟, is a number between −1 and +1, inclusively, that measures the strength and direction of a linear relationship between the two types of quantities. If 𝑟 = 1 or 𝑟 = −1, then the graph of data points of the bivariate data set lie on a line of positive or negative slope.)

Descriptive Statistics (All materials here)

Topic A: Shapes and Centers of Distributions

Focus Standards

S-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots)

S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Overview

In Topic A, students observe and describe data distributions. They reconnect with their earlier study of distributions in Grade 6 by calculating measures of center and describing overall patterns or shapes.

Students deepen their understanding of data distributions recognizing that the value of the mean and median are different for skewed distributions and similar for symmetrical distributions. Students select a measure of center based on the distribution shape to appropriately describe a typical value for the data distribution. Topic A moves from the general descriptions used in Grade 6 to more specific descriptions of the shape and the center of a data distribution.

Lessons

Topic B: Describing Variability and Comparing Distributions

Focus Standards

S-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Overview

In Topic B, students reconnect with methods for describing variability first seen in Grade 6. Topic B deepens students’ understanding of measures of variability by connecting a measure of the center of a data distribution to an appropriate measure of variability. The mean is used as a measure of center when the distribution is more symmetrical. Students calculate and interpret the mean absolute deviation and the standard deviation to describe variability for data distributions that are approximately symmetric. The median is used as a measure of center for distributions that are more skewed, and students interpret the interquartile range as a measure of variability for data distributions that are not symmetric.

Students match histograms to box plots for various distributions based on an understanding of center and variability. Students describe data distributions in terms of shape, a measure of center, and a measure of variability from the center.

Lessons

Topic C: Categorical Data on Two Variables

Focus Standards

S-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

S-ID.C.9 Distinguish between correlation and causation.

Overview

In Topic C, students reconnect with previous work in Grade 8 involving categorical data. Students use a two-way frequency table to organize data on two categorical variables. Students calculate the conditional relative frequencies from the frequency table. They explore a possible association between two categorical variables using differences in conditional, relative frequencies. Students also come to understand the distinction between association between two categorical variables and a causal relationship between two variables. This provides a foundation for work on sampling and inference in later grades.

Lessons

Topic D: Numerical Data on Two Variables

Focus Standards

S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit the function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals. c
. Fit a linear function for a scatter plot that suggests a linear association. S-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

S-ID.C.9 Distinguish between correlation and causation.

Overview

In Topic D, students analyze relationships between two quantitative variables using scatterplots and by summarizing linear relationships using the least squares regression line. Models are proposed based on an understanding of the equations representing the models and the observed pattern in the scatter plot. Students calculate and analyze residuals based on an interpretation of residuals as prediction errors.

Lessons

Module 3: Linear and Exponential Functions

(24 Hours)

Focus Standards

F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦 = 𝑓(𝑥).

F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F-IF.A.35 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by 𝑓(0) = 𝑓(1) = 1, 𝑓(𝑛 + 1) = 𝑓(𝑛) + 𝑓(𝑛 − 1) for 𝑛 ≥ 1.

F-IF.B.46 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛) gives the number of person-hours it takes to assemble 𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function.

F-IF.B.67 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Overview

In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

Unit Vocab

Function (A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched16 to one and only one element of 𝑌. The set 𝑋 is called the domain; the set 𝑌 is called the range.)

Domain (Refer to the definition of function.)

Range (Refer to the definition of function.)

Linear Function (A linear function is a polynomial function of degree 1.)

Average Rate of Change (Given a function 𝑓 whose domain includes the closed interval of real numbers [𝑎, 𝑏] and whose range is a subset of the real numbers, the average rate of change on the interval [𝑎, 𝑏] is 𝑓(𝑏)−𝑓(𝑎) 𝑏−𝑎 .)

Piecewise Linear Function (Given non-overlapping intervals on the real number line, a (real) piecewise linear function is a function from the union of the intervals to the set of real numbers such that the function is defined by (possibly different) linear functions on each interval.)

Linear and Exponential Functions (All materials here)

Topic A: Linear and Exponential Sequences

Focus Standards

F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦 = 𝑓(𝑥).

F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by 𝑓(0) = 𝑓(1) = 1, 𝑓(𝑛 + 1) = 𝑓(𝑛)+ 𝑓(𝑛 − 1) for 𝑛 ≥ 1. F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-BF.A.1a Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Overview

In Topic A, students explore arithmetic and geometric sequences as an introduction to the formal notation of functions (F-IF.A.1, F-IF.A.2). They interpret arithmetic sequences as linear functions with integer domains and geometric sequences as exponential functions with integer domains (F-IF.A.3, F-BF.A.1a).

Students compare and contrast the rates of change of linear and exponential functions, looking for structure in each and distinguishing between additive and multiplicative change (F-IF.B.6, F-LE.A.1, F-LE.A.2, F-LE.A.3).

Lessons

Topic B: Functions and Their Graphs

Focus Standards

F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦 = 𝑓(𝑥).

F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

F-IF.C.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

Overview

In Topic B, students connect their understanding of functions to their knowledge of graphing from Grade 8. They learn the formal definition of a function and how to recognize, evaluate, and interpret functions in abstract and contextual situations (F-IF.A.1, F-IF.A.2).

Students examine the graphs of a variety of functions and learn to interpret those graphs using precise terminology to describe such key features as domain and range, intercepts, intervals where the function is increasing or decreasing, and intervals where the function is positive or negative. (F-IF.A.1, F-IF.B.4, F-IF.B.5, F-IF.C.7a).

Lessons

Topic C: Transformation of Functions

Focus Standards

A-REI.D.11 Explain why the 𝑥-coordinates of the points where the graphs of the equations 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥) = 𝑔(𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

F-IF.C.7b Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

F-BF.B.3 Identify the effect on the graph of replacing 𝑓(𝑥) by (𝑥)+𝑘, 𝑘𝑓(𝑥), 𝑓(𝑘𝑥), and 𝑓(𝑥 + 𝑘) for specific values of 𝑘 (both positive and negative); find the value of 𝑘 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Overview

In Topic C, students extend their understanding of piecewise functions and their graphs including the absolute value and step functions. They learn a graphical approach to circumventing complex algebraic solutions to equations in one variable, seeing them as f(x) = g(x) and recognizing that the intersection of the graphs of f(x) and g(x) are solutions to the original equation (A-REI.D.11).

Students use the absolute value function and other piecewise functions to investigate transformations of functions and draw formal conclusions about the effects of a transformation on the function’s graph (F-IF.C.7, F-BF.B.3).

Lessons

Topic D: Using Functions and Graphs to Solve Problems

Focus Standards

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-SSE.B.3c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15𝑡 can be rewritten as (1.151/12) 12𝑡 ≈ 1.01212𝑡 to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Overview

In Topic D, students apply and reinforce the concepts of the module as they examine and compare exponential, piecewise, and step functions in a real-world context (F-IF.C.9).

They create equations and functions to model situations (A-CED.A.1, F-BF.A.1, F-LE.A.2), rewrite exponential expressions to reveal and relate elements of an expression to the context of the problem (A-SSE.B.3c, F-LE.B.5), and examine the key features of graphs of functions, relating those features to the context of the problem (F-IF.B.4, F-IF.B.6).

Lessons

Module 4: Polynomial and Quadratic Expressions, Equations, and Functions

(24 Hours)

Focus Standards

N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.2 b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret 𝑃(1 + 𝑟) 𝑛 as the product of 𝑃 and a factor not depending on 𝑃.

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥 4 − 𝑦 4 as (𝑥 2 ) 2 − (𝑦 2 )^ 2 , thus recognizing it as a difference of squares that can be factored as (𝑥 2 − 𝑦 2 )(𝑥 2 + 𝑦 2 ).^ 3

A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials

A-REI.B.4 Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in 𝑥 into an equation of the form (𝑥 − 𝑝) 2 = 𝑞 that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for 𝑥 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏.

Overview

In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.

Unit Vocab

Axis of symmetry of the graph of a quadratic function (Given a quadratic function in standard form, 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, the vertical line given by the graph of the equation, 𝑥 = − 𝑏 2𝑎 , is called the axis of symmetry of the graph of the quadratic function.)

Cube root function (The parent function 𝑓(𝑥) = √𝑥 3 .)

Cubic function (A polynomial function of degree 3.)

Degree of a monomial term (The degree of a monomial term is the sum of the exponents of the variables that appear in a term of a polynomial.)

Degree of a polynomial (The degree of a polynomial in one variable in standard form is the highest degree of the terms in the polynomial.)

Discriminant (The discriminant of a quadratic function in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 is 𝑏 2 − 4𝑎𝑐. The nature of the roots of a quadratic equation can be identified by determining if the discriminant is positive, negative, or equal to zero.)

End behavior of a quadratic function (Given a quadratic function in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 (or 𝑓(𝑥) = 𝑎(𝑥 − ℎ) 2 + 𝑘), the quadratic function is said to open up if 𝑎 > 0 and open down if 𝑎 < 0.)

Factored form for a quadratic function (A quadratic function written in the form 𝑓(𝑥) = 𝑎(𝑥 − 𝑛)(𝑥 − 𝑚).) Leading coefficient (The leading coefficient of a polynomial is the coefficient of the term of highest degree.)

Parent function (A parent function is the simplest function in a “family” of functions that can each be formed by one or more transformations of another.)

Quadratic formula (The quadratic formula is the formula that emerges from solving the general form of a quadratic equation by completing the square, 𝑦 = −𝑏±√𝑏 2−4𝑎𝑐 2𝑎 . It can be used to solve any quadratic equation.)

Quadratic function (A polynomial function of degree 2.)

Roots of a polynomial function (The domain values for a polynomial function that make the value of the polynomial function equal zero when substituted for the variable.) 𝑎𝑛𝑥 𝑛 + 𝑎𝑛−1𝑥 𝑛−1 + ⋯ + 𝑎1𝑥 + 𝑎0, where 𝑛 is a non-negative integer, and 𝑎0, 𝑎1, 𝑎2, …, 𝑎𝑛 are constant coefficients with 𝑎𝑛 ≠ 0.)

Square root function (The parent function 𝑓(𝑥) = √𝑥.)

Standard form for a quadratic function (A quadratic function written in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.)

Standard form of a polynomial in one variable (A polynomial expression with one variable symbol 𝑥 is in standard form if it is expressed as,

Vertex form (Completed-square form for a quadratic function; in other words, written in the form 𝑓(𝑥) = 𝑎(𝑥 − ℎ) 2 + 𝑘.)

Vertex of the graph of a quadratic function (The point where the graph of a quadratic function and its axis of symmetry intersect is called the vertex. The vertex is either a maximum or a minimum of the quadratic function, depending on whether the leading coefficient of the function in standard form is negative or positive, respectfully.)

Polynomial and Quadratic (All materials here)

Topic A: Quadratic Expressions, Equations, Functions and Their Connection to Rectangles

Focus Standards

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret 𝑃𝑃(1 + 𝑟𝑟)𝑛𝑛 as the product of 𝑃𝑃 and a factor not depending on 𝑃𝑃.

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥𝑥4 − 𝑦𝑦4 as (𝑥𝑥2)2 − (𝑦𝑦2)2, thus recognizing it as a difference of squares that can be factored as (𝑥𝑥2 − 𝑦𝑦2)(𝑥𝑥2 + 𝑦𝑦2).

A-SSE.B.3a Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.

A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Overview

Topic A introduces polynomial expressions. They analyze, interpret, and use the structure of polynomial expressions to multiply and factor polynomial expressions (A-SSE.A.2). They understand factoring as the reverse process of multiplication. In this topic, students develop the factoring skills needed to solve quadratic equations and simple polynomial equations by using the zero-product property (A-SSE.B.3a).

Students transform quadratic expressions from standard form to factored form, and then solve equations involving those expressions. Students apply symmetry to create and interpret graphs of quadratic functions (F-IF.B.4, F-IF.C.7a). Using area models, students explore strategies for factoring more complicated quadratic expressions, including the product-sum method and rectangular arrays. They create one- and two-variable equations from tables, graphs, and contexts and use them to solve contextual problems represented by the quadratic function (A-CED.A.1, A-CED.A.2). Students then relate the domain and range for the function to its graph and the context (F-IF.B.5).

Lessons

Topic B: Using Different Forms of Quadratic Functions

Focus Standards

N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret 𝑃𝑃(1 + 𝑟𝑟)𝑛𝑛 as the product of 𝑃𝑃 and a factor not depending on 𝑃𝑃.

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥𝑥4 − 𝑦𝑦4 as (𝑥𝑥2)2 − (𝑦𝑦2)2, thus recognizing it as a difference of squares that can be factored as (𝑥𝑥2 − 𝑦𝑦2)(𝑥𝑥2 + 𝑦𝑦2).

A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

Overview

Students apply their experiences from Topic A as they transform quadratic functions from standard form to vertex form in Topic B. The strategy known as completing the square is used to solve quadratic equations when the quadratic expression cannot be factored (A-SSE.B.3b). Students recognize that this form reveals specific features of quadratic functions and their graphs, namely the minimum or maximum of the and the line of symmetry of the graph (A-APR.B.3, F-IF.B.4, F-IF.C.7a).

Students derive the quadratic formula by completing the square for a general quadratic equation in standard form and use it to determine the nature and number of solutions for equations when y equals zero (A-SSE.A.2, A-REI.B.4). For quadratics with irrational roots, students use the quadratic formula and explore the properties of irrational numbers (N-RN.B.3).

Lessons

Topic C: Function Transformations and Modeling

Focus Standards

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-IF.C.7b Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

F-IF.C.8a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F-BF.B.3 Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) by 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘𝑘𝑘(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), and 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) for specific values of 𝑘𝑘 (both positive and negative); find the value of 𝑘𝑘 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them

Overview

In Topic C, students explore the families of functions that are related to the parent functions, specifically for quadratic (f(x) = x2), square root (f(x) = the square root of x), and cube root (f(x) = cube root of x), to perform horizontal and vertical translations as well as shrinking and stretching (F-IF.C.7b, F-BF.B.3). They recognize the application of transformations in vertex form for a quadratic function and use it to expand their ability to efficiently sketch graphs of square and cube root functions.

Students compare quadratic, square root, or cube root functions in context and represent each in different ways (verbally with a description, as a table of values, algebraically, or graphically). In the final two lessons, students examine real-world problems of quadratic relationships presented as a data set, a graph, a written relationship, or an equation. They choose the most useful form for writing the function and apply the techniques learned throughout the module to analyze and solve a given problem (A-CED.A.2), including calculating and interpreting the rate of change for the function over an interval (F-IF.B.6).

Lessons

Module 5: A Synthesis of Modeling with Equations and Functions

(9 Hours)

Focus Standards

N-Q.A.22 Define appropriate quantities for the purpose of descriptive modeling.

N-Q.A.33 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

A-CED.A.14 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F-IF.B.45 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛𝑛) gives the number of person-hours it takes to assemble 𝑛𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function.

F-IF.B.66 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Overview

In Module 5, students synthesize what they have learned during the year about functions to select the correct function type in a series of modeling problems. Students no longer have the benefit of a module or lesson title that includes function type to guide them in their choices. Skills and knowledge from the previous modules will support the requirements of this module, including writing, rewriting, comparing, and graphing functions and interpretation of the parameters of an equation. Students must also draw on their study of statistics in Module 2, using graphs and functions to model a context presented with data and/or tables of values. In this module, the modeling cycle is used as the organizing structure, rather than function type.

Unit Vocab

Analytic Model (A model that seeks to explain data based on deeper theoretical ideas. For example, by using an algebraic equation. This is sometimes referred to as a symbolic model.)

Descriptive Model (A model that seeks to describe phenomena or summarize them in a compact form. For example, by using a graph.)

Modeling with Equations and Functions (All materials here)

Topic A: Elements fo Modeling

Focus Standards

N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛𝑛) gives the number of person-hours it takes to assemble 𝑛𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function.

F-BF.A.1 Write a function that describes a relationship between two quantities.

a. Determine an explicit expression

Overview

Topic A focuses on the skills inherent in the modeling process: representing graphs, data sets, or verbal descriptions using explicit expressions (F-BF.A.1a) when presented in graphic form in Lesson 1, as data in Lesson 2, or as a verbal description of a contextual situation in Lesson 3. They recognize the function type associated with the problem (F-LE.A.1b, F-LE.A.1c) and match to or create 1- and 2-variable equations (A-CED.A.1, A-CED.2) to model a context presented graphically, as a data set, or as a description (F-LE.A.2).

Function types include linear, quadratic, exponential, square root, cube root, absolute value, and other piecewise functions. Students interpret features of a graph in order to write an equation that can be used to model it and the function (F-IF.B.4, F-BF.A.1) and relate the domain to both representations (F-IF.B.5). This topic focuses on the skills needed to complete the modeling cycle and sometimes uses purely mathematical models, sometimes real-world contexts.

Lessons

Topic B: Completing the Modeling Cycle

Focus Standards

N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛𝑛) gives the number of person-hours it takes to assemble 𝑛𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function

Overview

Students choose and define the quantities of the problem (N-Q.A.2) and the appropriate level of precision for the context (N-Q.A.3). They create 1- and 2-variable equations (A-CED.A.1, A-CED.A.2) to model the context when presented as a graph, as data and as a verbal description. They can distinguish between situations that represent a linear (F-LE.A.1b), quadratic, or exponential (F-LE.A.1c) relationship.

For data, they look for first differences to be constant for linear, second differences to be constant for quadratic, and a common ratio for exponential. When there are clear patterns in the data, students will recognize when the pattern represents a linear (arithmetic) or exponential (geometric) sequence (F-BF.A.1a, F-LE.A.2). For graphic presentations, they interpret the key features of the graph, and for both data sets and verbal descriptions they sketch a graph to show the key features (F-IF.B.4). They calculate and interpret the average rate of change over an interval, estimating when using the graph (F-IF.B.6), and relate the domain of the function to its graph and to its context (F-IF.B.5).

Lessons