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Algebra 1-2 Scope and SequenceGeneral Information

Suggested Pacing is 1 lesson per day. Each lesson has an exit ticket that may be used as a formative assessment.

Suggestions Teachers may create additional assessments as they feel necessary. Modules (student materials) may be printed and bound for students to use as a workbook. Common Core belief is to provide students with answer keys to practice correctly.

Required Materials – Per Common Core Graphing Calculators

1Dysart USD – engageny 2014 ~ 2015

Algebra 1-2 Scope and SequenceQuarter 1

Module 1: Relationship Between Quantities and Reasoning with Equations and Their GraphsTopic A: Introduction to Functions Studied this YearIn 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 Description – Student Outcome(s) Mathematical Practice(s)

1: Graphs of Piecewise Linear Functions

Students define appropriate quantities from a situation (a “graphing story”), choose and interpret the scale and the origin for the graph, and graph the piecewise linear function described in the video. They understand the relationship between physical measurements and their representation on a graph.

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2: Graphs of Quadratic Functions

Students represent graphically a non-linear relationship between two quantities and interpret features of the graph. They will understand the relationship between physical quantities via the graph.

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3: Graphs of Exponential Functions

Students choose and interpret the scale on a graph to appropriately represent an exponential function.

Students plot points representing number of bacteria over time, given that bacteria grow by a constant factor over evenly spaced time intervals.

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4: Analyzing Graphs –Water Usage During a Typical Day at School

Students develop the tools necessary to discern units for quantities in real-world situations and choose levels of accuracy appropriate to limitations on measurement. They refine their skills in interpreting the meaning of features appearing in graphs.

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5: Two Graphing Stories Students interpret the meaning of the point of intersection of two graphs and use analytic tools to find its coordinates.

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Algebra 1-2 Scope and Sequence

Topic B: The Structure of ExpressionsIn middle school, students applied the properties of operations to add, subtract, factor, and expand expressions (6.EE.3, 6.EE.4, 7.EE.1, 8.EE.1). Now, 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). The Mid-Module Assessment follows Topic B.


6: Algebraic Expressions –The Distributive Property

Students use the structure of an expression to identify ways to rewrite it.

Students use the distributive property to prove equivalency of expressions.MP.7

7: Algebraic Expressions –The Commutative and Associative Prop.

Students use the commutative and associative properties to recognize structure within expressions and to prove equivalency of expressions.

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8: Adding and Subtracting Polynomials

Students understand that the sum or difference of two polynomials produces another polynomial and relate polynomials to the system of integers; students add and subtract polynomials.

9: Multiplying Polynomials Students understand that the product of two polynomials produces another polynomial; students multiply polynomials.

Mid – Module Assessment Task – Topics A through B (recommended assessment 2 days, return and remediation or further application 4 days)


Algebra 1-2 Scope and SequenceTopic C: Solving Equations and InequalitiesThroughout middle school, students practice the process of solving linear equations (6.EE.5, 6.EE.7, 7.EE.4, 8.EE.7) and systems of linear equations (8.EE.8). Now, 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).


10: True and False Equations

Students understand that an equation is a statement of equality between two expressions. When values are substituted for the variables in an equation, the equation is either true or false.

Students find values to assign to the variables in equations that make the equations true statements.

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11: Solution Sets for Equations and Inequalities

Students understand that an equation with variables is often viewed as a question asking for the set of values one can assign to the variables of the equation to make the equation a true statement. They see the equation as a “filter” that sifts through all numbers in the domain of the variables, sorting those numbers into two disjoint sets: the Solution Set and the set of numbers for which the equation is false.

Students understand the commutative, associate, and distributive properties as identities, e.g., equations whose solution sets are the set of all values in the domain of the variables.

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12: Solving Equations Students are introduced to the formal process of solving an equation: starting from the assumption that the original equation has a solution.

Students explain each step as following from the properties of equality. Students identify equations that have the same solution set.

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13: Some Potential Dangers when Solving Equations

Students learn “if-then” moves using the properties of equality to solve equations. Students also explore moves that may result in an equation having more solutions than the original equation.

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14: Solving Inequalities Students learn if-then moves using the addition and multiplication properties of inequality to solve inequalities and graph the solution sets on the number line.

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Algebra 1-2 Scope and Sequence15: Solution Sets of Two or More Equations (Inequalities) Joined by “And” or “Or”

Students describe the solution set of two equations (or inequalities) joined by either “and” or “or” and graph the solution set on the number line.

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16: Solving and Graphing Inequalities Joined by “And” and “Or”

Students solve two inequalities joined by “and” or “or,” then graph the solution set on the number line.

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17: Equations Involving Factored Expressions

Students learn that equations of the form (𝑥−𝑎)(𝑥−𝑏)=0 have the same solution set as two equations joined by “or:” 𝑥−𝑎=0 or 𝑥−𝑏=0. Students solve factored or easily factorable equations.

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18: Equations Involving a Variable Expression in the Denominator

Students interpret equations like 1𝑥 = 3 as two equations “1𝑥= 3” and “𝑥 ≠ 0” joined by “and.” Students find the solution set for this new system of equations.

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19: Rearranging Formulas Students learn to think of some of the letters in a formula as constants in order to define a relationship between two or more quantities, where one is in terms of another, for example holding V in V = IR as constant, and finding R in terms of I.

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20: Solution Sets to Equations and Inequalities with 2 Variables - Part 1

Students recognize and identify solutions to two-variable equations. They represent the solution set graphically. They create two variable equations to represent a situation. They understand that the graph of the line ax + by = c is a visual representation of the solution set to the equation ax + by = c.

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21: Solution Sets to Equations and Inequalities with 2 Variables - Part 2

Students recognize and identify solutions to two-variable inequalities. They represent the solution set graphically. They create two variable inequalities to represent a situation.

Students understand that a half-plane bounded by the line 𝑎𝑥 + 𝑏𝑦 = 𝑐 is a visual representation of the solution set to a linear inequality such as 𝑎𝑥 + 𝑏𝑦 < 𝑐. They interpret the inequality symbol correctly to determine which portion of the coordinate plane is shaded to represent the solution.

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22: Solution Sets to Simultaneous Equations P1

Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically.

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Algebra 1-2 Scope and Sequence23: Solution Sets to Simultaneous Equations P2

Students create systems of equations that have the same solution set as a given system. Students understand that adding a multiple of one equation to another creates a new system

of two linear equations with the same solution set as the original system. This property provides a justification for a method to solve a system of two linear equations algebraically.

24: Applications of Systems of Equations and Inequalities

Students use systems of equations or inequalities to solve contextual problems and interpret solutions within a particular context.

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Topic D: Solving Equations and InequalitiesIn 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). The End-of-Module Assessment follows Topic D.


25: Solving Problems in 2 ways – Rates and Algebra

Students investigate a problem that can be solved by reasoning quantitatively and by creating equations in one variable.

They compare the numerical approach to the algebraic approach.

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26: Recursive Challenge Problem – Double and Add 5 Game – Part 1

Students learn the meaning and notation of recursive sequences in a modeling setting. Following the modeling cycle, students investigate the double and add 5 game in a simple

case in order to understand the statement of the main problem.

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27: Recursive Challenge Problem – Double and Add 5 Game – Part 2

Students learn the meaning and notation of recursive sequences in a modeling setting. Students use recursive sequences to model and answer problems. Students create equations and inequalities to solve a modeling problem. Students represent constraints by equations and inequalities and interpret solutions as viable

or non-viable options in a modeling context.

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28: Federal Income Tax Students create equations and inequalities in one variable and use them to solve problems. Students create equations in two or more variables to represent relationships between

quantities and graph equations on coordinate axes with labels and scales. Students represent constraints by inequalities and interpret solutions as viable or non-viable

options in a modeling context.

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End – Module Assessment Task – Topics A through D (recommended assessment 2 days, return and remediation or further applications 4 days)


Algebra 1-2 Scope and SequenceFocus Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them. Students are presented with problems that require them to try special cases and simpler forms of the original problem in order to gain insight into the problem. MP.2 Reason abstractly and quantitatively. Students analyze graphs of non-constant rate measurements and reason from the shape of the graphs to infer what quantities are being displayed and consider possible units to represent those quantities. MP.3 Construct viable arguments and critique the reasoning of others. Students reason about solving equations using “if-then” moves based on equivalent expressions and properties of equality and inequality. They analyze when an “if-then” move is not reversible. MP.4 Model with mathematics. Students have numerous opportunities in this module to solve problems arising in everyday life, society, and the workplace from modeling bacteria growth to understanding the federal progressive income tax system. MP.6 Attend to precision. Students formalize descriptions of what they learned before (variables, solution sets, numerical expressions, algebraic expressions, etc.) as they build equivalent expressions and solve equations. Students analyze solution sets of equations to determine processes (like squaring both sides of an equation) that might lead to a solution set that differs from that of the original equation. MP.7 Look for and make use of structure. Students reason with and about collections of equivalent expressions to see how all the expressions in the collection are linked together through the properties of operations. They discern patterns in sequences of solving equation problems that reveal structures in the equations themselves: 2𝑥 + 4= 10, 2(𝑥 − 3) + 4= 10, 2(3𝑥 − 4) + 4= 10, etc. MP.8 Look for and express regularity in repeated reasoning. After solving many linear equations in one variable (e.g., 3𝑥 + 5=8𝑥 − 17), students look for general methods for solving a generic linear equation in one variable by replacing the numbers with letters: 𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑. They have opportunities to pay close attention to calculations involving the properties of operations, properties of equality, and properties of inequality as they find equivalent expressions and solve equations, noting common ways to solve different types of equations.


Algebra 1-2 Scope and SequenceModule 2: Descriptive StatisticsTopic A: Shapes and Centers of DistributionIn 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. (S-ID.1, 2, 3).


1: Distributions and Their Shapes

Students use informal language to describe the shape, center, and variability of a distribution based on a dot plot, histogram, or box plot. Students recognize that a first step in interpreting data is making sense of the context. Students make meaningful conjectures to connect data distributions to their contexts and the questions

that could be answered by studying the distributions.

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2: Describing the Center of a Distribution

Students construct a dot plot from a data set. Students calculate the mean of a data set and the median of a data set. Students observe and describe that measures of center (mean and median) are nearly the same for

distributions that are nearly symmetrical. Students observe and explain why the mean and median are different for distributions that are skewed. Students select the mean as an appropriate description of center for a symmetrical distribution and the

median as a better description of center for a distribution that is skewed.

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3: Graphs of Exponential Functions

Students estimate the mean and median of a distribution represented by a dot plot or a histogram. Students indicate that the mean is a reasonable description of a typical value for a distribution that is

symmetrical but that the median is a better description of a typical value for a distribution that is skewed.

Students interpret the mean as a balance point of a distribution. Students indicate that for a distribution in which neither the mean nor the median is a good description

of a typical value, the mean still provides a description of the center of a distribution in terms of the balance point.

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Algebra 1-2 Scope and SequenceTopic B: Describing Variability and Comparing DistributionsIn 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. (S-ID.1, 2, 3) .


4: Summarizing Deviations from the Mean

Students calculate the deviations from the mean for two symmetrical data sets that have the same means.

Students interpret deviations that are generally larger as identifying distributions that have a greater spread or variability than a distribution in which the deviations are generally smaller.

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5: Measuring Variability for Symmetrical Distributions

Students calculate the standard deviation for a set of data. Students interpret the standard deviation as a typical distance from the mean.

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6: Interpreting the Standard Deviation

Students calculate the standard deviation of a sample with the aid of a calculator. Students compare the relative variability of distributions using standard deviations.

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7: Measuring Variability for Skewed Distributions (Interquartile Range)

Students explain why a median is a better description of a typical value for a skewed distribution.

Students calculate the 5-number summary of a data set. Students construct a box plot based on the 5-number summary and calculate the interquartile

range (IQR). Students interpret the IQR as a description of variability in the data. Students identify outliers in a data distribution.

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8: Comparing Distributions Students compare two or more distributions in terms of center, variability, and shape. Students interpret a measure of center as a typical value. Students interpret the IQR as a description of the variability of the data. Students answer questions that address differences and similarities for two or more

distributions.

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Mid-Module Assessment - Topics A through B (assessment 1 day, return 0 days, remediation or further applications 1 day)9

Dysart USD – engageny 2014 ~ 2015


Topic C: Categorical Data on Two VariablesIn 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. (S-ID.5,9)


9: Summarizing Bivariate Categorical Data

Students distinguish between categorical data and numerical data. Students summarize data on two categorical variables collected from a sample using a two-

way frequency table.

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10: Summarizing Bivariate Categorical Data with Relative Frequencies

Students summarize data on two categorical variables collected from a sample using a two-way frequency table.

Given a two-way frequency table, students construct a relative frequency table and interpret relative frequencies.

11: Conditional Relative Frequencies and Association

Students calculate and interpret conditional relative frequencies from two-way frequency tables.

Students evaluate conditional relative frequencies as an indication of possible association between two variables.

Students explain why association does not imply causation.


Algebra 1-2 Scope and SequenceTopic D: Numerical Data on Two Variables


Algebra 1-2 Scope and SequenceIn 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. (S-ID.6, 7, 8, 9)




12: Relationships between Two Numerical Variables P1

Students distinguish between scatter plots that display a relationship that can be reasonably modeled by a linear equation and those that should be modeled by a nonlinear equation.

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Algebra 1-2 Scope and Sequence13: Relationships between Two Numerical Variables P2

Students distinguish between scatter plots that display a relationship that can be reasonably modeled by a linear equation and those that should be modeled by a nonlinear equation.

Students use an equation given as a model for a nonlinear relationship to answer questions based on an understanding of the specific equation and the context of the data.

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14: Modeling Relationships with a Line

Students determine the least-squares regression line from a given set of data using technology.

Students use the least-squares regression line to make predictions.

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15: Interpreting Residuals from a Line

Students use the least-squares line to predict values for a given data set. Students use residuals to evaluate the accuracy of predictions based on the least-squares line.

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Algebra 1-2 Scope and Sequence16: More on Modeling Relationships with a Line

Students use the least-squares line to predict values for a given data set. Students use residuals to evaluate the accuracy of predictions based on the least-squares line.

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17: Analyzing Residuals Students use a graphing calculator to construct the residual plot for a given data set. Students use a residual plot as an indication of whether the model used to describe the

relationship between two numerical variables is an appropriate choice.

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18: Analyzing Residuals Students use a graphing calculator to construct the residual plot for a given data set. Students use a residual plot as an indication of whether the model used to describe the

relationship between two numerical variables is an appropriate choice.

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19: Interpreting Correlation Students use technology to determine the value of the correlation coefficient for a given data. Students interpret the value of the correlation coefficient as a measure of strength and

direction of a linear relationship. Students explain why correlation does not imply causation.

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Focus Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them. Students choose an appropriate method of analysis based on problem context. They consider how the data were collected and how data can be summarized to answer statistical questions. Students select a graphical display appropriate to the problem context. They select numerical summaries appropriate to the shape of the data distribution. Students use multiple representations and numerical summaries and then determine the most appropriate representation and summary for a given data distribution.MP.2 Reason abstractly and quantitatively. Students pose statistical questions and reason about how to collect and interpret data in order to answer these questions. Students form summaries of data using graphs, two-way tables, and other representations that are appropriate for a given context and the statistical question they are trying to answer. Students reason about whether two variables are associated by considering conditional relative frequencies. MP.3 Construct viable arguments and critique the reasoning of others. Students examine the shape, center, and variability of a data distribution and use characteristics of the data distribution to communicate the answer to a statistical question in the form of a poster presentation. Students also have an opportunity to critique poster presentations made by other students.


20: Analyzing Data Collected on Two Variables

Students use data to develop a poster that involves the focus standards. Students construct a scatter plot of the data. Students analyze their data, examining the residual plot, and interpreting the correlation coefficient.

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End-of-Module Assessment - Topics A through D (assessment 1 day, return 1 day, remediation or further applications 1 day)

Algebra 1-2 Scope and SequenceMP.4 Model with mathematics. Students construct and interpret two-way tables to summarize bivariate categorical data. Students graph bivariate numerical data using a scatterplot and propose a linear, exponential, quadratic, or other model to describe the relationship between two numerical variables. Students use residuals and residual plots to assess if a linear model is an appropriate way to summarize the relationship between two numerical variables. MP.5 Use appropriate tools strategically. Students visualize data distributions and relationships between numerical variables using graphing software. They select and analyze models that are fit using appropriate technology to determine whether or not the model is appropriate. Students use visual representations of data distributions from technology to answer statistical questions. MP.6 Attend to precision. Students interpret and communicate conclusions in context based on graphical and numerical data summaries. Students use statistical terminology appropriately.

Quarters 2 & 3

Module 3: Linear and Exponential FunctionsTopic A: Linear and Exponential SequencesIn 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).


1: Integer Sequences—Should You Believe in Patterns?

Students examine sequences and are introduced to the notation used to describe them.

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2: Recursive Formulas for Sequences

Students write sequences with recursive and explicit formulas. MP.6MP.8

3: Arithmetic and Geometric Students learn the structure of arithmetic and geometric sequences.


Algebra 1-2 Scope and SequenceSequences

4: Why Do Banks Pay YOU to Provide Their Services?

Students compare the rate of change for simple and compound interest and recognize situations in which a quantity grows by a constant percent rate per unit interval.

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5: Exponential Growth Students are able to model with and solve problems involving exponential formulas. MP.4

6: Exponential Growth—U.S. Population and World Population

Students compare linear and exponential models of population growth.

7: Exponential Decay Students describe and analyze exponential decay models; they recognize that in a formula that models exponential decay, the growth factor b is less than 1; or, equivalently, when b is greater than 1, exponential formulas with negative exponents could also be used to model decay.

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Topic B: Functions and Their GraphsIn 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).


8: Why Stay with Whole Numbers?

Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Students create functions that represent a geometric situation and relate the domain of a function to its graph and to the relationship it describes.

90: Representing, Naming, and Evaluating Functions

Students understand that a function from one set (called the domain) to another set (called the range) assigns each element of the domain to exactly one element of the range.


10 Representing, Naming, Students understand that a function from one set (called the domain) to another set (called the range) assigns each element of the domain to exactly one element of the


Algebra 1-2 Scope and Sequenceand Evaluating Functions range and understand that 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.


11: The Graph of a Function Students understand set builder notation for the graph of a real-valued function: {(x , f ( x ) )|x∈D }.

Students learn techniques for graphing functions and relate the domain of a function to its graph.

12: The Graph of the Equation y=f (x )

Students understand the meaning of the graph of y=f (x ), namely {( x , y )|x∈D∧ y=f (x)}.

Students understand the definitions of when a function is increasing or decreasing.

13: Interpreting the Graph of a Function

Students create tables and graphs of functions and interpret key features including intercepts, increasing and decreasing intervals, and positive and negative intervals.

14: Linear and Exponential Models—Comparing Growth Rates

Students compare linear and exponential models by focusing on how the models change over intervals of equal length.

Students observe from tables that a function that grows exponentially will eventually exceed a function that grows linearly.

15: Piecewise Functions Students examine the features of piecewise functions including the absolute value function and step functions.

Students understand that the graph of a function 𝑓 is the graph of the equation 𝑦=𝑓(𝑥).Mid-Module Assessment: Topics A through B (assessment 2 days, return 1 day, remediation or further applications 2 day)


Algebra 1-2 Scope and SequenceTopic C: Transformation of FunctionsIn 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).


15: Piecewise Functions Students examine the features of piecewise functions including the absolute value function and step functions.

Students understand that the graph of a function f is the graph of the equation y=f (x ).

16: Graphs Can Solve Equations Too

Students discover that the multi-step and exact way of solving |2 x−5|=¿3 x+1∨¿ using algebra can sometimes be avoided by recognizing that an equation of the form f ( x )=g (x) can be solved visually by looking for the intersection points of the graphs of y=f (x ) and y=g (x).

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17: Four Interesting Transformations of Functions.

Students examine that a vertical translation of the graph of y=f (x ) corresponds to changing the equation from y=f (x ) to y=f ( x )+k.

Students examine that a vertical scaling of the graph of y= f (x ) corresponds to changing the equation from y=f (x ) to y=kf (x ).


Students examine that a horizontal translation of the graph of y=f (x ) corresponds to changing the equation from y= f (x ) to y=f ( x−k ).


Students examine that a horizontal scaling with scale factor k of the graph of

y=f (x ) corresponds to changing the equation from y=f (x ) to y=f (1k x).20: Four Transformations of Functions.

Students apply their understanding of transformations of functions and their graphs to piecewise functions.


Algebra 1-2 Scope and SequenceTopic D: Using Functions and Graphs to Solve Problems

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).


21: Comparing Linear and Exponential Models Again

Students create models and understand the differences between linear and exponential models that are represented in different ways.

22: Modeling and Invasive Species Population

Students apply knowledge of exponential functions and transformations of functions to a contextual situation.

23: Piecewise and Step Functions in Contest

Students apply knowledge of exponential functions and transformations of functions to a contextual situation.

24: Comparing Linear and Exponential Models Again

Students create piecewise and step functions that relate to real-life situations and use those functions to solve problems.

Students interpret graphs of piecewise and step functions in a real-life situation.

End-or-Module Assessment: Topics A through B (assessment 2 days, return 1 day, remediation or further applications 2 day)

Focus Standards for Mathematical Practice20




Module 4: Polynomial and Quadratic Expressions, Equations, and Functions21


Algebra 1-2 Scope and SequenceTopic A: Quadratic Expressions, Equations, Functions, and Their Connection to RectanglesTopic A introduces polynomial expressions. In Module 1, students learned the definition of a polynomial and how to add, subtract, and multiply polynomials. Here, their work with multiplication is extended and connected to factoring polynomial expressions and solving basic polynomial equations (A-APR.A.1, A- REI.D.11). 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, 𝑎𝑥2 + 𝑏𝑥 + 𝑐, to factored form, (𝑥) = (𝑥 −𝑛)(𝑥 −𝑚), and then solve equations involving those expressions. They identify the solutions of the equation as the zeros of the related function. Students apply symmetry to create and interpret graphs of quadratic functions (F-IF.B.4, F-IF.C.7a). They use average rate of change on an interval to determine where the function is increasing or decreasing (F-IF.B.6). 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 Description – Student Outcome(s) Practice(s)1: Multiplying and Factoring Polynomial Expressions Part 1

Students use the distributive property to multiply a monomial by a polynomial and understand that factoring reverses the multiplication process.

Students use polynomial expressions as side lengths of polygons and find area by multiplying. Students recognize patterns and formulate shortcuts for writing the expanded form of

binomials whose expanded form is a perfect square or the difference of perfect squares.

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2: Multiplying and Factoring Polynomial Expressions Part 2

Students understand that factoring reverses the multiplication process as they find the linear factors of basic, factorable quadratic trinomials.

Students explore squaring a binomial, factoring the difference of squares, and finding the product of a sum and difference of the same two terms.

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3: Advanced Factoring Strategies for Quadratic Expressions Part 1

Students develop strategies for factoring quadratic expressions that are not easily factorable, making use of the structure of the quadratic expression.

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4: Advanced Factoring Strategies for Quadratic Expressions (part 2)

Students factor quadratic expressions that cannot be easily factored and develop additional strategies for factorization, including splitting the linear term, using graphing calculators, and using geometric or tabular models.

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5: The Zero Product Property

Students solve increasingly complex one-variable equations, some of which need algebraic manipulation, including factoring as a first step and using the zero product property.

6. Solving Basic One- Students use appropriate and efficient strategies to find solutions to basic quadratic equations. MP 2


Algebra 1-2 Scope and SequenceVariable Quadratic Equations

Students interpret the verbal description of a problem and its solutions in context and then justify the solutions using algebraic reasoning.

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7. Creating and Solving Quadratic Equations in One Variable

Students interpret word problems to create equations in one-variable and solve them (i.e., determine the solution set) using factoring and the zero product property.

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8. Exploring the Symmetry in Graphs of Quadratic Functions

Students examine quadratic equations in two variables represented graphically on a coordinate plane and recognize the symmetry of the graph. They explore key features of graphs of quadratic functions: 𝑦-intercept and 𝑥-intercepts, the vertex, the axis of symmetry, increasing and decreasing intervals, negative and positive intervals, and end behavior. They sketch graphs of quadratic functions as a symmetric curve with a highest or lowest point corresponding to its vertex and an axis of symmetry passing through the vertex.

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9. Graphing Quadratic Functions from Factored Form, 𝒇(𝒙)=𝒂(𝒙−𝒎)(𝒙−𝒏)

Students use the factored form of a quadratic equation to construct a rough graph, use the graph of a quadratic equation to construct a quadratic equation in factored form, and relate the solutions of a quadratic equation in one variable to the zeros of the function it defines.

Students understand that the number of zeros in a polynomial function corresponds to the number of linear factors of the related expression and that different functions may have the same zeros but different maxima or minima.

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10. Interpreting Quadratic Functions from Graphs and Tables

Students interpret quadratic functions from graphs and tables: zeros (𝑥𝑥-intercepts), 𝑦𝑦-intercept, the minimum or maximum value (vertex), the graph’s axis of symmetry, positive and negative values for the function, increasing and decreasing intervals, and the graph’s end behavior.

Students determine an appropriate domain and range for a function’s graph and when given a quadratic function in a context, recognize restrictions on the domain.

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Mid-Module Assessment: Assessment 1 day, return 1 day, remediation or further applications 1 day




Topic B: Using Different Forms for Quadratic FunctionsStudents apply their experiences from Topic A as they transform quadratic functions from standard form to vertex form, (𝑥) = (𝑥 −ℎ)2 + 𝑘 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 function (i.e., the vertex of the graph) 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, 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, 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). With the added technique of completing the square in their toolboxes, students come to see the structure of the equations in their various forms as useful for gaining insight into the features of the graphs of equations (A-SSE.B.3). Students study business applications of quadratic functions as they create quadratic equations and graphs from tables and contexts, and then use them to solve problems involving profit, loss, revenue, cost, etc. (A-CED.A.1, A- CED.A.2, F-IF.B.6, F-IF.C.8a). In addition to applications in business, students solve physics-based problems involving objects in motion. In doing so, students also interpret expressions and parts of expressions in context and recognize when a single entity of an expression is dependent or independent of a given quantity (A-SSE.A.1).Lessons Description – Student Outcome(s) Practice(s)11: Completing the Square (part 1)

Students rewrite quadratic expressions given in standard form, 𝑎𝑥2+𝑏𝑥+𝑐 (with 𝑎=1), in the equivalent completed-square form, 𝑎(𝑥–ℎ)2+𝑘, and recognize cases for which factored or completed-square form is most efficient to use.

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12: Completing the Square (part 2)

Students rewrite quadratic expressions given in standard form, 𝑎𝑥2+𝑏𝑥+𝑐 (with 𝑎≠1), as equivalent expressions in completed square form, (𝑥−ℎ)2+𝑘. They build quadratic expressions in basic business application contexts and rewrite them in equivalent forms.

13: Solving Quadratic Equations by Completing the Square

Students solve complex quadratic equations, including those with a leading coefficient other than 1, by completing the square. Some solutions may be irrational. Students draw conclusions about the properties of irrational numbers, including closure for the irrational number system under various operations

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14: Deriving the Quadratic Formula

Students derive the quadratic formula by completing the square for a general quadratic equation in standard form, 𝑎𝑥2+𝑏𝑥+𝑐=0, and use it to verify the solutions for equations from the previous lesson for which they have already factored or completed the square.

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15: Using the Quadratic Formula

Students use the quadratic formula to solve quadratic equations that cannot be easily factored.

Students understand that the discriminant, 𝑏2−4𝑎𝑐, can be used to determine whether a quadratic equation has one, two, or no real solutions


Topic C: Function Transformations and ModelingIn Topic C, students explore the families of functions that are related to the parent functions, specifically for quadratic (𝑓(𝑥) = 𝑥2), square root((𝑥) = √𝑥), and cube root ((𝑥) = √𝑥3), 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).

Lesson Description Standards18: Graphing Cubic, Square Root, and Cube Root Functions.

Students compare the basic quadratic (parent) function, 𝑦=𝑥2, to the square root function and do the same with cubic and cube root functions. They then sketch graphs of square root and cube root functions, taking into consideration any constraints on the domain and range.

19: Translating Functions Students recognize and use parent functions for linear, absolute value, quadratic, square root, and cube root to perform vertical and horizontal translations. They identify how the graph of 𝑦=(𝑥) relates to the graphs of 𝑦=𝑓(𝑥)+ 𝑘 and 𝑦=𝑓(𝑥+𝑘) for any specific values of 𝑘, positive or negative, and find the constant value, 𝑘, given the parent functions and the translated graphs. Students write the function representing the translated graphs.

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16: Graphing Quadratic Equations from the Vertex Form, 𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌

Students graph simple quadratic equations of the form 𝑦 = (𝑥 −ℎ)2 + 𝑘 (completed-square or vertex form), recognizing that (ℎ, 𝑘) represents the vertex of the graph and use a graph to construct a quadratic equation in vertex form.

Students understand the relationship between the leading coefficient of a quadratic function and its concavity and slope and recognize that an infinite number of quadratic functions share the same vertex.

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17: Graphing Quadratic Functions from the Standard Form, 𝒇(𝒙𝒙)=𝒂𝒙𝟐+𝒃𝒙+𝒄

Students graph a variety of quadratic functions using the form (𝑥)=𝑎𝑥2+𝑏𝑥+𝑐 (standard form). Students analyze and draw conclusions about contextual applications using the key features of a

function and its graph.

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Algebra 1-2 Scope and Sequence20: Stretching and Shrinking Graphs of Functions

Students recognize and use parent functions for absolute value, quadratic, square root, and cube root to perform transformations that stretch and shrink the graphs of the functions. They identify the effect on the graph of 𝑦=𝑓(𝑥) when 𝑓(𝑥) is replaced with 𝑘𝑓(𝑥) and 𝑓(𝑘𝑥), for any specified value of 𝑘, positive or negative, and identify the constant value, 𝑘, given the graphs of the parent functions and the transformed functions. Students write the formulas for the transformed functions given their graphs.

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21: Transformations of the Quadratic Parent Function, f(x) = x2

Students make a connection between the symbolic and graphic forms of quadratic equations in the completed-square (vertex) form. They efficiently sketch a graph of a quadratic function in the form, (𝑥)=𝑎(𝑥−ℎ)2+𝑘, by transforming the quadratic parent function, 𝑓(𝑥)=𝑥2, without the use of technology. They then write a function defined by a quadratic graph by transforming the quadratic parent function.

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22: Comparing Quadratic, Square Root, and Cube Root Functions Represented in Different Ways

Students compare two different quadratic, square root, or cube root functions represented as graphs, tables, or equations. They interpret, contextualize and abstract various scenarios to complete the comparative analysis.

23: Modeling with Quadratic Functions Part 1

Students write the quadratic function described verbally in a given context. They graph, interpret, analyze, check results, draw conclusions, and apply key features of a quadratic function to real-life applications in business and physics.

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24: Modeling with Quadratic Part 2

Students create a quadratic function from a data set based on a contextual situation, sketch its graph, and interpret both the function and the graph in context. They answer questions and make predictions related to the data, the quadratic function, and graph.

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End-of-Module Assessment: Topics A through C (assessment 1 day, return 1 day, remediation or further applications 1 day)



MP.1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. In Module 4, students make sense of problems by analyzing the critical components of the problem, a verbal description, data set, or graph and persevere in writing the appropriate function to describe the relationship between two quantities. MP.2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. This module alternates between algebraic manipulation of expressions and equations and interpretation of the quantities in the relationship in terms of the context. Students must be able to decontextualize―to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own without necessarily attending to their referents, and then to contextualize―to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning requires the habit of creating a coherent representation of the problem at hand, considering the units involved, attending to the meaning of quantities (not just how to compute them), knowing different properties of operations, and flexibility in using them. MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In this module, students create a function from a contextual situation described verbally, create a graph of their function, interpret key features of both the function and the graph (in the terms of the context), and answer questions related to the function and its graph. They also create a function from a data set based on a contextual situation. In Topic C, students use the full modeling cycle. They model quadratic functions presented mathematically or in a context. They explain the reasoning used in their writing or using appropriate tools, such as graphing paper, graphing calculator, or computer software. MP.5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Throughout the entire module, students must decide whether to use a tool to help find a solution. They must graph functions that are sometimes difficult to sketch (e.g., cube root and square root functions) and functions that are sometimes required to perform procedures that, when performed without technology, can be tedious and distract students from thinking mathematically (e.g., completing the square with non-integer coefficients). In such cases, students must decide when to use a tool to help with the calculation or graph so they can better analyze the model. MP.6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. When calculating and reporting quantities in all topics of Module 4, students must be precise in choosing appropriate units and use the appropriate level of precision based on the information as it is presented. When graphing, they must select an appropriate scale. MP.7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. They can see algebraic expressions as single objects, or as a composition of several objects. In this Module, students use the structure of expressions to find ways to rewrite them in different but equivalent forms. For example, in the expression 𝑥2 + 9𝑥 + 14, students must see the 14 as 2 ×7 and the 9 as 2 +7 to find the factors of the quadratic. In relating an equation to a graph, they can see 𝑦 = −3(𝑥 − 1)2 + 5 as 5 added to a negative number times a square and realize that its value cannot be more than 5 for any real domain value.



Module 5: A Synthesis of Modeling with Equations and FunctionsTopic A: Elements of ModelingTopic A deals with some foundational skills in the modeling process. With each lesson, students build a “toolkit” for modeling. They develop fluency in analyzing graphs, data sets, and verbal descriptions of situations for the purpose of modeling; recognizing different function types (e.g., linear, quadratic, exponential, square root, cube root, and absolute value); and identifying the limitations of the model. From each graph, data set, or verbal description, students will recognize the function type and formulate a model, but stop short of solving problems, making predictions, or interpreting key features of functions or solutions. This topic focuses on the skill building required for the lessons in Topic B, where students will take a problem through the complete modeling cycle. This module will deal with both "descriptive models" (such as graphs) and "analytic models" (such as algebraic equations). (N-Q.A.2, A-CED.A.2, F-IF.B.4, F-IF.B.5, F-BF.A.1a, F-LE.A.1b, F-LE.A.1c, F-LE.A.2


1: Analyzing a Graph From a graphic representation, students recognize the function type, interpret key features of the graph, and create an equation or table to use as a model of the context for functions addressed in previous modules (i.e., linear, exponential, quadratic, cubic, square root, cube root, absolute value, and other piecewise functions).

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2: Analyzing a Data Set Students recognize linear, quadratic, and exponential functions when presented as a data set or sequence, and formulate a model based on the data.

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3: Analyzing a Verbal Descriptions

Students make sense of a contextual situation that can be modeled with linear, quadratic, and exponential functions when presented as a word problem. They analyze a verbal description and create a model using equations, graphs, or tables.

MP.1MP.4


Algebra 1-2 Scope and SequenceTopic B: Completing the Modeling CycleTopic B follows a similar progression as Topic A, in that students create models for contexts presented as graphs, data, and as a verbal description. However, in this topic students complete the entire modeling cycle, from problem posing and formulation to validation and reporting. In Lesson 4, students use the gamut of functions covered in the Algebra I course for modeling purposes. They interpret the functions from their respective graphs: linear, quadratic, exponential, cubic, square root, cube root, absolute value, and other piecewise functions, including a return to some graphs from Topic A. Students build on their work from those lessons to complete the modeling cycle. Additionally, students will determine appropriate levels of numerical accuracy when reporting results.( N-Q.A.2, N-Q.A.3, A-CED.A.1, A-CED.A.2, F-IF.B.4, F-IF.B.5, F-IF.B.6, F-BF.A.1a, F-LE.A.1b,F-LE.A.1c, F-LE.A.2)


4: Modeling a Context from a Graph

Students create a two-variable equation that models the graph from a context. Function types include linear, quadratic, exponential, square root, cube root, and absolute value. They interpret the graph and function and answer questions related to the model, choosing an appropriate level of precision in reporting their results.

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5: Modeling from a Sequence

Students recognize when a table of values represents an arithmetic or geometric sequence. Patterns are present in tables of values. They choose and define the parameter values for a function that represents a sequence.

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6: Modeling a Context from Data Part 1

Students write equations to model data from tables, which can be represented with linear, quadratic, or exponential functions, including several from Lessons 4 and 5. They recognize when a set of data can be modeled with a linear, exponential, or quadratic function and create the equation that models the data.

Students interpret the function in terms of the context in which it is presented, make predictions based on the model, and use an appropriate level of precision for reporting results and solutions.

Students use linear, quadratic, and exponential functions to model data from tables, and choose the regression most appropriate to a given context. They use the correlation coefficient to determine the accuracy of a regression model and then interpret the function in context. They then make predictions based on their model, and use an appropriate level.

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Algebra 1-2 Scope and Sequence7: Modeling a Context from Data Part 2

Students write equations to model data from tables, which can be represented with linear, quadratic, or exponential functions, including several from Lessons 4 and 5. They recognize when a set of data can be modeled with a linear, exponential, or quadratic function and create the equation that models the data.

Students interpret the function in terms of the context in which it is presented, make predictions based on the model, and use an appropriate level of precision for reporting results and solutions.

Students use linear, quadratic, and exponential functions to model data from tables, and choose the regression most appropriate to a given context. They use the correlation coefficient to determine the accuracy of a regression model and then interpret the function in context. They then make predictions based on their model, and use an appropriate level of precision for reporting results and solutions.

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8: Modeling a Context from a Verbal Description Part 1

Students model functions described verbally in a given context using graphs, tables, or algebraic representations.

Students interpret the function and its graph and use them to answer questions related to the model, including calculating the rate of change over an interval, and always using an appropriate level of precision when reporting results.

Students use graphs to interpret the function represented by the equation in terms of the context, and answer questions about the model using the appropriate level of precision in reporting results.

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9: Modeling a Context from a Verbal Description Part 2

Students model functions described verbally in a given context using graphs, tables, or algebraic representations.

Students interpret the function and its graph and use them to answer questions related to the model, including calculating the rate of change over an interval, and always using an appropriate level of precision when reporting results.

Students use graphs to interpret the function represented by the equation in terms of the context, and answer questions about the model using the appropriate level of precision in reporting results.

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