Algebra 1 Scope and Sequence Standards Trajectorycrwolff.weebly.com/uploads/2/2/0/5/22059726/alg1... · 1: Statistics, Take 1 18 days August 26–September 19, 2013 2: Proportional

Algebra 1 Scope and Sequence Standards Trajectory

Denver Public Schools 2013–2014 1

Course Name Algebra 1 Grade Level High School

Algebra 1 Common Core State Standards

Conceptual Category Domain Clusters

Number and Quantity The Real Number System (N-RN) Use properties of rational and irrational numbers. (Additional)

Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting)

Algebra

Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major)

Write expressions in equivalent forms to solve problems. (Supporting)

Arithmetic with Polynomials and Rational Expressions (A-APR)

Perform arithmetic operations on polynomials. (Major)

Understand relationship between zeros and factors of polynomials. (Supporting)

Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major)

Reasoning with Equations and Inequalities (A-REI)

Understand solving equations as a process of reasoning and explain reasoning. (Major)

Solve equations and inequalities in one variable. (Major)

Solve systems of equations. (Additional)

Represent and solve equations and inequalities graphically. (Major)

Functions

Interpreting Functions (F-IF)

Understand concept of a function and use function notation. (Major)

Interpret functions that arise in applications in terms of context. (Major)

Analyze functions using different representations. (Supporting)

Building Functions (F-BF) Build functions that model relationships between two quantities. (Supporting)

Build new functions from existing functions. (Supporting)

Linear, Quadratic, and Exponential Models(F-LE)

Construct and compare linear, quadratic, and exponential models and solve problems. (Supporting)

Interpret expressions for functions in terms of situations they model. (Supporting)

Statistics and Probability Interpreting Categorical and Quantitative Data (S-ID)

Summarize, represent, and interpret data on single count or measurement variables. (Additional)

Summarize, represent, and interpret data on two categorical and quantitative variables. (Supporting)

Interpret linear models. (Major)

Modeling Standards

Algebra 1 Scope and Sequence Standards Trajectory


Colorado 21st Century Skills

Critical Thinking and Reasoning: Thinking Deeply, Thinking Differently

Information Literacy: Untangling the Web

Collaboration: Working Together, Learning Together

Self-Direction: Own Your Learning

Invention: Creating Solutions

Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique others’ reasoning.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Unit of Study Length of Unit Time Frame

1: Statistics, Take 1 18 days August 26–September 19, 2013

2: Proportional Reasoning 14 days September 20–October 9, 2013

3: Solving and Writing Linear Equations 17 days October 10–November 5, 2013

4: Statistics, Take 2 26 days November 6–December 20, 2013

5: Systems of Equations and Inequalities 13 days January 7–24, 2014

6: Exponential Growth 15 days January 27–February 14, 2014

7: Functions 18 days February 19–March 14, 2014

8: Transformations 14 days March 17–April 11, 2014

9: Quadratic Models 33 days April 14–June 5, 2014

End-of-Year Fluency Recommendations

Solve characteristic problems involving the analytic geometry of lines. (i.e., write equations of lines given point and slope) (A/G)

Add, subtract, and multiply polynomials. (A-APR.1)

Transform expressions and chunking (see parts of expressions as single objects). (A-SSE.1b)

Invention

Algebra 1, Unit 1: Statistics, Take 1


Unit of Study 1: Statistics, Take 1 Length of Unit 18 days (August 26–September 19, 2013)

Focusing Lens Modeling

Standards

Content Standards

Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting) N-Q.1: Use units as a way to understand problems and guide the solution of multi-step problems; choose and interpret units consistently in formulas;

choose and interpret scale and origin in graphs and data displays. Interpreting Categorical and Quantitative Data (S-ID) Summarize, represent, and interpret data on single count or measurement variables. (Additional) S-ID.1: Represent data with plots on the real number line (dot plots, histograms, box plots). S-ID.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.3: Interpret differences in shape, center, and spread in context of data sets, accounting for possible effects of extreme data points (outliers). Summarize, represent, and interpret data on two categorical and quantitative variables. (Supporting) S-ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in context of data (including joint,

marginal, and conditional relative frequencies). Recognize possible associations and trends in data.

Standards for Mathematical Practice

4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.

Fluency Recommendations

N/A

Inquiry Questions Is there really a difference?

How different is really different?

ELGs Calculate and use measures of center and spread (range and interquartile range) to interpret differences in data sets. (ELG.MA.HS.S.1)

Represent and interpret data displayed in dot plots, histograms, and box plots. (ELG.MA.HS.S.1)

Concepts Shape, center, spread, data representations, outliers, statistical measures



Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding

My students Understand that… Factual Conceptual

Knowledge of shape, center, and spread facilitates comparison of two data sets. (S-ID.2-3)

What is the difference between mean and median? What is the relationship between mean and median in skewed data? How can we use technology to find center and spread for data sets? What can be inferred about two data sets with large differences in measures of spread?

Why is mean by itself not a complete summary of data sets? How can summary statistics or data displays be accurate but misleading?

Analyzing a variety of data representations helps determine appropriate measures of center and spread to describe data sets. (S-ID.1)

What is the best way to display data? How does our data display choice affect which information will be conveyed?

When would median be a more appropriate measure of center than mean? How can summary statistics or data displays be accurate but misleading? Why is it important to analyze data spread?

Outlier influence is an important consideration when selecting and interpreting statistical measures. (S-ID.3)

What is an outlier? Why do outliers affect some measures of center more than others? Why do outliers affect some measures of spread more than others?

Two-way frequency tables provide the necessary structure to make conclusions about the association of categorical variables. (S-ID.5)

What is categorical data? What do joint, marginal, and conditional relative frequencies mean?

Why is it appropriate to use a two-way frequency table with categorical data?

Key Knowledge and Skills (Procedural Skill and Application)

My students will be able to (Do)…

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

Use statistics appropriate to data distributions’ shapes to compare centers (median, mean) and spreads (range, interquartile range, standard deviation) of two or more different data sets. (S-ID.2)

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

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



Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse

Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples.

Given two data sets, compare and contrast the data by describing shape, center, and spread.

Academic Vocabulary Represent, quantities, fit, assess, accuracy, recognize, trends, interpret, shape, center, spread, comparison, data, representations, communicate, difference, findings, predictions

Technical Vocabulary Variables; relative frequency; joint, marginal, and conditional frequencies; mean, median, interquartile range, dot plot, histogram, box plot, two-way frequency tables, categorical, association, outliers, statistical measures, skewed distribution, skewed, quartiles, range, standard deviation, measures of center, measures of spread

Resources

Textbook Core Lessons

Discovering Algebra 1.1: Dot plots 1.2: Measures of center 1.3: Box plots and interquartile range 1.4: Histograms

Additional Core Lessons

1.0a: Two-Way Tables 1.4a: Describing Distributions 1.4b: Standard Deviation

Core Instructional Task Representing Data Using Box Plots

Technology Advanced Data Grapher: http://illuminations.nctm.org/ActivityDetail.aspx?ID=220

Performance/ Learning Task

Haircut Costs: http://www.illustrativemathematics.org/illustrations/942 (S-ID.2)

Misconceptions

Students might not recognize the difference between categorical and quantitative data and confuse when to use bar graphs, histograms, dot plots, and box plots.

Students might not account for differences in scales when comparing distributions of two data sets.

Students might not sort data values before finding the median.

Students might think box plot interval lengths are related to the number of subjects in each interval.

Notes

As you plan for the unit, focus on comparing data sets.

Outliers are addressed in Lesson 1.4a: Describing Distributions. Note that outliers are defined in Discovering Algebra, on page 58, question 10.

If you need additional background about the statistics in this unit, refer to the Draft High School Progression on Statistics and Probability, an excellent resource for understanding the statistics in the CCSS: http://ime.math.arizona.edu/progressions/.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=220

http://www.illustrativemathematics.org/illustrations/942

http://ime.math.arizona.edu/progressions/

Algebra 1, Unit 2: Proportional Reasoning


Unit of Study 2: Proportional Reasoning Length of Unit 14 days (September 20–October 9, 2013)

Focusing Lens Relationships

Standards

Content Standards

The Real Number System (N-RN) Use properties of rational and irrational numbers. N-RN.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. Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting) N-Q.1: Use units as a way to understand problems and guide the solution of multi-step problems; choose and interpret units consistently in formulas;

choose and interpret the scale and origin in graphs and data displays. N-Q.2: Define appropriate quantities for the purpose of descriptive modeling. N-Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales.

Reasoning with Equations and Inequalities (A-REI) Represent and solve equations and inequalities graphically. (Major) A-REI.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).


2. Reason abstractly and quantitatively. 7. Look for and make use of structure.


N/A

Inquiry Question How do ratios connect with unit conversion?

ELGs Use units as a way to understand problems. (ELG.MA.HS.N.3)

Create equations to model and solve proportional relationships, including data presented in tables. (ELG.MA.HS.A.7)

Concepts Proportion, ratio, variation, unit conversion, direct variation, inverse variation





Proportional relationships can be modeled with linear equations of the form y = kx or y = k/x (A-CED.2)

What is a proportional relationship? How do we know when a proportional relationship involves direct variation or inverse variation?

Why are proportional relationships important?

Precision with units is key to dimensional analysis. (N-Q.1)

What is dimensional analysis? How is dimensional analysis used in science?

How is dimensional analysis related to precision?

Sums and products of rational numbers remain in the set of rational numbers. (N-RN.3)

What is the sum of two irrational numbers? What is the sum of a rational and an irrational number? What is the product of a rational and irrational number? What is the product of two rational numbers? What is the product of two irrational numbers?

Why is the sum or product of two rational numbers always rational? Why are the sum and products of irrational numbers with rational numbers always irrational?



Change measurement units through conversion factors and dimensional analysis. (N-Q.1)

Analyze proportional relationships to determine whether they represent direct variations or inverse variations.

Write equations for proportional relationships representing direct variations or inverse variations. (A-CED.2)

Graph proportional relationships. (A-CED.2)

Explain why the sum or product of two rational numbers is rational. (N-RN.3)

Explain why the sum of a rational number and an irrational number is irrational. (N-RN.3)

Explain why the product of a nonzero rational number and an irrational number is irrational. (N-RN.3)



Compare and contrast the constant of variation, the graph, and the equation of a direct variation and an inverse variation.

Academic Vocabulary Compute, identify, represent, explain, simulate, random

Technical Vocabulary Proportion, constant of variation, ratio, proportional relationship, coordinate graphs, direct variation, directly proportional, inverse variation, inversely proportional, rational, irrational



Resources


Discovering Algebra 2.1: Proportions 2.2: Capture-Recapture 2.3: Proportions and Measurement Systems 2.4: Direct Variation 2.5: Inverse Variation


N/A

Instructional Task N/A

Technology N/A

Performance/ Learning Tasks

Runner’s World: http://www.illustrativemathematics.org/illustrations/19 (N-Q)

Operations with Rational and Irrational Numbers: http://www.illustrativemathematics.org/illustrations/690 (N-RN)

Misconceptions

Students might think all in/out tables operate the same as direct variation tables.

Students might extend properties of rational and irrational numbers and conclude (through limited examples) that the sum of any two irrational numbers is also irrational (e.g., (2 + √3) + (2 - √3) = 4, a rational number).

Students might not realize the importance of unit conversions along with the computation when solving problems involving measurement.

Students might express answers to a greater degree of precision than required when using a calculating device’s display of eight to 10 decimal places.

Notes

Emphasize the graph of an equation in two variables is the set of all the equation’s solutions.

The performance/learning task for N-RN, Operations with Rational and Irrational Numbers, is essential for student engagement because it addresses the necessary understandings for N-RN.3. Consider beginning the unit with this task. Students studied both rational and irrational numbers in middle school and will work more with rational numbers in this unit.



Algebra 1, Unit 3: Solving and Writing Linear Equations


Unit of Study 3: Solving and Writing Linear Equations Length of Unit 17 days (October 10–November 5, 2013)

Focusing Lens Structure

Standards

Content Standards

Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.1: Interpret expressions that represent a quantity in terms of its context.

b. Interpret complicated expressions by viewing one or more of their parts as single entities. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.1: Create equations and inequalities in one variable and use them to solve problems. A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales. A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Reasoning with Equations and Inequalities (A-REI) Understand solving equations as a process of reasoning and explain reasoning. (Major) A-REI.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 viable arguments to justify solution methods. Solve equations and inequalities in one variable. (Major) A-REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Building Functions (F-BF) Build functions that model relationships between two quantities. (Supporting) F-BF.1: Write functions that describe relationships between two quantities.

a. Determine explicit expressions, recursive processes, or steps for calculations from context.


2. Reason abstractly and quantitatively. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.


Solve characteristic problems involving analytic geometry of lines (i.e., write equations of lines given point and slope). (A/G)

Transform expressions and chunking (see parts of expressions as single objects). (A-SSE.1b)

Inquiry Questions Why do some equations have a unique solution while others have no solution?

How many solutions can a linear equation have?

ELGs

Interpret complicated expressions by analyzing structures of expressions to solve equations; evaluate expressions. (ELG.MA.HS.A.1)

Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (ELG.MA.HS.A.7)

Solve linear equations in one variable; justify processes and solutions. (ELG.MA.HS.A.8)

Write arithmetic sequences both recursively and explicitly, use to model situations, and translate between the two forms. (ELG.MA.HS.F.4)

Concepts Solving equations, linear equations, recursive patterns, rate of change





Creating equivalent algebraic equations provides the necessary foundation for solving linear equations in one variable. (A-REI.1)

What is an example of a one-variable linear equation with no solution? What is an example of a one-variable linear equation with infinite solutions? How does creating equivalent expressions lead to solving one-variable linear equations?

How can a one-variable linear equation have no solutions or infinite solutions? How does the context of the problem affect the reasonableness of a solution?

Linear relationships can be described using multiple representations. (A-CED.2)

Which methods can be used to represent linear relationships? How does the context of the linear relationship help to interpret the rate of change and initial value of the linear function? How can graphs, tables, and equations determine the rate of change and initial value?

Why do we represent linear relationships using different representations?



Solve linear equations, including equations with coefficients represented by letters. (A-REI.3)

Write recursive routines or formulas to describe linear relationships. (F-BF.1a)

Write linear equations in intercept form. (A-CED.2)

Graph linear equations. (A-CED.2)

Rearrange formulas to highlight quantities of interest. (A-CED.4)

Explain each step in solving equations. (A-REI.1)



Solve linear equations in one variable and explain the logic in each step.

Academic Vocabulary Evaluate, convert, undoing process, recursive, rule, intercept, rate of change

Technical Vocabulary Order of operations, recursive sequence, sequence, term, recursive routine, starting value, linear relationship, additive inverse, like terms, y-intercept



Resources


Discovering Algebra 2.7: Evaluating Expressions 2.8: Undoing Operations 3.1: Recursive Sequences 3.2: Linear Plots 3.3: Time-Distance Relationships 3.4: Linear Equations and the Intercept Form 3.5: Linear Equations and Rate of Change 3.6 Solving Equations Using the Balancing Method


N/A


Technology Algebra Balance Scales: http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.html?open=instructions&from=topic_t_2.html


Same Solutions?: http://www.illustrativemathematics.org/illustrations/613 (A-REI.1–2)

Misconception Students may struggle with the arithmetic of negative numbers due to over-memorization of rules for integer operations.

Notes

Emphasize that graphs of equations in two variables are the set of all the equations’ solutions.

Include an informal discussion of functions throughout Lessons 3.4 and 3.5 using the vocabulary: relation, input, output, function. A formal definition of function and function notation is introduced in Unit 7.

Include problems focused on A-CED.4; see Equations and Formulas: http://www.illustrativemathematics.org/illustrations/393 and question 11, page 202, in Discovering Algebra.

http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.html?open=instructions&from=topic_t_2.html





Unit of Study 4: Statistics, Take 2 Length of Unit 26 days (November 6–December 20, 2013)

Focusing Lenses Communication and Modeling

Standards

Content Standards

Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting) N-Q.2: Define appropriate quantities for the purpose of descriptive modeling. Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.2: Use the structure of an expression to identify ways to rewrite it. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales. A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch

graphs showing key features given verbal descriptions of the relationships. Key features include: intercepts; intervals where function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.6: Calculate and interpret average rates of change of functions (presented symbolically or as tables) during specified intervals. Estimate rates of change from graphs.

Linear, Quadratic, and Exponential Models(F-LE) Interpret expressions for functions in terms of situations they model. (Supporting) F-LE.5: Interpret parameters in linear functions or exponential functions in terms of the context. Interpreting Categorical and Quantitative Data (S-ID) Summarize, represent, and interpret data on two categorical and quantitative variables. (Supporting) S-ID.6: Represent data on two quantitative variables on scatter plots and describe how variables are related.

a. Fit functions to data; use functions fitted to data to solve problems in context of data. Use given functions or choose functions suggested by context. Emphasize linear, quadratic, and exponential models.

b. Informally assess fits of functions by plotting and analyzing residuals. c. Fit linear functions for scatter plots that suggest linear associations.

Interpret linear models. (Major) S-ID.7: Interpret slopes (rates of change) and intercepts (constant term) of linear models in the context of data. S-ID.8: Compute (using technology) and interpret correlation coefficients of linear fits. S-ID.9: Distinguish between correlation and causation.


1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically.



Fluency Recommendation

Solve characteristic problems involving analytic geometry of lines (i.e., write equations of lines given point and slope). (A/G)

Inquiry Question How can mathematics help us make predictions and decisions?

ELGs

Use structures of equations and expressions to identify ways to write equivalent ones (e.g., point-slope, slope-intercept). (ELG.MS.HS.A.1)

Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (ELG.MA.HS.A.7)

Interpret key features (e.g., slopes, intercepts) of functions from equations, graphs, situations, or tables. (ELG.MA.HS.F.2)

Represent data on two quantitative variables on scatter plots and write linear functions to model data. (ELG.MA.HS.S.2)

Concepts Representations, correlation, causation, association, slope, y-intercept, residual



Correlation does not imply causation. (S-ID.9) What is the difference between correlation and causation?

How can results of statistical investigations be used to support arguments?

Correlation coefficients can determine linear model usefulness for describing data and making predictions. (S-ID.8)

How do we find correlation coefficients on a graphing calculator? What are residuals and how do we calculate them? How do we determine whether we have strong or weak linear correlations? How do we quantify the strength of correlations?

Why is it important to know correlation strength for data sets? Why does correlation not imply a causal relationship? Linear models are not always the best choice for all data sets. Why?

Mathematicians focus on slope and y-intercept when interpreting linear models in the context of data. (S-ID.7)

What do slope and intercept of linear models mean? How do slope and y-intercept help interpret linear models?

Linear models describe situations with constant rates of change (slope). (S-ID.6c)

What is slope? How can we tell if a situation has a constant rate of change?

Why can we only model situations having constant rates of change with linear functions?





Graph linear functions and show intercepts. (A-CED.2)

Represent data on two quantitative variables on scatter plots and describe how variables are related. (S-ID.6)

Informally assess fits of functions by plotting and analyzing residuals. (S-ID.6b)

Fit linear functions for scatter plots that suggest linear associations. (S-ID.6c)

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

Compute (using technology) and interpret correlation coefficients of linear fits. (S-ID.8)

Distinguish between correlation and causation. (S-ID.9)

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



Correlation does not imply causation.

Academic Vocabulary Represent, quantities, fit, recognize, trends, interpret, data, representations, communicate, findings, predictions

Technical Vocabulary Variables, scatter plot, correlation, causation, association, correlation coefficient, linear, slope, y-intercept, parameter

Resources


Discovering Algebra 4.1: A Formula for Slope 4.2: Writing a Linear Equation to Fit Data 4.3: Point-Slope Form of a Linear Equation 4.4: Equivalent Algebraic Equations 4.5: Writing Point-Slope Equations to Fit Data 4.6: More on Modeling (Using Q-points is an optional method to determine lines of fit.) 4.7: Applications of Modeling


4.5a: Correlation 4.5b: Residuals 4.7a: Correlation and Causation

Instructional Task Fast Food Sandwiches

Technology Advanced Data Grapher: http://illuminations.nctm.org/ActivityDetail.aspx?ID=220 (use to create scatter plot)

Line of Best Fit: http://illuminations.nctm.org/ActivityDetail.aspx?ID=146


Coffee and Crime: http://www.illustrativemathematics.org/illustrations/1307 (S-ID.6–9)






Misconceptions

Students might apply the distributive property inappropriately, so emphasize the distributive property of multiplication over addition.

Students might think residual plots should show patterns.

Students might think 45-degree lines in scatter plots of two numerical variables always indicate a slope of 1, but this is only when the two variables have the same scaling.

Notes

Average rates of change could be discussed when creating lines of fit by calculating rates of change, if different points were used to create the line of fit.

Use property vocabulary as students solve equations.

F-LE.5 refers to parameters of linear functions. Parameters are constants that can take on various values in function families. For example, given y = mx + b, constants m and b are called parameters. The term “parameter” should be used in classroom discourse.

Limit calculator use of linear regression until Lesson 4.5a.

When students are asked to create mathematical models, they should create equations to represent the mathematical situations and define appropriate variables, even when students are not prompted to do so.

Algebra 1, Unit 5: Systems of Equations and Inequalities


Unit of Study 5: Systems of Equations and Inequalities Length of Unit 13 days (January 7–24, 2014)


Standards

Content Standards

Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales. A-CED.3: Represent constraints by equations and/or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or

nonviable options in a modeling context. Reasoning with Equations and Inequalities (A-REI) Solve systems of equations. (Additional) A-REI.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.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically. (Major) A-REI.10: Understand that graphs of equations in two variables are the set of all the solutions plotted in the coordinate plane, often forming a curve

(which could be a line). A-REI.11: Explain why x-coordinates of the points where graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation

f(x) = g(x); find solutions approximately (e.g., using technology to graph 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.

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


1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 5. Use appropriate tools strategically.


N/A

Inquiry Question How might we determine when a hybrid car would be a better buy, compared to a less expensive, nonhybrid car?

ELGs

Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or nonviable options in terms of the context. (ELG.MA.HS.A.7)

Solve systems of two linear equations exactly (algebraically) and approximately (graphically) and justify processes and solutions. (ELG.MA.HS.A.10)

Graph solutions to linear inequalities in two variables as half-planes and identify solutions to systems of two linear inequalities graphed in the plane. (ELG.MA.HS.A.11)

Concepts Constraint, equations, inequalities, solutions, intersection, systems of equations and inequalities





The points on graphs of equations represent the set of all solutions for a context. (A-REI.10)

How can we determine from a graph if an ordered pair is part of the solution set of an equation? How do graphs represent all solutions to an equation?

Why is it important to understand units of the problem variables when determining solutions to the problem?

When solving systems of linear equations, the type of solution set (one solution, no solutions, or infinite solutions) can be determined both graphically and algebraically. (A-REI.6)

What do different types of solutions for a system of linear equations look like on a graph? How are solutions to systems of equations visualized or approximated on a graph? Is it possible for a system of equations to have no solution? What would it look like on a graph, and what would it look like when algebraically solving the system?

How does the graph of a pair of lines describe the possible solution sets for a system of a pair of linear equations?

Characteristics of equations in systems determine the most efficient strategy for finding solutions. (A-REI.6)

What are the different types of solution processes for solving systems of linear equations? How can we use a calculator to determine the solution to systems of equations?

How do we decide which method to use when given a system of equations? What are the advantages to each method and what are the limitations?

Why is substitution sometimes more efficient than elimination for solving a system of linear equations algebraically and vice versa?

The intersection of two half-planes provides a means to visualize and represent the solution to a system of linear inequalities. (A-REI.12)

What would a graph showing a system of linear inequalities with no solution look like?

Why are solutions to linear inequalities better represented graphically than algebraically?

Mathematicians evaluate mathematical solutions for their relevance to a model; not all solutions to a system are viable in context. (A-CED.3)

What are characteristics of nonviable solutions? How do we know when a solution will be viable?

Why would we model a context with an inequality rather than an equation? Why is it important to evaluate all solutions within the original context?





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.5)

Solve systems of linear equations in two variables exactly and approximately. (A-REI.6)

Explain why x-coordinates of the points where graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find approximate solutions using technology to graph functions, make tables of values, or find successive approximations of linear functions. (A-REI.11)

Create linear equations in two variables to represent relationships between quantities and graph equations on coordinates with labels and scale. (A-CED.2)

Graph solutions to linear inequalities in two variables as a half-plane. (A-REI.12)

Graph solution sets to systems of two linear inequalities in two variables as the intersection of the corresponding half-planes. (A-REI.12)

Represent constraints by equations and inequalities, or by systems of equations and inequalities, and interpret solutions as viable or nonviable within a modeling context. (A-CED.3)



The intersection of two linear equations is their solution set. If the lines do not intersect, there are no solutions. If the lines are the same, an infinite number of solutions exist.

Academic Vocabulary Intersection, efficiency, characteristics, solutions, one solution, no solutions, infinite solutions, viable, nonviable, approximation, constraints, relevance, context

Technical Vocabulary Systems of equations, linear equations, solution set, graphically, algebraically, inequalities, system of inequalities, half-plane, model, elimination, substitution, function, linear

Resources


Discovering Algebra 5.1: Solving Systems of Equations 5.2: Solving Systems of Equations Using Substitution 5.3: Solving Systems of Equations Using Elimination 5.5: Inequalities in One Variable 5.6: Graphing Inequalities in Two Variables 5.7: Systems of Inequalities


N/A

Instructional Task Notebooks and Pens

Technology Linear Programming: http://www.nctm.org/standards/content.aspx?id=32704 (scroll down page to application)


Fishing Adventures 3: http://www.illustrativemathematics.org/illustrations/644 (A-REI)

http://www.nctm.org/standards/content.aspx?id=32704




Misconceptions

Students might confuse the rule of changing a sign of an inequality when multiplying or dividing by a negative number with changing the sign of an inequality when one or two sides of the inequality is negative.

Students might believe the graph of a function is simply a line or curve “connecting the dots,” without recognizing that the graph represents all solutions to the equation.

Notes Students should recognize when linear systems have exactly one solution, no solutions, or infinitely many solutions from graphs and from the

systems’ equations (see Discovering Algebra, page 279, question 11).

Algebra 1, Unit 6: Exponential Growth


Unit of Study 6: Exponential Growth Length of Unit 15 days (January 27–February 14, 2014)


Standards

Content Standards

Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.1: Interpret expressions that represent a quantity in terms of its context.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(a + r)n as the product of P

and a factor not depending on P. Write expressions in equivalent forms to solve problems. (Supporting) A-SSE.3: Choose and produce equivalent forms of expressions to reveal and explain properties of the quantity represented by the expressions.

c. Use properties of exponents to transform expressions for exponential functions. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales. Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of the context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch


Building Functions (F-BF) Build functions that model relationships between two quantities. (Supporting) F-BF.1: Write functions that describe relationships between two quantities.

a. Determine explicit expressions, recursive processes, or steps for calculations from context. Linear, Quadratic, and Exponential Models(F-LE) Construct and compare linear, quadratic, and exponential models and solve problems. (Supporting) F-LE.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.2: Construct linear and exponential functions, including arithmetic and geometric sequences, with graphs, descriptions of relationships, or two input-output pairs (include reading these from tables).

F-LE.3: Observe, using graphs and tables, that quantities increasing exponentially eventually exceed quantities increasing linearly, quadratically, or (more generally) as polynomial functions.

Interpret expressions for functions in terms of situations they model. (Supporting) F-LE.5: Interpret parameters in linear or exponential functions in terms of context. (F-LE.5)




1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically.


N/A

Inquiry Question What are the consequences of a population that grows exponentially?

ELGs

Use properties of exponents to transform expressions for exponential functions. (ELG.MA.HS.A.2)

Graph exponential functions showing key features, including intercepts and end behavior. (ELG.MA.HS.F.3)

Determine explicit expressions, recursive processes, or steps for calculation from context. (ELG.MA.HS.F.4)

Recognize situations in which one quantity grows or decays by a constant percentage rate over equal intervals. (ELG.MA.HS.F.6)

Construct exponential functions including geometric sequences with graphs, descriptions, or tables. (ELG.MA.HS.F.6)

Observe, with graphs and tables, that quantities increasing exponentially eventually exceed quantities increasing linearly. (ELG.MA.HS.F6)

Concepts Growth, decay, constant rate of change, constant rate of growth, linear functions, exponential functions, arithmetic and geometric sequences



Linear and exponential functions provide the means to model constant rates of change and growth, respectively. (F-LE.1)

How do we determine whether situations can be modeled by linear functions, exponential functions, or neither? How do we determine from equations whether exponential functions model growth or decay? What are typical situations modeled by linear and exponential functions? How many points of data do we need to determine whether functions are linear or exponential?

How are differences between linear and exponential functions visible in equations, tables, and graphs? Why does a common difference indicate a linear function and a common ratio indicate an exponential function?

Quantities increasing exponentially eventually exceed quantities increasing linearly or quadratically. (F-LE.3)

How does the rate of growth in linear and exponential functions differ? How can we determine when an exponential function will exceed a linear function?

Why can so many situations be modeled by exponential growth? Why is it important to consider limitations of exponential models?

Linear and exponential functions model arithmetic and geometric sequences, respectively. (F-LE.2)

How do we know whether sequences are arithmetic or geometric? How can we determine slope and y-intercept of arithmetic sequences? How can we determine ratios for geometric sequences?

Why do linear and exponential functions model so many situations? Why is the domain of a sequence a subset of integers?



Parameters of equations interpretation must consider real-world contexts. (F-LE.5)

What is a coefficient? How do we choose coefficients given a set of data?

Why are coefficients sometimes represented with letters? Why does changing coefficients affect a model?

The generation of equivalent exponential functions by applying properties of exponents sheds light on a problem context. (A-SSE.3)

How do properties of exponents simplify exponential expressions? Why does a number raised to the power of zero equal one?

How do exponential patterns explain negative exponents?



Use properties of exponents to transform expressions for exponential functions with integer exponents. (A-SSE.3c)

Create equations in one variable and use them to solve problems; include equations arising from linear and exponential functions with integer exponents. (A-CED.2)

Recognize situations in which quantities grow or decay by constant percentage rate per unit interval relative to another. (F-LE.1c)

Recognize situations in which one quantity changes at a constant rate per unit internal relative to another. (F-LE.1b)

Construct linear and exponential functions, including arithmetic and geometric sequences, with graphs, relationship descriptions, or two input-output pairs. (F-LE.2)

Interpret parameters in linear or exponential (domain of integers) functions in terms of real-world context. (F-LE.5)

Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. (F-LE.1a)



A linear function has a constant rate of change, while an exponential function has a constant rate of growth.

Academic Vocabulary Transform, model, create, interpret, situations, real-world contexts, growth, decay, relationships, tables, graphs

Technical Vocabulary Quantity, constant rate of change, constant rate of growth, linear functions, exponential functions, exponentially, linearly, quadratically, arithmetic sequence, geometric sequence, explicit, recursive, initial value, parameter, common difference, common ratio, parameter, coefficient, key features

Resources


Discovering Algebra 6.1: Recursive Routines 6.2: Exponential Equations 6.3: Multiplication and Exponents 6.4: Scientific Notation for Large Numbers 6.5: Looking back with Exponents 6.6: Zero and Negative Exponents 6.7: Fitting Exponential Models to Data

Additional Core N/A



Lessons

Instructional Task Comparing Investments

Technology Modeling: http://www.nctm.org/standards/content.aspx?id=32704 (scroll down page to application)


Comparing Exponentials: http://www.illustrativemathematics.org/illustrations/213 (F-LE)

Misconceptions Students might believe arithmetic and geometric sequences are the same, and they might not be able to recognize the difference.

Students might believe the laws of exponents work for all operations.

Students might believe any number to the zero power is zero.

Notes In working with sequences, use the language of arithmetic sequence (linear function) and geometric sequence (exponential function).

http://www.nctm.org/standards/content.aspx?id=32704


Algebra 1, Unit 7: Functions


Unit of Study 7: Functions Length of Unit 18 days (February 19–March 14, 2014)

Focusing Lens Representation

Standards

Content Standards

Creating Equations (A-CED) Creating equations that describe numbers or relationships (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales. Interpreting Functions (F-IF) Understand the concept of a function and use function notation. (Major) F-IF.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.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of context. F-IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Interpret functions that arise in applications in terms of context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch


F-IF.5: Relate the domain of a function to its graph and, where applicable, to quantitative relationship it describes. F-IF.6: Calculate and interpret average rates of change of functions (presented symbolically or as tables) during specified intervals. Estimate rates of

change from graphs. Analyze functions using different representations. (Supporting) F-IF.7: Graph functions expressed symbolically and show key features of graphs, 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.9: Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).


2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. 7. Look for and make use of structure.


N/A

Inquiry Question Can all real-world situations be modeled with a function?



ELGs

Identify functions from graphs or tables and their domains and ranges. (ELG.MA.HS.F.1)

Evaluate functions using function notation and interpret statements that use function notation in terms of context. (ELG.MA.HS.F.1)

Sketch graphs showing key features given verbal descriptions of relationships, relating domains of functions to their graphs and context. (ELG.MA.HS.F.2)

Graph absolute value functions and show key features of graphs. (ELG.MA.HS.F.3)

Concepts Function, domain, range, key features of functions, rate of change, functional representations, continuous, discrete



Functions describe contexts where one quantity, an input, determines another, the output. (F-IF.1)

Given an input and output, how do we determine a rule? Which notation is used to write a function? What is the meaning of f(x)? How do we know if data in tables, graphs, or equations are functions?

Why is only one output permissible for every input in a function? Why are functions an important tool in mathematical modeling?

Limiting domains of functions ensures both the domains and ranges make sense in given contexts. (F-IF.5)

What is another name for the inputs of a function? What is another name for the outputs of a function? For a relation to be a function, what must be true for a domain and range? How do we express domain and range?

How do we determine a reasonable domain and range for a context? How do we know whether it is discrete or continuous? Why is it necessary to constrain domains and ranges of function models?

Functions model relationships between quantities using a variety of representations (tables, graphs, and equations). (F-IF.4)

What are examples of linear, quadratic, and exponential contexts? How do graphs, equations, and tables show similarities and differences of functions?

Why are two variable equations helpful in modeling a variety of contexts? Why do linear and exponential functions model so many situations? Why is it important to interpret differences and similarities of functions through multiple representations?

Visualizing a variety of functions helps interpret key features, such as domain, range, maxima, minima, intercepts, symmetry, end behavior, and average rate of change. (F-IF.4)

What are important characteristics of functions that can be seen on graphs? What do the graphs of linear, exponential, square root, cube root, step, and absolute value functions look like?

How does visualizing functions help interpret the relationship between two variables? How is the graph of an equation related to its solutions?





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

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

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (F-IF.5)

Interpret key features of graphs and tables for functions that model relationships between two quantities. Key features include: intercepts, intervals where function is increasing, decreasing, positive, negative; relative maximums and minimums; and symmetries, including continuous and discrete. (F-IF.4)

Calculate and interpret average rates of change of functions during specified intervals and estimate rates of change from graphs. (F-IF.6)

Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (F-IF.9)



When using functions to model real-world phenomena, it is important to constrain the domain and range to values that make sense in the context of the model.

Academic Vocabulary Input, output, relationship, limit, contexts, explain, determine, generate, tables, model, representations, graphs, increasing, decreasing

Technical Vocabulary Coordinate plane, axes, set, function, domain, range, equivalent, functions: absolute-value, step, piecewise defined, square root and cube root, functional representations, maxima, minima, average rate of change, end behavior, vertical line test, discrete, continuous

Resources


Discovering Algebra 7.1: Secret Codes 7.2: Functions and Graphs 7.3: Graphs of Real-World Situations 7.4: Function Notation 7.5: Defining the Absolute-Value Function 7.6: Squares, Squaring, and Parabolas


7.5a: Piecewise Functions 7.6a: More Functions


Technology N/A


Using Function Notation I: http://www.illustrativemathematics.org/illustrations/598 (F-IF)

Pizza Place Promotion: http://www.illustrativemathematics.org/illustrations/578 (F-IF)

Interpreting the Graph: http://www.illustrativemathematics.org/illustrations/636 (F-IF)

Misconceptions Students might believe all relationships having an input and output are functions, and therefore, misuse function terminology.

Students might believe the notation f(x) means to multiply some value f times another value x.






Students might believe it is reasonable for any x-value to be in the domain.

Students might believe that each family of functions is independent of the others and not recognize commonalities among all functions and their graphs.

Notes Emphasize function notation.

Revisit sequences to establish that sequences are functions.

Limit the study of functions in this unit to linear, square root, cube root, piecewise, step, absolute value, and exponential (integer domains).

Algebra 1, Unit 8: Transformations


Unit of Study 8: Transformations Length of Unit 14 days (March 17–April 11, 2014)

Focusing Lens Representation

Standards

Content Standards

Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch


F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Analyze functions using different representations. (Supporting) F-IF.7: Graph functions expressed symbolically and show key features of graphs, by hand in simple cases and using technology for more complicated

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

Building Functions (F-BF) Build new functions from existing functions. (Additional) F-BF.3: Identify effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find values

of k given the graph. Experiment with cases and illustrate explanations of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.


5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.


N/A

Inquiry Question How are geometric transformations the same as, or different from, transformations with functions?

ELGs


Graph absolute value functions and show key features of graphs. (ELG.MS.HS.F.3)

Identify effects of transformations (translations, reflections, stretches, or shrinks) on parent functions. (ELG.MA.HS.F.5)

Concepts Functions, translations, scaling, key features, transformations, reflection





Functions, translated and scaled, enhance the application of similar functions to multiple situations. (F-BF.3)

How do we shift a function up or down? How do we shift a function right or left? How do we stretch a function? How do we reflect a function?

Which features of functions remain the same when translated and scaled? Which features of functions change when translated and scaled?

Visualizing various functions on a coordinate plane helps interpret key features, such as domain, range, maxima, minima, intercepts, symmetry, end behavior, and average rate of change. (F-IF.4)

What are important key features of functions that can be seen on graphs? Give an example of a graph equation where the domain stays the same but the range changes. Give an example of a graph equation where the range stays the same but the domain changes. Give an example of a graph equation where the range and domain are the same.

After reflecting around the x-axis, how do the key features stay the same? How are they different? How do we change location of x-intercepts? How does each transformation change the number and location of x-intercepts? Which transformations switch increasing and decreasing intervals? Which transformations switch minima and maxima? How do transformations affect domain and range?

One can predict effects on a graph when replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative). (F-BF.3)

What is the impact of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative)?

Why are effects on a graph predictable when replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative)?



For functions (linear, quadratic, square root, cube root, piecewise, exponential with domain in integers) modeling relationships between two quantities, interpret key features of graphs and tables in terms of quantities and sketch graphs showing key features given verbal descriptions of the relationships; key features include: intercepts; intervals where function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (F-IF.4)

Identify effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find values of k given the graph and experiment with cases and illustrate explanations of effects on the graph using technology for linear and quadratic functions. (F-BF.3)



Key features of graphs include domain, range, maxima, minima, intercepts, and rate of change. Functions can be shifted, stretched, or shrunk.

Academic Vocabulary Relationship, increasing, decreasing, graph, line, key features, prediction, effects, identity, compare, calculate, interpret, estimate, illustrate

Technical Vocabulary Functions, scale, translate, translations, scaling, equations, coordinate plane, curve, domain, range, maxima, minima, intercepts, symmetry, average rate of change, transformation, algebraically, graphically, verbally, axes, intervals, reflections

Resources




Discovering Algebra 8.1: Translating Points 8.2: Translating Graphs 8.3: Reflecting Points and Graphs 8.4: Stretching and Shrinking Graphs 8.6: Introduction to Rational Functions


N/A


Technology N/A


Transforming the graph of a function: http://www.illustrativemathematics.org/illustrations/742 (F-BF.3)

Misconceptions Students might believe each family of functions is independent of the others, so they may not recognize commonalities among all functions

and their graphs.

Students might believe the graph of y = (x – 4)2 is the graph of y = x

2 shifted 4 units to the left (due to the subtraction symbol).

Notes Use this unit to reinforce function families introduced in Unit 7.


Algebra 1, Unit 9: Quadratic Models


Unit of Study 9: Quadratic Models Length of Unit 33 days (April 14–June 5, 2014)

Focusing Lens Structure

Standards

Content Standards

Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.1: Interpret expressions that represent quantities in terms of context.

a. Interpret parts of expressions, such as terms, factors, and coefficients. A-SSE.2: Use the structure of an expression to identify ways to rewrite it. Write expressions in equivalent forms to solve problems. (Supporting) A-SSE.3: Choose and produce equivalent forms of expressions to reveal and explain properties of the quantity represented by the expressions.

a. Factor quadratic expressions to reveal the zeros of the function they define. b. Complete the square in quadratic expressions to reveal maximum or minimum value of the function they define.

Arithmetic with Polynomials and Rational Expressions (A-APR) Perform arithmetic operations on polynomials. (Major) A-APR.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-APR.3: Identify zeros of polynomials when suitable factorizations are available and use zeros to construct rough graphs of functions defined by the

polynomials. Reasoning with Equations and Inequalities (A-REI) Understand solving equations as a process of reasoning and explain reasoning. (Major) A-REI.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 viable arguments to justify solution methods. Solve equations and inequalities in one variable. (Major) A-REI.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.

Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch


F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. F-IF.6: Calculate and interpret average rates of change of functions (presented symbolically or as tables) during specified intervals. Estimate rates of

change from graphs. Analyze functions using different representations. (Major) F-IF.7: 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. F-IF.8: Write functions defined by expressions in different but equivalent forms to reveal and explain different properties of the functions.

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


2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure.

Fluency Recommendation


Inquiry Question Which situations are best modeled by a quadratic? Are there limitations?

ELGs

Write quadratic functions in factored form and/or vertex form given tables, situations, or graphs. (ELG.MA.HS.A.2)

Convert between standard, vertex, and factored forms. (ELG.MA.HS.A.2)

Solve quadratic equations by inspection, taking square roots, completing the square, quadratic formula, and factoring. (ELG.MA.HS.A.9)


Graph quadratic functions and show intercepts, maxima, minima, and symmetry and interpret in terms of context. (ELG.MA.HS.F.3)

Concepts Quadratic functions, roots and zeros, factoring, expanding, completing the square, quadratic, polynomial



Polynomials form a closed system under the operations of addition, subtraction, and multiplication. (A-APR.1)

Which operations can be performed to two polynomials that result in another polynomial?

Why are polynomials not closed under division? Give an example. How do we know if an expression or equation is a polynomial?

Use the structure of an expression to identify ways to rewrite it (A-SSE.2)

Which patterns exist when factoring quadratic equations?

How do we know if rewriting an expression will provide the information needed to solve the contextual problem? What are the benefits of simplifying complicated expressions?

The transformation of quadratic expressions and equations reveals underlying structures and solutions. (A-SSE.3)

What are the different ways to solve quadratic equations? How is factoring used to solve a polynomial with a degree greater than two?

Why would we use a particular method to solve a quadratic equation?



The choice of an appropriate way to rewrite quadratic expressions can aid efficiency and accuracy when solving quadratic equations. (A-REI.4)

What is the difference between the methods for solving quadratic equations? What information does completing the square for quadratic functions reveal? How do we know when a quadratic has a maximum or minimum?

Why is it beneficial to write quadratics in different forms? What does it mean if a function is not factorable?

Quadratic functions and their graphs model real-world applications by helping visualize symmetry and extreme values. (F-IF.4)

What do the zeros of quadratic equations represent in terms of a model? How can we see the symmetry of a quadratic in its equation?

Why is a quadratic a good model for projectile motion, and are there limits to its application? Why might we want to solve for the zeros of a quadratic?




Interpret parts of expressions, such as terms, factors, and coefficients in terms of context. (A-SSE.1a)

Use the structure of an expression to identify ways to rewrite it. (A-SSE.2)

Factor quadratic expressions to reveal the zeros of the functions they define and use these to construct rough graphs. (A-APR.3)

Complete the square in quadratic expressions to reveal maximum or minimum value of the function they define. (A-SSE.3b)

Explain each step in solving simple quadratic equations. (A-REI.1)

Solve quadratic equations in one variable using the method of completing the square to derive the quadratic formula. (A-REI.4a)

Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. (A-REI.4b)

Use the processes of factoring and completing the square in quadratic functions to show zeros, extreme values, and symmetry of the graph and interpret these in terms of context. (F-IF.8a)

Graph quadratic functions and show intercepts, maxima, and minima. (F-IF.7a)



Completing the square of quadratic equations reveals the vertex of the parabola and the axis of symmetry. Factoring quadratic equations reveals the zeros or roots.

Academic Vocabulary Identify, symmetry, reveal, interpret, justify, explain, structure, graph, model, real-world applications

Technical Vocabulary Quadratic, parabola, complete the square, factor, expression, zeros, roots, square root, polynomial, extreme values, maximum, minimum, closed, vertex, equivalent, functions, equations, solutions, axis of symmetry

Resources

Textbook Core Lessons Discovering Algebra 9.1: Solving Quadratic Equations



9.2: Finding the Roots and the Vertex 9.3: From Vertex to General Form 9.4: Factored Form 9.5: Activity Day: Projectile Motion 9.6: Completing the Square 9.7: The Quadratic Formula

Additional Core Lesson 9.7a: Polynomial Operations


Technology N/A


Profit of a Company: http://www.illustrativemathematics.org/illustrations/434 (A-SSE)

Misconceptions

Students might believe that factoring and completing the square are isolated techniques without understanding their use in solving quadratic equations.

Students might think the minimum (the vertex) of the graph of y = (x + 5)2 is shifted to the right of the minimum (the vertex) of the graph y = x

2

because of the addition sign.

Students might think the minimum of the graph of a quadratic function always occurs at the y-intercept.

Students might believe all functions have a first common difference and need to explore common differences for quadratic functions.

Notes

Include the language of maxima and minima in Lesson 9.6.

Ask students to explain each step when solving quadratic equations.

Recognize situations when solutions to quadratics are not real (Discovering Algebra, page 534, question 6).

Emphasize function notation with quadratics.


Documents

Algebra 1 Scope and Sequence Standards Trajectorycrwolff.weebly.com/uploads/2/2/0/5/22059726/alg1... · 1: Statistics, Take 1 18 days August 26–September 19, 2013 2: Proportional