Sequenced Units for Arizona s College and Career Ready Standards MA27 Algebra I

Transcription

1 Sequenced Units for Arizona s College and Career Ready Standards MA27 Algebra I Year at a Glance Semester 1 Semester 2 Unit 1: Solving Linear Equations (12 days) Unit 2: Solving Linear Inequalities (12 days) Unit 3: Graphing Linear Functions (14 days) Unit 4: Writing Linear Functions (14 days) Unit 5: Solving Systems of Linear Equations and Linear Inequalities (15 days) Unit 6: Exponential Functions and Sequences (17 days) Unit 7: Polynomial Equations and Factoring (19 days) Unit 8: Graphing Quadratic Equations (16 days) Unit 9: Solving Quadratic Equations (16 days) Unit 10: Radical Functions and Equations (11 days) Unit 11: Data Analysis and Displays (14 days)

2 In the three years prior to Algebra I, students have already begun their study of algebraic concepts. They have investigated variables and expressions, solved equations, constructed and analyzed tables, used equations and graphs to describe relationships between quantities, and studied linear equations and systems of linear equations. The Algebra I course outlined in this scope and sequence document begins with connections back to that earlier work, efficiently reviewing algebraic concepts that students have already studied while at the same time moving students forward into the new ideas described in the high school standards. Students contrast exponential and linear functions as they explore exponential models using the familiar tools of tables, graphs, and symbols. Finally, they apply these same tools to a study of quadratic functions. Throughout, the connection between functions and equations is made explicit to give students more ways to model and make sense of problems. This document reflects our current thinking related to the intent of Arizona s College and Career Ready Standards and assumes 160 days for instruction, divided among 11 units. The number of days suggested for each unit assumes 45-minute class periods and is included to convey how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the standards; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit. Throughout Algebra I, students should continue to develop proficiency with Arizona s College and Career Ready Standards eight Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning. These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. When using this document to help in planning your instructional program, you will also need to refer to the Mesa Public Schools Standards Implementation document, relevant progressions documents for the standards, and the appropriate assessment consortium framework. Mesa Public Schools 1 May 2015

3 Unit 1: Solving Linear Equations Suggested number of days: 12 Unit 1 presents the foundational skills related to solving linear equations and the connected skills of solving absolute value equations and rewriting equations and formulas. Most students will have prior experience with the Properties of Equality and techniques presented in the first three sections. It will sound familiar that whatever operation is performed on one side of the equation, the same operation must be performed on the other side of the equation to keep equality, or balance. During the unit students apply the techniques of equation solving to the context of absolute value equations, understanding absolute value as a function concept and not simply two vertical lines can be challenging for students and solving literal equations which requires students to see the structure of equations. Quantities N-Q A. Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Creating Equations A-CED A. Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities A-REI A. Understand solving equations as a process of reasoning and explain the reasoning 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. B. Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. N-Q These standards are integrated throughout both the Algebra I and Algebra II course. Most notably in modeling tasks. For example, in a situation involving data, the student might autonomously decide that a measure of center is a key variable in a situation, and then choose to work with the mean. A-CED.A.1 Focus on linear, quadratic, or exponential equations with integer exponents. A-REI.A.1 Focus on quadratic equations. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. Mesa Public Schools 2 May 2015

4 Unit 2: Solving Linear Inequalities Suggested number of days: 12 Students have just finished a unit on solving linear equations simple and multi-step equations along with absolute value equations. The techniques used in solving linear equations are applied to linear inequalities in this unit. In grade 7, students solved and graphed linear inequalities; so many of the topics in this unit should be familiar to them. The unit begins with an introduction to writing and graphing inequalities. Color coding and verbal models are used to help students develop confidence in writing inequalities, a necessary skill for the unit. The graphs are used to display and check solutions. As the unit progresses students focus on solving increasingly complex inequalities. Tools used in developing facility with these problems include symbolic manipulation, tables, and spreadsheets. Practice with real number operations is integrated throughout. The unit introduces compound inequalities, which are necessary in solving absolute value inequalities. Look for the helpful teaching strategies offered in the lessons within this unit. Formative assessment tips are offered in many of the lessons, and tips from the previous unit are referenced throughout the notes at point of use. Creating equations A-CED A. Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Reasoning with Equations and Inequalities A-REI B. Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-CED.A.1 Focus on linear, quadratic, or exponential equations with integer exponents. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. Mesa Public Schools 3 May 2015

5 Unit 3: Graphing Linear Functions Suggested number of days: 14 Students should have a conceptual understanding of functions from a prior grade. Their understanding may be of a function machine where there is an input, a function is performed, and an output results. There is a pairing of the input and output, and each input is associated with exactly one output. Unit 3 extends this introductory understanding of functions and presents the notation of functions. Consistent use of the notation and language of functions will help students become more confident. The unit purposefully focuses on function notation, representing functions, discrete and continuous functions, and evaluating functions. Students may be resistant to using function notation, preferring the simpler y = notation. It is hard for students to appreciate what the broader notation enables us to do because they have not learned enough at this stage. When two equations are graphed on the same axes, we can clearly refer to f and g, versus saying the first y = and the second y =. We compose functions and have functions with multiple inputs, two examples where function notation is useful. The unit also introduces two forms of linear equations standard and slope-intercept. Students should be able to do quick sketches of each by inspecting information from the equation. The unit looks at transformations of linear and absolute value functions and these same transformations will be applied to other types of functions, such as quadratic and trigonometric. Function notation is used to describe the transformations. Creating Equations A-CED A. Create equations that describe numbers or relationships 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 D. Represent and solve equations and inequalities graphically 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). Interpreting Functions F-IF A. Understand the concept of a function and use function notation 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). 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. B. Interpret functions that arise in applications in terms of the context 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F-IF.B.4 Include problem-solving opportunities utilizing a real-world context. Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F- IF.7, and F-IF.9. Mesa Public Schools 4 May 2015

6 Unit 3: Graphing Linear Functions Suggested number of days: 14 C. Analyze functions using different representations 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. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions F-BF B. Build new functions from existing functions 3. Identify the effect 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 the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear, Quadratic, and Exponential Models F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. B. Interpret expressions for functions in terms of the situation they model 5. Interpret the parameters in a linear or exponential function in terms of a context. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F- IF.7, and F-IF.9. F-IF.C.9 Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. F-BF.B.3 Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on F-LE.B.5 Include problem-solving opportunities utilizing a real-world context. Focus on exponential functions with domains in the integers. Mesa Public Schools 5 May 2015

7 Unit 4: Writing Linear Functions Suggested number of days: 14 Students have just finished a unit on graphing linear equations in various forms, including transformations of graphs of linear functions. This unit continues with lessons on writing linear equations in slope-intercept form and standard form. These forms are extended in to include the cases of parallel and perpendicular lines. Many real-life applications involve data that can be modeled by a linear equation and another focus is on graphing scatter plots and writing a best-fit line for the data. Students first approximate a line of best fit and then use graphing calculators (or spreadsheets) to generate the regression equation of the line of best fit. Connections to linear equations are made in this unit. Students should make the connection between the common difference in an arithmetic sequence and the slope of a linear equation. Further, a 0 is the y-intercept of a linear equation and write the linear equations for each part of a piecewise function. Graphing calculator techniques are presented as helpful teaching strategies. Quantities N-Q A. Reason quantitatively and use units to solve problems. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Creating Equations A-CED A. Create equations that describe numbers or relationships 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 D. Represent and solve equations and inequalities graphically 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). Interpreting Functions F-IF A. Understand the concept of a function and use function notation 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. C. Analyze functions using different representations 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Building Functions F-BF A. Build a function that models a relationship between two quantities 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. N-Q These standards are integrated throughout both the Algebra I and Algebra II course. Most notably in modeling tasks. For example, in a situation involving data, the student might autonomously decide that a measure of center is a key variable in a situation, and then choose to work with the mean. F-IF.A.3 This standard is part of the Major content in Algebra I. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F- IF.9. F-BF.A.1a Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on Mesa Public Schools 6 May 2015

8 Unit 4: Writing Linear Functions Suggested number of days: 14 Linear, Quadratic, and Exponential Models F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). B. Interpret expressions for functions in terms of the situation they model 5. Interpret the parameters in a linear or exponential function in terms of a context. Interpreting Categorical and Quantitative Data S-ID B. Summarize, represent, and interpret data on two categorical and quantitative variables 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. C. Interpret linear models 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. 9. Distinguish between correlation and causation. F-LE.A.2 Focus on constructing linear and exponential functions in simple context (not multi-step). F-LE.B.5 Include problem-solving opportunities utilizing a real-world context. Focus on exponential functions with domains in the integers. S-ID.B.6a Include problem-solving opportunities utilizing a real-world context. Focus on exponential functions with domains in the integers. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. Mesa Public Schools 7 May 2015

9 Unit 5: Solving Systems of Linear Equations Suggested number of days: 15 There are three common techniques for solving a system of equations: graphing, substitution, and elimination. These techniques are presented in this unit. Students are introduced to the definition of a linear system, and they learn to check their solutions. Students look at special linear systems, where there is no solution because the lines are parallel or there are infinitely many solutions because the lines coincide. Students learn that solving a system by graphing can be used to solve an equation with variables on both sides. This is actually a technique that students will use extensively in future mathematics courses. Creating equations A-CED A. Create equations that describe numbers or relationships 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Reasoning with Equations and Inequalities A-REI C. Solve systems of equations 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. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. D. Represent and solve equations and inequalities graphically 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 12. Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. A-REI.C.6 Include problem-solving opportunities utilizing a real-world context. Tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.). A-REI.D.11 Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piece-wise. Focus on exponential functions with domains in the integers. Mesa Public Schools 8 May 2015

10 Unit 6: Exponential Functions and Sequences Suggested number of days: 17 Having finished work with linear equations, this unit involves polynomials and work with quadratics. This unit introduces students to exponential functions and sequences. Students will revisit exponential functions and learn about logarithmic functions in Algebra 2. The properties of exponents that are presented should be a review for students. Many of the problems involve numeric expressions, although there are algebraic expressions as well. The same properties of exponents are applied to radicals and rational exponents. The unit moves into exponential functions and the attributes of exponential growth and decay functions. Exponential equations are solved using the properties of exponents initially and then graphically with a graphing calculator. Geometric sequences are introduced with the connection made to exponential functions. Additionally, recursively defined sequences looks at both arithmetic and geometric sequences, writing them as recursive rules. Seeing Structure in Expressions A-SSE B. Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Creating Equations A-CED A. Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 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 A. Understand solving equations as a process of reasoning and explain the reasoning 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. D. Represent and solve equations and inequalities graphically 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Interpreting Functions F-IF A. Understand the concept of a function and use function notation 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. A-SSE.B.3c Include problem-solving opportunities utilizing a real-world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation. Focus on expressions with integer exponents. A-CED.A.1 Focus on linear, quadratic, or exponential equations with integer exponents. A-REI.A.1 Focus on quadratic equations. A-REI.D.11 Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piece-wise. Focus on exponential functions with domains in the integers. F-IF.A.3 This standard is part of the Major content in Algebra I. Mesa Public Schools 9 May 2015

11 Unit 6: Exponential Functions and Sequences Suggested number of days: 17 B. Interpret functions that arise in applications in terms of the context 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. C. Analyze functions using different representations 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions F-BF A. Build a function that models a relationship between two quantities 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. B. Build new functions from existing functions 3. Identify the effect 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 the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear, Quadratic, and Exponential Models F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems 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. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F-IF.B.4 Include problem-solving opportunities utilizing a real-world context. Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewisedefined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F- IF.9. F-IF.C.9 Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on F-BF.A.1a Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on F-BF.B.3 Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. F-LE.A.2 Tasks are limited to constructing linear and exponential functions in simple context (not multistep). Mesa Public Schools 10 May 2015

12 Unit 7: Polynomial Equations and Factoring Suggested number of days: 19 This unit is about polynomial equations and factoring. It is positioned here in the sequence in preparation for upcoming work with quadratics. To begin the unit, the vocabulary and representation of polynomials is introduced, along with operations with polynomials. Operations of addition, subtraction, and multiplication are then presented. The unit focuses on solving polynomial equations, which can be done when the polynomial is written in factored form. Students will use the Zero-Product Property to solve polynomial equations in factored form. Students will learn a series of techniques for factoring polynomials, aided by visual explorations using algebra tiles. Seeing Structure in Expressions A-SSE A. Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its content. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x2) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). B. Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. Arithmetic with Polynomial and Rational Expressions A-APR A. Perform arithmetic operations on polynomials 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. B. Understand the relationship between zeros and factors of polynomials 3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Reasoning with Equations and Inequalities A-REI B. Solve equations and inequalities in one variable 4. Solve quadratic equations in one variable. 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. A-SSE.A.2 Focus on numerical expressions and polynomial expressions in one variable. Examples: Recognize as a difference of squares and see an opportunity to rewrite it in the easier-to-evaluate form (53 47)(53 47). See an opportunity to rewrite a 2 9a 14 as (a 7) (a 2). A-APR.B.3 Focus on quadratic and cubic polynomials in which linear and quadratic factors are available. For example, find the zeros of (x - 2) (x 2-9). A-REI.B.4b Excluding solutions for quadratic equations that have roots with nonzero imaginary parts. However, include cases that recognize when a quadratic equation has no real solutions. Note, solving a quadratic equation by factoring relies on the connection between zeros and factors of polynomials (cluster A-APR.B). Mesa Public Schools 11 May 2015

13 Unit 8: Graphing Quadratic Functions Suggested number of days: 16 This unit continues with general work about polynomials and more specifically about quadratics. In the last unit, students factored quadratics and used the Zero-Product Property to solve quadratics. The unit begins with graphing quadratic functions. Connections to transformations of functions are made, just as students had seen with linear functions earlier. In addition to graphing, the concept of zeros of functions is introduced and connected to the x-intercept of a graph. There are three forms of quadratic equations: standard, vertex, and intercept. Students should become familiar with what the parameters of each equation tell about the graph of the function. Students compare the behavior of linear, exponential, and quadratic functions. This is done by looking at the data numerically and graphically, a theme emphasized throughout the unit. Seeing Structure in Expressions A-SSE B. Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. Arithmetic with Polynomial and Rational Expressions A-APR B. Understand the relationship between zeros and factors of polynomials 3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Creating Equations A-CED A. Create equations that describe numbers or relationships 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 B. Interpret functions that arise in applications in terms of the context 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. C. Analyze functions using different representations 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. A-APR.B.3 Focus on quadratic and cubic polynomials in which linear and quadratic factors are available. For example, find the zeros of (x - 2)(x 2-9). F-IF.B.4 Include problem-solving opportunities root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.B.6 Include problem-solving opportunities root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.C.7 Include problem-solving opportunities root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F-IF.9. Mesa Public Schools 12 May 2015

14 Unit 8: Graphing Quadratic Functions Suggested number of days: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions F-BF A. Build a function that models a relationship between two quantities 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. B. Build new functions from existing functions 3. Identify the effect 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 the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear, Quadratic, and Exponential Models F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F-IF.C.9 Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on F-BF.A.1a Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on F-BF.B.3 Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. Mesa Public Schools 13 May 2015

15 Unit 9: Solving Quadratic Equations Suggested number of days: 16 In previous units, students factored quadratics and graphed quadratic equations in different forms: standard, vertex, and intercept. This unit is about solving quadratic equations, and depending upon the form in which the equations are written, different techniques are used. Students review square roots and how to simplify them. These skills are needed for work later in the unit. Students are presented different ways in which a quadratic can be solved: graphing, using square roots, completing the square, and using the Quadratic Formula. Students learn to distinguish between finding the zero of a function and finding the x-intercept of a graph. Students solve systems of nonlinear equations by applying the techniques they used in Unit 5. The Real Number System N-RN B. Use properties of rational and irrational numbers. 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. Seeing Structure in Expressions A-SSE B. Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Creating Equations A-CED A. Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities A-REI B. Solve equations and inequalities in one variable 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. A-CED.A.1 Focus on linear, quadratic, or exponential equations with integer exponents. A-REI.B.4b Excluding solutions for quadratic equations that have roots with nonzero imaginary parts. However, include cases that recognize when a quadratic equation has no real solutions. Note, solving a quadratic equation by factoring relies on the connection between zeros and factors of polynomials (cluster A-APR.B). Mesa Public Schools 14 May 2015

16 Unit 9: Solving Quadratic Equations Suggested number of days: 16 D. Represent and solve equations and inequalities graphically 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Interpreting Functions F-IF C. Analyze functions using different representations 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. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. A-REI.D.11 Focus on the following function types: linear, quadratic, square root, cube root, exponential, and piece-wise. Focus on exponential functions with domains in the integers. F-IF.C.7 Include problem-solving opportunities root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F- IF.7, and F-IF.9. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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. Mesa Public Schools 15 May 2015

17 Unit 10: Radical Functions and Equations Suggested number of days: 11 This unit introduces radical functions, both square root and cubic functions. The transformations of the parent functions give students another opportunity to examine horizontal and vertical translations, horizontal reflections in the x-axis, and vertical stretches and shrinks. Solving radical equations is a nice connection to solving equations in general. Students have used the Properties of Equalities and have taken the square root of each side of an equation, now they learn that squaring both sides of an equation is another technique for solving equations, although extraneous roots may be introduced. This unit presents inverse relations and inverse functions. It connects the functions studied in this unit with functions studied in earlier units. Creating Equations A-CED A. Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 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 B. Interpret functions that arise in applications in terms of the context 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. C. Analyze functions using different representations 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. A-CED.A.1 Focus on linear, quadratic, or exponential equations with integer exponents. F-IF.B.4 Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.B.6 Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on F-IF.C.7 Include problem-solving opportunities root, exponential, and piecewise-defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.C.9 Include problem-solving opportunities root, exponential, and piecewise- defined (including step functions and absolute value functions. Focus on exponential functions with domains in the integers. Mesa Public Schools 16 May 2015