Sequenced Units for Arizona s College and Career Ready Standards MA40 Algebra II

Transcription

1 Sequenced Units for Arizona s College and Career Ready Standards MA40 Algebra II Year at a Glance Semester 1 Semester 2 Unit 1: Linear Functions (10 days) Unit 2: Quadratic Functions (10 days) Unit 3: Quadratic Equations and Complex Numbers (14 days) Unit 4: Polynomial Functions (20 days) Unit 5: Rational Exponents and Radical Functions (13 days) Unit 6: Exponential and Logarithmic Functions (17 days) Unit 7: Rational Functions (12 days) Unit 8: Sequences and Series (13 days) Unit 9: Trigonometric Ratios and Functions (19 days) Unit 10: Probability (16 days) Unit 11: Data Analysis and Statistics (16 days)

2 Algebra II extends the knowledge students have of algebraic and statistical concepts. They have investigated linear, exponential, and quadratic functions in previous years. Algebra II further develops important mathematical ideas introduced in Algebra I by extending techniques to solve equations and students knowledge of functions by studying inverses and new function families: polynomial, radical, trigonometric, and rational functions. Students will also spend a significant portion of the school year studying probability and statistics. There are some (+) standards that are included in this course because the standards naturally support the assessed Algebra II content. This document reflects our current thinking related to the intent of Arizona s College and Career Ready Standards for Mathematics 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 II, 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 district's instructional program, you will also need to refer to the standards document, relevant progressions documents for the standards, and the appropriate assessment consortium framework. Mesa Public Schools 1 May 2015

3 Unit 1: Linear Functions Suggested number of days: 10 Unit 1 presents topics that were studied in Algebra 1. Transformations of linear, quadratic, and absolute value functions are explored. The parent functions are established and then transformed functions are compared to the parent. Students will review modeling with linear functions involving writing linear functions from given information and fitting a line to data. Results from performing a linear regression are compared to the model determined by hand. Students extend prior work with systems of equations to solving linear systems in three variables. Students may well be a bit rusty with the algebra skills. The review content should be familiar to students, with new content introduced at an appropriate level. It is assumed that students will be using graphing technology in this book. In this first unit, many fundamental calculator skills are integrated in the lessons. 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. 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 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Interpreting Functions F IF 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 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). A-REI.C.6 Include 3x3 systems. F-IF.B.9 Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-LE.A.2 Extend to include solving multi-step problems by constructing linear and exponential functions. Mesa Public Schools 2 May 2015

4 Unit 1: Linear Functions Suggested number of days: 10 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. S-ID.B.6a Include problem-solving opportunities utilizing a real-world context. Extend to include all exponential functions. 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 2: Quadratic Functions Suggested number of days: 10 Students have studied quadratic functions in Algebra 1. Their background should include factoring quadratic expressions, graphing quadratic equations written in three forms, and solving quadratic equations using a variety of approaches. Students will extend their knowledge of quadratic functions in this unit. Transformations on quadratic functions are introduced as well as characteristics of quadratic functions. Understanding the connection between the characteristics of a quadratic and its equation can help students apply their knowledge when working with a real-life application. Arithmetic with Polynomials and Rational Expression 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. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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-APR.B.3 Include quadratic, cubic, and quartic polynomials and polynomials for which factors are not provided. For example, find the zeros of (x 2-1)( x 2 + 1). F-IF.B.4 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F- IF.6, F-IF.7, and F-IF.9. F-IF.B.6 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.C.9 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. Mesa Public Schools 4 May 2015

6 Unit 2: Quadratic Functions Suggested number of days: 10 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. Expressing Geometric Properties with Equations G GPE A. Translate between the geometric description and the equation for a conic section 2. Derive the equation of a parabola given a focus and directrix. 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. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. S-ID.B.6a Include problem-solving opportunities utilizing a real-world context. Extend to include all exponential functions. 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 5 May 2015

7 Unit 3: Quadratic Equations and Complex Numbers Suggested number of days: 14 The strategies for solving quadratic equations presented previously were introduced at the end of Algebra 1. The difference now is that solutions are not restricted to real numbers. The technique of completing the square so that the Quadratic Formula can be derived is introduced. Students will use five strategies for solving quadratic equations: graphing, square rooting, factoring, completing the square, and using the Quadratic Formula. As the number of strategies increases in the unit, students should be making informed choices as to which strategy to use given the equation. The last two lessons extend work with solving quadratic equations to solving nonlinear systems and solving quadratic inequalities. It is important throughout the unit to be clear with students about your expectation of the role of technology versus computational and analytical skills. Quantities N-Q A. Reason qualitatively and use units to solve problems. 2. Define appropriate quantities for the purpose of descriptive modeling. The Complex Number System N CN A. Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. 2. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. C. Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions. Seeing Structure in Expressions A SSE A. Interpret the structure of expressions 2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). 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. 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 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. N-Q These standards are integrated throughout both and Algebra I and Algebra II course. Most notably in modeling tasks. For example, in a situation involving periodic phenomena, the student might autonomously decide that amplitude is a key variable in a situation, and then choose to work with peak amplitude. A-SSE.A.2 Focus on polynomial, rational, or exponential expressions. Examples: See x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). In the equation x 2 + 2x y 2 = 9, see an opportunity to rewrite the first three terms as (x+1) 2, thus recognizing the equation of a circle with radius 3 and center ( 1, 0). See (x 2 + 4)/(x 2 + 3) as ( (x 2 +3) + 1 )/(x 2 +3), thus recognizing an opportunity to write it as 1 + 1/(x 2 + 3). A-CED.A.1 Extend to exponential equations with rational or real exponents and rational functions. Include problem-solving opportunities utilizing a realworld context. A-REI.B.4b Include all solution cases. In the case of equations that have roots with nonzero imaginary parts, students write the solutions as a ± bi for real numbers a and b. Mesa Public Schools 6 May 2015

8 Unit 3: Quadratic Equations and Complex Numbers Suggested number of days: 14 C. Solve systems of equations 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. 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. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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. 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. 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.D.11 Include any of the function types mentioned in the standard. Extend to include all exponential functions. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. Mesa Public Schools 7 May 2015

9 Unit 4: Polynomial Functions Suggested number of days: 20 Linear and quadratic functions are two types of polynomials, so connections to earlier work are easily made and polynomial functions are defined and graphed. The notation and vocabulary can be overwhelming for students, though some of the vocabulary was used in Algebra 1. End behavior of even- and odd-degree polynomials is explored. Prior work with factoring is extended to third- and fourth-degree expressions. Synthetic division is used to efficiently check for possible rational roots when rewriting polynomials in factored form in order to solve polynomial equations. All of the work with operations on polynomials, factoring, and solving leads to the Fundamental Theorem of Algebra in this unit. This unit also deals with polynomial functions, in particular the graphs of these functions. Earlier work with transformations is applied to polynomials. The Complex Number System N CN C. Use complex numbers in polynomial identities and equations. 8. Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i). 9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Seeing Structure in Expressions A SSE A. Interpret the structure of expressions 2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Arithmetic with Polynomials and Rational Expression 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 2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). 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. C. Use polynomial identities to solve problems 4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. 5. Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) D. Rewrite rational expressions 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-SSE.A.2 Focus on polynomial, rational, or exponential expressions. Examples: See x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). In the equation x 2 + 2x y 2 = 9, see an opportunity to rewrite the first three terms as (x+1) 2, thus recognizing the equation of a circle with radius 3 and center ( 1, 0). See (x 2 + 4)/(x 2 + 3) as ( (x 2 +3) + 1 )/(x 2 +3), thus recognizing an opportunity to write it as 1 + 1/(x 2 + 3). A-APR.B.3 Include quadratic, cubic, and quartic polynomials and polynomials for which factors are not provided. For example, find the zeros of (x 2-1)( x 2 + 1). Mesa Public Schools 8 May 2015

10 Unit 4: Polynomial Functions Suggested number of days: 20 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. 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. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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. F-IF.B.4 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. F-IF.C.7 Include problem-solving Opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. Mesa Public Schools 9 May 2015

11 Unit 5: Rational Exponents and Radical Functions Suggested number of days: 13 This unit introduces radicals and nth roots and how these may be written as rational exponents. A connection is made to the properties of exponents studied in Algebra 1, noting that now exponents can be rational numbers and are no longer restricted to being nonzero integers. Radical expressions, also written in rational exponent form, are represented as functions and are graphed. This leads to a look at the difference between even- and odd-degree functions and what the domains are for each function type. Even and odd functions are defined. The graphs of radical functions are used to help students think about solutions of radical equations and inequalities. Certainly, one goal is for students to recognize that solving radical equations is an extension of solving other types of functions. The difference, however, is that sometimes extraneous solutions are introduced when solving radical equations, so it is necessary to check apparent solutions. Inverse functions are presented finding the inverse of linear, simple polynomial, and radical functions, and noting that the graphs of inverse functions are reflections in the line y = x. The Real Number System N RN A. Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal Rewrite expressions involving radicals and rational exponents using the properties of exponents. Creating equations A CED A. Create equations that describe numbers or relationships 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. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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. 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. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. A-REI.A.1 Extend to simple rational and radical equations. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. Mesa Public Schools 10 May 2015

12 Unit 5: Rational Exponents and Radical Functions Suggested number of days: 13 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. 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x 1) for x 1. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. Mesa Public Schools 11 May 2015

13 Unit 6: Exponential and Logarithmic Functions Suggested number of days: 17 This unit presents two new types of functions, exponential and logarithmic. Students should have some prior experience with exponential functions from Algebra 1, particularly with growth and decay models. The natural base e, an irrational number, is introduced and students write and graph exponential functions for base e and other bases. Compound interest and continuous compounding are two of the many applications explored. The logarithmic function, which is the inverse of the exponential function, is introduced, and the connection to properties of exponents is made. The unit also looks at solving exponential and logarithmic equations using different approaches: analytical, numerical, and graphical. The unit uses mathematical modeling - given a set of data, an exponential or logarithmic equation is fit to the data. Seeing Structure in Expressions A SSE A. Interpret the structure of expressions 2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). 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 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. 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. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. A-SSE.A.2 Focus on polynomial, rational, or exponential expressions. Examples: See x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). In the equation x 2 + 2x y 2 = 9, see an opportunity to rewrite the first three terms as (x+1) 2, thus recognizing the equation of a circle with radius 3 and center ( 1, 0). See (x 2 + 4)/(x 2 + 3) as ( (x 2 +3) + 1 )/(x 2 +3), thus recognizing an opportunity to write it as 1 + 1/(x 2 + 3). 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. Extend to include expressions with real number exponents. A-REI.A.1 Extend to simple rational and radical equations. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9. Mesa Public Schools 11 May 2015

14 Unit 6: Exponential and Logarithmic Functions Suggested number of days: 17 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. 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x 1) for x 1. Linear, Quadratic, and Exponential Models F LE A. Construct and compare linear, quadratic, and exponential models and solve problems 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). 4. For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 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. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-LE.A.2 Extend to include solving multi-step problems by constructing linear and exponential functions. F-LE.B.5 Include problem-solving opportunities utilizing a real-world context. Extend to include all exponential functions. 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 12 May 2015

15 Unit 7: Rational Functions Suggested number of days: 12 Unit 7 introduces rational functions, a new type of function for students to work with. The simplest of rational functions, inverse variation, is introduced. The inverse variation function is distinguished from the direct variation function, and it also provides the introduction to rational functions and their graphs. Students learn to identify the horizontal and vertical asymptotes by inspecting the equations. Simple transformations of rational functions are also performed. Connections are made to operations with fractions, and symbolic manipulation skills are necessary to perform the operations. Arithmetic with Polynomials and Rational Expression A APR D. Rewrite rational expressions 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 7. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 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. 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. 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. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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. A-CED.A.1 Extend to exponential equations with rational or real exponents and rational functions. Include problem-solving opportunities utilizing a real-world context. A-REI.A.1 Extend to simple rational and radical equations. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. Mesa Public Schools 13 May 2015

16 Unit 8: Sequences and Series Suggested number of days: 13 This unit builds on skills from Algebra 1, where arithmetic and geometric sequences were first introduced, by extending the work students have previously done. New in this unit is the skill of adding terms of a sequence. Partial sums and sums of infinite geometric series are explored numerically and graphically. This unit involves recursively defined functions by reviewing knowledge of arithmetic and geometric sequences with connections to linear and exponential functions. Seeing Structure in Expressions A SSE B. Write expressions in equivalent forms to solve problems4. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. 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. 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. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Linear, Quadratic, and Exponential Models F LE A. Construct and compare linear, quadratic, and exponential models and solve problems 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.A.3 This standard is Supporting work in Algebra II. This standard should support the Major work in F-BF.2 for coherence. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-LE.A.2 Extend to include solving multi-step problems by constructing linear and exponential functions. Mesa Public Schools 14 May 2015

17 Unit 9: Trigonometric Ratios and Functions Suggested number of days: 19 Right triangle trigonometry that students learned in geometry is reviewed. Students are introduced to radian measure, and the six trigonometric functions are defined in terms of a unit circle and lessons focus on graphing the six trigonometric functions. The graphs of sine and cosine are developed by plotting functional values for benchmark angles, and the concept of periodic functions is introduced. The graphs of the remaining four trigonometric functions are deduced from knowing the relationship between these functions and sine and cosine. Knowledge of transformations is used to plot graphs beyond the parent functions. The unit introduces students to trigonometric identities and sum and difference formulas. 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 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. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 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. Trigonometric Functions F TF A. Extend the domain of trigonometric functions using the unit circle 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. B. Model periodic phenomena with trigonometric functions 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. C. Prove and apply trigonometric identities 8. Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. F-IF.C.7 Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. The function types listed here are the same as those listed in the Algebra II column for standards F-IF.6, F-IF.7, and F- IF.9. F-BF.A.1a Include problem-solving opportunities utilizing a real-world context. Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. F-BF.B.3 Function types extend to include polynomial, radical, logarithmic, simple rational, and trigonometric. Extend to include all exponential functions. Mesa Public Schools 15 May 2015