Algebra 2 Mathematics Curriculum Guide

Transcription

1 Unit 4: Rational and Radical Functions Time Frame: 13 Days Primary Focus Students will build on their understanding of functions and their inverses. They will restrict the domain of quadratic functions so that the inverse of the function exists. Students will investigate the characteristics of square root and cube root functions and apply basic transformations to these functions. They will also investigate rational functions with linear and quadratic functions and their graphs. They will explore the behavior of the graphs of both rational and radical functions. Students will solve rational and radical equations and use them to model real world situations. Common Core State Standards for Mathematical Practice Standards for Mathematical Practice MP4 - Model with mathematics. MP6 - Attend to precision. MP7 - Look for and make use of structure. How It Applies to this Topic Use a variety of methods to model, represent, and solve real-world problems. Calculate answers efficiently and accurately and label them appropriately. Use patterns or structure to make sense of mathematics and make connections between radical and rational functions and their corresponding graphical features. Unit 1 Clover Park School District Page 1

2 Stage 1 Desired Results Transfer Goals Students will be able to independently use their learning to Apply properties of radical and rational functions to real world applications. UNDERSTANDINGS Students will understand that Informally verify that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression. Students know a square root function has a restricted domain and range A rational function is a ratio of polynomial functions. If a rational function is in simplified form and the polynomial in the denominator is not constant, the graph of the rational function features asymptotic behavior. It looks quite different from the graphs of either of its polynomial components Meaning Goals ESSENTIAL QUESTIONS What do the key features of the graphs of square root and cube roots tell you about the function? When should you check for extraneous solutions? What is the relationship between a graph and its inverse? How can asymptotes relate to real world situations? How do you use rational and radical functions to model real world situations? Acquisition Goals Students will know and will be skilled at Students must identify key features of the graphs of square root and cube root functions by hand or using technology. How to solve radical equations and inequalities in one variable showing how extraneous solutions may arise. Simplify expressions involving positive and negative exponents Simplify expressions with rational exponents Add, subtract, multiply, and divide rational expressions Determine the domain of a rational function. Determine the domain of a radical function. Solve radical equations in one variable. Solve rational equations in one variable. Graph rational functions including vertical and horizontal asymptotes Stage 1 Established Goals: Common Core State Standards for Mathematics Unit 1 Clover Park School District Page 2

3 Rewrite rational expressions. A.APR.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.APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Explanations, Examples, and Comments This cluster is the logical extension of the earlier standards on polynomials and the connection to the integers. Now, the arithmetic of rational functions is governed by the same rules as the arithmetic of fractions, based first on division. In particular, in order to write a(x) r(x) in the formq(x) +, students need to work through the long division described for A.APR.2-3. This is merely writing the result b(x) b(x) of the division as a quotient and a remainder. For example, we can rewrite 75/8 in the form 9 + 3/8. Note that the fraction 75/8 is interpreted as the division 75 8, so that 75 is the dividend and 8 is the divisor. The result indicates that 9 is the quotient and 3 is the remainder. Note that for division of integers, we expect the remainder to be between 0 and the divisor, which in this case is 8. (If the remainder were greater than or equal to 8, we could subtract another 8, and increase the quotient by 1.) In order to rewrite simple rational expressions in different forms, students need to understand that the rules governing the arithmetic of rational expressions are the same rules that govern the arithmetic of rational numbers (i.e., fractions). To add fractions, we use a common denominator: a b + c d = aa bb + bb aa + bb = bb bb as long as b, d 0. Although in simple situations, a, b, c, and d would each be whole numbers, in fact they can be polynomials. Understand solving equations as a process of reasoning and explain the reasoning. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Explanations, Examples, and Comments Investigate the solutions to equations such as 3 = x + 2x 3. By graphing the two functions, y = 3 and y = x + 2x 3, students can visualize that graphs of Unit 1 Clover Park School District Page 3

4 the functions only intersect at one point. However, subtracting x = x from the original equation yields 3 x = 2x 3, which when both sides are squared produces a quadratic equation that has two roots x = 2 and x = 6. Students should recognize that there is only one solution (x = 2) and that x= 6 is generated when a quadratic equation results from squaring both sides; x = 6 is x extraneous to the original equation. Some rational equations, such as result in extraneous solutions as well. = (x 2) (x 2) x Begin with simple, one-step equations and require students to write out a justification for each step used to solve the equation. Ensure that students are proficient with solving simple rational and radical equations that have no extraneous solutions before moving on to equations that result in quadratics and possible solutions that need to be eliminated. Provide visual examples of radical and rational equations with technology so that students can see the solution as the intersection of two functions and further understand how extraneous solutions do not fit the model. It is very important that students are able to reason how and why extraneous solutions arise. F-IF.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. Given the graph if a function, determine the practical domain of the function as it relates to the numerical relationship it describes. Students may explain orally or in written format, the existing relationships. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*(modeling standard) b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (+) Explanations, Examples, and Comments For F.IF.7d focus on using graphing technology to develop understanding of critical points (intercepts), vertical asymptotes and end behavior (horizontal asymptotes). F.BF.4 Find inverse functions. Unit 1 Clover Park School District Page 4

5 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) = 2 x3 or f(x) = (x+1)/ (x- 1) for x 1. For F.BF.4a focus on linear functions but consider simple situations where the domain of the functions must be restricted in order for the invers to exist, such as f(x) = x2, x al, simple 0. This work radical will and be simple extended exponential in Algebra functions. 2 to include simple ration Exchange the x and y values in a symbolic functional equation and solve for y to determine the inverse function. Recognize that putting the output from the original function into the input of the inverse results in the original input value. Students may believe that all functions have inverses and need to see counter examples, as well as examples in which a non-invertible function can be made into an invertible function by restricting the domain. For example, f(x) = x 2 has an inverse f 1 = x provided that the domain is restricted to x 0. EXAMPLE MATERIALS: Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Holt Algebra 2: Chapter 1 Lessons 3 & 5 Holt Algebra 2: Chapter 8 Lessons 1-3 & 5-8 Holt Algebra 2: Chapter 9 Lesson 5 Supplemental Text- Discovering Advanced Algebra: Performance Tasks NCTM Illuminations: Light it Up problems 1-8 NCTM Illuminations: Domain Representations MVP Course 3, Module 1 Lessons 1-3: Developing Understanding of Inverse Functions Rational Functions and Equations EngageNY: Algebra 2 Module 1: Lessons Radical Functions and Equations EngageNY: Algebra 2 Module 1: Lessons 28&29 Unit 1 Clover Park School District Page 5

6 Evaluative Criteria/Assessment Level Descriptors (ALDs): Claim 1 Clusters: Target G: A-CED.1 & 2 Target H: A-REI.2 Target M: F-IF.7c Sample Assessment Evidence Stage 2 - Evidence Concepts and Procedures Target G Level 3 Students should be able to create and use linear, quadratic, and rational equations and inequalities and exponential equations with an integer base and a polynomial exponent in multiple variables to model an unfamiliar situation and to solve an unfamiliar problem. They should be able to graph an equation in two variables and be able to rearrange a linear, a quadratic, an absolute, a rational, or a cubic multi-variable formula for a particular given quantity. Target H Level 2 students should be able to look for and make use of structure to solve simple radical equations and simple rational equations in one variable in which the variable term is in the numerator and should understand the solution steps as a process of reasoning. They should be able to understand and explain solution steps for solving linear equations in one variable as a process of reasoning. Level 3 students should be able to look for and make use of structure to solve simple radical and rational equations in one variable presented in various forms. They should be able to understand and explain solution steps for solving quadratic, radical, and rational equations in one variable as a process of reasoning. Target M Level 2 Students should be able to graph linear and quadratic functions by hand; graph square root, cube root, piecewise defined, polynomial, exponential, and logarithmic functions by hand or by using technology Claim 2 Clusters: Claim 3 Clusters: Level 3 Students should be able to analyze and compare properties of two functions of different types represented in different ways and understand equivalent forms of functions. Problem Solving Level 3 students should be able to map, display, and identify relationships, use appropriate tools strategically, and apply mathematics accurately in everyday life, society, and the workplace. They should be able to interpret information and results in the context of an unfamiliar situation. Level 4 students should be able to analyze and interpret the context of an unfamiliar situation for problems of increasing complexity and solve problems with optimal solutions. Communicating Reasoning Level 3 students should be able to use stated assumptions, definitions, and previously established results and examples to Unit 1 Clover Park School District Page 6

7 test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students should be able to break an argument into cases to determine when the argument does or does not hold. Go here for Sample SBAC items Level 4 students should be able to use stated assumptions, definitions, and previously established results to support their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of logic to justify or refute a proposition or conjecture and to determine the conditions under which an argument does or does not apply. Go here for more information about the Achievement Level Descriptors for Mathematics: Unit 1 Clover Park School District Page 7

8 Stage 3 Learning Plan: Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning LEARNING ACTIVITIES: The suggested Unit Progression Inverse Functions Holt Algebra 2: Chapter 9 Lesson 5 (May need to supplement for deeper understanding) Rational Functions and Equations Performance Task (Light it up) Holt Algebra 2: Chapter 8 Lesson 1,2,3,4,5 (omit Inequalities) Radical Functions and Equations Holt Algebra 2: Chapter 1 Lessons 3 & 5 and Chapter 8 Lessons 6,7,8 (omit inequalities) Daily Lesson Components Learning Target Warm-up Activities Whole Group: Small Group/Guided/Collaborative/Independent: Whole Group: Checking for Understanding (before, during and after): Assessments NOTES: The Engage NY Algebra 2 Module 1 Lessons may substitute for the Holt Algebra 2 Rational Functions should be in the kept in the form f(x) = 1 + k. The emphasis is on (x h) introductions the characteristics of the function and will be built upon in Pre-Calculus. You should introduce the vertical and horizontal asymptotes. Radical Functions should be kept in the form n f(x) = a x h + k where a is 1 or -1 and n is 2 or 3 For Rational and Radical Equations it is important to show cases where there is an extraneous solution and connect it to the concept of domain. Unit 1 Clover Park School District Page 8