Unit 3 Radical and Rational Functions Algebra 2

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1 Number of Days: 29 11/28/16 1/20/17 Unit Goals Stage 1 Unit Description: A theme of Unit 3 is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Connecting to the properties of exponents learned in Algebra 1, students now see that exponents can be rational numbers and are no longer restricted to being nonzero integers. Graphs help to illustrate the solutions to radical equations and inequalities. Even and odd functions and domains are investigated and defined. Function operations lead to solving for the inverses of functions where possible. The graphs of functions compared to the graphs of their inverses add a visual component to understanding inverse relationships. From direct variation in middle school, the students in Algebra 2 move on to rational functions, the simplest of which is inverse variation. Graphs play an important role in understanding rational functions as students are introduced to asymptotes and note the effect of simple transformations. Operations with rational expressions are primarily symbolic manipulation, but graphs can be used to confirm results. Materials: Graphing calculators, Desmos Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content Clusters Addressed [m] A-APR.D Rewrite rational expressions. [m] A-CED.A Create equations that describe numbers or relationships. [m] A-REI.A Understand solving equations as a process of reasoning and explain the reasoning. [m] A-REI.B Solve equations and inequalities in one variable. Transfer Goals Students will be able to independently use their learning to Make sense of never-before-seen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will 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. Each step in solving a simple equation follows from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Functions can be represented in multiple ways including algebraic, graphical, verbal and tabular representations. Links between these representations allow for deeper understanding of relationships and change. KNOWLEDGE Students will know The connection between radical notation and rational exponents. The definition of even and odd functions. ESSENTIAL QUESTIONS Students will keep considering What effects do the parameters in the function f(x) = a f(x h) + k have on the graph of the parent function? Functions each have their own characteristics that lend themselves to modeling different real-world phenomena. What characteristics differentiated the functions in this unit from the functions that you have previously encountered? What are the conditions under which functions have inverses? Where would you see an example of an asymptote in the real world? Acquisition SKILLS Students will be skilled at and/or be able to Rewrite simple rational expressions in different forms. Create equations and inequalities in one variable and use them to solve problems Reposted 1/9/17

2 [s] F-IF.C Analyze functions using different representations. [m] F-BF.A Build a function that models a relationship between two quantities. [a] F-BF.B Build new functions from existing functions. Unit Goals Stage 1 The shapes and domains of the parent graphs for radical and rational functions. Which functions have inverses. Inverse variation is a rational function. The difference between direct and inverse variation. Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with labels and scales. 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. Rearrange formulas to highlight a quantity of interest. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Graph functions expressed symbolically and show key features of the graph. Combine standard function types using arithmetic operations. 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. Transform the graphs of parent functions. Identify and use asymptotes to limit a domain and/or range, and to graph rational functions Reposted 1/9/17

3 Standards for Mathematical Practice Assessed Grade Level Standards SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content [s] A-APR.D 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. [ACC] 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. [m] A-CED.A Create equations that describe numbers or relationships. A-CED.1 A-CED.2 A-CED.3 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. A-CED.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. [m] A-REI.A Understand solving equations as a process of reasoning and explain the reasoning. A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [s] F-IF.C Analyze functions using different representations. 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. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [m] F-BF.A Build a function that models a relationship between two quantities. F-BF.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 Reposted 1/9/17

4 Assessed Grade Level Standards [a] F-BF.B Build new functions from existing functions. F-BF.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-BF.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) =2 x 3 or f(x) = (x+1)/(x 1) for x 1. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters; [ACC] = Algebra 2 ACC only Reposted 1/9/17

5 Assessment Evidence Unit Assessment Evidence of Learning Stage 2 Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: [m] A-APR.D Rewrite simple rational expressions in different forms. [m] A-CED.A Create equations in one variable and use them to solve problems. Create equations in two variables to represent relationships between quantities. Graph equations on coordinate axes with labels and scales. 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. Rearrange formulas to highlight a quantity of interest. [m] A-REI.A Explain each step in solving an 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. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [s] F-IF.C Graph functions expressed symbolically showing key features of the graph. Combine standard function types using arithmetic operations. [m] F-BF.A Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. [a] F-BF.B Identify the effect on the graph when f(x) is replaced by f(x) + k, k f(x), f(kx), or f(x + k) for specific values of k both positive and negative; find the value of k given the graph. Find inverse functions. 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 Reposted 1/9/17

6 Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: A.CED.A A.REI.A F.IF.C F.BF.A Other Evidence Formative Assessment Opportunities Evidence of Learning Stage 2 Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: APR.D A.REI.A F.IF.C F.BF.B Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: A.CED.A A.REI.A A.REI.B F.IF.C Opening Tasks Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Modeling Lessons (SMP 4) Tasks Formative Assessment Lessons (FAL) Quizzes/Chapter Tests Big Ideas Math Performance Tasks SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents Reposted 1/9/17

7 Days Learning Target Expectations 1 day I will use my knowledge of graphing and working with rational expressions to solve a realworld application in the Opening Task. Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction OPENING TASK The Gift This Opening Task provides a real-life application to which the students can relate. The more people contributing to the price of a gift, the less each person needs to spend. What happens if only one person contributes? What happens if one-million people contribute? This task provides opportunity to begin a conversation about asymptotes and rational functions from a realistic perspective. Graphing technology (Desmos) can support the students observations and generalizations. Big Ideas Math Algebra 2 (Activities and Lessons) Curriculum Intranet Application: The Gift Task 5-6 I will visualize and evaluate radical functions by Using a rational exponent to represent a power involving a radical. Finding n th roots of numbers. Evaluating expressions with rational exponents. Solving equations using n th roots. Using properties of rational exponents to simplify expressions with rational exponents. Using properties of radicals to write radical expressions in simplest form. Identifying the domain and range of a radical function. Graphing and transforming radical functions, parabolas and circles. Identifying the function given a graph of a function that has been transformed. Answering questions such as o How can you use a rational exponent to represent a power involving a radical? 3 o Why does 64 = 4, but 4 81 has no solution? o How can you use properties of exponents to simplify products and quotients of radicals? o How can you identify the domain and range of a radical function? o What effects do the parameters in the function f(x) = a f(x h) + k have on the graph of the parent function? Section 5.1 Section 5.2 Section 5.3 Conceptual Understanding: FAL: Evaluating Statements about Rational and Irrational Numbers Which One Doesn t Belong: Square Roots and Cubes Procedural Skills and Fluency: Graphing Radical Functions Matching Activity Desmos: Square Root Functions Reposted 1/9/17

8 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Days Learning Target Expectations Algebra 2 (Activities and Lessons) I will solve and Section 5.4 simplify radical Section 5.5 expressions, STEM Video: The equations and Heartbeat inequalities by Hypothesis Section Using techniques to solve radical equations that were used to solve other types of equations. Isolating the radical to one side of an equation and raising both sides of the equation to the same power to eliminate the radical. Checking apparent solutions to avoid extraneous solutions. Performing the four operations on radical functions using a symbolic, numeric or graphical approach. Defining the domain of a function so that the domain of the result of an operation on two functions consists of the domains of both functions being operated on and the denominator of a quotient does not equal 0. Intuiting, solving and graphing to find the inverse of a function. Verifying that functions are inverses of each other. Restricting the domain of a function in order to have an inverse that is a function. Answering questions such as o What are ways to solve a radical equation? When would you use each of those techniques? o How do you know if a solution is extraneous? o How can you use the graphs of two functions to combine those two functions? o Is it possible to write two functions whose sum contains radicals, but whose product does not? o How can you sketch the graph of the inverse of a function? o Does every function have an inverse? o How can you verify that two functions are inverses of each other? o Is the inverse of a linear function always linear? Can your answer be generalized to functions of other degrees? Curriculum Intranet Procedural Skills and Fluency: Working with Radical Equations Dynamic Quiz Application: STEM Performance Task: The Heartbeat Hypothesis 2-3 I will check my understanding of radical expressions by participating in the FAL. FORMATIVE ASSESSMENT LESSON Evaluating Statements about Radicals Conceptual Understanding: FAL: Evaluating Statements about Radicals Reposted 1/9/17

9 4-5 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Days Learning Target Expectations Algebra 2 (Activities and Lessons) I will understand rational functions by 5-6 I will operate on and solve rational functions by Distinguishing between direct and inverse variation. Writing equations to define inverse variation. Knowing that inverse variation is an example of a rational function. Graphing and transforming graphs of rational functions. Making connections between symbolic, numeric and graphical representations of rational functions. Using asymptotes to help graph a rational function. Answering questions such as o What are some of the characteristics of the graph of a rational function? o What are some real-life examples in which two quantities vary inversely? o How can you recognize when two quantities vary directly or inversely? o How does exponential decay compare with inverse variation? o How can asymptotes help to graph a rational function? o How many asymptotes can a rational function have? Simplifying rational expressions. Multiplying, dividing, adding and/or subtracting rational expressions. Rewriting a rational expression in different forms to illustrate characteristics of the related function. Simplifying complex fractions. Using reciprocals and common denominators to solve rational equations. Section 7.1 Section 7.2 STEM Video: 3D Printing Section 7.3 Section 7.4 Section 7.5 Curriculum Intranet Conceptual Understanding: What Does It Mean to Be Rational Desmos: Building Rational Functions Illuminations: Do I Have to Mow the Whole Thing? Which One Doesn t Belong: Rational Graphs Procedural Skills and Fluency: Desmos: Polygraph Rational Functions Desmos: Marbleslides Rationals Sorting Functions Activity Application: STEM Performance Task: The Price is Right Illuminations: Light It Up Procedural Skills and Fluency: Illustrative Mathematics: Domains Illustrative Mathematics: A Sum of Functions Reposted 1/9/17

10 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Days Learning Target Expectations Algebra 2 (Activities and Lessons) Answering questions such as o Will rational expressions always have excluded values? o Is it possible to write two rational functions whose product, when graphed, is a parabola? Whose quotient, when graphed, is a hyperbola? o How are adding, subtracting, multiplying and/or dividing rational expressions similar to adding, subtracting, multiplying and/or dividing simple fractions with like denominators? Different denominators? o How can you determine the domain of the sum or difference of two rational expressions? o How do a, h, and k effect the graph of f( x) = a + k? x h o What are techniques you can use to solve a rational equation? o Do all functions have an inverse? How do you know if a function has an inverse? o What are some real-life examples that can be modeled with rational functions? Curriculum Intranet Practice Solving Rational Equations Dynamic Quiz Application: Illustrative Mathematics: Summer Intern Illustrative Mathematics: Combined Fuel Efficiency 2-3 I will prepare for the unit assessment on radical and rational functions by... Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Procedural Skills and Fluency: Rational Function Review Illuminations: Domain Representations Illuminations: Carousel Card Game Function Review 1-2 Unit Assessment (LBUSD Math Intranet, Assessment) Reposted 1/9/17