Unit 1 Linear and Quadratic Functions Algebra 2

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1 Number of Days: 24 9/5/17 10/6/17 Unit Goals Stage 1 Unit Description: In Unit 1, students explore and transform linear, absolute value and quadratic parent functions. Characteristics of the functions are given both abstract and contextual meaning. Applying their knowledge of these functions in a real world situations, including writing equations based on given data, using lines of best fit to approximate data and solving 3-by-3 systems of equations, will give significance and relevance to these functions, their graphs, and their equations. 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.B Understand the relationship between zeros and factors of polynomials. [m] A-CED.A Create equations that describe numbers or relationships. [s] F-IF.B Analyze functions using different representations. [m] F-IF.C Interpret functions that arise in applications in terms of the [m] F-BF.A context. Build a function that models a relationship between two quantities. [a] F-BF.B Build new functions from existing functions. [s] F-LE.A Construct and compare linear, quadratic, and exponential models 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. UNDERSTANDINGS Students will understand that In the function f(x) = a f(x b) + c, a, b, and c have the same effect on the shape of the graph in every graph family. Math models, including lines of regression, allow us to estimate solutions to real world situations. Representing a function in different ways does not change the function itself, although different representations may highlight different characteristics of that function. KNOWLEDGE Students will know The basic shapes of parent functions. How the parameters a, b, and c, affect the graph of f(x) =a f(x b) + c. The graph of a linear equation in three variables x, y, and z is a plane in threedimensional space. The solution of a system of linear equations in three variables is either a point, a line or no solution. Making Meaning ESSENTIAL QUESTIONS Students will keep considering In the function f(x) =a f(x b) +c, how do a, b, and c effect the shape of the graph? Is this the same for all graph families? How do you know how many solutions a system of equations has? In a contextual situation, what information can you derive from the characteristics of the graph of a function? What is the connection between the characteristics of a quadratic and its equation? Acquisition SKILLS Students will be skilled at and/or be able to Identify function families. Describe transformations of these parent functions: linear, absolute value and quadratic. Write functions representing rigid and non-rigid transformations. Write linear functions given a point and a slope. Find and interpret the characteristics of a line of best fit. and solve problems Reposted 10/11/17

2 [a] G-GPE.A Translate between the geometric description and the equation for a conic section. [a] S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables. Unit Goals Stage 1 The vocabulary of the parabola: vertex, axis of symmetry, minimum value, maximum value, y-intercept, and x-intercept. The properties of the graphs of: f(x) = ax 2 + bx + c, f(x) = a(x - p)(x - q), and f(x) = a(x - h) 2 + k. Use technology to solve for a regression line through a given set of data. Interpret a correlation coefficient and the parameters of a linear function. Visualize and solve a system of linear equations in three variables. Quantify, graph, and describe the characteristics of a parabola. Write the equation for a parabola. Given vertices, points and x-intercepts, write the equation for a quadratic function. Apply functions to solve real-world problems Reposted 10/11/17

3 Assessed Grade Level Standards 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 [m] A-APR.B Understand the relationship between zeros and factors of polynomials. A-APR.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. [m] A-CED.A Create equations that describe numbers or relationships. A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.3 Represent constraints by equations 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. [m] F-IF.B Interpret functions that arise in applications in terms of the context. F-IF.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. F-IF.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. [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. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F-IF.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. [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. a. Determine an explicit expression, a recursive process, or steps for calculation from a context Reposted 10/11/17

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. [s] F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems. F-LE.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] G-GPE.A Translate between the geometric description and the equation for a conic section. G-GPE.2 Derive the equation of a parabola given a focus and directrix. [a] S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables. S-ID.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. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters; [ACC] = Algebra 2 ACC only Indicates a modeling standard linking mathematics to everyday life, work, and decision-making CA Indicates a California-only standard Reposted 10/11/17

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: A-APR.B Students will identify zeros of quadratics when suitable factoring is possible. Students will use the zeros of a quadratic to construct a rough graph of the function defined by the quadratic. A-CED.A Students will create equations in two or more variables to represent relationships between quantities. Students will graph equations which represent relationships between quantities on a coordinate axis with labels and scales. Students will represent constraints by systems of equations. Students will interpret solutions as viable or non-viable options in a real-world context. F-IF.B Students will interpret the key features of graphs and tables in terms of the context of the problem. Students will sketch graphs showing key features given a verbal description of the relationship between two quantities. Students will calculate and interpret the average rate of change of a quadratic function over a specified interval. F-IF.C Students will graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Students will calculate and interpret the average rate of change of a function over a specified interval. Students will compare properties of two functions each represented in a different way. F-BF.A Students will determine an explicit expression or steps for calculation from a context. F-BF.B Students will 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. Students will find the value of k given transformed graphs. F-LE.A Students will construct linear functions when given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). G-GPE.A Students will derive the equation of a parabola given a focus and directrix Reposted 10/11/17

6 Evidence of Learning Stage 2 S-ID.B Students will represent data of two quantitative variables on a scatter plot and describe how the variables are related. Students will fit a function to a set of data. Students will use functions fitted to data to solve problems in the context of the data. 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 F-IF.B F-IF.C F-BF.A 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: A-APR.B 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 F-IF.B F-IF.C F-BF.A F-LE.A Other Evidence Formative Assessment Opportunities Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Tasks Modeling Lessons (SMP 4) Formative Assessment Lessons (FAL) Quizzes/Chapter Tests SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents Reposted 10/11/17

7 Days Learning Target Expectations I will use my knowledge of linear equations and graphs to solve a real-world application in the Opening Task. I will transform linear, absolute value, and quadratic functions by Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction OPENING TASK Movie Lines This Opening Task is a review of linear equations and graphs from Algebra 1. The student is asked to analyze the cost of DVD rentals and interpret the results. Students can begin this task at a variety of levels: plotting data points or solving for the equation directly. Students will be asked to: graph, solve an equation, interpret the characteristics of the graph, and use the equation or graph to speculate on costs of DVD rentals. Whole-class discussion should bring Movie Lines to a close as students are asked to discuss their solutions in the context of the problem. Identifying basic parent functions. Identifying the effect a, b, and c have on the function f(x) = a f(x b) + c. (SMP 7) Graphing functions and their transformations using technology. (SMP 5) Describing the transformations of the parent functions. (SMP 3) Writing functions that represent rigid and non-rigid transformations. Answering questions such as o What are the basic characteristics of each parent function? o How do the graphs of y = f(x) + k, y = f(x h), and y = f(x) compare to the graph of the parent function f? Big Ideas Math Algebra 2 (Activities and Lessons) Section 1.1 STEM Video: Dirt Bike Trajectory Section 1.2 Supplemental Resources Application: Illuminations: Movie Lines Task Movie Lines PowerPoint Conceptual Understanding: Frayer Model Functions Template Desmos: Marbleslides: Lines Desmos: Absolute Value Translations Desmos: Marbleslides: Parabolas Which One Doesn t Belong: Graphs Procedural Skills and Fluency: Jeopardy: Linear Functions and Graphs Quizlet: Matching a Quadratic Function with its Graph Desmos: Polygraph: Absolute Value Desmos: Polygraph: Parabolas Reposted 10/11/17

8 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Algebra 2 Days Learning Target Expectations (Activities and Lessons) Supplemental Resources Khan Academy: Shifting Functions Khan Academy: Stretching Functions Application: Illuminations: There Has to be a System for This Sweet Problem STEM Performance Task: Changing the Course 2-3 I will represent the data in a scatter plot by Using an ordered pair and slope to write a linear function to represent a set of data. Creating a scatter plot, sketching a line that appears to fit the data, and solving for the equation of that line. Using technology to plot data, and calculate a regression line and correlation coefficient. (SMP 5) Interpreting the characteristics of a line of best fit in the context of a problem. (SMP 3) Answering questions such as o How can you use a linear function to model and analyze a real-life situation? Section 1.3 Procedural Skills and Fluency: Illuminations Using a Calculator for Finding the Equation of a Function Application Victim s Height Activity Illuminations: Exploring Linear Data Illuminations Determining Functions Using Regression: What's the Function? Illustrative Mathematics: Coffee and Crime Illustrative Mathematics: Used Subaru Foresters Illustrative Mathematics: Temperature Change Quarterback Salaries and Regression Lines Reposted 10/11/17

9 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Days Learning Target Expectations Big Ideas Math Algebra 2 (Activities and Lessons) I will find the Section 1.4 solutions to a system of three linear equations by I will graph, transform and interpret the characteristics of quadratic functions by Visualizing solutions to a system of linear equations in three variables. Writing equations in three variables to represent a real-life situation. Solving a three-variable system using elimination. (SMP 6) Answering questions such as o How can you determine the number of solutions of a linear system? o How would you describe the solutions of a threevariable linear system? Transforming parabolas using rigid and non-rigid transformations. Writing the equation for a transformed quadratic function. (SMP 7) Describing the transformation of a quadratic function. Graphing quadratic functions using technology. (SMP 5) Applying a quadratic function to a real-world situation. Interpreting the solution, and other characteristics, of a quadratic equation in the context of a real-world application. (SMP 3) Solving for the vertex and/or intercepts of a parabola. Graphing a parabola from one of the four common forms in which a quadratic function is written. Writing the equation of a parabola given the focus and directrix or a vertex and a point. (SMP 6) Answering questions such as o How do the constraints a, h, and k affect the graph of the quadratic function g(x) = a(x h) 2 + k? Section 2.1 Section 2.2 Section 2.3 STEM Video: Solar Energy Using a Parabolic Mirror Section 2.4 Supplemental Resources Conceptual Understanding: Open Middle: Systems of Equations Multiple Solutions Open Middle: Systems of Equations One Solution Procedural Skills and Fluency: Solving Systems of Three Variables Khan Academy: Systems with Three Variables Conceptual Understanding: Open Middle: Create a Quadratic Equation, Given Constraints Illustrative Mathematics: Throwing Baseballs Geogebra: Parabola Given a Focus and Directrix Interactive Tool Procedural Skills and Fluency: Double Bubble Map: Comparing Equations of Parabolas Activity Khan Academy: Quadratic Vertex Form Khan Academy: Quadratic Standard Form Khan Academy: Forms and Features of Quadratic Functions Reposted 10/11/17

10 Days Learning Target Expectations I will check my understanding of functions by participating in the FAL. I will prepare for the unit assessment on linear and quadratic functions by... o o o Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction What type of symmetry does the graph of f(x) = a(x h) 2 + k have, and how can you describe this symmetry? What is the focus of a parabola? How can you use a quadratic function to model a real-life situation? FORMATIVE ASSESSMENT LESSON Representing Quadratic Functions Graphically (SMP 1, 3, 6) Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Big Ideas Math Algebra 2 (Activities and Lessons) Unit Assessment Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit. Supplemental Resources Application: Desmos Activity: Will It Hit the Hoop? Illuminations: Egg Launch Contest Task STEM Performance Task: Solar Energy Conceptual Understanding: FAL: Representing Quadratic Functions Graphically Reposted 10/11/17