4/2/2014 6:02:01 PM3/29/2014 9:17:13 PM Adapted from UbD Framework

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1 Approximate Time Frame: 2-3 weeks Connections to Previous Learning: In previous units, students have worked with quadratic functions in various forms to determine their roots and vertices. They have used the factored form to determine if and where zeros exist. They have examined the equations of quadratic functions to determine the function's end behavior, and used all of this information to produce a graph of a quadratic. Focus of the Unit: This unit will call on students' understanding of modeling and various functions, including linear, exponential, piecewise and absolute value, polynomial, and rational to compare and choose appropriate models. Students will use all of these functions to create and compare models of real-life situations. Use of technology to facilitate the inspection of data and potential models will again become essential. Connections to Subsequent Learning: In the next unit, students will build on their knowledge of quadratics to include higher degree polynomial functions. They will use the factored form to determine zeros and the sign of the leading coefficient to determine end behavior. Multiplicity of factors will be used to help determine the behavior of the graph at its roots. These notions will all be extended to rational functions, which will initially be viewed as the reciprocal of a polynomial, and ultimately as the ratio of two polynomials. From the Grade 8, High School, Function Progression Document p.9: Analyze functions using different representations: Functions are often studied and understood as families, and students should spend time studying functions within a family, varying parameters to develop an understanding of how the parameters affect the graph of function and its key features. F-IF.7 Within a family, the functions often have commonalities in the qualitative shapes of their graphs and in the kinds of features that are important for identifying functions more precisely within a family. This standard indicates which function families should be in students repertoires, detailing which features are required for several key families. It is an overarching standard that covers the entire range of a student s high school experience; in this part of the progression we merely indicate some guidelines for how it should be treated. From the High School, Algebra Progression p. 12: Just as the algebraic work with equations can be reduced to a series of algebraic moves unsupported by reasoning, so can the graphical visualization of solutions. The simple idea that an equation can be solved (approximately) by graphing y=fx y=gx and finding the intersection points involves a number of pieces of conceptual understanding. A-REI.11 This seemingly simple method, often treated as obvious, involves the rather sophisticated move of reversing the reduction of an equation in two variables to an equation in one variable. Rather, it seeks to convert an equation in once variable,, to a system of equation in two variables, and, by introducing a second variable y and setting it to equal to each side of the equation. If x is a solution to the original equation then ) and are equal, and thus is a solution to the new system. This reasoning is often tremendously compressed and presented as obvious graphically; in fact following it graphically in a specific example can be instructive.

2 Desired Outcomes Standard(s): 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 function in 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.* 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 zeroes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 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. Extend the domain of trigonometric functions using the unit circle. F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F.TF.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. Transfer: Students will apply

3 Understandings: Students will understand that Functions have key features that can be represented on a graph and can be interpreted to provide information to describe relationships of two quantities. These functions can be compared to each other or other functions to model a situation. The average rate of change can be estimated, calculated, or analyzed from a polynomial or rational function or a graph. Transformations on the function modify the key features of the function. The trigonometric functions sin(x) and cos(x) can be used to model real-life situations that exhibit periodic behavior. The unit circle allows all real numbers to work in trigonometric functions. Key features in graphs and tables shed light on relationships between two quantities. Essential Questions: How is the domain of a function related to its graph? What do the key features of a graph represent in a modeling situation? What are key characteristics to identify when choosing a function to model a given situation? How can the graphs of functions be modified to best fit the situations being modeled? How do you use/read a unit circle (using radians)? Highlighted Mathematical Practices: 1. Make sense of problems and persevere in solving them. Students will consider a real-life situation, determine if it is best modeled by a linear, * exponential, quadratic, polynomial, rational, logarithmic, exponential, or trigonometric function, and answer questions and solve problems based on this model. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. Students will use graphs, tables, and equations to model functions. Students will interpret the appropriateness of the models for a given data set or situation. * 5. Use appropriate tools strategically. Use graphing technology to explore graphs of various functions; use technology (graphs, tables) to solve for a point of intersection of two functions. * 6. Attend to precision. Graph a function while carefully indicating zeros, asymptotes, y-intercept, and other significant features. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

4 Prerequisite Skills/Concepts: Determine domain and range of functions involving simple rational expressions. Choose and interpret appropriate models limited to linear, exponential and quadratic functions. Recognize how functions can be represented on a graph and in a table and how scale and labels can change the look of a representation. Identify the effects of the transformations f(x) + k, k f(x), f(kx), and f(x + k) on linear, exponential, and quadratic functions. Advanced Skills/Concepts: Fit a function to the data; use polynomial functions fitted to data to solve problems in the context of the data. Use the given functions or choose a function suggested by the context. Distinguish between linear, exponential, quadratic, polynomial, rational, logarithmic, exponential, and trigonometric functions from multiple representations. Sketch a reasonable graph given the key features. Extend the domain of trigonometric functions using the unit circle F.TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Knowledge: Students will know All standards in this unit go beyond the knowledge level. Skills: Students will be able to do Construct viable arguments to justify a solution method. Calculate and interpret the average rate of change of a function. Identify the effect on the graph of replacing by and. Use a model of a function to interpret information about a real-life situation. Define appropriate quantities for the purpose of descriptive modeling. Translate among representations of functions including tables, graphs, and real-life situations. 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. Find the radian measure of an angle. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

5 WIDA Standard: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: Academic Vocabulary: Critical Terms: x-intercept y-intercept Maximum Minimum Symmetry Interval Increasing Decreasing End behavior Periodicity Midline Amplitude Exponential function Logarithmic function Radian Unit Circle Supplemental Terms: domain range Tables Graphs Real-life situations Equations Limit Arc Trigonometric Ratio Right Triangle Sine, Cosine, Tangent

6 Assessment Pre-Assessments Formative Assessments Summative Assessments Self-Assessments Popcorn Roller Coaster Key Features Describe Key Features of Functions Matching Game Graphs from the Unit Circle Scavenger Hunt Degrees and Radians Lesson 2 Summative Lesson 2 Self-Assessment Sample Lesson Sequence 1. T.TF.1, T.TF.2 Unwrap the unit circle using radian measure to extend the trigonometric functions to all real numbers. See (3 4 days) 2. F.IF.4, F.IF.6, F.IF.7 Analyze the key features of functions from their graphs (polynomial, exponential, square root, logarithmic, rational, trigonometric). (3 4 days) 3. F-BF.3 Transform functions f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for positive and negative values of k. (3 4 days)