SKILLS Students will be skilled at and/or be able

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1 Unit Goals Stage 1 Number of Days: MS 32 1/8/18 2/23/18 HS 32 1/8/18 3/2/18 (1/22/18 to 1/26/18 have been omitted for finals) Unit Description: Students build on their understanding of integer exponents to consider exponential functions with integer domains. They compare and contrast linear and exponential functions, looking for structure in each and distinguishing between additive and multiplicative change. They expand their understanding of arithmetic sequences as linear functions to interpret geometric sequences as exponential functions. Materials: graph paper, 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 [a] N-RN.A Extend the properties of exponents to rational exponents. [m] A-SSE.A Interpret the structure of expressions. [s] A-SSE.B Write expressions in equivalent forms to solve problems. [m] A-CED.A Create equations that describe numbers or relationships. [m] A-REI.D Represent and solve equations and inequalities graphically. [m] F-IF.A Understand the concept of a [m] F-IF.B function and use function notation. Interpret functions that arise in applications in terms of the context. [s] F-IF.C Analyze functions using different representations. 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 A quantity that increases exponentially will eventually exceed a quantity that increases linearly. The process of transforming graphs of exponential functions is the same as it is for other functions. The end behavior of exponential functions is unique in that the domain can be restricted by the context it represents while the range can be restricted by asymptotes. Acquisition KNOWLEDGE Students will know The definition of academic vocabulary words, such as asymptotes, exponential decay, exponential growth, and geometric sequence. Linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals. Common types of exponential functions are exponential growth and decay. ESSENTIAL QUESTIONS Students will keep considering How do exponential functions compare to linear functions? How do transformations affect the graph of an exponential function? How is the end behavior of an exponential function unique? SKILLS Students will be skilled at and/or be able to Extend the properties of integer exponents to rational exponents. Distinguish between situations that can be modeled with linear functions and with exponential functions. Construct exponential functions given a graph, a description of a relationship, or a table. LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 6/19/17

2 [s] F-BF.A 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 and solve problems. [s] F-LE.B Interpret expressions for functions in terms of the situation they model. Unit Goals Stage 1 Exponential growth has a common ratio that is more than 100%, while exponential decay has a common ratio that is greater than 0% and less than 100%. Graph exponential functions showing intercepts and end behavior and interpret the parameters in terms of a context. Use transformations to build new exponential functions from existing functions. Recognize situations in which a quantity grows or decays by a common ratio. Write geometric sequences both recursively and explicitly. LONG BEACH UNIFIED SCHOOL DISTRICT 2 Posted 6/19/17

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 [a] N-RN.A Extend the properties of exponents to rational exponents. N-RN.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 5. N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. [m] A-SSE.A Interpret the structure of expressions [linear, exponential, quadratic.] A.SSE.1 Interpret expressions that represent a quantity in terms of its context. [s] A-SSE.B Write expressions in equivalent forms to solve problems. [Quadratic and exponential.] A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expressions. c. Use properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be written as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. [m] A-CED.A Create equations that describe numbers or relationships. [Linear, quadratic, and exponential.] A-CED.1 Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [m] A-REI.D Represent and solve equations and inequalities graphically. [Linear and exponential; learn as general principle.] A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI.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. [m] F-IF.A Understand the concept of a function and use function notation. [Learn as general principle; focus on linear and exponential and on arithmetic and geometric sequences.] F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. [m] F-IF.B Interpret functions that arise in applications in terms of the context. [Linear, quadratic, and exponential.] 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 LONG BEACH UNIFIED SCHOOL DISTRICT 3 Posted 6/19/17

4 Assessed Grade Level Standards the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. [s] F-IF.C Analyze functions using different representations. [Linear, exponential, quadratic, absolute value, step, piecewise-defined.] 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. e. Graph exponential and logarithmic functions, showing intercepts and end behavior. F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use 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. 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. [s] F-BF.A Build a function that models a relationship between two quantities. [Linear, quadratic, and exponential.] 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. F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. [a] F-BF.B Build new functions from existing functions. [Linear, quadratic, exponential, and absolute value.] 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. [s] F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems. F-LE.1 F-LE.2 F-LE.3 Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 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). Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. [s] F-LE.B Interpret expressions for functions in terms of the situation they model. F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. Key: [m] = major clusters; [s] = supporting clusters, [a] = additional clusters Indicates a modeling standard linking mathematics to everyday life, work, and decision-making CA Indicates a California-only standard LONG BEACH UNIFIED SCHOOL DISTRICT 4 Posted 6/19/17

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: N-RN.A The student rewrites expressions in radical form into an equivalent expression with rational exponents. The student rewrites expressions with rational exponents into an equivalent expression in radical form. The student uses the properties of exponents to write equivalent expressions involving radicals and rational exponents. The student solves equations that require an understanding of the definitions of radicals and rational exponents. A-SSE.A The student uses the structure of an expression to identify ways of rewriting it. A-SSE.B The student uses the properties of exponents to transform exponential expressions. A-CED.A The student graphs equations on coordinate axes with labels and scales to represent the solution to a contextual problem. The student creates equations in two variables to represent a relationship between quantities. A-REI.D The student understands that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). The student finds solutions (either exact or approximate as appropriate) to the equation f ( x) = g( x). F-IF.A The student recognizes any necessary restrictions that need to be placed on the domain in order to represent a function. The student understands that the graph of f is the graph of f (x). The student recognizes that sequences are functions whose domain is a subset of the integers. F-IF.B The student interprets key features of a graph or a table representing an exponential function modeling a relationship between two quantities. The student sketches graphs showing key features given a verbal description of an exponential relationship between two quantities. The student relates the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. F-IF.C The student graphs functions expressed symbolically and shows key features of the graph. The student writes an exponential function defined by an expression in an equivalent form using the properties of exponents to reveal and explain different properties of the functions and to classify them as representing growth or decay. The student compares properties of two functions each represented in a different way (e.g., as equations, tables, graphs, or written descriptions). F-BF.A The student writes explicit or recursive functions to describe relationships between two quantities in a context. The student translates between explicit formulas and recursively defined functions. The student writes geometric sequences both recursively and with an explicit formula. LONG BEACH UNIFIED SCHOOL DISTRICT 5 Posted 6/19/17

6 Evidence of Learning Stage 2 Assessment Evidence F-BF.B The student identifies the effects of transformations on the graphs of exponential functions by replacing f(x) with f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative) and finds the value of k given the graphs. F-LE.A The student distinguishes between situations that can be modeled with linear functions and exponential functions. The student compares linear and exponential models. The student constructs linear and exponential models given a graph, a description of a relationship, or input-output pairs (including reading these from a table). The student understands that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. F-LE.B The student interprets the parameters in an exponential function in terms of a context. 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-SSE.A A-SSE.B A.CED.A A-REI.D F-IF.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: N-RN.A A-REI.D F-IF.B F-IF.C 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-SSE.B A-CED.A F-IF.C F-BF.A F-LE.A F-LE.B 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 LONG BEACH UNIFIED SCHOOL DISTRICT 6 Posted 6/19/17

7 Days Learning Target Expectations 1 day I will explore exponential functions by participating in the Opening Task. Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction OPENING TASK Football Drills This Opening Task is a Solo-Team-Teach activity. First, pass out the activity, Football Drills, and have students work independently. Next, have students work with their team. After working together, facilitate a class discussion about their method(s) for choosing the football drill utilizing Talk Moves. This task is a gateway into the entire unit on exponential functions. Big Ideas Math Algebra 1 (Explorations and Lessons) Supplemental Resources Conceptual Understanding: Football Drills 4-5 I will use the properties of rational exponents by Using patterns to create conjectures for the rules of zero and integer exponents. (SMP 7) Using expanded form to look for patterns to create conjectures for multiplying powers with the same base, dividing powers with the same base, and raising a power to a power. (SMP 7) Evaluate expressions with rational exponents. Rewriting expressions involving rational exponents (fractional exponents) and radicals using properties of exponents. Explaining how the meaning of fractional exponents extends from the properties of integer exponents. (SMP 3) Answering questions such as o Justify each exponent property using expanded form. o Why is the value of the base irrelevant (except for zero) when it is raised to the zero power? o How do you write and evaluate an n th root of a number? o How are radicals related to exponents? o How do you rewrite expressions involving radicals and rational exponents using the properties of exponents? o Synergy Item Bank: Item ID 63724, Section 6.1 Section 6.2 Conceptual Understanding: Evaluating a Special Exponential Expression Task Evaluating Exponential Expressions Task Rational Exponents Task Procedural Skills and Fluency: Positive Exponents Product Game Negative Exponents Product Game Exponents Math-O Fractional Exponents Solo-Team-Teach Application: How High Can You Count on Your Fingers? Video LONG BEACH UNIFIED SCHOOL DISTRICT 7 Posted 6/19/17

8 7-8 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Days Learning Target Expectations Big Ideas Math Algebra 1 (Explorations and Lessons) I will explore Section 6.3 exponential functions by Distinguishing between linear and exponential functions. Explaining that a quantity increasing exponentially will eventually exceed a quantity increasing linearly. (SMP 3) Identifying exponential functions in different representations, such as graphically, algebraically, numerically in tables, or by verbal descriptions. Graphing simple exponential functions by hand and using technology for more complicated cases. (SMP 5) Identifying key features of exponential functions, such as intercepts and end behavior, and interpreting the parameters in terms of a context, including asymptotes. Interpreting the domain and range of an exponential function and relating them to the relationship it describes. Graphing exponential functions in the form x h f ( x) = ab + k, where b > 0. Identifying the effect of transformations on the x graph of the exponential function f ( x) = b by replacing f(x) with f(x) + k, k f(x), and f(x + k) for specific values of k (both positive and negative). (SMP 7) Answering questions such as o What are some characteristics of an exponential function? o How are the graphs of linear and exponential functions similar? How are they different? o How do you graph an exponential function? o How does changing the values of a, h, and k affect the graph of an exponential function? o Synergy Item Bank: Item ID 58365, 59090, 58736, Supplemental Resources Conceptual Understanding: Desmos: Avi and Benita s Repair Shop Pay It Forward Sample Lesson with Video Clip Power of Two Activity The Penny Experiment Activity and Graph Exponential Functions Which One Doesn t Belong? Exponential Parameters Task Procedural Skills and Fluency: Desmos: Polygraph Exponentials Desmos: What Comes Next? Desmos: Marbleslides Exponentials Exponential Curves Matching Activity Application: Desmos: Penny Circle Linear and Exponential Functions Task Population and Food Supply Task Exponential vs. Linear Growth Task Linear or Exponential? Task Modeling: Domino Skyscraper (SMP 4) LONG BEACH UNIFIED SCHOOL DISTRICT 8 Posted 6/19/17

9 Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Days Learning Target Expectations Big Ideas Math Algebra 1 (Explorations and Lessons) I will write, graph, Section 6.4 and interpret exponential growth and decay functions by 4-5 Distinguishing between exponential growth and decay from inspecting an equation, a table, or graph. Describing situations that represent exponential growth or exponential decay, such as compound interest and half-life. Identifying key features of exponential growth and decay functions, such as intercepts and end behavior, and interpreting the parameters in terms of a context, including asymptotes. (SMP 2) Interpreting the domain and range of exponential growth and decay functions from equations, graphs, and tables. Graphing simple exponential growth and decay functions by hand and using technology for more complicated graphs. (SMP 5) Answering questions such as o How do you write, graph, and interpret exponential growth and decay functions? o How is the formula for exponential decay similar to the formula for exponential growth? o How is compound interest different from simple interest? o Synergy Item Bank: Item ID 54980, Supplemental Resources Conceptual Understanding: Compound Interest Simulator Procedural Skills and Fluency: Desmos: Card Sort Exponentials Graphing Exponential Growth and Decay PowerPoint Application: Desmos: iphone 6s Opening Weekend Sales (SMP 4) Interesting Interest Rates Task Predicting Your Financial Future Task Saving for College Activity Modeling: Fry s Bank (SMP 4) Modeling: How Much Did Peterson Loose by Not Cashing his Check? (SMP 4) 2-3 I will solve exponential equations Algebraically. Graphically (i.e. system of a linear and exponential function). Answering questions such as o How can you solve exponential equations with the same base? With unlike bases? o How can you solve exponential equations by graphing? o Synergy Item Bank: Item ID Section 6.5 Procedural Skills and Fluency: Solving Exponential Equations Card Sort LONG BEACH UNIFIED SCHOOL DISTRICT 9 Posted 6/19/17

10 Days Learning Target Expectations 2-3 I will check my understanding of exponential functions by participating in the FAL. Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction FORMATIVE ASSESSMENT LESSON (choose 1) Representing Linear and Exponential Growth (SMP 1, 2, 4,5, 7, 8) Modeling Population Growth: Having Kittens (SMP 1, 2, 3, 4, 5, 6, 7, 8) Big Ideas Math Algebra 1 (Explorations and Lessons) Supplemental Resources 4-5 I will discover geometric sequences by Writing an explicit rule for a geometric sequence. Writing a recursive rule for a geometric sequence. Comparing the graph of a geometric sequence to the graph of an exponential function. Translating between explicit formulas and recursively-defined functions. Answering questions such as o How is a geometric sequence different from an arithmetic sequence? o Compare and contrast recursive and explicit rules. o How are geometric sequences related to exponential functions? o How can you tell whether a common ratio will be greater than one or less than one? o Synergy Item Bank: Item ID 51336, 55686, Section 6.6 Section 6.7 Procedural Skills and Fluency: Recursive and Exponential Rules Activity Application: Arithmetic vs. Geometric Sequences Task Seeing Music Task day I will prepare for the unit assessment on exponential functions by Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Ch. 6 Review (p ) Unit Assessment Students will take the Synergy Online Unit Assessment. Application: Basketball Bounces Task Desmos: 2015 NBA Draft Salary Prediction (SMP 4) Exponential Functions Application Problems National Debt and Wars Task Modeling: XBOX Xponential (SMP 4) Unit Assessment Resources (Word or PDF) can be used throughout the unit. LONG BEACH UNIFIED SCHOOL DISTRICT 10 Posted 6/19/17