Math 3 Unit 6: Radical, Exponential, and Logarithmic Representations and Modeling Approximate Time Frame: 5-6 Weeks Connections to Previous Learning: In Math 2, students graphed square root and cube root functions and simplified exponential expressions with rational exponents; here students will solve radical equations with an unknown in the radicand. In previous years, students have studied exponential functions, but have only been able to solve equations with an unknown in the exponent that can be solved by inspection or the use of exponent rules. Logarithms open the door to students being able to solve more of these equations, not restricted to trivial cases. This unit builds on students’ understanding of linear, quadratic, exponential, polynomial, and rational functions to include logarithmic functions as well.

Focus of this Unit: In this unit, students will learn how to solve radical equations, exponential equations, and the features of logarithmic functions, focusing on those with base 2, 10, and e. The relationship between logarithms and exponentials and their graphs will be a key understanding. Another key understanding is that a logarithm is itself an exponent, and therefore it can be the solution to an exponential equation. When the solution is rational, students will tie this understanding to previous learning with rational exponents. Students will build exponential and logarithmic functions to solve problems in areas including (but not limited to) finance and the physical sciences (such as light intensity, ph values, radioactive decay). Students will create logarithmic and exponential functions based on data and the properties of a situation, and students will use their knowledge of these and previously studied functions to determine what function can be best used to model a given situation or solve a given problem.

Connections to Subsequent Learning: In subsequent units students will continue to learn new types of functions that can be used in problem-solving and modeling, and will continue to build on the problem-solving and modeling skills developed here and in previous units.

From the High School, Algebra Progression document p.5: Much of the ability to see and use structure in transforming expressions comes from learning to recognize certain fundamental techniques. One such technique is recognizing internal cancellations, as in the expansion (푎 − 푏)(푎 + 푏) = 푎2 − 푏2 An impressive example of this is (푥 − 1)(푥푛−1 + 푥푛−2 + ⋯ + 푥 + 1) = 푥푛 − 1 in which all the terms cancel except the end terms. This identity is the foundation for the formula for the sum of a finite geometric series.A-SSE.4 From the High School, Function Progression document p. 13: When it comes to inverse functions,F-BF.4a the expectations are 푓(푥) = 푐. The point is to provide an informal sense of determining the input when the output is known. Much of this work can be done with specific values of c. Eventually, some generality is 1 푐 warranted. For example, if푓(푥) = 2푥3, then solving 푓(푥) = 푐 leads to 푥 = ( )3, which is the general formula for finding an input from a specific output, c, for this 2 function, f . At this point, students need neither the notation nor the formal language of inverse functions, but only the idea of “going backwards” from output to input. This can be interpreted for a table and graph of the function under examination. Correspondences between equations giving specific values of the functions,

11/17/2014 1:12:31 PM Adapted from UbD® framework Page 1 Priority Standards Supporting Standards Additional Standards Math 3 Unit 6: Radical, Exponential, and Logarithmic Representations and Modeling table entries, and points on the graph can be noted (MP.1). And although not required in the standard, it is reasonable to include, for comparison, a few examples where the input cannot be uniquely determined from the output. For example, if푔(푥) − 푥2, then 푔(푥) = 5 has two solutions, 푥 = ±√5.

From the Grade 8, High School, Functions Progression document p.15: Students note the correspondence between rise and run on a graph and differences of inputs and outputs in a symbolic form of the proof (MP1). A symbolic proof has the advantage that the analogous proof showing that exponential functions grow by equal factors over equal intervals begins in an analogous way. The process of going from linear or exponential functions to tables can go in the opposite direction. Given sufficient information, e.g., a table of values together with information about the type of relationship represented, F-LE.4, students construct the appropriate function. For example, students might be given the information that the table below shows inputs and outputs of an exponential function, and asked to write an expression for the function. Input Output 0 5 8 33

For most students, the logarithm of x is merely shorthand for a number that is the solution of an exponential equation in x.F-LE.4

Desired Outcomes Standard(s): Create equations that describe numbers or relationships.  A.CED.1 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.  A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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 number 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. Interpret functions that arise in application 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. Analyze functions using different representations.  F.IF.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (by hand in simple cases and using technology for more complicated cases) 11/17/2014 1:12:31 PM Adapted from UbD® framework Page 2 Priority Standards Supporting Standards Additional Standards Math 3 Unit 6: Radical, Exponential, and Logarithmic Representations and Modeling  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. 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.4a 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. Write expressions in equivalent forms to solve problems.  A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and use the formula to solve problems. For example, calculate mortgage payments. Construct and compare linear, quadratic, and exponential models and solve problems.  F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Reason quantitatively and use units to solve problems.  N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. Represent and solve equations and inequalities graphically.  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.★ Summarize, represent, and interpret data on two categorical and quantitative variables.  S.ID.6a 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.

Transfer: Students will apply concepts and procedures in solving exponential equations with multiple representations in order to justify a method of solution and interpret key features of the graph. Example: Given a logarithmic function, students will explain the existence of restrictions on the domain. Students will apply concepts and procedures regarding a relationship between two quantities in order to model situations with logarithmic functions. Example: Given a scenario, students will define quantities, create a logarithmic equation, obtain a result using technology, and interpret and validate the result.

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Understandings: Students will understand …  The logarithm of a number is the exponent that another value (the base) must be raised to produce the given number. x  That logb y  x is another way of expressing b  y and that this logarithmic expression can be used to determine the solution of an equation where the unknown is in the exponent with bases 2, 10, and e.  Exponential functions and equations can be rewritten as logarithmic functions and equations, and vice versa.  Convenient values for representing a logarithmic function on a table are powers of the base of the logarithm.  Exponential functions can be determined from data and used to represent many real-life situations (population growth, compound interest, depreciation, etc.).  Exponential and logarithmic equations can be solved graphically through the use of technology.  Logarithmic functions (and logarithmic scales) can be useful to represent numbers that are very large or that vary greatly and are used to describe real- world situations (Richter scale, Decibels, pH scale, etc.).

Essential Questions:  What is a logarithm?  How can exponential equations be solved for an unknown in the exponent?  What are the key features of the graph of a logarithmic function?  How can a logarithmic function be represented numerically or in a table?  How do you create or interpret a scatter plot from data and fit a function to this data?  How do logarithms relate to exponential models, and how can they be used to solve exponential equations?

Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. Students will develop this practice as they work from the context of an applied situation, put it into exponential and logarithmic terms, and then manipulate those terms to answer the question. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. Students will explore and model real-life situations and phenomena using exponential and logarithmic functions. * 5. Use appropriate tools strategically. Students will develop this mathematical practice as they determine which logarithmic expressions can be evaluated with and without the use of technology. 6. Attend to precision. * 7. Look for and make use of structure. Students will develop this mathematical practice as they further investigate the use of transformations on functions (logarithmic) and relate a function to its graph and a table of values. 8. Look for and express regularity in repeated reasoning.

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Prerequisite Skills/Concepts: Advanced Skills/Concepts: Students should already be able to: Some students may be ready to:  Write, graph, and interpret exponential functions.  Use logarithms to give exact solutions to exponential equations in any base.  Solve equations with unknowns in the exponent by inspection or  Understand logarithms and exponentials as inverse functions and use this to use of exponent rules. explain the relationships between domain, range, asymptotes, and average rate of  Identify the effects of the transformations f(x) + k, k f(x), f(kx), and change. f(x + k).  Use properties of logarithms to rewrite expressions.  Create a scatter plot and fit linear and quadratic models.  F.BF.4 Find Inverse functions.  Find the intersection of two functions graphically or numerically. b. (+) Verify by composition that one function is the inverse of  Model applied situations using linear, quadratic, exponential, another. polynomial, and rational functions. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.  F.BF.5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Knowledge: Skills: Students will know… Students will be able to …  The definition of a logarithm.  Solve radical and rational equations in one variable, checking for extraneous  The graph of f(x) = log (x) has a domain of x>0, a vertical asymptote solutions. at x=0, and an x-intercept at x = 1.  Use logarithms to solve exponential equations in base 2, 10, or e.  Change of base formula.  Evaluate logarithms based on the definition for simple cases.  Evaluate logarithms using the change of base formula with technology.  Graph exponential functions, identifying intercepts and end behavior.  Graph logarithmic functions, identifying intercepts and end behavior.  Construct a viable argument to justify a solution method.  Translate back and forth between logarithmic and exponential representations.  Compare properties of two functions each represented in a different way.  Derive the formula for the sum of a finite geometric series.  Determine the best function to fit a certain situation or set of data.  Use technology to fit exponential models to data.  Model applied situations using exponential and logarithmic functions and answer

11/17/2014 1:12:31 PM Adapted from UbD® framework Page 5 Priority Standards Supporting Standards Additional Standards Math 3 Unit 6: Radical, Exponential, and Logarithmic Representations and Modeling questions using those models.  Reason quantitatively and use units to solve problems.  Define appropriate quantities for the purpose of descriptive modeling.  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  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.  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.  Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.  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.  For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. WIDA Standard: (English Language Learners) English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from:  Explicit vocabulary instruction with regard to the types and components of function representations.  Guided conversations regarding the connections between graphic, algebraic, tabular and verbal descriptions of functions. Academic Vocabulary: Critical Terms: Supplemental Terms: Radical Common Logarithm Exponential Form Inverse function Index Base Logarithmic Form Limit Radicand Asymptote Rational Extraneous Natural logarithm (ln or loge) Irrational Exponent Logarithmic scales

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Assessment Summative Assessments Unit 5 Summative Assessment Pre-Assessment Formative Assessments Self-Assessments Activating Prior Knowledge Magnitude of Revenue Lesson 3 Self Assessment I Have, Who Has? Solve-Justify-Switch Dr. Terrible’s Bacteria Epidemic Sample Lesson Sequence 1. Working with radical equations. A.REI.1, A.REI.2 2. Exponential Models in the Real World: creating models, using equations to solve problems, solving graphically when the solution is irrational, Reading Reflection – Understanding e, Population Growth A.CED.1, A.CED.2, A.SSE.4, S.ID.6a, N.Q.2, A.REI.11 3. Logarithms: Self-Assessment, Magnitudes of Revenue (F.LE.4), I Have, Who Has? (F.LE.4), Solve-Justify-Switch (F.LE.4, A.REI.1), Dr. Terrible’s Bacteria Epidemic (F.LE.4, A.CED.1, F.BF.4a) (Model Lesson 4. Graphs of Exponential and Logarithmic Functions F.IF.7e, F.IF.4, F.IF.6, F.IF.9, F.BF.3, F.BF.4a, A.REI.11

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