Math 3 Unit 4b: Logarithms and Exponents - Modeling



Approximate Time Frame: 3 – 4 WeeksConnections to Previous Learning: In previous units and years, students have learned to create functions based on given properties or data, and to create functions to model real-life situations. Students will build on that understanding in this unit as they apply this prior knowledge to the functions studied in the previous unit.Focus of this Unit: 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 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.7Within 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 fx=gx can be solved (approximately) by graphing ?=?? ?=?? and finding the intersection points involves a number of pieces of conceptual understanding.A-REI.11This 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,fx=gx, to a system of equation in two variables, y=fx and y=gx, 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 thenf(x) and gxare equal, and thus x,y 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. Desired OutcomesStandard(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. Reason quantitatively and use units to solve problems. N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. 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. 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 casesGraph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period,midline, and amplitude.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.6 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. Emphasize linear, quadratic, and exponential models. 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.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 regarding key features of graphs, tables and equations.Guidance with using technology tools for function representations.Understandings: Students will understand …Exponential functions can be determined from data and used to represent many real-life situations (population growth, compound interest, depreciation, etc.).Logarithms can be used to solve the exponential equations modeling many real-life situations.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.).The properties of a situation or data set determine what type of function (linear, quadratic, exponential, polynomial, rational, or logarithmic) should be used to model it.Essential Questions:What can be modeled using exponential functions?What can be modeled using logarithmic functions?What type of function is best to model a given situation?How can logarithmic and exponential equations be solved?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, 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 solve problems numerically, algebraically, or with graphs and the use of technology (i.e. 2x=16; 100ex=200; 3log10(x+2)=6.4)6. Attend to precision. *7. Look for and make use of structure. Students will compare the key characteristics of exponential and logarithmic functions to other function types in order to determine the best model for various situations.8. Look for and express regularity in repeated reasoning. Prerequisite Skills/Concepts: Students should already be able to:Create a scatter plot and fit linear and quadratic models.Find the intersection of two functions graphically or numerically.Model applied situations using linear, quadratic, exponential, polynomial, and rational functions.Advanced Skills/Concepts:Some students may be ready to:Use properties of logarithms to rewrite exponential models with different bases (base 2, 10, or e).Knowledge: Students will know…All standards in this unit go beyond the knowledge level.Skills: Students will be able to …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 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 relationshipCalculate 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.Academic Vocabulary:Critical Terms:Logarithmic scalesSupplemental Terms:base ................
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