Rochester City School District
Algebra II Module 3 Exponential and Logarithmic FunctionsTopic AReal Numbers9Topic BLogarithms14Topic C Exponential and Logarithmic Functions and their Graphs9Topic DUsing Logarithms in Modeling Situations10Topic EGeometric Series and Finance3OverviewIn this module, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). Students identify appropriate types of functions to model a situation. They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as, “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions,” is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations. This module builds on the work in Algebra I, Modules 3 and 5, where students first modeled situations using exponential functions and considered which type of function would best model a given real world situation. The module also introduces students to the extension standards relating to inverse functions and composition of functions to further enhance student understanding of logarithms. Topic E culminates in lessons where students consider applying their knowledge to financial literacy. Students find the sum of a geometric series.LessonBig IdeaEmphasizeSuggestedProblems(Problem Set)Exit Ticket# of DaysTOPIC A1Students review and practice applying the properties of exponents for integer exponents. Students model a real-world scenario involving exponential growth and decay. Properties of exponents learned in previous courses.Example 2Exercises 1-5Problems 4,6,7,10,11,12 HYPERLINK "" eMathInstruction Unit 1 Lesson 4No22Students review place value and scientific notation. Students use scientific notation to compute with large numbers. Operations with numbers in scientific notation.Examples 1,2Exercises 1-9Problems 1-2Yes13Students will calculate quantities that involve positive and negative rational exponents. See N-RN.A.1 and N-RN.A.2Opening ExerciseExample 1Exercises 1-12Problems 1-14 HYPERLINK "" eMathInstructionUnit 4 Lesson 2Yes34Students rewrite expressions involving radicals and rational exponents using the properties of exponents. See N-RN.A.1 and N-RN.A.2Opening ExerciseExamples 1-3Exercises 1-6Problems 2-3, 6-8 HYPERLINK "" eMathInstructionUnit 8 Lesson 4Yes36Students are introduced to Euler’s number ?.e≈2.72… is an irrational number used in exponential modeling.The Lesson Notes introduce the irrational number e and discuss its history. You might use a different resource to introduce the number e or wait until it is used later in Topic C.No0LessonBig IdeaEmphasizeSuggestedProblems(Problem Set)Exit Ticket# of DaysTOPIC B7Students solve simple exponential equations numerically. Solving exponential equations by finding common bases.Opening ExerciseExample 1Exercises a-nProblems 1-3 HYPERLINK "" eMathInstructionUnit 4 Lesson 5Yes28Students calculate a simple logarithm using the definition. A logarithm is an exponent.Opening ExerciseExercises 1-10Examples 1-8Problems 1-9 HYPERLINK "" eMathInstructionUnit 4 Lesson 8Yes310Students construct a table of logarithms base 10 and observe patterns that indicate properties of logarithms. You might use alternative resources to address the following properties of logarithms:742956032500No111Students construct a table of logarithms base 10 and observe patterns that indicate properties of logarithms. You might use alternative resources to address the following properties of logarithms:-450859842500No1LessonBig IdeaEmphasizeSuggestedProblems(Problem Set)Exit Ticket# of Days12Students justify properties of logarithms using the definition and properties already developed.Properties of logarithms.Opening ExerciseExercises 1-10Problems 1-5, 9, 10, 12 HYPERLINK "" eMathInstructionUnit 4 Lesson 10Yes313Students understand how to change logarithms from one base to another. Change of base formula.Example 1Exercises 1-6Problems 1-3, 5-10, 14-15Yes214Students solve simple logarithmic equations using the definition of logarithm and logarithmic properties.Solving logarithmic equations.Opening ExerciseExamples 1-3Exercises 1-3, 5-9Problems 1,2,4 Yes2TOPIC C17Students graph logarithmic functions.Key features of logarithmic graphs.Opening ExerciseExercises 1-3Problems 4-6Yes118Students compare the graph of an exponential function to the graph of its corresponding logarithmic function.The corresponding graphs are reflections of each other over the line y = x.Opening ExerciseExercises 1-3Problems 1-2 HYPERLINK "" eMathInstructionUnit 4 Lesson 9Yes219Students will understand that the logarithm function base ? and the exponential function base ? are inverse functions. See Lesson Summary.Opening ExerciseExercises 1-3, 6Problem 4a-e HYPERLINK "" eMathInstructionUnit 2 Lesson 6Yes2LessonBig IdeaEmphasizeSuggestedProblems(Problem Set)Exit Ticket# of Days20Students study transformations of the graphs of logarithmic functions. Horizontal shift, vertical shift, reflections.Opening Exercise (a)Exploratory ChallengeExamples 1-2Exercises 1-4Yes221Students graph the natural logarithm function and understand its relationship to other base ? logarithm functions. The inverse of y = ex is y = ln x.y = ln x is equivalent to y = loge x.Exploratory ChallengeExamples 1-2 HYPERLINK "" eMathInstructionUnit 4 Lesson 12Yes2TOPIC D24Students apply properties of logarithms to solve exponential equations. Use logarithms to solve an equation with a variable exponent.Opening ExerciseExercises 1-5Problems 1, 8,9, 12a-b HYPERLINK "" eMathInstructionUnit 4 Lesson 11Yes225Students use geometric sequences to model situations of exponential growth and decay. Geometric sequence is an exponential function because it has a common ratio.Opening ExerciseExercises 1-3Problems 1ab, 2ab, 3,7,10,12 HYPERLINK "" eMathInstruction Unit 4 Lesson 6Yes226Students develop a general growth/decay rate formula in the context of compound interest. Formulas for calculating compound interest, including continuously compounded interest.Examples 1-3Problems 4-5 HYPERLINK "" eMathInstructionUnit 4 Lesson 13Yes227Students create exponential functions to model real-world situations. Finding the equation of an exponential function algebraically.Opening ExerciseExercises 1-14Problem 3,6 HYPERLINK "" EMathInstructionUnit 4 Lesson 4Yes228Students apply knowledge of exponential and logarithmic functions and transformations of functions to a contextual situation. Newton’s Law of Cooling.Exercises 1-2Problem 1 HYPERLINK "" eMathInstrucionUnit 4 Lesson 14Yes2TOPIC E29Students derive the sum of a finite geometric series formula. Formula for finding the sum of the terms in a geometric sequence.Opening ExerciseExamples 1-3Exercises 1-4Problems 1-5,15,23-25 HYPERLINK "" eMathInstructionUnit 5 Lesson 5No3 ................
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