Course 4 Unit 3 - Logarithmic Functions and Data Models
1st Edition

In previous courses of Contemporary Mathematics in Context, students have modeled situations with linear, exponential, power, quadratic, absolute value, square root, and three trigonometric (sine, cosine, and tangent) functions. In Course 4, they will extend their families of functions toolkits to include common and natural logarithmic functions; secant, cosecant, and cotangent functions; and polynomial and rational functions. (See the descriptions of Course 4 Units.)

Unit Overview

Unit 3, Logarithmic Functions and Data Models, develops student understanding of logarithmic functions and their use in modeling and analyzing problem situations and data patterns. Students will develop an understanding of inverse functions, the operation of composing functions, and logarithmic scales in the process of meeting the objectives of the unit. The properties of logarithms are developed and used to rewrite expressions and solve equations involving logarithmic expressions. Students then use logarithmic transformations to linearize bivariate data and find an appropriate algebraic model for the data.

Unit Objectives
  • To explore, understand, and represent inverse relationships in algebraic and numerical settings
  • To understand when a function has an inverse
  • To produce and use inverses for y = ax + b and y = a(bx)
  • To understand the inverse relationship between logarithms and exponents, and to use this relationship to simplify complex computations and solve exponential equations
  • To linearize bivariate (x, y) data by transforming one or both variables
  • To use linearizing as a tool in finding an appropriate model for bivariate data

Sample Overview

The sample material from Unit 3 is comprised of the first two investigations of Lesson 3, "Linearizing Data." In the first investigation, students examine a transformation of the y-values that does not linearize the data and then examine some transformations of y-values that do linearize the data. In the second investigation, students review the idea of least squares regression and learn that the exponential regression function on the calculator does not produce the exponential function that is the best fit using the criterion of minimizing the sum of the squared residuals.

Instructional Design

Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a common pattern as elaborated under Instructional Design.

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How the Algebra and Functions and
Statistics and Probability Strands Continue

In Course 4, Units 5 and 6, students intending to pursue college majors in the mathematical, physical, and biological sciences or engineering extend their ability to use polynomial and rational functions to solve problems and extend their ability to manipulate symbolic representations of exponential, natural logarithmic, and trigonometric functions.

Students intending to pursue college programs in social, management, and some of the heath sciences or humanities will continue their study of statistics and probability by studying Unit 5, Binomial Distributions and Statistical Inference, and Unit 10, Problem Solving, Algorithms, and Spreadsheets.

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