Course 1 Unit 6 - Exponential Models
1st Edition

Exponential Models is the sixth unit in Course 1 of the Contemporary Mathematics in Context program. By the time students begin this unit, they will have developed the ability to make sense of real-world data through the use of graphical displays and summary statistics. They will be able to recognize important patterns of change between related variables and use linear equations to model real-world problems. Students will also have developed skills in algorithmic problem solving and learned how to model a variety of situations with vertex-edge graphs. They will have enhanced their visualization skills and developed a deeper understanding of two- and three-dimensional shapes and their properties. (See the descriptions of Course 1 Units.)

Unit Overview

Exponential Models develops student ability to use exponential functions to model and solve problems in situations that exhibit exponential growth and decay.

Objectives of the Unit
  • To recognize and give examples of situations in which exponential models are likely to match the patterns of change that are observed or expected. This model-recognition skill should apply to information given in data tables, graphs, or verbal descriptions of related changing variables
  • To find exponential rules to match patterns of change in exponential model situations. This should include rules in the "y = ..." and "NOW-NEXT" forms
  • To use exponential rules and graphing calculators or computer software to produce tables and graphs to answer questions about exponential change of variables
  • To interpret an exponential function rule in order to sketch or predict the shape of its graph and the pattern of change in tables of values
  • To describe major similarities and differences between linear and exponential patterns of change

Sample Overview

In Lesson 1 of this unit, students investigate and model exponential growth using both explicit and recursive rules. (For the initial development of recursive rules for linear models, see page 112-115; for recursive exponential rules see page 422.) The sample material consists of the three short investigations from Lesson 2, "Exponential Decay." Students determine and explore exponential models of the form y = a(bx), where 0 < b < 1, through tables, graphs, and algebraic rules. They then compare these models to exponential growth and linear models.

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 Strand Continues

In early units in Course 2, students develop matrix and linear combination methods for solving systems of two linear equations. In Unit 4, Power Models, students develop the ability to recognize and model data patterns and problem conditions that involve direct or inverse power variation and quadratic models. They also study their applications.

Course 3 contains three units devoted to extending students' ability to represent and solve problems using algebraic methods. Students develop the ability to construct, reason with, and solve equations involving several variables and constraints in Unit 1, Multiple-Variable Models. Unit 3, Symbol Sense and Algebraic Reasoning, formalizes the function concept, introduces polynomial and rational functions, extends the solution of equations and inequalities by methods including factoring and the quadratic formula, and develops student ability in algebraic proof. The final algebra and functions unit, Families of Functions, reviews and extends student understanding of the basic function families and develops student ability to adjust these basic functions to match patterns in tables, graphs, and problem conditions.

Four units in Course 4 extend student understanding of algebra and function concepts in preparation for post-secondary education. Students develop understanding of the fundamental concepts underlying calculus, develop understanding of logarithmic functions and their use in modeling and analyzing problem situations, extend their ability to use polynomial and rational functions to solve problems, and extend their ability to manipulate symbolic representations of exponential, logarithmic, and trigonometric functions.

A unit that develops understanding and skill in the use of standard spreadsheet operations while reviewing and extending many of the basic algebra topics from Courses 1-3 is included for students intending to pursue programs in social, management, and some of the health sciences or humanities.

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