Course 1 Unit 3 - Linear Models
1st Edition

Linear Models is the second unit from the algebra and functions strand of the Contemporary Mathematics in Context program. In Unit 2, Patterns of Change, attention was focused on the variety of types of change inherent in real situations. These patterns of change were represented using tables of numerical data, coordinate graphs, verbal descriptions, and equations. (See the description of Course 1 Units.)

Unit Overview

In this unit, students develop confidence and skill in using linear functions to model and solve problems that exhibit constant (or nearly constant) rate of change or slope.

Unit Objectives
  • To recognize patterns in tables and graphs of data that are modeled well by linear equations
  • To write equations in the form y = a + bx to model linear patterns in graphs or numerical data
  • To use table, graph, or symbolic representations of linear models to answer questions about the modeled situation: (1) Find y for a given x; (2) Find x for a given y (i.e., solve equations and inequalities); and (3) Describe the rate at which y changes as x changes

Sample Overview

The first investigation of Lesson 3, "Linear Equations and Inequalities," emphasizes the formulation and solution of linear equations and inequalities from a function point of view. This investigation and homework (MORE) set focus mainly on how to write, solve, and check equations and inequalities, using tables and graphs. Later investigations in this lesson introduce "undoing" and "balancing" methods to solve simple linear equations, analyzing and solving a system of linear equations, and writing equivalent linear expressions and equations.

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

A third unit in Course 1, Exponential Models, develops student ability to use exponential functions to model and solve problems in situations that exhibit exponential growth or decay.

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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