More Complicated Functions:
Introduction to Linear Functions

Abstract

This lesson is designed to introduce students to the idea of functions composed of two operations, with specific attention to linear functions and their representations as rules and data tables, including the mathematical notions of independent and dependent variables.

Objectives

Upon completion of this lesson, students will:

• have been introduced to functions
• have learned the terminology used with linear functions
• have practiced describing linear functions in English sentences, data tables, and with simple algebraic expressions.

Activities

This lesson introduces students to linear functions through the following activity:

• Linear Function Machine

Standards

The activities and discussions in this lesson address the following Standards:

• Number sense, number operations and number relationships
• Patterns, relationships and functions
• Algebra

Key Terms

This lesson introduces students to the following terms through the included discussions:

• linear function
• function
• slope
• intercept

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and fractional arithmetic
• Algebraic: Students must be able to:
  • work with simple functions having one operation
• Technological Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• access to a browser
• pencil and paper
• copies of supplemental materials for the activities:
  • Linear Function Machine Worksheet

Lesson Outline

This lesson assumes that the student is already familiar with the material in the Introduction to Functions Lesson. These activities can be done individually or in teams of as many as four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class.

1. Lead a discussion on building more complicated functions using composition.

2. Have the students practice a few of these more complicated functions by hand by filling in a few tables. Give them some functions in English, some as tables and some as algebra. Have them write the functions in all the forms. For example:
   1. Find the function that adds one and then multiplies the result by 2
   2. y = 4 - x/2

   3. x	-2	-1	0	1	2
      y	-7	-4	-1	2	5

   Note: The function rule for these more complicated functions can be much harder to guess from just the data table.

3. Lead a discussion on functions of the special form y = ___ * x + ___.

4. Have students practice their linear function skills by using the Linear Function Machine.

   Be sure to have students record how many numbers they needed to look at before correctly guessing the function structure. Have them write the functions they w orked with in three ways:

   • English sentence
   • Table of values
   • Algebra rule

5. Have them try to think of situations in their lives that might be governed by some of the functions they worked with.

Alternate Outlines

This lesson can be rearranged in several ways.

• Omit the information on more complicated functions, discussing only functions of the form y = mx + b.
• Add a "name that function" contest (modeled on name that tune) in which teams of students compete to figure out the function. Here is a set of possible rules for such a game:
  • Show two input/output pairs to both teams - two students on a team works very well.
  • Have each team state how many more pairs they think that they would need to see to "name that function." The team that claims the fewest needed pairs goes first.
  • If a team guesses wrong the other team gets to try, after seeing one more pair. Teams alternate turns until one guesses correctly.
  This game can be played in about 10 minutes per pair of teams, making it time consuming if the entire class is to have a turn.
• Introduce more complicated non-linear functions by allowing exponentiation (whole numbers to start) and division by x.

Extensions

After these discussions and activities, students will have an intuitive understanding of functions and will have seen many examples of linear functions. The next lesson The Coordinate Plane will introduce students to plotting points on the coordinate plane.