Introduction to Functions

Abstract

This lesson is designed to introduce students to the idea of 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 functions
• have practiced describing functions with one operationin English sentences, data tables, and with simple algebraic expressions.

Activities

This lesson introduces students to the idea of functions through the following activity:

• 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:

• function
• input
• output

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and fractional arithmetic
• 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

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

Lesson Outline

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. Introduce the idea of functions as machines, by leading a class discussion on functions.

2. Have the students construct machines to test each other with. Start them with single operation machines, and suggest that they build tables for the input/output pairs. Reinforce the convention that mathematicians avoid confusion by always putting the input first in an ordered pair enclosed in parentheses and separated by commas:
   ( x , y )

   Give them one or two tables with a few outputs for them to fill in. Ask them to describe in words what the function does. For example:

   INPUT	OUTPUT		INPUT	OUTPUT
   5	-1		-5	-15
   3	-3		2	6
   -1	-7		4	12
   4	-2		0	0
   -5	-11		3	9
   2	-4		-2	-6
   10			10
   -9			-7

3. After they practice describing functions in English sentences, discuss the convention of letting a letter (often but not always x) stand in for the input and another (often but not always y) stand in for the output.Have them write all their earlier functions as algebra rules with x as input and y as output.

   Formalize the terminology:

   Variable:

   A letter standing in for an unknown or changeable number.

   Independent Variable:

   The input into a function, often represented by x.

   Dependent Variable:

   The output from a function, often represented by y.

   Function:

   A process that takes one or more numbers as input and producesa single number as output.

4. Have the students practice their new function building and pattern recognition skills with the Function Machine Game. 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 worked 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. For example, y = x + 1 might be the function describing growing one year older on your birthday. y = 2 * x might be the function "everything tastes twice as good during the holiday."

Alternate Outlines

This lesson can be rearranged in several ways.

• Add a "name that function" contest (modeled on the game show, "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 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 More Complicated Functions will introduce students more general linear functions.