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 operation in English sentences, data tables, and with simple algebraic expressions.

### Standards

The activities and discussions in this lesson address the following NCTM standards:

Algebra

Understand patterns, relationships and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
• relate and compare different forms of representation for a relationship;
• identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.
Represent and analyze mathematical situations and structures using algebraic symbols.
• develop an initial conceptual understanding of different uses of variables;
• explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
• use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;
• recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
use mathematical models to represent and understamd quantitative relationships
• model and solve contextualized problems using various representations, such as graphs, tables, and equations
Analyze change in various contexts
• use graphs to analyze the nature of changes in quantities in linear relationships.

### 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

Students will need:

• pencil and paper
• Copies of supplemental materials for the activities:

### Key Terms

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

### 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. Focus and Review

Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

2. Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

• Today, class, we are going to learn about functions and their representations.
• We are going to use the computers to learn about functions and their representations, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

3. Teacher Input

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

4. Guided Practice

• 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:

INPUTOUTPUTINPUTOUTPUT
5-1-5-15
3-326
-1-7412
4-200
-5-1139
2-4-2-6
10
` `
10
```
```
-9
` `
-7
```
```

• 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 produces a single number as output.

5. Independent Practice

• 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

• 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."

6. Closure

• You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

### Alternate Outlines

This lesson can be rearranged in several ways.

• 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 who 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.

### Suggested Follow-Up

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.