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.

### 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 varius 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
• 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:

• 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

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. 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:

• Who remembers what a function is?
• Can someone give me an example of a function?
• Can someone give me an example of something that is not a function?

2. Objectives

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

3. Teacher Input

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

4. Guided Practice

• Have the students practice "pumping" 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 y -2 -1 0 1 2 -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.
• Lead a discussion on functions of the special form y = ___ * x + ___.

5. Independent Practice

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

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.

• 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 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 more complicated 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 The Coordinate Plane will introduce students to plotting points on the coordinate plane.