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More Complicated Functions: Introduction to Linear Functions
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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.

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.

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 Addressed:

Student Prerequisites

  • Arithmetic: Student 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 for experimenting with the activities
  • Algebraic: Students must be able to:
    • work with simple functions having one operation

Teacher Preparation

Key Terms

interceptSee x-intercept or y-intercept
linear functionA function of the form f(x) = mx + b where m and b are some fixed numbers. The names "m" and "b" are traditional. Functions of this kind are called "linear" because their graphs are straight lines
slope of a linear functionThe slope of the line y = mx + b is the rate at which y is changing per unit of change in x. The units of measurement of the slope are units of y per unit of x (cf. Linear Functions Discussion).

Lesson Outline

  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:

    • Today, class, we are going to learn more about functions.We are going to use the computers to learn more about functions, 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

    • 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 -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.
    • 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 worked with in three ways:
      • English Sentence
      • Table of Values
      • Algebraic 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 Outline

  • 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.
  • 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 Graphing and the Coordinate Plane will introduce students to plotting points on the coordinate plane.

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