Abstract

The following discussions and activities are designed to lead the students to explore the the vertical line test for functions. Plotting points and drawing simple piecewise functions are practiced along the way.

Objectives

Upon completion of this lesson, students will:

• be able to recognize functions from graphs
• be able to recognize functions as formulas
• have learned how to use the vertical line test to verify if a curve is a function
• have practiced their point and function plotting skills

Standards

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

Algebra

Understand patterns, relations, and functions

• identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

• use coordinate geometry to represent and examine the properties of geometric shapes

Links to other standards.

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and fractional arithmetic
  • plot points on the Cartesian coordinate system
• Algebraic: Students must be able to:
  • work with simple algebraic expressions
• 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
• two dice, preferably of different colors

Key Terms

constants constant functions continuous graph discontinuous graph function

graph input origin output

Lesson Outline

This lesson can be done individually or in groups of any size. It is brief, the short version taking as little as 30 minutes.

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:

   • We have been practicing plotting points on the cartesian coordinate plane. (Draw a line on a graph on the board.) Does any one have any ideas on how we could tell people how to draw this exact same line on another graph without showing it to them?

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 graphing functions.
   • We are going to use the computers to learn about graphing 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 class discussion on the vertical line test.

4. Guided Practice

   • Practice with the students the Simpleplot exercise so that they can practice plotting ordered pairs.
   • Have the students then practice graphing skills on graph paper using the tables of values they generated in the Functions and Linear Functions lessons, using the vertical line test to verify that the graphs represent functions.

5. Independent Practice
   • Have the students try the computer version of the Vertical Line Test activity to practice applying the vertical line test.

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.

• Do only the vertical line discussion and function checker activity.
• Add a discussion about fractional movement on the coordinate plane.
• Limit the exercise to the positive domain only.

Suggested Follow-Up

After these discussions and activities, the students will have a good foundation for simple function, function notation, and the vertical line test.