Impossible Graphs

Abstract

This lesson is devoted to impossible graphs. Users of the module can learn to distinguish between possible and impossible graphs of functions, and to learn why some graphs are impossible.

Objectives

Upon completion of this lesson, students will:

• have practiced plotting functions on the Cartesian coordinate plane
• be able to read a graph, answering questions about the situation described by the graph
• be able to look at a graph and decide if it makes sense

Activities

This lesson introduces students to impossible graphs through the following activity:

• Possible or Not?

Standards

The activities and discussions in this lesson address the following Standards:

• Number sense, number operations and number relationships
• Patterns, relationships and functions

Key Terms

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

• function
• discontinuous
• continuous
• output
• input

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and fractional arithmetic
  • plot points on the Cartesian coordinate system
  • read the coordiates of a point from a graph
• Algebraic: Students must be able to:
  • evaluate algebraic expressions in order to plot points
• 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
  • use a graphing utility to plot simple algebraic expressions

Teacher Preparation

Students will need:

• access to a browser
• copies of supplemental materials for the activities:
  • Impossible Graphs Worksheet

Lesson Outline

These activities together give a brief lesson that can be completed in as little as 30 minutes class-time, depending on how many teams need to share their ideas. The discovery process takes about 15 minutes, and each presentation about 5 minutes.

1. Lead a discussion on impossible graphs.

2. Next have a "live" discussion while going through the Possible or Not? activity. Give each group of students a different graph from the database, and have them present their ideas and findings to the entire class.

Alternate Outlines

This lesson can be rearranged in several ways.

• This lesson can be extended to include not only impossible graphs, but also non-function graphs (those that do not pass the vertical line test).
• This lesson can be extended to include having each team of students discuss a situation in which the impossible graph could be possible. This is a good place to discuss how time is not the only possible independent variable.

Extensions

After these discussions and activities, students will have more experience with functions and relationship between the English description, graphical and algebraic representations - including what cannot occur.