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

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

• use graphs to analyze the nature of changes in quantities in linear relationships

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
  • 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. 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:

   • Ask the students if they remember how to read graphs.
   • Provide them with several graphs and ask them to interpret them.
   • Draw a graph on the board for example distance covered in (x)amount of time. Place a break in the graph making it an impossible graph and ask the students if they can explain what is "wrong" with it.

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 impossible graphs and how to determine if a graph is impossible.
   • We are going to use the computers to learn about impossible graphs , 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 impossible graphs.

4. Guided Practice
   • 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.
5. Independent Practice
   • If you choose to pass out the impossible graphs worksheet have the students work independantly or in small groups to complete it.
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.

• 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.

Suggested Follow-Up

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.