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. 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.

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 Addressed:

Grade 10

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Grade 9

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Functions

Interpreting Functions

Understand the concept of a function and use function notation

Interpret functions that arise in applications in terms of the context

Analyze functions using different representations

Linear, Quadratic, and Exponential Models

Interpret expressions for functions in terms of the situation they model

Grades 9-12

Algebra

Represent and analyze mathematical situations and structures using algebraic symbols

Understand patterns, relations, and functions

Algebra 1

Algebra

Competency Goal 4: The learner will use relations and functions to solve problems.

Algebra I

Algebra

Competency Goal 4: The learner will use relations and functions to solve problems.

Grade 8

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Introductory Mathematics

Algebra

COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.

COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

8th grade

Geometry

The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane.

The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation

Elementary Algebra

Elementary Algebra

Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.

Intermediate Algebra

Algebra

The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

Secondary

Algebra II

AII.12 The student will represent problem situations with a system of linear equations and solve the system, using the inverse matrix method. Graphing calculators or computer programs with matrix capability will be used to perform computations.

AII.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions.

AII.12

AII.14

Student Prerequisites

Arithmetic: Student 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

Access to a browser

Copies of supplemental materials for the activities:

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.

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.

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.

Independent Practice

If you choose to pass out the impossible graphs
worksheet have the students work independantly or in small groups to complete it.

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

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.