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. 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 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.

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.

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

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.

Algebra I

  • Algebra

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

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
    • 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.

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: 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

Key Terms

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:

    • 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

  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 independently 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 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.