Non-Euclidean Geometry Lesson Plan
Abstract

This lesson is designed to improve students understanding of geometry and measurement concepts. The lesson also introduces students to basic Non-Euclidean Geometry.

Standards (NCTM)

Geometry Standard

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
  • understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
  • create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship
Student Prerequisites

  • Educational:

    Students should know:

    • basic concepts of measurement (area, perimeter)
    • basic geometrical concepts (angles, parallel lines)

  • 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

Teachers will need:
  • notecards with the Euclidian Postulates written on them.
  • Lenart spheres, oranges, craft styrofoam spheres, or any other sphere shaped object that can be drawn on with a marker.
  • 2 paper squares of different colors for each student to be used during the triangle activity
  • 2 paper squares of different colors for the Pythagorean Theorem proof.
      • Depending on the age of your students and the time alloted for this lesson, you may want to draw the templates on the squares before class.
  • Poincare Disk Model Applet .
  • Taxicab Geometry Applet .
Students will need:
  • access to a browser


Lesson Outline

  1. Focus and Review

    • Ask students to prove that the two shaded areas in this picture are equal.
  1. Objectives

    • Introduce students to basic spherical geometry.
    • Introduce students to taxi-cab geometry.
    • Teach a few simple geometrical proofs in Euclidean geometry.
  1. Guided Practice

    • Divide the students into small groups.
    • Ask each group to define "straight" using pencils and paper.
    • After each group seems to have a suffient answer, hand out markers and spheres to each group.
    • Ask each group to define "straight" using the sphere and markers.
    • Let each group draw on the sphere.
    • Have students write down any trends or ideas they have about the definition of straight. Including properties of straight lines on a plane and a sphere.
  1. Teacher Input

    • Discuss the properties each group found.
    • Identify the similarities and diferences between geometry on a plane and geometry on a sphere.
    Guided Practice

    • Split the class into 5 groups.
    • Give each group a notecard containing one of Euclid's Postulates.
    • Have the students determine what the postulate means on a plane.
    • Next, have the students decide if the postulate holds true on a sphere.
    Teacher Input

    • Allow each group to present on their postulate.
    • Fill in any information the students may have missed.
    • Point out why sphereical geometry is important and why people study it.
    • If we live on a sphere, why do people use planar geometry when doing calculations?
    Guided Practice

    • Hand out the paper squares for the triangle proof.
    • Each student should draw a triangle on one of the paper squares.
    • Distribute scissors to the class.
    • Students should cut out their triangles.
    • After labeling the angles - 1, 2, 3 - tear off each corner.
    • On the other paper square, arrange the corners in such away to make a straight line.
    • Hand out tape and let the students attach the corners and triangle to the other paper square.
    Teacher Input

    • Discuss properties of triangles in Euclidean geometry.
    • What about triangles in other geometries?
    Guided Practice

    • Split the students back up into groups and hand out the spheres.
    • Allow students to explore properties of triangles on a sphere.
    • Students should record trends they notice on a piece of paper.
    • When students become bored with the spheres, bring up the Poincare Disk Applet.
    • Allow student to explore properties of triangle on the Poincare Disk.
    • Students should record trends they notice on a piece of paper.
    Teacher Input

    • Discuss some of the properties students discovered.
    • Explain the concepts of hyperbolic space and the Poincare Disk Model.
    • Recall the discussion about straight from earlier.
    • How does the concept of straight relate to triangles? The shortest distance between two points is a straight line.
    • Explain that the Pythagorean Theorem can be proved with paper and scissors.
    Guided Practice

    • Hand out the square pieces.
    • If the outline has not yet been drawn on the pieces, draw both squares up on the board.
    • Have students then copy the templates onto the pieces of paper.
    • Let students cut out the triangles from the first sheet.
    • Students must now arrange the triangles on the other square.
    • How does this prove the Pythagorean Theorem?
    Teacher Input

    • Lead the students through the proof of the Pythagorean Theorem.
    • Recall that some properties do not hold true in all geometries.
    • Introduce taxicab geometry.
    Guided Practice

    • Allow students to explore the TaxiCab Geometry Applet.
    • Be sure they record any properties they notice about this geometry on a sheet of paper.
    Teacher Input

    • Recall that the parrell postulate does not hold true in sphereical geometry.
    • What property doesn't hold true in TaxiCab geometry?
    • Discuss the concept of "straight" in this geometry.
    • Indicate how it relates to the statement, "The shortest distance between two paths is a straight line."
  1. Closure

    • Discuss applications of spherical geometry in real life.
    • Discuss applications of taxicab geometry.
    • Discuss possible applications of hyperbolic space.
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