Abstract
This lesson is designed to improve students understanding of geometry
and measurement concepts. The lesson also introduces students to
basic NonEuclidean Geometry.
Standards (NCTM)
Geometry Standard
Analyze characteristics and properties of two and threedimensional 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:
Students will need:
Lesson Outline
 Focus and Review
 Ask students to prove that the two shaded areas in
this picture are equal.
 Objectives
 Introduce students to basic spherical geometry.
 Introduce students to taxicab geometry.
 Teach a few simple geometrical proofs in Euclidean geometry.
 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.
 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 spherical 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."
 Closure
 Discuss applications of spherical geometry in real life.
 Discuss applications of taxicab geometry.
 Discuss possible applications of hyperbolic space.

