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:
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 taxi-cab 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.
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