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This lesson is designed for a 10th Grade Honors math course.
Prerequisite knowledge:
Fundamental knowledge of geometry concepts including definitions; experience with proving theorems in Euclidean geometry.
Experience proving and disproving ideas in Euclidean geometry.
Students have knowledge that in order to disprove they need only give a counterexample; however to prove, they must prove for in general that a statement is true. They have not had much practice in proving this way, but have seen examples.
Day 1
Aim: To introduce students to the concept of spherical geometry.
Materials: Each student needs a ball and rubber bands.
Teacher needs a large ball and rubber bands
Introduction: Up until now, everything we have learned in geometry has taken place in "Euclidean Space." Now we will look at a different space.
Introduce the sphere:
Definition: Sphere - a set of points in three dimensional space that are equidistant from one fixed point (the center).
Show students a ball. Explain that this is a sphere. The set of points in the definition above describes all points on the surface of the sphere.
Definition: Great Circles: Obtained by slicing through the center of the sphere with planes. The intersection of the plane and points on the surface of the sphere form a great circle.
The plane cutting through the sphere can be demonstrated using a rubber ball that has been cut in half. Put a piece of poster board in between the two halves and have students point out the intersection.
After a student is able to point out where the great circle should be, mark it off with a rubber band on the ball.
Ask the class: How many great circles can we form on the sphere?
elicit: infinitely many
Ask: Why can we form infinitely many:
elicit: There are infinitely many ways to cut through the center of a sphere.
Have several students come up and make more great circles using rubber bands.
On the sphere, these great circles are our "lines" or geodesics.
With that knowledge, have a student point out a line segment on the sphere
Explain that line segments are measure by lengths of string, and have students use string to show the measurement of two congruent line segments and two non congruent line segments below.
Line segment
Give students a few minutes to make their own great circles on their individual balls.
A B
Recall: When you first studied geometry in high school, you were told that you needed to accept certain postulates without proof. In this course we will work with postulates or axioms. An axiom is a known truth.
For example, in Euclidean space, we accept the following axiom, Axiom 1:
Axiom 1: Given any line l, the exist at least two distinct points on l (Draw a line on the board and demonstrate that you can find two distinct points).
Ask: Is Axiom 1 true on the sphere?
students can demonstrate with a ball and rubber bands, or with a sketch that Axiom 1 does hold in spherical space.
Axiom 2: There exist 3 distinct points with the property that no line is incident with all 3 of them.
Have students demonstrate this in Euclidean space and then on the sphere.
In Groups, students should work on the following handout.
This should be finished for homework. (sample answers provided)
Name__________________
Directions: Use your ball and rubber bands to model the following. When you finish each model, draw a sketch on paper.
1. Construct a great circle on the sphere.
2. Construct a triangle on the sphere.
3. A property that holds in Euclidean space is that two lines cannot intersect in more than one place? Does this property hold in spherical space? Demonstrate your answer with a sketch.
No. On the sphere, lines can intersect in more than one place.
Day 2
Aim: To explore axioms that hold in Euclidean space and determine if they hold in spherical space.
Materials: Balls, rubber bands
Go over worksheet from yesterday.
Definition: Antipodal points: Two endpoints of a diameter of a great circle.
Give the example of the earth's north and south poles.
Axiom 3: Given two distinct points P, Q, there exists a unique line incident with P, Q.
As a class determine whether Axiom 3 holds. Students should be able to recognize that there are an infinite number of great circles that can pass through two antipodal points. Have a student draw an example on the board or model this for the class using a ball and rubber bands.
Ask the class: Can we modify this axiom in any way to make it true?
Prompts: Remember that the reason it does not work is because all great circles intersect at each antipodal point.
If we can eliminate one antipodal point, we will not run into this problem. Change line to line segment. We say that Axiom 3 is true locally meaning we limit it to a certain area.
Homework assignment:
Assign the following questions for groups to work on.
1. Recall some of the first postulates you learned geometry:
There is a line between any two points.
A line segment can be extended indefinitely.
You can draw a circle about every point with any radius.
Demonstrate that the above postulates hold in Euclidean space and tell whether or not they hold on spherical space.
2. If possible, find two perpendicular "lines" on the sphere.
Day 3
Aim: To determine whether Playfair's postulate holds on the sphere.
Go over problems from yesterday's handout. Have students demonstrate on a ball that the 3 postulates will work on the sphere, and explain their thinking.
Playfair's Postulate states:
Given a line and a point, there exists a unique line passing through the point parallel to the first line.
Demonstrate in Euclidean space. Give students time to work on this in their groups. Discuss whether or not this will work in spherical space.
Students should try to verify this on the sphere and fail due to the fact that there are no parallel lines on the sphere. Every line on the sphere intersects in at least two points.
Have students work on the following questions:
1) In Euclidean space, given a line and a point P on that line, there is exactly one perpendicular to P. Demonstrate that this does not hold in spherical space.
2) Think of another property that holds in Euclidean space but does not hold in spherical space.
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