Introduction

The geometry you are familiar with is called Euclidean geometry, named for the famous Greek mathematician named Euclid (325-265 BC). Spherical geometry is one form of non-Euclidean geometry.

Task

1. Determine whether all lines of longitude on the globe are lines in spherical geometry.

2. Determine whether all lines of latitude on the globe are lines in spherical geometry.

3. In previous lessons, you learned that if 2 lines are perpendicular, they form four right angles. Is that true in spherical geometry? Explain and include a sketch.

4. Make a poster comparing a point, a line, and a plane in Euclidean geometry and spherical geometry. Include diagrams and sketches.

5. Develop a hands-on activity to demonstrate spherical geometry to the class using one or two volunteers.

Process

1. Use a globe, two large rubber bands, and the steps below to investigate lines on a sphere.

a. In Euclidean and spherical geometry, points are the same. A point is just a location that can be represented by a dot.

b. In Euclidean geometry, you can represent a plane by a sheet of paper. Remember that the paper is only part of the plane. The plane goes on forever in all directions. In spherical geometry, a plane is a sphere. The sphere is finite, that is, it does not go on forever. The globe will represent a plane in spherical geometry.

c. Place a large rubber band on the globe covering the equator. The equator is known as a line in spherical geometry. In Euclidean geometry, a line extends without end in both directions. In spherical geometry, a line is finite. In spherical geometry, a line is called a great circle, which divides a sphere into two congruent halves. On the globe, the equator is also called a line of latitude.

d. Place a second large rubber band on the globe so that it extends over both the North and South Poles. Position the band so that it also is a great circle. On the globe, a line like this is also called a line of longitude.

e. In Euclidean geometry you learned that when two lines intersect, they have only one point in common. Look at the rubber bands on your globe. How many points do they have in common?

2. Use the globe, removable tape (cut into long ¼ inch strips), and the steps below to investigate angle measures in spherical and Euclidean planes.

a. Select 2 points on the equator. Select another point close to the North Pole. Use 3 strips of removable tape to form a triangle as shown (leave no overhanging strips of tape). Use a protractor to estimate the measure of each angle of the triangle. Record your results.

b. Carefully remove each strip of tape from the sphere. Use the same 3 strips to form a triangle on a sheet of paper (connect endpoints of each strip to other endpoints). Use a protractor to estimate the measure of each angle of the triangle. Record your results.

c. You have formed two triangles with sides of the same length. The first was on a spherical plane. The second was on the Euclidean plane. How do the angle measures of the two triangles compare? Practice Problems

3. Prepare a 5 –10 minute presentation about your project. Teach the class some of the basics of what you have learned and design a quick problem for a volunteer to practice (Don’t pick a goof ball). DO NOT READ BORING PARAGRAPHS TO THE CLASS! Discuss it like you know what you are talking about! Use several visual aids to help the class understand the topic.

Resources

Explore Lunes and Spherical Triangles

Evaluation

Web Quest Evaluation Rubric

Presentation
	70	80	90	100
Were the presenters enthusiastic about this presentation?	Presenters may as well been asleep they were so boring.	Presenters were faking their enthusiasm or could have been more lively.	Presenters were reasonably excited about their topic.	Presenters were excited enough to make the audience want to know more.
Did the presenters make eye contact or just read to the class?	Read to the class long boring paragraphs	Some reading with a small bit of eye contact	Maintained eye contact throughout at least half the presentation	Great eye contact for most of the presentation. Made the listener feel that they were part of the presentation.
Did the presentation flow well or were there long pauses of silence?	Several long pauses of silence, very unorganized	Some pauses of silence, unorganized in parts	Very few pauses of silence, 1 part unorganized	No pauses; presentation flowed well
Did all team members have a role in delivering the presentation?	One person gave the entire presentation.	One person gave most of the presentation.	Both persons had equal speaking parts but they did not seem to function as a team.	Both persons had equal speaking parts and functioned well as a team.

Project Design 70 80 90 100

Did the project address all necessary information?	Left off several items	Left off a 2-3 of items	Left off one item	All required items included
Did the project have an interesting look?	Plain, no color, 1 picture	A little color, 2-3 small pictures	Colorful, a few pictures, but not very pleasing	Very interesting, pleasing look, good pictures
Did the visual aids offer variety and maintain interest?	Aids were uninteresting and did not pertain to subject.	Aids were too small to see but were appropriate.	Aids were interesting but were all the same type.	Interesting visual aids that were large enough to see.
How much effort seemed to be put into this project?	Looked and sounded like it was all done last night.	A little effort was put into the poster but none into the presentation (or vise-versa)	A respectable amount of work was obvious.	An outstanding job that Mrs. Papizan will show for future reference.

Presentation Avg: Project Avg: Overall Score:

Conclusion

Through this activity you have seen how mathematicians used previous knowledge of spheres to build up to today’s applications. You have had the opportunity to research some of the interesting mathematical concepts that relate to the mystical world of spheres. You have applied the mathematics that you know to see why spherical geometry has its own separate rules. Hopefully, you will be even more fascinated with the spheres and apply the knowledge you gained through this activity to real world problems.