Return to Mathematics IndexWilliam R. Colson - Morgan Park High SchoolSpherical Geometry: A Global Perspective

William R. Colson Morgan Park High School

1744 W. Pryor Ave.

CHICAGO IL 60643

(773) 535-2550Objective(s):

Suggested Grade Level: 3-12

1) Relate prior knowledge about the globe to definitions and properties in

spherical geometry.

2) Given a common definition or property in Euclidean geometry, make a

conjecture about the corresponding statement in spherical geometry.Materials Needed:

Clear, inflatable globe (optional: 1 small globe per group)

Index cards (1 per group)

Chalkboard/whiteboard with compass and meter stick

Optional:

Apples or white styrofoam balls (1 per group)

Paring knives or black markers (1 per group)

Lenart sphere (kit available from Key Curriculum Press)Strategy:

Begin with a review of terms and definitions from Euclidean

(conventional) geometry. This should be done through questioning, not

lecture, in order to assess prior knowledge. Students should at least have a

basic understanding of points, lines, and planes for this lesson to be

appropriate. Particular content, including properties to be investigated,

will be chosen according to the knowledge and grade level of the students.

Split the class into groups of 3-5 students. Produce a clear inflatable

globe containing latitude and longitude markings. Have a general discussion

about latitude and longitude. If available, give each group a small globe of

some type to use for individual reference. Compare to a flat map. What is

different about the latitude/longitude markings?

Eventually, someone should note that on the globe, latitude/longitude

markings are not lines, but circles; then, that latitude circles are of

different sizes, while longitude circles are all the same. Using the list of

terms developed in the opening discussion, identify corresponding parts on the

surface of a sphere and give their accepted names in spherical geometry (see

List #1 below).

When the class seems comfortable with the new terms, give each group an

index card containing a statement of a postulate or property in Euclidean

geometry and instructions to translate it into a corresponding statement in

spherical geometry (see List #2 below).

Depending on class level and time available, follow-up activities could

include such things as:

1) What would a spherical ruler/compass/protractor look like?

2) If parallelism does not exist in spherical geometry, can we still construct

figures that correspond to parallelograms? What would be their properties?

3) What about spherical "triangles"? What would correspond to acute, right,

or obtuse? What could we say about angle sums? Is there anything

corresponding to the Pythagorean theorem?

In my class, I gave each group an apple, a paring knife, and the

following instructions: "Cut your apple to represent a spherical 'triangle'.

Do this by scoring an 'equator' and one or two great circles through the

poles. Question: What is the possible range of the sum of the measures of the

angles of the triangle? (Answer: Greater than 180^{o}and less than or equal

to 360^{o}.) If they gave and explained a satisfactory conjecture, I gave them

a small cup of caramel dip and permission to slice and eat their apple. If

knives and food are inappropriate for your classroom, this activity (as well

as many others) may also be done using a white styrofoam ball and black

marker.List #1Corresponding terms (examples):EuclideanSphericalpoint same ("polar" points are endpoints of a diagonal of the sphere) line great circle plane sphere ray none line segment arc of a great circle angle angle (intersection of two arcs)List #2Corresponding statements (examples): 1) E: There is a unique straight line passing through any two points. S: There is a unique great circle passing through any pair of nonpolar points. 2) E: If three points are collinear, exactly one lies between the other two. S: If three points are collinear, any one of the three points is between the other two. 3) E: The intersection of two lines creates four angles. S: The intersection of two great circles creates eight angles. 4) E: If two lines are parallel to a given line, they are parallel to each other. S: There exist no parallel lines.Performance Assessment:

1) Individual responses when matching corresponding terms.

2) Group discussion and presentation of corresponding statements.

3) Group discussion and presentationorindividual write-up of conjecture

reached in follow-up activity.Conclusions:

Depends on particular content chosen. In general, they should conclude that

most, but not all, terms and properties in Euclidean geometry have

counterparts in spherical geometry. More advanced students may be asked to

discover properties unique to spherical geometry.