William R. Colson - Morgan Park High School
Spherical Geometry: A Global Perspective
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William R. Colson Morgan Park High School
1744 W. Pryor Ave.
CHICAGO IL 60643
Suggested Grade Level: 3-12
1) Relate prior knowledge about the globe to definitions and properties in
2) Given a common definition or property in Euclidean geometry, make a
conjecture about the corresponding statement in spherical geometry.
Clear, inflatable globe (optional: 1 small globe per group)
Index cards (1 per group)
Chalkboard/whiteboard with compass and meter stick
Apples or white styrofoam balls (1 per group)
Paring knives or black markers (1 per group)
Lenart sphere (kit available from Key Curriculum Press)
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 180o and less than or equal
to 360o.) 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
Corresponding terms (examples):
point same ("polar" points are endpoints of a
diagonal of the sphere)
line great circle
line segment arc of a great circle
angle angle (intersection of two arcs)
Corresponding 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
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
S: There exist no parallel lines.
1) Individual responses when matching corresponding terms.
2) Group discussion and presentation of corresponding statements.
3) Group discussion and presentation or individual write-up of conjecture
reached in follow-up activity.
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.