William R. Colson - Morgan Park High School  

Spherical Geometry: A Global Perspective

William R. Colson              Morgan Park High School  
                               1744 W. Pryor Ave.
                               CHICAGO IL 60643
                               (773) 535-2550  

Objective(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 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 
marker. 

List #1
Corresponding terms (examples):

Euclidean                     Spherical
point                         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 #2
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 
      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 presentation or individual 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.