Three-dimensional manipulatives
for undergraduate geometry classes
Donna A. Dietz
Mansfield University
ddietz@mansfield.edu
http://faculty.mansfield.edu/ddietz
Presented on March 27, 2004 at East Stroudsburg University, at the
Annual SSHE-MA Conference
Abstract
This article will cover the use of some basic
three-dimensional manipulatives, such as balloons and
paper polyhedra, which can enhance undergraduate
exploration of non-euclidean geometry. Students' intuition of non-euclidean spaces is strengthened, and their enjoyment of
the subject is augmented by teamwork in a laboratory-style setting. Copies of sample laboratories and manipulatives are available through the author's website or
by email from the author.
1. Introduction
Students in my Spring 2004 undergraduate geometry course, Modern Geometry I, were having trouble absorbing geometric concepts in a traditionally-styled lecture. Most of these students were math or computer science majors who had taken calculus, and many of them were also secondary education majors. The first examination results demonstrated a lack of learning and a discussion with the students revealed a deepening dislike for the subject area, so I decided to take a more hands-on approach for teaching this course and started giving them group work in the form of laboratories relying heavily on three-dimensional manipulatives. The second examination results and the student's feedback both indicated that this approach was much more effective at cultivating both enjoyment of and deeper understanding of the subject matter. The sections which follow summarize the various laboratories which were completed by the students.
I
highly recommend experimenting with the laboratories in addition to or in lieu
of reading this document for those interested in learning more about non-euclidean geometry.
This document is intended for those interested in collecting ideas for
teaching the topic but fails to reflect the curiosity and exploratory tone of
the laboratory environment.
2. Triangles on an
elliptic surface
The objectives were:
1) Demonstrate to the students that the sum of angles on a
sphere is more than 180 degrees.
2) Get students thinking about what a straight line is.
3) Convince students that areas of the triangles on the
sphere have something to do with the sum of the
degrees.
Students were each
individually given a length and two angular measurements and were told to draw
the appropriate triangle on a balloon using a ruler and a protractor. Rulers from Lenart
spheres were used as a guide to help make all the balloons about the same size,
and students were asked to draw the triangle on the most sphere-like portion of
the balloon. As an interesting note,
they all drew geodesics instinctively, without any discussion about what
geodesics were. They were then told to
estimate the area of their triangle using the familiar plane formula. Answers for all students were tabulated on
the board.
This
laboratory took one class hour, and some assistance was given. The results of the laboratory were
encouraging, even though the plane formula for the triangle's area is rather
inaccurate for many triangles on the sphere.
Getting accurate measurements on a balloon also proved to be tricky, but
the students were mature enough to understand the drawbacks of the tools being
used.
3. Areas of triangles
on an elliptic surface
The objectives were:
1) Students should recall or learn that the area of the
sphere is 4
.
2) Students learn that a triangle on a sphere arises from
three great circles.
3) Derive formula for area of triangle on sphere.
4) Generalize formula for area of triangle on sphere to area
of any polygon on sphere.
5) Remove dependence on number of sides in polygon from
formula for area of polygon on sphere.
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Students
were again given balloons to work with, pretending the balloons to be perfect
spheres. They were asked to first draw
two different great circles on their balloons and find the proportion of the
area of the whole sphere contained in each of the lunes
formed. The eight resulting angles
were labeled as a
or a', with a and a' being supplementary angles.
Next, they were walked through a proof of the formula for the surface
area of the sphere. (Most of the
students had difficulty with this part of the laboratory.) This picture shows a styrofoam model of the eight triangles the students drew on their balloons. |
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Next, a
third great circle was drawn so as to split each of the four lunes into two triangles.
Resulting angles were labeled as b,
b', c, and c', with b and b' (and c and c') being supplements and so that one
triangle would have the angles a, b,
and c. The eight resulting areas have for different
area measures, so the sections were labeled A,
B, C, and D so that areas of same
area had the same label, and the triangle with the angles a, b, and c was labeled A. (Different labels will be suggested
next time, as students had quite a bit of trouble distinguishing the small
lowercase letters from one another as the balloon was rotated. A lesser amount of frustration resulted from
the use of the capital letters matching the lowercase letters.) Now, knowing how to calculate the surface
area of a lune and the surface area of a hemisphere
allowed for algebraic manipulations to solve for A in terms of a, b, and c. (A
and B , A and C, and A and D each form lunes.) Students found that A = 4 (a + b + c -) where R is the radius of the sphere.
This result for the area of triangles on a sphere was extended to find the area of any convex polygon on a sphere, using induction. Finally, dependence upon the number of sides in the polygon was removed by measuring the supplements of the internal angles (called turning angles or angles of deflection) rather than the internal angles. This final formulation could be slightly altered to achieve a continuous representation of this result.
This
laboratory took about two to three class hours and required much more
assistance than the first laboratory. The
results of this laboratory were good, but it would have been more effective to
have it split into two laboratories rather than one two or three class hour laboratory. Students sometimes became discouraged with
their lack of progress on a laboratory that would span over two or three class
hours. After the first time this occurred, however, they did not mind as much.
4. Cone points
The objectives were:
1) Students create models of hyperbolic and elliptic cone
points.
2) Students learn that adding (removing) angles at the cone
points
causes the triangle's internal angle sum to decrease
(increase)
by the amount of the added (removed) angle.
Students were shown samples of elliptic and hyperbolic cone points made from paper with equilateral triangular gridlines, and they were given materials to make their own. There was a classroom discussion about what was or was not developable, and a working definition was given that "a developable surface can be made out of paper". Students drew triangles (three intersecting geodesics) on the models and found the sum of the angles in them. They realized that any angles which they added to their models to create hyperbolic cone points would decrease the sum of the triangle's interior angles, but that any angles they removed from their models to create elliptic cone points would increase the sum of the triangle's interior angles.
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This
laboratory was completed in one class hour with some assistance. Results of this laboratory were as
expected, and the students enjoyed the exercise. |
5. Total Gaussian
curvature on various polyhedra
The objectives were:
1) Students learn that the sum of the angle deficits for a
convex polyhedron is 4.
2) Students learn that even for an object with hyperbolic
cone points, the angle deficits formed by the
introducing the hyperbolic cone points are balanced out by the elliptic cone points which also must
arise.
Various
regular convex paper polyhedra such as a tetrahedron,
cube, trunctetrahedron, snubcube
(shown below), and one paper bathtub-like shape were given to the students
pre-assembled as well as in flat pattern format (as shown below). Students were asked to calculate the angle
deficit for each cone point in each object.
They added up the angle deficit for each object and found that they all
totaled 4 radians.
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This
laboratory was completed in one class hour with some assistance. Results of this laboratory were as expected,
and the students enjoyed the exercise.
Some students asked for a proof of the fact that this type of object
would always have an angle deficit of 4 radians, indicating that they were ready to be presented
with the proof!
6. Proof of total
curvature for polyhedra
The objectives were:
1) Learn about Euler characteristics.
2) Prove that the total Gauss curvature of a topologically
spherical object composed of polygons is 4.
A model
of a 'square donut' was provided, that is, a cube with a square 'tunnel'
through it. The two outer faces with
missing squares were split in half along the diagonal so that the object had 12
faces, each topologically equivalent to a disc.
Students found that the total Gaussian curvature for this object was
zero.
Next,
the Euler characteristic, or 
(where v represents the number of vertices in
the object, f represents the number
of faces in the object, and e
represents the number of edges in the object) was considered for triangular
tessellations of surfaces of topologically spherical objects. The proof started
with a planar case, where the Euler characteristic is one and moved to the
spherical case (where it is two) by adding one additional face. (No proof was investigated for objects
topologically equivalent to a torus.) The Euler characteristic, the total radians
in a planar triangle, and some additional clues were given to the students, who
were then able to arrive at the result that for any topologically spherical
object composed of polygons, the total Gauss curvature is 4, which is 'coincidentally' also the surface area of a unit
sphere.
This
laboratory was completed in two to three class hours with considerable
assistance. The results of this laboratory
varied greatly by student. Many students
understood the entire laboratory completely, but many students were (unfortunately)
satisfied to believe the results and borrow work from their classmates.
7. Gauss-Bonnet Theorem
The objectives were:
1) Learn that the sum of angle deficits within a curve on a polyhedron and angle defects on the curve is
2 (Gauss-Bonnet theorem).
2) Cause students to suspect that this result holds for
non-polyhedral surfaces as well.
First, students were asked to draw a straight line on a piece of paper and observe the line as the paper was distorted into a cone or cylinder or other arrangement. Next, photographs of an icosidodecahedron with closed curves drawn on it were provided in the laboratory (for which the model was available at the front of the room). In the first photograph, the closed curve encircled only one elliptic cone point, for which the students calculated the angle deficit. For regular polyhedra, it is convenient to create closed curves by connecting the centers of adjacent faces. The closed curves so created will often cross an edge between faces perpendicularly to the edge, making it easy to conclude that the path is indeed a geodesic. Students were able to figure out the total turning angle for the closed curve on the icosidodecahedron. The second photograph showed a triangle drawn on an icosidodecahedron. The internal angles for that triangle were labeled for the students. From this, they were able to find the total turning angle for the curve, and deduce the angle deficit at each of the three identical cone points enclosed by the curve. This result matched their previous results for the same polyhedron. A third photograph gave a closed pentagon-like curve on a truncated icosahedron. Analogous values were calculated for this shape as for the icosidodecahedron.
Students
were given photos of three objects (a sphere, a cylinder, and a star-like
object) each encircled with curves (ribbons) lying flat along their
surfaces. They correctly concluded that
the total Gauss curvature enclosed by either side of the curve must be 2 since each curve experienced no angular deflection. Details on the geometry of the star-like
object were given to students so they could verify the result using techniques
from previous laboratories. Finally,
students were given diagrams representing surfaces and were asked to use
algebra with the Gauss-Bonnet theorem to find missing information such as the
Gauss curvature at a single point.
This
laboratory was completed in two to three class hours with considerable
assistance. The results of this laboratory
were mostly positive, but some students could have used more time on the laboratory.
8. Examination
The
examination for this sequence of laboratory exercises relied heavily on the use
of digital photographs, some of which were taken directly from the laboratory
exercises. Students were given a review
sheet which highlighted some of the concepts they would be responsible for. The examination emphasized later material
more than earlier material and did not emphasize supplemental material, such as
the proof for the surface area of the sphere.
Objects which appeared in the examination as photographs were available
at the front of the room for students to borrow during the examination.
9. Conclusions
After
reviewing the students' examination results, reflecting on their verbal
responses, and considering their overall attitudes about geometry, I have to
conclude that the course was much more successful after introducing hands-on
laboratories. Another important change
which contributed to the improved results was that I decided to cover less
material but cover material more deeply, so the students would have the chance
to think over more important concepts for longer periods of time. It seems that even with a decreased emphasis
on covering material, more material is being absorbed. The most important change to report is that
my students literally went from 'hating' geometry to saying that they looked
forward to the class and were enjoying the material.
Acknowledgements
I wish to thank my
colleague, Howard Iseri, and my husband, Michael
Robinson, both of whom have been indispensable to me in this endeavor, both by
their encouragement and by their helping me formulate geometric ideas clearly
and accurately.
References
Paul Bourke and Matt Storz, website with patterns for paper polyhedra, Swinburne Centre for Astrophysics and Supercomputing, Melbourne, Australia. http://astronomy.swin.edu.au/~pbourke/polyhedra/polycuts/ David Henderson, website with patterns for equilateral triangular grid paper, Cornell University, Ithaca, NY. http://www.math.cornell.edu/~dwh/books/eg00/supplements/39c0d5b7.jpg