STUDENT PROJECTS IN GEOMETRY

STUDENT PROJECTS IN GEOMETRYAuthor(s): ANDREW A. ZUCKERSource: The Mathematics Teacher, Vol. 70, No. 7 (OCTOBER 1977), pp. 567-570Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960968 .

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STUDENT PROJECTS IN GEOMETRY

Looking for ideas and references to start or expand your list of student projects?

By ANDREW A. ZUCKER

Milton Academy Milton, MA 02186

In addition to the variety of lectures, dis

cussions, reading assignments, and prob lems that constitute most of a high school

mathematics course, many of us try to in clude at least a few "special" activities each

year. These may be films or games, for ex

ample. We include them for their usefulness in motivating students and perhaps because

they serve some purpose that cannot easily be met in any other way.

My geometry classes have experimented with several activities that are out of the

ordinary in a mathematics class: laboratory activities, reading and discussing material other than the textbook, and student proj ects. I doubt that these ideas are unique. Nevertheless, it may be of interest to others to read about some of the ideas and re sources that my students have used, espe

cially since more than half the difficulty in

organizing projects is collecting practical ideas and good references.

Geometry lends itself beautifully to stu

dent projects that involve design, construc

tion, photographs, puzzles, posters, and

other hands-on activities. For that reason,

my students attempt projects that are not

primarily "bookish." Although open to

suggestions from students, I generally do not expect to see book reports, or even

written reports on particular mathematical

topics.

Every student is given a list of possible project topics similar to the one shown in this article. During the course of several

days, I give the class brief, capsule descrip tions of each topic on the list and answer

any questions they may have. Then the stu dents are asked to choose several topics that interest them most. It is wise to ask for three choices if you want to have most of the students in a class do substantially dif ferent projects.

Projects are due three weeks after the choices are made. For several days during this interval, no homework is assigned. Otherwise, no special arrangements are

made, except that references for virtually all the projects on the list are arranged to be available to the students.

This latter point is important, since not

only is it time-consuming for students to hunt for references, it can also be very frus

trating. For example, I once suggested as a

project creating a poster that would show the geometric construction of several sym bols used in magic or alchemy and describe the supposed function of each symbol (e.g., five- and six-pointed stars). Although I did not have a reference for this project, I didn't think it would be difficult to find one.

Several days before the due date, the stu

dent who had chosen this project reported that no useful references could be found. It became necessary for that student to shift

suddenly to a new project, since I was too

busy to search for the references myself. Later I discovered two good references for

the project: Creative Constructions (Dale

Seymour and Reuben Schadler, Creative

Publications, Palo Alto, Calif.) and Man,

October 1977 567



Myth and Magic (an encyclopedia pub lished by the Marshall Cavendish Corpora tion, New York, 1970).

Spring is a good time to schedule proj ects. Both teachers and students are ready for something different at this time of year. Several of the topics in the following list are

especially suitable for spring or early sum

mer, such as kite-building or photograph ing geometric forms in nature. Grades on the projects are usually quite

high, averaging and B+. This seems ap propriate to me for projects that the stu dents have chosen themselves and to which

they often devote more attention and effort than they do to their usual assignments. This high degree of student interest and effort is probably the feature that makes student projects most worthwhile.

Topics for Student Projects

1. Make and fly either a five-pointed star

kite, a six-pointed star kite, or a tetrahedral kite (fig. 1). (1)

Fig. 1. Five-pointed star kite, by Charles Truslow

2. Make a large poster showing as many geometric symbols as possible that are used

as logos for corporations or that symbolize events (Bicentennial, Olympics).

3. Make a variety of "flexagons" and decorate them with geometric designs. In

vestigate their properties. (2) 4. Try to find all the convex tangrams

that can be made from the seven tangram pieces. (There are less than twenty.) If pos sible, prove that you have found them all.

(3) 5. Make a model Wankel engine and ex

plain its novel geometry. (4) 6. Construct models of the five "perfect

solids" and explain why there are only five.

(5) 7. Construct one of the more challenging

models in the book Polyhedra Models for the Classroom, by Wenninger. (6)

8. Write and run one or more computer programs that calculate pi. You might try using the Monte Carlo, or dart-throwing, method, which is based on probability. (7)

9. Put together a small photographic ex hibit of "geometry in nature," using flow ers, shells, crystals, and the like (fig. 2). (8)

Fig. 2. Sand dollar, by Arnold Cohen

10. Make a model, exhibit, or demon stration of an invention based on unusual

geometric configurations or properties. (Examples: Rolamite bearing; new internal combustion engine; holograms.) (9)

11. Construct a model geodesic dome

(fig. 3). (10)

568 Mathematics Teacher



Fig. 3. Geodesic dome, by Alvaro Montealegre

12. Produce one or more "periodic drawings" of the Escher variety, or learn how to analyze Escher's drawings to find

the unit cell, et cetera. (11) 13. Make a poster illustrating the prob

lem of tiling the plane with similar figures (tessellations?fig. 4). (12)

Fig. 4. Tessellation by Kip States

14. Work through the problems in the booklet on "Mosaics." (13)

15. Learn to draw stereograms, using the

Houghton Mifflin booklet on the subject.

(14) 16. Make an original drawing, painting,

or photograph that is "anamorphic" (dis torted, for example, so that it looks normal

only when viewed in a cylindrical mirror).

(15) 17. "Stitch" a line design with colored

thread or yarn. (Line designs look like they are constructed from curved segments, but

in fact consist of straight segments.) (16) 18. Build an accurate sundial. (17)

NOTES AND REFERENCES

1. Leslie L. Hunt, 25 Kites That Fly (New York: Dover Publications, 1971.

2. Martin Gardner, The Scientific American Book

of Mathematical Puzzles & Diversions (New York:

Simon & Schuster, 1959).

Karen Billings, Carol Campbell, and Alice

Schwandt, Art V Math (Action Math Associates, Eu

gene, Oregon, 1975).

3. Martin Gardner, "Mathematical Games," Sci

entific American, August 1974, p. 98 and September 1974, p. 187.

4. Two different plastic Wankel engine kits are

available from Edmund Scientific Co., Barrington, NJ

08007.

5. See, for example, Courant and Robbins's article

"Topology" in vol. 1 of The World of Mathematics

(New York: Simon & Schuster, 1956), pp. 581-99.

6. Marcus Wenninger, Polyhedra Models for the

Classroom (Reston, Va.: National Council of Teachers

of Mathematics, 1975). 7. This is a standard elementary program; if you're

not familiar with it, ask your computer system man

ager. 8. John Wahl and Stacey Wahl. / Can Count the

Petals of ? Flower (Reston, Va.: National Council of

Teachers of Mathematics, 1976). P?ter S. Stevens, Patterns in Nature (Boston:

Little, Brown & Co.). 9. Edmund Scientific would be a good source (see

Science ("Rotary V Engine Uses Fixed Piston," Feb

ruary 1976, p. 106; "ROVAC: Now It Can Heat and Cool your House," August 1976, p. 84). Also see "The Wankel Engine," Scientific American, August 1972,

p. 15.

10. An inexpensive kit is available from ikoso-kits, Route 3, Box 480, Eugene, Oregon 97405. Instructions for building a paper or cardboard dome are given in

Applications in Mathematics, Course (Scott, Fores man & Co.), pp. 57-60. Also see Domebook 2 (Shelter Publications, P.O. Box 279, Bolinas, California, 3H

ed., 1974).

11. Joseph L. Teeters, "How to Draw Tessellations

of the Escher Type," Mathematics Teacher, April

another would be the magazine Popular

October 1977 569



1974; and Caroline MacGillavry, H. Fantasy & Sym metry: The Periodic Drawings of M. C. Escher (New York: H. N. Abrams, 1976).

Sheila Haak, "Transformation Geometry and the Artwork of M. C. Escher," Mathematics Teacher 69 (December 1976):647-52.

12. See Steinhaus's Mathematical Snapshots, pp. 78-83. Also: Martin Gardner, "Mathematical

Games," Scientific American, July 1975, p. 112 and

August 1975, p. 112.

13. Donald W. Stover, Mosaics, in the Houghton Mifflin Mathematics Enrichment Series (Boston: Houghton Mifflin Co., 1966).

14. Donald W. Stover, Stereograms, in the Hough ton'Mifflin Mathematics Enrichment Series (Boston:

Houghton Mifflin Co., 1966). 15. "Mathematical Games," Scientific American,

January 1975, p. 110; and Fred Leeman, Joost Elffers, and Mike Schuyt, Hidden Images (Harry Abrams Inc., New York, 1976).

16. Dale Seymour, Linda Silvey, and Joyce Snider, Line Designs (Palo Alto: Creative Publications, 1974).

17. Charles T. Wolf, "The Design, Proof, and

Placement of an Inclined Gnomon Sundial Accurate for Your Locality," Mathematics Teacher 68 (May 1975):438-41.

Editor's Note: Readers interested in this article

might also wish to explore a new NCTM publication entitled Mathematics Projects Handbook, by Adrien L.

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STUDENT PROJECTS IN GEOMETRY