ENRICHMENT ACTIVITIES FOR GEOMETRYAuthor(s): ZALMAN USISKINSource: The Mathematics Teacher, Vol. 76, No. 4, Gifted Students (April 1983), pp. 264-266Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963464 .

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ENRICHMENT ACTMTIES FOR GEOMETRY

By ZALMAN USISKIN University of Chicago

Chicago, IL 60637

Little reason exists for engaging in en richment for enrichment's sake or in prob lem solving merely for mental practice unless one wishes to waste precious time. The curriculum is too crowded to allow such luxuries even for gifted students, be cause we want these students to be aware of a much greater range of mathematical

concepts than their less knowledgeable peers. Consequently, the activities one selects must do at least double-duty; they

must instruct about mathematics as they in struct in mathematics. Enrichment activi ties ought to involve the most important content, and appeal should be given prefer ence over difficulty.

Regarding enrichment in geometry in

particular, the main course itself should in clude significant work with coordinates and transformations, some mention of the

principles and ideas behind non-Euclidean

geometry, a little work with vectors, and at least an introduction to the ways com

puters can deal with geometric figures. The activities given here should be considered for use only after appropriate attention has been given to the basic content. The activi ties are grouped by the facet of geometry they are meant to convey.

Facet 1: Geometry's continuing growth

A tessellation is a covering of the plane by nonoverlapping regions each congruent to a region called the fundamental unit. Any triangle and any convex quadrilateral can be a fundamental unit for a tessellation, but no one has yet identified all convex penta gons that tessellate.

1. Trace shapes (a), (b), and (c) in figure 1 and show that each can be a fundamental unit for a tessellation.

2. Draw a convex pentagon that will not tessellate.

3. In 1975, Marjorie Rice, a California

housewife, discovered a hitherto unknown

type of tessellating pentagon. A pentagon ABCDE is of this type if 2mLA + m LB = 2m LD + mLC = 360? and AB =

BC = CD = EA. Prove that such a penta gon will tessellate.

4. Find as many types of tessellating pentagons as you can. Compare your re

sults with those in the referenced article by Doris Schattschneider (1978). If you find a new type, write to her (Moravian College, Bethlehem, PA 18018). You will have con

tributed to the growth of mathematics.

Facet 2: Geometrical applications to

nongeometric endeavors

A round robin tournament is one in

264 Mathematics Teacher

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which each team plays every other team

exactly once. For the activities below it is assumed that teams play once a week; so it takes at least ? 1 weeks for teams to

complete the tournament.

5. Devise a schedule for a round robin tournament involving seven teams.

6. Here is a way to do activity 5. Con sider each team as a vertex of a regular heptagon. (Haven't you always wanted a real-world application of a regular hepta gon?) When two teams play each other, connect the corresponding vertices. For the first week, schedule teams so that the seg ments connecting them have different

lengths. Then each week thereafter, rotate the segments one-seventh of the way around the heptagon and use the resulting segments to make up the schedule. Extend this idea to schedule a round robin tourna

ment for nine teams.

7. Adapt the result of activity 6 to sched ule a round robin tournament for ten teams. (Hint: When you do so, you will be able to prove the surprising result that it takes no longer to schedule In teams than it does to schedule 2n ? 1 teams.)

Facet 3: Some surprisingly beautiful geometric theorems

One can choose from literally thousands of theorems (see the references of Altshiller Court [1952] and Johnson [1968]). The following theorems are notable because

symmetry appears (usually in the form of an equilateral triangle) even though the

given triangle may be scalene. One should first make an accurate drawing and verify the theorem by measurement. A person who can prove even one of these theorems is doing quite well.

8. Prove that, in any triangle, the three intersection points of adjacent angle insec tors of the three angles are vertices of an

equilateral triangle. (This is Morley's the orem, first proved by Frank Morley around

1900.) 9. Let ABC be a triangle with no angle

greater than 120?. Construct equilateral

triangles ABN, BCL, and ACM on the sides of ABC, exterior to ABC. Show that

AL, BM, and CN are concurrent and that

they form six 60? angles at their point of intersection (known as the Ferm?t point of the triangle, named after Pierre Ferm?t, the French mathematician [1601-1665]).

10. Show that the centers of the equi lateral triangles ABN, BCL, and ACM of

activity 9 are themselves vertices of an

equilateral triangle (known as the outer Na

poleon triangle of ABC, named after Napol eon Bonaparte?an amateur geometrist,

but probably not good enough to have dis covered this).

11. Construct equilateral triangles ABN', BCL', and ACM' as in activity 9 but over

lapping the interior of ABC. Show that the centers of these three triangles also are ver tices of an equilateral triangle (the inner Na

poleon triangle of ABC). 12. And now?the pi?ce de r?sistance

?show that the difference between the areas of the outer and inner Napoleon triangles of activities 10 and 11 is equal to the area of the original triangle! This theo rem is known as Napoleons theorem.

Facet 4 : Intimate connections between

geometry and complex numbers

The great mathematician Carl Friedrich Gauss was among the first to recognize the

potential of geometrizing the complex numbers. Here the complex number

= a + bi is considered equal to the point = (a, b).

13. Prove that the average of the com

plex numbers and z2 is the midpoint of the points zx and z2.

14. Prove that the four distinct complex numbers zl9 z2, z3, and z4 are vertices of a

square in counterclockwise order if and

only if z1 + z2i = z3 + z4i. What modifi cation needs to be made to get the vertices of a square in clockwise order?

15. Use the results of activities 13 and 14 to prove the following amazing theorem :

Let ABCD and A'BCD' be any two squares with vertices in the same order. Then the

April 1983 265

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midpoints of A?, BE', CC, and DD' are

themselves vertices of a square.

16. Generalize the result of activity 15.

BIBLIOGRAPHY

Altshiller-Court, Nathan. College Geometry. New

York: Barnes & Noble, 1952.

Coxeter, H. S. M., and Samuel Greitzer. Geometry Revisited. New Mathematical Library, vol. 19.

Washington, D.C.: Mathematical Association of

America, 1967.

Gardner, Martin. "On Tessellating the. Plane with Convex Polygon Tiles." Scientific American 233

(July 1975). Honsberger, Ross. Mathematical Gems. Dolciani

Mathematical Expositions, nos. 1 and 2. Wash

ington, D.C.: Mathematical Association of

America, 1973,1976.

Johnson, R. A. Advanced Euclidean Geometry. New

York: Dover Publications, 1968.

Klee, Victor. "Some Unsolved Problems in Plane Ge

ometry." Mathematics Magazine 52 (May 1979): 131-45.

Posamentier, Alfred S., and Charles T. Salkind.

Challenging Problems in Elementary Geometry, 2

vols. New York: Macmillan, 1970.

Schattschneider, Doris. "Tiling the Plane with

Congruent Pentagons." Mathematics Magazine 51

(January 1978) : 29-44.

Yaglom, I. M. Complex Numbers in Geometry. Trans

lated by Eric J. F. Primrose. New York: Academic

Press, 1968.

-. Geometric Transformations, vols. 1-3. New Mathematical Library, vols. 8, 21, and 24. Wash

ington, D.C.: Mathematical Association of

America, 1962, 1969, 1973.

( Continued from page 218)

115110| 101511,

which, when converted to standard notation, becomes

1 50

1000 10000 50000 100000 161051 = II5.

Similarly, the seventh row will read

11 ? 1151201151 ? 11, and this is equivalent to 1 771 561, or II6.

Edwin A. Rosenberg Western Connecticut

State College Danbury, CT 06810

Calendar comments I would like to begin by saying that I have enjoyed

your publications for the past several years, and I have found the mathematics calendar to be a very useful means to direct students to pursue problem solving. However, I do feel that it is necessary to offer two corrections to the October problems.

The problem for 14 October asked, What number can you add to both 164 and 100 to produce two

perfect squares? One answer is 125, but ~64is also a

correct solution. The problem for 16 October stated, Which is larger,

the sum of the cubes of the elements in the set {1, 2, 2, 3,4, 6} or the cube of the sum? Readers were told to

read "Number Patterns?Sets with the Square-Cube Property" by Robert G. Stein in that same issue to find the solution. I must take strong exception to the abuse of set notation. In all the resources on set

theory at my disposal (Suppes's Axiomatic Set Theory or Lipschutz's Set Theory and Related Topics, for ex

ample), {1, 2, 2, 3,4, 6} =

{1, 2, 3, 4, 6}. Unfortu

nately, in Stein's notation these must represent two

distinctly different sets. Although Stein's ideas are in

teresting and lend some insight into number patterns, this nonstandard use of set notation can do little

more than confuse students. I would suggest that some other notation be chosen that does not conflict with accepted practice.

William K. Tomhave

University of Minnesota

Morris, MN 56267

Pythagorean serendipity V I recently spent some time with my second-year

algebra classes talking about the Pythagorean theo rem. After explaining the a2 + b2 = c2 relationship and pointing out the (3, 4, 5) and (5, 12, 13) triples, I asked the students to try to come up with three

more prime triples for the next day. Two hours lat er one of the sophomores, Michael Wyatt, came

running in quite excited about the discovery of the "Wyatt theorem." He found that if one picks a prime number for a, then a, (a2

- l)/2, and

(a2 + l)/2 will form a prime triple. He also showed me that it will always work, since a2 +

((a2 -

l)/2)2 = ((a2 + l)/2)2. Inspired by this new piece of information, I attempted to come up with a

relationship to find triples that do not start with a

prime number and "fell" upon the "Bode theo rem":

//a is an even number, than a, (a214) -

/, and

(a214) + 1 will always yield a Pythagorean triple.

Two very important results have sprung forth from this lesson. (1) Now the students are busy try ing to come up with a rule that will derive triples like (20, 21, 29). (2) When a new concept is intro duced and a student asks how anyone could possi bly have come up with "something like that," I show them the "Wyatt theorem" and refer them to

Michael.

James L. Bode Bradenton Christian School

Bradenton, FL 33529

266 Mathematics Teacher

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