Pythagorean serendipity IIAuthor(s): Leonard GillmanSource: The Mathematics Teacher, Vol. 74, No. 8, Microcomputers (November 1981), p. 674Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962673 .

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o. 61 st Annual Meeting, 13-16 April 1982, De

troit, Michigan Stephen S. Willoughby, New York, New

York, Convention Chairman Hamilton Blum, Greenvale, New York, Pro

gram Chairman

(to be appointed), Chairman of Local Ar

rangements

19. Mathematics Education Trust Committee

(Terms conclude 31 May.) Develops, implements, and monitors the pro cedures for collecting and distributing funds that will be devoted to special projects outside the normal activities of the Council.

G?raldine Green, Royal Oak, Michigan, Chair

man, 1982; 1982

Shirley A. Hill, Kansas City, Missouri; 1983 John C. Egsgard, Orillia, Ontario; 1984

Shirley M. Frye, Scottsdale, Arizona; 1985 James D. Gates, Reston, Virginia (Staff liaison)

20. Headquarters Office Executive Staff James D. Gates, Executive Director Charles R. Hucka, Director of Publication Serv

ices James R. Tewell, Director of Financial Services

Joseph R. Caravella, Director of Professional Services

Jane M. Hill, Managing Editor, Arithmetic Teacher

Harry B. Tunis, Managing Editor, Mathematics

Teacher, and Director of Research Services

Robert Murphy, Computer Services Manager Marna J. Petersen, Director of Convention Serv

ices

Computer league

There is a new computer activity in many schools: it's the American Computer Science League (ACSL). It administers monthly contests for junior and senior

high school students and awards prizes to outstanding students and schools on local and regional levels. Last

year, in its third year of operation in New England, the top-scoring individual from among the 600 partic ipants (representing eighty schools) was awarded a

microcomputer at the year's-end All-Star Contest fes tivities!

ACSL provides a unique and exciting educational

opportunity for aspiring computer enthusiasts. Con test problems motivate students to study computer topics not covered in their schools' curricula and to

pursue classroom topics in depth. The league is en

dorsed by the New England Association of Secondary School Principals as an integral part of its diversified

program and has been the focal point for extracurric ular clubs as well as for entire computer courses.

Contests are held at each participating school, and an unlimited number of students from all grade levels

may compete at each school. A school's score is the sum of the scores of its five highest-scoring students. In each competition, students are given short theoreti cal and applied questions and then a practical prob lem to solve within the following two days using their schools' computer facilities. After the contest is ad

ministered by the faculty advisor, each school's results are returned to the league for tabulation. Complete study materials and annotated solutions are provided.

For more information, contact Marc H. Brown, Di

rector, American Computer Science League, P. O. Box 2417A, Providence, RI 02906, (401)863-3300.

(Continued from page 598)

teresting discussion of such symbolic mathematics

systems by Stoutmeyer appeared in the August 1979 issue of Byte. Such systems will probably begin to have a major impact on education in the next decade.

William Squire West Virginia University Morgantown, WV 26506

Pythagorean serendipity II In a letter on finding a leg of a right triangle ("Pyth

agorean serendipity," January 1981), Jo Ann Sim mons reported a "wrong method" that turned out to

give the correct answer (if one leg is and the hypote nuse is + 1, then the other leg is y/n + (n + 1)); then she went through some algebra to verify that the

method works. There is an opportunity here to dis cover more. We notice that to get from l2 to 22 we add

674 Mathematics Teacher

1 + 2; to get from 22 to 32 we add 2 + 3; and so on. After some unstructured "messing around," we come

up with a generalization: To go from l2 to 32 we add twice (1 +3); to go from 22 to 42 we add twice (2 + 4); and so on. To go from l2 to 42 we add three times (1 +

4) and so on. Why does this work? Time for some notation. Say a> b>0. If a ? b= 1,

then a2 - b2 = a + b; if a - b = 2, then a2 - b2 =

2(a + b); and so on. All these results are instances of the general formula

a2-b2 = (a- b)(a + b).

Since this identity holds for all numbers a and b without restriction and irrespective of which of the two is the greater?positive, negative, zero; integral, fractional, irrational?so does the description in terms

of "getting from b2 to a2."

Leonard Gillman

University of Texas

Austin, TX 78712

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