1776Author(s): HELEN CUNNINGHAMSource: The Mathematics Teacher, Vol. 69, No. 4 (APRIL 1976), pp. 310-311Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960464 .

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Celebrate 1776 in your mathematics class!

By HELEN CUNNINGHAM

University of Louisville

Louisville, Kentucky

THE year 1776 will be an important date in English and history classes in 1976. Will it also be a part of mathematics classes?

There are many mathematical activities for

kindergarten through high school that can be based on this historical date. For junior and senior high school students, 1776 can be used for lessons in general mathematics,

algebra, number theory, and geometry. Cre ative teachers can devise many interesting mathematical problems as a part of the Bi centennial celebration.

The examples suggested here can be

adapted and extended to fit the differing abilities of students. With encouragement, pupils will enjoy creating different types of mathematical activities using the events of America's historical beginning.

The digits 1, 7, 7, and 6 can be used in a

variety of problems. How many different one-, two-, three-, or four-digit numerals can be written using only 1, 7, 7, and 6? Of

these, how many are prime numbers? How

many are divisible by 2, 3, 4, 6, 8, or 16? Is there an easy way?not by dividing?that can be used to show that any number formed by 1,7, 7, and 6 is not divisible by 9?

Probabilities can be computed using the four digits 1, 7, 7, and 6. What is the proba bility of obtaining the sum 7 or 6 in the roll of a pair of dice? What is the probability of tossing a 7 with a cube that has faces la beled U., S., 1, 7, 7, and 6? What would be the probability of landing on a 1? What

happens to the probability if two such cubes are used?

In algebra, the digits 1, 7, 7, and 6 can be the basis for many interesting lessons. Used as thie coefficients in equations, they will

provide examples of different types of roots. In equations such as x2 -f Ix + 6 = 0 and 2 - 6x - 1 = 0, roots will be rational. But what happens if Ix2 - Ix + 6 = 0 or

2 + 6x + 7 = 0 are formed? Using the same

numbers for coefficients, what other equa tions can we write that have rational roots; how many quadratic equations are possible with imaginary roots? Can a cubic equation be formed with rational roots if the coefficients are from the famous four digits? Such exercises can be used for introducing different types of solutions or for review.

Multiplying factors and factoring can in

corporate 1, 7, 7, and 6 in conceptual devel

opmental exercises. Using only the 7,

(x -

7)(jc -

7) and (x -

7)(x + 7), we get contrasting products. These examples, com bined with such products as (x

- 1)(jc 4- 6)

or (x ?

6)( : - 7), illustrate important as

pects of multiplication that must be recog nized in factoring. A unit on factoring can be designed quite easily using just 1, 7, 7, and 6 with variables. The determination of

quadratic or cubic equations when roots are given can be introduced using the patri otic digits.

There are many permutation or com bination problems that can be based on 1776 for algebra classes. If the governor of the state is issued a license tag in 1976 with the letters U.S. and 1, 7, 7, 6 for the digits, how many other officials could receive

plates with U.S. and the same digits with out repeats except for 7? If digits can be

repeated, how many license plates are pos sible? The same idea can be extended to

U.S.A., with three digits from the 1, 7, 7, and 6 group.

310 Mathematics Teacher

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The Bicentennial celebration can be eas

ily incorporated in many different types of lessons. The numbers 1, 7, and 6 as entities are fascinating. The number 1 has many unique characteristics that can generate stu dent projects. Many interesting observa tions about 6 appeared in the Arithmetic Teacher (Hoffer 1973; Hohlfeld and Schwandt 1975). The idea of luck is often associated with 7. However, it affords other

intriguing studies. In what ways do the mul

tiplication tables for modulo 7 and modulo 6 differ? The modulo 7 arithmetic can be

put in a one-to-one correspondence with the days of the week, using Sunday as 0,

Monday as 1, and so on. What day of the week is 30 days after Friday? Today is Tues

day, and there are 32 days until Christmas; what day of the week will Christmas fall on this year?

Some students might enjoy doing re

search on an earlier use of 7 in the Rhind

papyrus of Egypt. The problem 79 in that ancient work is similar to the one in a

Mother Goose rhyme that is also a puzzle:

As I was going to Saint Ives, I met a man with seven wives;

Every wife had seven sacks;

Every sack had seven cats;

Every cat had seven kits.

Kits, cats, sacks, and wives? How many were going to Saint Ives?

The hexagon is a fascinating figure. It is the basis for the Star of David, the familiar

daisy design, and other interesting construc

tions. The heptagon can be inspected in a

new fashion with an emphasis on the num

ber 7. Students' use of 6 and 7 in geometric designs creates new interest in those figures.

Thought-provoking area and volume

problems arise from 1776. What regular

polygons can have perimeters of 1776 units?

Determine figures with an area of 1776

square units or solids with volumes of 1776

cubic units. Is it possible for cylinders or

spheres to have volumes of 1776 cubic

units? Creative mathematical measurement

lessons evolve from this one important date to add interest to a class.

Surveys and statistical studies can be de

signed using 1776. The interest of the stu dents will determine such activities as sur

veys involving the number of students

having those four digits in their telephone numbers, house addresses, zip codes, and so on. This type of endeavor can be ex

panded to include school or neighborhood surveys. Ways of studying the data and ex

pressing the results can be mathematical but will have a historical celebration flavor.

Students can use 1776 in varied creative activities. A cross-number puzzle based on

finding products or sums of combinations of 1, 7, 7, and 6 can easily be made. Stu dents will enjoy designing puzzles for gen eral mathematics classes. They will find that 1776 can be exciting in the mathematics classroom if they use it to create practice exercises, factor trees, puzzles, or other en richment activities.

Throughout the year 1976, the Bicenten nial will be celebrated in many ways. Let students discover that 1776 is a date of mathematical interest. It is a number that can transform many ordinary lessons into

exciting activities.

REFERENCES

Hofier, Alan R. "What You Always Wanted to Know about Six but Have Been Afraid to Ask." Arith metic Teacher 20 (March 1973): 173-80.

Hohlfeld, Joe, and Lynn Schwandt. "Six Is a Fascinat

ing Number." Arithmetic Teacher 22 (April 1975): 269-70.

Answers Answers "Calculator Charades" (p. 281):

1. 91851345 2. 5537061

3. 804158055 4. 5517840 5. 93877.1

6. 918517718 7. 5304557735 8. 573344614 9. .09771

10. 7714.00791

11. .378193771

SHEISB1G ILLGOLESS BOY IS BOSS OHBL1SS I'LLBEG BILLISB1G SELLSSHOES HIGHHEELS ILLGO' IGLOO'HILL ILLEGIBLE'

April 1976 311

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