Aesthetics in elementary mathematics

Aesthetics in elementary mathematicsAuthor(s): EDWIN A. ROSENBERGSource: The Arithmetic Teacher, Vol. 15, No. 4 (APRIL 1968), pp. 333-336Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185771 .

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Aesthetics in elementary mathematics

EDWIN A. ROSENBERG Danbury, Connecticut Dr. Rosenberg is a professor of mathematics aï Western Connecticut State College. He has been active in mathematics education and was a member of the New York Madison Project Workshop last summer.

jfVppreciation of mathematics is repeat- edly and properly touted as one objective of instruction in mathematics. Formally, appreciation may be interpreted as sensi- tive awareness, especially of aesthetic qual- ities. Those who pursue mathematical stud- ies often obtain aesthetic satisfactions in perceiving relationships, structures, and the "elegant" nature of certain solutions.

What about others, however, who lack an extensive background in mathematics? For them, a comprehensible illustration of what constitutes aesthetics must be based upon a simple question with a reasonably simple answer. One way to elicit the respect and wonder which is the beginning of appreciation for the infinite variety of mathematics is to answer a single question by different techniques of increasing but not great sophistication. Particularly for elementary school teachers does this ap- proach appear suited.

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Figure 1

Such students, whose view of mathematics cannot yet be labeled wholly unjaun- diced, have evidenced some curiosity at the diverse methods by which the answer may be obtained to the following query: Of the 100 sums in the addition table for 0-9, how many have two digits? (See Fig. 1.) Below will be shown some of the ways, many suggested in class by the students, to arrive at the answer.

1. The query can be answered most simply, if tediously, by counting the two- digit sums in the table:

1, 2, 3, . . .,45. 2. But a shortcut is available to reduce

the drudgery and potential errors involved in counting, namely, addition. Two-digit sums (from here on called "elements") appear in the columns headed "1" through "9," with the number of elements in each column indicated by the column heading. So we add: 1 +2 + 3 +4 + 5+6 + 7+8 + 9 = 45.

3. Now when we write out the above addition, we may notice four pairs of numbers which sum to ten, with the number in the middle - the 5 - remaining alone. Extended mental application of associa- tivity, commutativity, and distributivity result in the introduction of multiplication, and we reach the answer by computing:

(4 x 10) + 5 = 45. 4. Next, the consecutive integers 1

through 9 provide a basic example of an arithmetic series whose sum is the product of the average and the quantity of numbers. Since the average is most easily

April 1968 333



obtained by dividing the sum of the first and last numbers by 2, we calculate:

(1 + 9) X 9 = 45. 2

5. In a different vein, we might observe that the entire table has 100 entries and that the main diagonal (lower left to upper right) contains ten 9's. Removing this diagonal leaves 90 sums, of which half are one-digit and half two-digit. Thus:

1 -X 90 = 45. 2

6. Finally, where counting of one sort or another is used, we could introduce set concepts and point out that the quantity we seek is precisely the complement, with respect to 100, of the quantity of one- digit sums. To obtain this latter quantity, we need merely ... .

Turning from counting, we realize that the elements whose quantity so intrigues us occupy a particular region of the table. If we consider each element as centered in a unit square, we could determine the desired result by calculating, somehow, the total area taken up by all the elements.

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Figure 2

7. So we note (Fig. 2) that our elements considerately form a right isosceles triangle, with legs 9 units long. And, apply- ing the appropriate formula, we have:

A = - bh 2 1 = _ x 9 x 9 2

1 = 40 -. 2

A miss, a very palpable miss! But a closer examination of the triangu- lar region reveals that we discarded the

upper left half of each of the 9 squares along the hypotenuse (Fig. 3). Conse- quently, we must add

2 2 to our previous computation and so have

1 i 40 - + 4 - = 45 .

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8. We can make use of the more ad- vanced area formula.

A = Vs(s - a)(s - b)(s - c), where a, b, and с represent the lengths of the three sides of the triangle and s is one-half their sum. Since a = b = 9 с = 9лД. Then

1 л/2 s = - (9 + 9 + 9v/2) = 9(1 + -), 2 2 and

/ V2" V2" y/1 VT l A =V9(1 + -)(9 -)(9 -)(9 - 9 -) = 40-. 2 2 2 2 2 As in (7), we must add 4VŽ.

9. A new sketch (Fig. 4) compensates for the AV2 units previously lost by draw-

334 The Arithmetic Teacher



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ing the diagonal in such a manner as to include, for each square along the hypotenuse, a small isosceles right triangle (legs Vi unit) which is not properly part of the square but is congruent to an excluded area which is part of the square. Now we have a trapezoidal region, acde, which exactly represents the area encompassing our elements.

To calculate this area, we first break it into two subregions: a triangle, abe, with area

1 1 _X9X9 = 40-, 2 2

and a rectangle, bcde, with area 1 1 _ x 9 = 4-. 2 2

10. However, since there is available a perfectly good formula for the area of a trapezoid, we can more directly calculate:

A = -(bx + b*)h 2 1 1 1 = _(_ + 9 _) x 9 2 2 2

= 45 11. We might also extend lines ae and

cd until they meet at /, producing a new isosceles right triangle, acf. Its area, since / is one-half unit to the left of d, is then

1111 _ x 9- X 9- = 45- . 2 2 2 8

Resisting the temptation to sweep the Vs under the rug, we notice that we have introduced an extraneous isosceles right

triangle, edf, whose legs are one-half unit long. So we must subtract

1111 _ X _ X _ = _ 2 2 2 8

and voilà, 45, again. 12. It is possible to utilize also the

similarity of triangles edf and acf to calculate the area of the latter. First computing the area of edf, as above, to be Vs, we then determine that the equal sides of acf, 9lA units, are just 19 times those of edf. Then the area of acf will be 192 = 361 times the area of edf.

1 1 361 X _ = 45 -,

8 8 from which the area of edf must of course once more be subtracted.

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Figure 5

13. A rearrangement of the elements (Fig. 5) produces an isosceles triangle with base and height both equal to 9. This triangle's area is 40 Vi, and as in (7), we are short 4 V2 square units. The dia- gram reveals that one quarter of a square is cut off by the triangle's sides, on both ends of each row, and of course,

1 1 2 x 9 x _ = 4- .

4 2 Once again,

1 1 40_ + 4_ = 45 .

2 2 14. Further, the above triangle provides

an instance where careful construction of the figure on, say, heavy paper will permit the triangle to be cut in two so that the

April 1968 335



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Figure 6

pieces, when properly aligned, produce a rectangle (Fig. 6) of area 5 x 9 = 45. (Question: Does Figure 6 accurately represent the resultant rectangle?)

15. Last and most impressively, al- though it will be beyond the present train- ing of most elementary teachers, a solution can be reached by integral calculus. In- troducing a coordinate system (Fig. 7) readily enables us to determine that the line ea may be represented by the equation

y-X + ',

and we may obtain the area under the curve by integration from 0 to 9. Thus:

A =/ (x + Vi)dx

= 45.

Undoubtedly the available techniques for the solution of the original simple in-

quiry have not been exhausted. However, even without here expanding on the der- ivation of the formulae used, it should be clear that there is a wealth of arithmetic, algebraic, and geometric concepts involved in the approaches shown. A class asked to devise as many ways as possible to answer the question might well find it- self reviewing a wide range of fundamental material as well as, hopefully, becoming sufficiently involved in the ramifications of the problem to begin indeed appreciating the aesthetics of mathematics. And if it appears as though we may have used a cannon to dispose of a gnat, is that not, after all, one aspect of elegance?

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Figure 7

Napier's Bones

An arithmetic teacher named Jones Was reduced by the new math to groans And shortly expired. Since he has not retired, He now serves as Napier's Bones.

Edwin A. Rosenberg Western Connecticut State College

Danbury, Connecticut

336 The Arithmetic Teacher



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Aesthetics in elementary mathematics