An easy formulaAuthor(s): Andrew BerrySource: The Mathematics Teacher, Vol. 95, No. 6 (SEPTEMBER 2002), p. 406Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871084 .

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REACTIONS TO ARTICLES AND POINTS OF VIEW ON TEACHING MATHEMATICS

An easy formula While I was teaching the concept of the inverse of a function in my precalculus class, I created an

example on the board. I knew that it was a one-to-one function,

thereby ensuring that it did in fact have an inverse. The function that I haphazardly scribbled on the board was

(1) f(x) = y = 3x + l

(We can easily prove?algebraical ly or geometrically?that functions of the form

ax + b y =-3 cje + a

are all one-to-one.)

Finding the inverse of function (1) in the usual way gives us

5y-3

5xy-3x = 3y + l9

y(5*-3) = 3x+l.

Since

y=f-\x\ we have

3*+l

5x-3

We notice here that fix) = f~\x). I mentioned to the class that I had not planned this interesting result. A student asked the ques tion, "How does one create a function of the form

ax+b y

=

cx + d

that is its own inverse?" Knowing that this student was particularly mathematically able, I asked her to try to come up with her own answer to the question and then present it to the class the

following day. Her solution was surprisingly

simple and elegant?she simply inverted the general expression

y=f(x) _ ax+b

cx + d

and arrived at

-xd+b

r'W= cx-a

which implies that a must equal -d for the given function to be its own inverse.

For example, the functions

and

y = -

_6x-5

x-6

-4*+ 2

x+4

both have this property, since in both cases a = -d. If such a rational function and its inverse are

graphed (where a = -cf), as well as the line y = x, the graph can

readily be seen to be symmetric with respect to the line y = x, which gives us a geometric inter pretation of the phenomenon. The graph can be created by hand or with a graphing utility. Recall that f(x) and f~\x) are

symmetric about the line y-x. Andrew Berry ajberry@onepine.com LaGuardia Community

College Long Island City, NY 11101

David M. Collison's trimagic square Shortly before he died, David M. Collison of California sent me his papers, and I have gleaned the best parts over the years and published them wherever possible in his name. Figure 1 (Hendricks) is the world's

smallest known trimagic square. The sums of its rows, columns and both diagonals are each 9,520. If you square all the numbers, then the sums are

8,228,000. If you cube all the numbers, then the sums are

7,946,344,000. John R. Hendricks hypercube@shaw. ca Victoria, BCV8V2M9

Mental arithmetic?

finding squares This "Reader Reflection" shares a mental arithmetic trick that I learned in elementary school while reading the delightful book Cheaper by the Dozen, by Frank Gilbreth and Ernestine G. Carey (New York: T. Y Crowell Co., 1948). In chapter 6 of the book, the authors explained a method for "quickly" computing the square of each integer from 1 to 100. Evidently, their father had required Frank, Ernestine, and their siblings to learn the technique as part of their home education.

I have used the method and its extensions ever since I learned it, both for personal calculations and at the blackboard during class. Students often ask how I am able to perform a calculation

so quickly. The tricks should be part of any advanced first-year algebra class because they give students the opportunity to prac tice mental arithmetic and because of the connections that students can make between men tal arithmetic and algebra. I have encountered many algebra stu dents who know that a2 - b2 =

(a + b) (a - b) but do not have the foggiest idea how to use that information to calculate 52 48.

The first step is to be able to square integers easily. We can use a simple technique to find, for example, 572. The technique requires that we first calculate the distances that 57 is from both 25 and from 50. The result of 57 - 25 is 32, and 32 forms the first two digits of the product of 57 times 57. The last two digits of the product are then the square of the distance of 57 from 50, that is, 72 = 49, so we see that 57 57 =

3249. Similarly, to find 462, we think (46-25) = 21 and (46-50)2 = 16. Thus, 462 = 2116.

The proof that this method

always works is a nice exercise for algebra students. They simply need to show that x2 = (x-25)- 100 + (*-50)2.

Students must take extra care

(Continued on page 432)

1160 1189 539 496 672 695 57 10 11 58 631 654 515 558 1123 1152

531 560 675 632 43 66 1179 1132 1133 1180 2 25 651 694 494 523

1155 1089 422 379 831 767 92 45 91 44 790 808 403 360 1118 1126

832 766 99 56 1154 1090 415 368 414 367 1113 1131 80 37 795 803

1106 1135 411 454 716 739 27 74 75 28 757 780 473 430 1143 1172

409 438 717 760 19 42 1115 1162 1163 1116 60 83 779 736 446 475

999 1007 192 235 977 995 164 211 163 210 1018 954 173 216 1036 970

982 990 175 218 994 1012 181 228 180 227 1035 971 156 199 1019 953

183 191 991 1034 195 213 963 1010 962 1009 236 172 972 1015 220 154

200 208 974 1017 178 196 980 1027 979 1026 219 155 955 998 237 171

715 744 20 63 1107 1130 418 465 466 419 1148 1171 82 39 752 781

18 47 1108 1151 410 433 724 771 772 725 451 474 1170 1127 55 84

101 35 1153 1110 423 359 823 776 822 775 382 400 1134 1091 64 72

424 358 830 787 100 36 1146 1099 1145 1098 59 77 811 768 387 395

667 696 46 3 1165 1188 550 503 504 551 1124 1147 22 65 630 659

38 67 1168 1125 536 559 686 639 640 687 495 518 1144 1187 1 30

Fig. 1 (Hendricks) Trimagic square

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406 MATHEMATICS TEACHER

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