Mathematics in the Real World, ReallyAuthor(s): DAN KENNEDYSource: The Mathematics Teacher, Vol. 78, No. 1 (JANUARY 1985), pp. 18-22Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964342 .

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Mathematics in the Real

World, Really By DAN KENNEDY, Baylor School, Chattanooga, TN 37401

Just

in case the title of this article is not a sufficient warning to the reader that

the contents fall short of being profound, let me assure all who have read this far that you are not likely to learn much math ematics in these pages. Indeed, it is the very triviality of these applications that prompt ed me to write this paper.

I have found that mathematicians in

general, and mathematics teachers in par ticular, tend to refer rather blithely to the Real World when pointing out a known ap

plication of a given morsel of mathematics

when, in fact, the world they speak of might be fairly remote from their own acquaint ance with reality. The unreal qualities of some of our classic Real World problems have been widely discussed by more capable pundits than I, so let me simply capture the essence of their case with a favorite rhe torical question from Zalman Usiskin: "Where is this guy who rows upstream at a constant rate for two hours?" The rhetorical

answer, of course, is that he is in the Real World. Sure he is.

Once one begins to question the reality of these Real World applications, however, it is difficult to stop. Mixing three pounds of nuts at $2.00 a pound with pounds of nuts at $4.50 a pound to make a mixture worth $3.00 a pound, for example, seems

like a perfectly reasonable task for a nut

vendor, but I will confess to never having done it. Indeed, I have solved scores of mix ture problems in the classroom (presum ably, the Unreal World), yet never once

have I had the occasion to solve such a

problem for real. Neither have I cut squares from the corners of a rectangular piece of cardboard and folded up the sides to con

tain a maximum volume, nor launched a

rocket straight up with velocity ;0, nor slid a ladder down a vertical wall by pulling the

18

bottom away at a constant rate?even

though, as a mathematics teacher, I can tell

you what to expect when I do any of these

things. Frankly, most of the mathematical ap

plications that crop up in my Real World are rather pedestrian, especially when com

pared to the challenges confronted daily by the folks in the exercises in textbooks. The same is probably true of a great many math ematics teachers and even of a great many research mathematicians outside their areas of research. Yet I would wager that each of us, at one time or another, has ex

perienced that "Magic Moment" when the Real World has presented, directly in our

path, an unanticipated locked door for which our training has serendipitously sup plied us with the key. Since I am old

enough to have acquired some interesting examples of this phenomenon, and young enough to remember them, I shall offer a few here as one mathematician's view of the Real World.

The Magic Retail Numbers

This example is so simple that I feel it must

happen all the time. If it has not happened to you, I hope it does someday, under cir cumstances as favorable as these.

The father of one of my former students owns a retail store that sells wine, et

cetera, and one afternoon I happened to be

purchasing a fifth of et cetera. Realizing that I taught mathematics and concluding incorrectly that I would be interested in a discourse on the travails of retail pricing, the proprietor began telling me how diffi cult it was to compute the discounted shelf

price of a bottle on the basis of the whole sale cost. To make his point, he demon strated the problem with a case of wine that cost him $37.50. Dividing by 12, he arrived

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at a price of $3,125 for each bottle. He com

puted that the 20 percent markup would be

$.625, which he added to the base price to

get $3.75. He then computed the state tax and added it on, the local tax and added it on, and a 10 percent discount and sub tracted it, finally arriving at a shelf price for this particular wine. Oh, he lamented, what he would not give for a magic formula that would enable him to do this in one

step! You see what I mean by the "Magic

Moment."

Praying that I could demonstrate this formula without making it appear too easy, I stroked my chin thoughtfully and said

that, yes, I thought I could help him. Was this always the procedure for wines? Yes, he said. Same numbers but different whole sale costs. And for other beverages? Almost the same, but the markup was a different fixed percentage. Stroking my chin a few more times for effect, I savored the moment, much as a poker player must savor a royal flush, and then unsheathed my calculator. Now then, I said, trying to look serious, what were those percentages again ?

When the calculator had multiplied two

products of the form

(l+p^l+PaXl+PaXl-pJ

for the percentages corresponding to the various markups, taxes, and discounts, I

presented the wide-eyed merchant with two numbers written on a paper bag, assuring him that multiplication by these numbers would do the trick in the dreamed-of single step. When he tried the wine number on his $3,125 burgundy and discovered that it worked to the penny, I thought the man was going to cry. I was unaccustomed to

seeing high school algebra bring this much

joy to anyone, and I was rather unprepared for the reverence that I had suddenly earned from this man, but since I had no intention of charging him for my services, I rationalized that allowing him to regard me as the second coming of Isaac Newton would be fair compensation.

There was no charge for my order that

day, and I still cannot walk into that store

within a month of Christmas without being forced to accept a free bottle of champagne. The paper bag on which I had written the

magic numbers is still displayed on the wall of the office.

The Bypass Some bits of mathematical knowledge are so uninteresting in themselves that one feels obliged to embellish reality to find an

application that is worth talking about. Such is the case with the relatively trivial observation that the circumference of a circle varies directly as the radius, that is,

=( ) 0

for a circle of any size. Many puzzle fans are probably familiar with the classic appli cation of this principle to the case of a steel band wrapped tightly around the equator of the earth. First you hypothesize the exis tence of such a band ; then you hypothesize the addition of an extra ten feet, after which the enlarged band is spaced evenly above ground level all the way around the earth. The problem, then, is to decide whether you could crawl under it. If you have heard this problem before, then you can probably recall your initial disbelief on

discovering that, yes, you could crawl under it, and that, in fact, you could do so while a person with a sixty-inch waist line heaved through right beside you. This

no-longer-quite-so-uninteresting formula

shows the clearance (Ar) to be a roomy 10/2 feet.

Obviously, nobody is about to suggest that this classic problem is from the Real

World, but, of course, that is precisely its charm. Once you know the answer you must still perform a small leap of faith just to accept it, and performing the experiment on a grapefruit will do little to ease your incredulity. However, when the same prin ciple unlocks one of those doors in your Real World, the solution takes on a differ ent level of significance.

I was driving along an interstate high way one day, approaching a major city. As with most major cities, a circular bypass went around the center of town, with signs

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inviting drivers to save time by going around the city. The alternative, as usual, was to follow the interstate through town

along a diameter, proceeding at a slower pace because of increased traffic, and thus

losing time despite the shorter distance. I

usually do what is expected of me in these instances and take the bypass, but this time the signs gave me sufficient advanced warn

ing to ask myself, Just how much slower would the traffic need to move along the diameter to make the bypass a time-saver? I

thought about the problem for a moment, decided that I needed to know the diameter of the city, concluded that I would surely crash if I tried to extract that information from the road map while cruising along a crowded interstate, and almost abandoned the problem?when the Magic Moment sud

denly intervened: the radius was irrelevant to the problem ! Since the ratio of distances involved (a semiperimeter and a diameter)

was /2, the traveling times would be the same if the ratio of average speeds were /2, regardless of the size of the circle.

Rounding 55 mph to 60 mph for compu tational convenience (an excuse that is un

likely to stand up in traffic court), I con cluded that I would save time going through town if I could average (2/ )(60)

mph?a little less than 40 mph, which I

figured I could beat with ease. With the

smug confidence of a mathematician who knows he has beaten the Real World, I sneered at the bypass exit and went barrel

ing along the diameter. Ten miles later I encountered a forty

five-minute delay on the downtown ex

pressway while a fleet of wreckers removed a stalled truck from the main bridge. Nobody ever said that the Real World plays fair.

The Elliptical Table

The following events actually happened to a friend of mine, Bryce Harris, and the

story is one of my favorites about the Real World.

Before he retired from teaching math ematics and acquired more time to pur sue his craft, Bryce had already become a skilled woodworker, turning out some very

beautiful pieces: clocks, cabinets, hob

byhorses, and all varieties of furniture. He had a fairly well equipped woodworking shop behind his home, near Gatlinburg, but

apparently he did not have a router that would perform all the tricks he needed. Fig uring it was worth a try, he stopped by one of the professional furniture-making estab lishments in Gatlinburg and asked if they

would allow him to use their router during slow periods. The man in charge was quick to reply that this would be impossible, for reasons that Bryce accepted. A conver sation about their craft ensued, during the course of which Bryce happened to com ment on a particularly fine elliptical table that was in the latter stages of production. The man agreed that he was proud of that

particular piece, but he assured Bryce that he would never have chosen that shape if it had not been specifically commissioned, due to the enormous difficulty of getting it to come out right. Bryce asked the man how he went about making an ellipse, and the man explained his method: First you fold a

paper into quarters, and then you very care

fully cut a smooth arc that starts and fin ishes at right angles to the fold. If it looks

good when you unfold it, you have your pat tern; otherwise, you must try again. It usu

ally takes several tries to get a smooth arc.

Bryce nodded sagely. After all those

years of teaching conic sections, he was ex

periencing a Magic Moment. He asked the furniture maker if he had some string and a

couple of thumbtacks. Sure, said the man. Well then, said Bryce, in his best poker face, let me show you a little trick.

Not many of us get the opportunity to walk into the jungle on the day of a total

eclipse and have a little fun with the na

tives, but the sensation is probably not unlike the one you experience when you fix two ends of a loose string to a board, pull it taut with a pencil, and twirl off a perfect ellipse in front of a skilled artisan who has wasted whole afternoons trying to luck into one using paper and scissors.

From that moment on, Bryce was ex tended full use of any machine in that Real

World shop, any time he wanted.

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The Senior-Trip Schedule

Sometimes a Magic Moment occurs when

you solve the problem at hand, but it can also occur when you save everyone the trouble of trying, by explaining that it cannot be solved. Part of the satisfaction one derives from this situation is that it is so rare in the Real World. Most of us, for

example, go through life armed with the

knowledge that you cannot trisect an angle with a compass and an unmarked straight edge, but then nobody in the Real World ever asks us to do it.

Once they determine that we are math

ematicians, however, people do ask us to do all kinds of things, especially to solve

scheduling problems. Some of these are

straightforward, and some are clearly im

possible (seven teams, each team plays every other team twice, three games a day for ten days), but the solution of the solv able and the declaration of the impossible in these instances are rarely satisfying enough to qualify as Magic Moments. I would not have considered a scheduling problem capable of such stature until the time they tried to reschedule the senior

trip.

For the past ten years, our seniors have closed out their high school days with a six

day outdoor experience known as the "senior trip." The format that has evolved over the years calls for the class to be divid ed into twelve heterogeneous groups, each with a faculty participant whose instinct for survival is somewhat deficient. Six dif ferent activities are scheduled for the six

days, and each group is paired with a com

panion group for the duration of the trip.

Six pairs, six days, six activities: what we veteran schedulers refer to as a "piece of cake." Indeed, this scheduling problem (called a Latin square) has so many differ ent solutions that one can impose such ad ditional niceties as the separation of any two river activities with a day on land. A

typical schedule appears in table 1. The schedule in table 1 was considered

to be ideal until someone suggested that it would be better if we could encourage more interaction between classmates without

sacrificing the special chemistry of the

groups. The obvious solution was to pair the groups differently each day. Thus was born the Senior Trip Scheduling Problem, a child of the Real World, and it did not waste much time finding its way to "some one who knows math," in this case, me. Since the director of the senior trip is also the chairman of our English department, I

clearly had no way to refuse the challenge. While he was still explaining the partic

ulars of the desired schedule, with the at tention to irrelevance that so characterizes the Real World, I had already redefined the

problem in my own mind: two different Latin squares with rows and columns la beled as in table 1, one for groups A, C, E, G, I, and K, the other for groups B, D, F, H, J, and L. When superimposed, no two pairs of overlapping groups could be the same.

By the time he finished issuing the

challenge, I had arrived at the Magic Moment. I told him it could not be done.

That I was able to reach this conclusion so suddenly was a definite shock to my col

league and, indeed, it even amazed me. Somewhere in the course of my mathemat ical studies I had encountered a Real World

TABLE 1

Ropes Rock Nature Canoe Day Course Rafting I Climbing Rafting II Hike Trip

1 A- CD E-F G-H l-J K-L 2 C-D E-F G-H l-J K-L A-B

3 E-F G l-J K-L A-B C-D 4 G-H I J K-L A-B C-D E-F

5 l-J L A-B C-D E-F G-H 6 K-L AB C-D E-F G-H l-J

January 1985 21

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problem concerning military officers

marching in a 6 6 formation, a problem that was allegedly posed to Leonhard

Euler, who could not solve it, but that was

finally resolved more than a century after his death. The solution called for two 6x6 Latin squares satisfying the same indepen dence condition (called orthogonality) as the one required for the senior trip sched ules. Euler, unable to find such a pair, con

jectured that the problem was impossible because 6 was twice an odd integer. He was unable to prove that conjecture, which was

understandable, since it was eventually shown to be false. The 6x6 problem, how

ever, did turn out to be impossible, and it is now known to be the only case (larger than

2x2) for which these orthogonal Latin

squares fail to exist (Liu 1968, 369-70). In some ways, this kind of deflating re

sponse to a Real World problem is more im

pressive to a champion of the liberal arts than the anticipated cranking out of the solution. My colleague could hardly believe I was serious. One more day or one fewer day on the senior trip and we could have had our schedule, but just because we hap pened to have six days, we were doomed? This was the stuff of classical tragedy, not

algebra. The beautiful irony seemed to miti

gate the disappointment of being unable to schedule the mixed groups, and it obviously amused the director to learn that a number could be so capricious; but a total defeat at the hands of the number 6 is Real World

mathematics at its dramatic best, and such a performance could not pass before an En

glish teacher unappreciated. I acquired a new mystique simply for having known of this sorcery, whereas I might have gotten a wink and a handshake for solving the prob lem if it had been solvable. For my own

part, naturally I was grateful that my train

ing in the Unreal World had prepared me for this particular Magic Moment; other wise I might be on my sixth ream of paper trying to schedule an impossible dream.

These anecdotes are surely not unique, but I hope they were worth relating in these pages. If they only serve to remind

you of your own Magic Moments, when

mathematics gave you the opportunity to

open unexpected doors in your Real World and step proudly through, then they will have served a valuable purpose. As math ematicians we must cherish those moments, and as teachers we must communicate their excitement to those who would learn math ematics from us.

You can only vicariously row upstream at a constant rate for so long before you become very, very tired.

REFERENCE

Liu, Chung L. Introduction to Applied Combinatorial Mathematics. New York: McGraw-Hill Book Co., 1968. m

[=Z| COMPUTING AND MATHEMATICS: THE IMPACT

[gSj ON SECONDARY SCHOOL CURRICULA, edited by

^fj^ James Fey. With the wide use of computers, what must change in the content, sequence, and emphasis of top ics in your mathematics programs? This analytical book spe cifically discusses the impact of computing on algebra, geometry, calculus, and the college preparatory curriculum. 100 pp., #337, $7.50. See the NCTM Materials Order Form in "New Publications."

Computers in Secondary School Mathematics A conference for teachers of secondary school mathematics to be held at Phillips Exeter Academy in Exeter. N.H., from June 23 to June 28. Participate in workshops led by experienced mathematics teachers that will help you integrate the computer with your

mathematics classes. Listen to prominent mathematics educators such as Stephen Maurer and Zalman Usiskin.

For more information write: Summer Computer Conference Phillips Exeter Academv Exeter. NH 03833

PROFESSIONAL DATES NCTM 63d Annual Meeting

17-20 April 1985, San Antonio, Tex.

NCTM 64th Annual Meeting 2-5 April 1986, Washington, D.C.

NCTM 65th Annual Meeting 8-11 April 1987, Anaheim, Calif.

For a listing of local and regional meetings, contact NCTM, Dept. PD, 1906 Association Dr., Reston, VA 22091, Tel: 703-620-9840; Compu Serve: 75445,1161; The Source: STJ228.

22 -?-Mathematics Teacher

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