THE NEED FOR ALGEBRA IN INDUSTRYKENNETH WALTERS

Colon, Michigan

Anyone who teaches algebra knows how often he hears studentsremark, ^Why should we study algebra when we will never need it inpractice?^ Many teachers have some convincing answers and thefollowing incident may suggest some new ones.One day I was approached by a member of the Bureau of Mines

with the following problem. During the war, a book had been foundin a bituminous coal mine in Germany explaining the theory of im-pact of an air hammer, which was very successful in mining bitumi-nous coal. The gentleman explained that he and his associates wereinterested in trying to adapt the air hammer (with modifications)to use with anthracite coal.

^But," he explained, ^we do not want to involve time, money, andeffort in the development ..until we have verified the concluding for-mula concerning the theory of impact of the hammer.57The author of the book had merely defined the symbols and stated

the important conclusion. The derivation is entirely algebraic andrequires an understanding of only those ideas developed early in afirst course of algebra�substitution, factoring, cancellation, specialproducts. It shows the advantage of knowing the special product(a?+H-c)2=a2+62+c2+2a6+ 2ac-{~2bc. Of course I am not suggest-ing that a student necessarily should have the maturity to work thisproblem at that stage of his training but it emphasizes the practicaluse of such training in a very real situation.The figure which follows gives ’the rough impression of the air

hammer.M (mass of piston)m (mass of tool)e (coefficient of elasticity)Lo (theoretical total kinetic energy)L (work done on coal face)Lb (loss by impact)

Before impact:A (velocity of piston)a (velocity of tool).

After impact:B (velocity of piston)b (velocity of tool)

tool "being sirucic(^

The author stated that since it is known that:

L^^MA^+^ma^L=^MB2+^mb2

212

THE NEED FOR ALGEBRA IN INDUSTRY 213

Lb^ LQ�L

)

M(A-a)(i+e)

A-B^-���-�� (1)

1+�m

,_._^<^M. p)m

^MThen it can be shown that:

{!-eYmM(A-aYLb=-

2(M+m)

Proof:

Lb= LQ�L

Z&= ^MA^ma2) -^MB^mb2). (3)

From (1)m{a-\-ae�Ae)-\-AM

M+mFrom (2)

M{A-\-Ae � ae) +amb=��

M-\-m

Substituting (4) and (5) in (3) and factoring out ^, we have,

^MA2+ma2)<iM+m)2--M[m(a+ae-Ae)+AM]21 -m[M(A+Ae-ae)+am}2

Lb==-� ��������������������������:���2 L (M+m)2

Expanding, collecting terms, and factoring out mM,

~MA2+mA2+Ma2+ma2-ma2e2-mA2e2mM

Lb^������ +2maAe2-2MaA-MA2e2-Ma2e22(M+m)2

+2MaAe2-2maA

Grouping for factoring:

mM \�M(iA2+a2-2aA-A2e2-a2e2+2aAe2) "1

^^(MTw^L +m(A2+a2-2aA-A2e2-a2e2+2aAe2)]’Factoring and cancelling:

214 SCHOOL SCIENCE AND MATHEMATICS

^mM , [(A-aY-e\A-aY}2{M+m)

(l-e^mM^A-a)2&-2(M+m)*

One thing that impressed my class was the fact that here were agroup of men, living in our city, who were finding real need in industryfor algebra. It serves as a warning to us and to the students howlittle we sometimes know of the use being made of the mathematicswe teach.

TAPE RECORDS IN MATHEMATICSThe Department of Mathematics of the University of Oklahoma has had

many requests for copies of our series of 15 minute radio broadcasts on mathe-matics. Arrangements have been made to tape record the talks. Anyone wishinga tape recording of one, or more, of these broadcasts may secure them in thefollowing manner:Your magnetic tape, in a single reel film shipping case, or packed between

sheets of corrugated cardboard and wrapped in heavy paper, can be mailed toTapes for Teaching, Educational Materials Services, University of Oklahoma,Norman, Oklahoma.A service change of 50 cents for each 15-minute broadcast, plus postage

charges, will be made for duplication of our master tape onto your tape. Be sureto state whether your recorder runs at 3} or 7j- inches per second! All recordingswill be single track; however they will play on double track recorders. Each15-minute broadcast will require a minimum of 300 feet of tape at 3i in./sec.or 600 feet at 7^ in./sec. Standard tape lengths are 600 feet (5 inch diameter)and 1200 feet (7 inch diameter) reels. Order tapes by giving the date on whichthe talk was given, the complete title, and the name of the speaker.As a convenience to those who do not wish to send in a tape, recordings may

be rented at $1.50 per 15-minute recording for a five day period.

PROGRAMSOctober 5�Mathematics�Our Great Heritage, C. E. SpringerOctober 12�How Mathematics Started, J. 0. HasslerOctober 19�The Struggle for a Number System, J. 0. HasslerOctober 26�Butter and Eggs Mathematics, R. C. DragooNovember 2�Mathematical Pastimes, J. C. BrixeyNovember 9�Codes and Ciphers, R. V. AndreeNovember 16�Proving the Impossible, A. F. BernhartNovember 23�Misinterpretation of Statistical Data, B. T. GoldbeckNovember 30�Quality Control, J. C. BrixeyDecember 7�Running Around in Circles, N. A. CourtDecember 14�Entertaining Mathematics, N. A. CourtDecember 21�Small Observatories, B. S. WhitneyDecember 28�New Light on an Old Problem, R. V. AndreeJanuary 4�What Should a High School Student Expect from Mathematics?,Miss McFarland

January 11�Mathematics and Stamp Collecting, W. N. HuffJanuary 18�Random Walk and Gambler’s Ruin, P. W. M. JohnJanuary 25�Why Study Geometry?, J. 0. Hassler