500 SCHOOL SCIENCE AND MATHEMATICS

finding the area of a triangle when the three sides are given. Letus take the triangle whose sides are 13, 20, 21.Let x -= CD.Then 20�x === DA.From the right triangles CDB and ABD we have

21�x2 =h2=== ^�^O�y)2or 441�x2 == 169�400+40;r�r2.Hence x==lQ%.Therefore h === 212�7o56^ =^^or h = 6%.Hence the area == %. ^%. 20 = 126.If we take a, b, c as the sides of the triangle we get the usual

Heronian formula,

area === ~\/s{s�a) (^�b) (s�c) where s == �’����/VOne reason why beginning algebra often becomes uninteresting

is that the teacher does not offer the student enough concreteillustrative material. It is hoped that the above simple applica-tions will be of use to the teacher and will incite him to thinkup more problems of a simple nature.

MATHEMATICAL EQUIPMENT AND ITS USES.

BY H. C. WRIGHT,University of Chicago High School.

In January, 1915, it was the privilege and the pleasure of thewriter to visit one of the large high schools in the vicinity ofChicago. The high-school district has an equalized assessedvaluation of approximately $8,000,000 and there is levied up-wards of $190,000 for the annual support of the high school.The equipment provided for the athletics, for science, for manualtraining, for English, for the classics, for music, for drawing, fordomestic science, for commercial instruction and for referencework seemed abundant and of good quality. But in the rooms ofthe seven mathematics teachers I found much the same equip-ment as was common in high schools a quarter of a century ago.Chalk, erasers, string, rulers, a portion of squared blackboard,several wooden protractors, and some blackboard compasses werein some of the rooms but not in all. The equipment in the pos-session of the pupils, aside from textbooks and pencils appearedto consist of rulers and loose paper. Although the school mayhave possessed equipment not in evidence at the time of my

MATHEMATICAL EQUIPMENT AND ITS USES 501

visit, such material was not mentioned in the remarks of theteachers with whom I talked about the matter.

Earlier in the fall of 1914, I visited the high school in an

Illinois city of some 26,000 inhabitants and found the mathe-matics teachers depending upon even less equipment than I sawin the Chicago suburban school.These two experiences have led me to recite in the following

pages the names and some of the uses of material at hand in theUniversity High School of the University of Chicago. I shouldstate at the beginning that if this school has considerable equip-ment it is principally because the mathematics teachers in theschool have asked for what they thought they needed and con-tinued to request the same until it was granted. They hope theend is not yet. Up to this time they have availed themselves ofthe custom in the school to ask for departmental requisitions inJune, the same to be acted upon during the following vacationand subsequent year. A special effort is made by each mathe-matics instructor to make his needs plainly known to those inposition to supply the desired material. And to a considerableextent the requisitions have been honored and the equipmentfurnished. Perhaps the department has aimed to ask for evenmore than it expected at any one time so that it would receiveall the benefit of any funds allotted to the mathematics depart-ment.One day in December, 1914, an inventory of the mathematical

apparatus in the rooms in the University High, School used formathematical instruction showed that each teacher has for hisuse or for that of the class: two large wooden protractors, two30°, 60°, 90° wooden triangles and two 45°, 45°, 90° trianglesof the same material; five wooden straight edges, several withhandles like those on a plasterer’s board; nine blackboard com-passes ; 14 foot-rulers; 10 lead pencils; two small brass protrac-tors ; 10 lead pencil compass holders. In one room was a sphericalblackboard. In another room hung a five foot slide rule. Eachroom has a cork bulletin board upon which the pupils5 writtenwork might be exhibited. Two of the three rooms containedglass doored cases for the storing of mechanical devices andmodels used in instruction or the result of the pupil’s construc-tive efforts. Besides these storage cases one room has an exhibitcase for written work. This case contains 25 revolving panelssimilar to the cases used in the Chicago Art Institute to displayphotographic copies of the masterpieces. Thus any year’s work

503 SCHOOL SCIENCE AND MATHEMATICS

of the entire mathematics department may be displayed thoughkept in a compact form. Then, too, each room had a section ofsquared blackboard for graphic problems and a supply of coloredcrayons. Scissors and string were also available.

Besides the equipment common to each room the mathematicsdepartment has a full complement of surveying instruments:two transits, level, leveling rod, steel chain, several steel tapes,wire pins, red and white sight rods, and engineer’s notebooks forfield work. There is also a high grade Duplex 10" slide rulefor use by either teachers or pupils.

In addition to the general room equipment the individual pupilcarries a notebook containing ruled and unruled paper, squaredpaper, and attached to the rings of the notebook is a ruler, abrass protractor, and a pencil compass. Thus, the pupil canundertake a variety of laboratory mathematics during the class-room exercise. For the written theorem assignments there is aprinted form that the pupil may buy and use if he chooses togo to that expense and thereby add to the neatness of his work aswell as save some time in doing it.The acquirement of all this varied material for the demonstra-

tion and the presentation of the mathematics taught in the highschool has to a large extent grown out o.f the unification of thefirst three years of the ordinary secondary-school course, the firsttwo years of which are developed from geometrical concepts.The first-year book in mathematics contains many of the factsand ideas of plane geometry and the algebraic quantities havea geometrical interpretation. In the second year the algebraicfacts acquired during the first year are again brought out byassigning algebraic values to the geometrical material presented.

This unification of the algebra and the geometry presents manyopportunities for the pupil to use mechanical drawing instru-ments, both to solve and to illustrate problems. In this way hehas constantly before him the practical application of algebraand geometry to each other, and can illustrate one subject interms of the other. For example, when he makes a study of theequation 3;r ^-4y == 9, his attention is called to the algebraic andgeometric representation of the same idea. Or again, when hefirst learns to multiply a+3 by a+4, it is by seeing a parallelo-gram whose area represents the product of these two factors.The parallelogram is divided so as to show that one part is a2’,a second part is 3a; a third part, 4a; and the fourth part, 12.Thus he has the area expressed geometrically and algebraically.In the geometry of the second year he is told, for example, that

MATHEMATICAL EQUIPMENT AND ITS USES 503

one of the equal angles of an isosceles triangle is Idr2 and theother 40.r�40. -He is then asked to find the value of x, the sizeof each angle, and to construct a triangle satisfying these condi-tions. In performing the required operation, the pupil needs tosolve the quadratic and to construct angles of 20° and 140°.If this be made a class exercise, each pupil works at his desk,using his individual equipment, and later some one pupil demon-strates his results to the class, making use of the room blackboardequipment and actually obtaining the required triangles. In casethe problem has not proved too difficult for any member of theclass, the instructor assigns a second problem, meanwhile pass-ing from pupil to pupil to supervise the class work.

This constant use of the drawing and measuring apparatus athand to solve and to illustrate the problems in mathematics seemsto create in the minds of the pupils the idea that mathematics isa single department of knowledge in which algebra, geometry,and trigonometry are but different modes of expressing the samefacts. For example, the pupil may have drawn a right-angledtriangle so that the hypotenuse is twice as long as. one of the sidesabout the right angle. He may have lettered the hypotenuse 2aand the one side a, and he may have expressed the fact that the

sine of angle A == �� = 1 �

/vCt

Then he may have drawn on squared paper or blackboard aright-angled triangle in which the hypotenuse is 8 units long andthe side opposite angle A 4 units long, from which figure hefound with the protractor angle A to be 30°. This enables himto find that the sine of 30°===^. He has in this operation usedarithmetic, geometry, algebra, and trigonometry, and meanwhilehe has been engaged in his second-year mathematics. Thegeometry furnishes space material to illustrate algebraic prin-ciples, which by means of his knowledge of algebra he is ableto carry a process beyond the limits set by the geometry.Another field for the profitable and pleasing use of the material

equipment is the study of scale drawing and similar triangles.This makes possible the solution of surveying problems involv-ing measurement, as the problem of finding, the distance fromeach of two land forts to a vessel off the harbor. In this workthe pupil may choose to use colored inks with the result that he,often succeeds in making a very attractive piece of drawingaccording to careful measurements, and a correct application ofgeometry and algebra to a concrete problem.

504 SCHOOL SCIENCE AND MATHEMATICS

Again, in third-year mathematics several days may be spent infield work. The height of a tall smokestack is found from dataobtained by establishing a base line and then measuring severalangles of elevation. This problem may be solved by a scaledrawing or by the use of the trigonometric formulas. A secondproblem is to run a line of levels from a given benchmark to a

second benchmark some hundreds of feet away. A profile draw-ing on squared paper will then be made from the data taken.Another interesting problem is to find the distance of one ofthe Lake Michigan water cribs from some shore point. The fol-lowing problem is apt to arouse considerable interest as well as

to illustrate the concrete use that is frequently made of an engi-neer’s notes. The class is divided into two sections, one takingfield notes which locate some particular object, the other usingthe notes of the first section to re-run the lines which locate theobject.The presence of all of the equipment thus far referred to

stimulates the pupils to make various models to illustrate the solidgeometry. By using wire, knitting needles, corks, and light lum-ber or cardboard the pupils make models to illustrate a numberof the theorems in solid geometry. The fact that these modelsmay be preserved from year to year in the glass cases installedin the rooms makes it possible for the teacher to have at handthe best constructive products from a large number of pupils ofthe preceding years.Each spring, generally in May, an exhibit of the best work of

all the mathematics classes is arranged for display to the wholeschool and those parents, friends of the school, and other personsinterested in the teaching of mathematics, who may come to viewthe collections. The drawings and written work are mountedaccording to some uniform plan and then hung on the walls ofsome room or of one of the corridors. The models are set outon tables or in glass cases. Thus the mathematics departmenttakes part in the education of the hand along with the training ofthe mind. The results obtained from a liberal use of the equip-ment mentioned above justify the possession and the applicationof all this varied assortment of material.

ERRATUM.Page 457 in May, 1915, number. Determinant should read:

[Sin a �(cosa+i^)"]Cos a�b Sin a j