ScHOOLSciENCEMATHEMATICS

VOL. XXIV, No. 4 APRIL, 1924 WHOLE No. 204

CLASS ROOM DEVICES IN TEACHING ALGEBRA ANDGEOMETRY.

BY JOSEFH A. NYBERG,Hyde Park High School, Chicago.

Before a carpenter inserts a screw into a .piece of wood he rubssoap on the threads of the screw. This is one of the tricks ofhis trade, learned as an apprentice from his master and notfound in any textbook. No newspaper is considered up-to-datewithout a column of ^Hints for the Busy Housewife/7 tellinghow to remove ink stains or how to make furniture polish.Every teacher likewise knows a few tricks from which we allcould profit. The present article deals with a few devices, notmentioned in the texts on pedagogy, which I have found usefulin teaching algebra and geometry.

1. The first device I call ^Showing Successive Stages of theWork.^ The idea for it was obtained from First YearMathematics by Marsh and Evans, page 52, where themethod is used in multiplying two approximate numbers. Thefigure below shows what has been written on the blackboardwhen the method is used to teach long division in algebra. Thesignificant feature is that each successive step is shown sep-arately. Steps 3, 6 are here shown below steps 1, 2 but in theclassroom the blackboard is usually large enough to permit allthe steps to be written alongside of each other.

3x______ 3x

1. 2x-5) Qx2-7x+15 2. 2x-5) 6x2-7x+15W-15x

3x______ 3x

3. 2x-5) 6x2-7x+15 4. 2x-5) 6X2-7X+15Qx2-!^ Gx2-^

8x 8x+15

346 SCHOOL SCIENCE AND MATHEMATICS

3x+4 3x+45. 2x-5) 6x2-7x+15 6. 2x-5 ) Gx^Px+lS

6x2--15x 6x2--15x

8x+15 8x+158x-20 8x-20

+35To show how the method is used let us suppose I am dividing

Gx^Tx+l^ by 2x�5. By comparing the work with arithmeticwe see that 6x2��2x, or 3x, is the first term of the quotient.Hence I write on the blackboard

_3x______2x-5) 6x2--7x+15

While I am next making some general remarks or answeringsome questions I make a copy on the board of what I have alreadywritten there.’ Then-when I multiply 2x--5 by 3x, I write theproduct, Gx2�-!^, in the second copy so that we have on theblackboard:

3x 3x

2x-5)6x2-7x+15 2x-5)6x2-7x+15G^-lSx

Apparently, as if the idea had just occurred to me, I then askthe pupils to get out their papers and pencils and do on paperthe same work that. I am doing at the board. One reason forasking the pupils to do this is merely to give me time to makeanother copy of what I have already written. The next stepis to subtract Gx2�]^ from 6x2�7x and this subtraction isperformed in my third copy, not in the second. Even whenbringing down the +15 in the next step I do this in a new copy,not in the one in which the subtraction"has been performed.Thus when the division is completed we have on the board

not merely the finished work but also a picture of each successivestage of the work. By looking at the blackboard, the pupilcan see what happened first, what happened after the subtrac-tion, etc., things that he can not see by looking at the last picturealone. Teachers frequently assume that they have the undividedattention of all the class when explaining any new process, butthe attention of even the best pupil varies considerably. Apupil may understand one step of a process and not anotherstep merely because his attention was momentarily distracted.The pupil then asks to have the process repeated. I find thatsuch pupils can usually answer their own questions by lookingat the successive stages of the work.

DEVICES IN ALGEBRA AND GEOMETRY 347

2. In teaching the construction of graphs this method is alsovery useful. The interpretation of a graph is so much easierthan the construction that teachers have assumed that drawinggraphs is subject matter suitable only for the less intelligentgroups. Nevertheless, the mere selecting of proper units onthe two axes calls for considerable good judgment and planningwhich are qualities that are found only in the more intelligentgroups. I have never actually seen a pupil draw the graph first,and then draw the axes but I should not be surprised if a pupilwere to do so. The frequency of the question, ^What do youdo first?^ shows that the pupil does not grasp the successivestages of the work and can not learn them by looking at thefinished picture. When solving an equation containing par-entheses and fractions, there is no fixed order in which we mustperform the operations. In drawing graphs, however, the stepsare not interchangeable. Hence my method is as follows:In a first picture I show

^only the horizontal axis witha scale clearly marked on it.Instead of adding a verticalscale to this picture, I drawthe horizontal axis a secondtime and in this picture showboth the horizontal and ver-tical axes. Again, on this pic-ture I do not draw the verticalbars or plot any points as theyare shown in the third pic-ture. The fourth pictureshows the graph. If the at-tention of one pupil was wan-dering when I was explainingthe second step and the atten-tion of another pupil was dis-tracted at the third step, I donot need to repeat any of theexplanations as the picture ofeach step is on the black-board. Of course, every textcontains rules but the abilityto understand printed rulesis not very common.

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Z. 3 ^ 5" 6 7 6

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348 SCHOOL SCIENCE AND MATHEMATICS

3. The work on graphs suggests that frequently in geometryalso we may err by putting too much into one picture; the detailsmayhide the mainidea or the manylinesmaybe distracting. Inthefigure below, forexample, we are given the common tangent totwotangent circles and. any two chords drawn through the point oftangency, and are to prove that CD and EF are parallel. Thisis a difficult exercise since the angle DAB is measured by halfof either of two apparently unequal arcs, AF or AD. But mostof the class is able to solve the problem (and incidentally learnsomething about the lengths of arcs) if the picture is presentedas follows. After the picture has been drawn on the blackboard,I erase the smaller circle and ask the class to prove anythingat all about the picture that they think might be useful. Eachpupil writes down his own discoveries so that there is no ex-

change of ideas. Then I restore the smaller circle to the pictureand erase the larger one, asking the pupils again to write downany discoveries they may make. Finally I erase the smallercircle again, leaving only the straight lines in the picture. Bythis time most of the pupils have overcome their difficulties .-Even better results can be obtained by showing on the boardfour pictures, (1) the figure shown above, (2) the same but withthe smaller circle erased, (3) the same as (1) but with the largercircle erased, and (4) with both circles erased.

Other similar exercises involving circles and tangents towhich this method may be applied are:

If two circles intersect, their common chord produced bisectsthe common tangents.

If two circles intersect in A and B and the two chords BD andBC are respectively tangents to the two circles, then AB is themean proportional between AD and AC.

DEVICES IN ALGEBRA AND GEOMETRY 349

If two circles intersect and from a point P in the prolongationof the common chord two secants are drawn, one intersectingone of the circles in A and B and the other secant intersectingthe other circle in C and D, then PAXPB = PCXPD.

4. Another and entirely different device I have found usefulwhen changing a fraction to an equivalent fraction in exercises

5 20on addition. If we ask a pupil why - = � he may answer that

9 3620 = 5X4 and 36 = 9X4.

If we ask him to find the missing numerator in the relation5 ?- == � and explain how he obtained it, he will surely say, ^9 goes9 36into 36 four times, and 5 times 4 is 20.^ The phrase goes intomay be a useful one in the lower grades but in the ninth gradethe word ^explain^ should mean a reference to the four funda-mental operations of arithmetic. In the present illustration thepupil divides 36 by 9 and then multiplies 5 by the quotient.

30 _ ?JF~ ~ ~2S^

3a ?In an exercise like �� = ���� the pupil should be conscious

5b2 20b2cof the fact that he is to divide 20b2c by 5b2 and multiply 3a bythe quotient. To encourage this way of thinking I keep on theblackboard for some time the figure below.The course of the arrow indicates the order in which we look

at the different terms and the operations performed.

To train natives of Alaska in the use of small power boats so that theycan earn their living by hunting seal, walrus, and whales, the United StatesBureau of Education has fitted up the U. S. S. Boxer as a floating school.Navigation and wireless telegraph operation will be among the subjectsof study. The Boxer is now used to carry teachers and supplies to thebureaus schools, reindeer stations, and hospitals in Alaska and to shipreindeer meat from Alaska to Seattle.