University or Texao ?ublioatione
University of Texas Bulletin
No. 2242: November 8, 1922
The Texas Mathematics Teachers' Bulletin
Volume VIII, No. 1
PUBLISHED BY
THE UNIVERSITY OF TEXAS
AUSTIN
Publications of the University of Texas
Publications Committee:
FREDERIC DUNCALF J. L. H ENDERSON KILLIS CAMPI3ELI, E. J. MATHEWS
F. w. GR.AFF H.J. MULLER
C. G. HATNES F. A. C. PERRIN HAL C. "\YEAVER
The University publishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue, the last two the position in the yearly series. (For example, No. 2201 is the first bulletin of the year 1922.) These comprise the official publications of the University, publications on humanistic and scientific subjects, bulletins prepared by the Bureau of Extension, by the Bureau of Economic Geology and 'rechnology, and other bulletins of general educational interest. With the exception of special numbers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publications should be addressed to University Publications, University of Te~as, Austin.
University of Texas Bulletin
No. 2242: November 8, 1922
The Texas Mathematics Teachers' Bulletin
Volume VIII, No. 1
PUBLieHED BY THE UNIVEHSJTY FOCH TIMES A MONTH AND
ENTERED AS SECOND-CLASS MATTER AT THE POSTOFFICE AT
AUSTIN, TEXAS. l.'NDER THE ACT OF AUGUST 24, 1912
The benefits of edn<:ation and of useful knowledge, generally diffused through a community, are essential to the preservation of a free government.
Sam Houston.
Cultivated mind is the guardian genius of democraey. It is the only dictator that freemen acknowledge and the only secnrity that freemen desire.
Mirabeau B. J,amar.
University of Te:x:as Bulletin
No. 2242: November 8, 1922
The Texas Mathematics Teachers' Bulletin
Volume VIII, No. 1
Edited by
MARY E. DECHERD
Instructor in Pure Mathematics,
and
JESSIE M. JACOBS
Instructor in Pure Mathematics
This Bulletin is open to the teachers of mathematics in Texas for the expression of their views. The editors assume no responsibility for statements of facts or opinions in articles not written by them.
PUBLH'HED BY THE UXIVERSJTY FOUR TIMES A :l!OXTH Al'\D
ENTERED AS SECOXD~CLASS MATTER AT THE POSTOFFICE AT
AUSTIN, 'TEXAS, UNDER THE AC.T OF AUGUST 24, 191 2
COXTE::\TS
The University of Texas Research Lecture· ship . .. .. . . ............. . . . .........Goldie P . Horton .. .. 6
The Trisection of an Angle . ..... ..... .. ...P. H. Underwood . .. . 7
Is Geometry a Drag? ......... .. ...... ... . R. A. Eads .........12
Learn to "Roll Your Own" Logs .. ..... . .. .. J . ·w. Calhoun ......15
A Suggesti-cn . . .... ... ... .. .... . ...... ...Anna H. Jones . ... .19
Teaching the Graph ........... ...........D. D. Batin ........ 22
The Brown Mathematical Prizes .. ... ........... . .... .. . .....2~
What is the ~latter With High School ~Iathernatics? ... ................ .......... . Lillian E. Tuttle ....24
The Straight Edge ..... .. ...... .. .. ... . .. . ... .. ............26
l\IATHE:JIATICS FAC"CLTY OF THE r:\IYERSITY OF TEXAS
P. :JI. Batchelder Phillis E. Henry
H. Y. Benedict Helma L. Holmes
A. A. Bennett Goldie P. H orton
J.E.
Burnam .Jessie :JI. .Jacohs
J.
W. Calhoun Renke G. Lubben
C.
JI. Cleveland .J. :\. :Jiichie
A.
E. Cooper R. L. }Ioore 1\fary Decherd }I. B. Porter
E.
L. Dodd C. D. Rice
H.
J. Ettlinger
FOREWORD
It is with reluctance that your new editors call attention to the fact that the hand that directs the fortunes of the Bulletin is no longer masculine. We should prefer to leave an unsuspecting public to fancy there had been no change, but honesty prevents us from resorting to such a policy. J\luch appreciation of the Bulletin has been expressed by the teachers of Texas. We hope to keep the magazine up to its former standard. We wish to express first of all our gratitude to those who are this time our contributors. Vve would especially commend, the selections of subjects treated in this issue, and would ask that some who read this foreword would gladden our hearts by themselves writing articles on these or other interesting subjects. Perhaps it is not known that this Bulletin may be received by any teacher upon request.
THE "C.:\IVERSITY OF TEX~.\S RESEARCH
LECTl-RESHIP
The Cninrsity of Texas has recently established what is called The C ni L'ersity of T e:rns Reswrch Ltctureship, with the object of encouraging research among the members of the faculty of the CniYersity and of impressing upon the minds of the students the importance of research. The lectureship is to rotate from year to year among Yarious groups of departments in the College of ~.\rts and Sciences. The lecturer is chosen each year by the Graduate Council of the Cni\·ersity after a most careful im·estigatirm of the qualific:atirins of the members of the faculties of the departments in the group in \1·hich the a\rnrd is made. The holder of the lectureship is to delinr in }farch of the year of award nvt kss than three and not more than fh·e lectures in a chosen field of inwstigation. These lectures and other research studies of the lecturer are to be published by the l-niYersity and be giYen appropriate publicity and distribution.
F or the year 1922-1923, this research lectureship fell to the Science Group, and \YaS awarded to Professor }lilton Brockett Porter, Professor of Pure :\Iathematics. Professor P orter is an alumnus of the Cninrsity of Texas, haYing graduated in 1892. He took his DrJctrJrate at Harrnrd CniYersity in 1891. He has heen PrrJfessor of Pun }fathematics at the Cninrsity of Texas since l~J02. From time to time he has pnblished in the leading mathematical periodicals results of research in the field of }fathrnrntical "\.nal~·sis, and in this field he stands preeminent among mathematicians in this crnmtry and abroad. The CniYersity has sho1red most exceilent judgment in honoring Professor Porter 1\·ith its first rcsearc:h lectureship, and in so doing, it has honored itself.
GOLDIE P. HoRTox. CniYersity rJf Tt-xas, October :31, 1922.
THE TRISECTION OF AN ANGLE
There are certain angles that can be trisected by Euclidean plane geometry.
(1) A right angle.
From the vertex A of the right angle DAB measure on one of its sides a segment AB and on this segment describe an equilateral triangle ABC, then the angle CAD is one-third of the right angle DAB.
A
I=" : r\. 2...
~~~~~~~~ ~~~~
(2) An angle of 108 ° can readily be constructed and trisected. The tenth proposition of the Fourth Book of Euclid, viz.: ''Construct an isosceles triangle having each of the base angles double the remaining angle'' does this. In fact, the problem occurs in our common school texts on geometry in the inscription of a regular decagon in a circle.
Assuming therefore that ABC (Fig. 2) is an isosceles triangle such that angle ABC equals angle BOA equals twice angle BAC
University of Texas Bulletin
produce CB, then angle ABD equals angles BCA and BAC and since angle BCA is twice angle BAC hence ABD equals three times angle BAC. Since B~lC is 36 °, and ABC 72° angle ABD is 108"'. If an c<1uilateral triangle EEC is described on BC the angle ECA=angle "\CB-angle ECB=72°-60°=12°. Consequently an angle of 36" can be trisected.
If an equilaterial triangle F AC is dEscribed on AC then angle F AB=angle F~lC-an~tle B~lC=60°-36°=24°. Hence angle F .AB equals one-third of angle ABC; that is an angle of 72° can be trisec:ted.
Our two constructions gin: angles of 36° and 30° and since the difference of t\\·o knO\Yn angles can be constructed hence an angle of 6'° can be constructed and one of 18° trisected.
·when certain necessary and sufficient conditions are given, the solutions of problems in geometry depend upon the determination of the position of points. ::\o,,· if one of the conditions for determining a point in a plane geometry problem is omitted and we are asked to find the p(Jint there will result a problem haYing an infinite number of solutions. The required points "·ill, ho\rewr, be restricted in position so as to lie on a line straight or cnrw d and this line is precisely what is meant by the locus of the point. As an illustration consider the problem: Construct a triangle, haYing giYen the base, the opposite angle and the difference of the other t\ro sides. Suppose ABC (Fig. 3)
is the triangle required, and make AD=AC then BD is the difference of the two sides. Also angle ADC+angle ACD=180°A and since angle ~lDC=angle ~.\CD therefore twice angle ADC =180"-A and angle ADC= 90°-1/2 A hence BDC=180°angle ADC
=180°-(90-1/2A)=90°+1/2A
Mathematics Teachers' Bulletin
In the triangle BDC we know BC, BD and angle BDC equals 90°+lfzA. Taking the first and last of these three conditions we have a locus for the point D, namely an arc of a circle at which BC subtends a known angle. As BD is known and B is a fixed point we have a second locus for D. Hence D is determined uniquely since angle BDC equals 90°+1/2A. As the point A is equally distant from D and C it lies on the perpendicular bisector of DC. It also lies on BD produced. Therefore A is determined and consequently the problem is solved.
The solution of a problem in geometry consists in an analysis of the given conditions and the application of a few known locus theorems.
Fiq 4
Reverting now to the isosceles triangle having each angle at the base double of the third.angle, let us drop one of the conditions and find the locus of the vertex of a triangle which has a given base and one of the base angles double of the vertical angle. Let ABC (Fig. 4) be a triangle having BC fixed in magnitude and position and B=2A, it is required to find the locus of the vertex A.
Denote .BA by r, BC by a:
r sin BCA
Law of Sines:
a sin A
sin ACD sin CB+A) sin A sin A
University of Texas Bulletin
3B sin2 =---, since A=1/2B B
Sln
2
B 3 sin B/ 2-4 sin3 2
sin 3/2
B
=3-4 sin2 2
B =3-2 (2 sin2 -) 2
= 3-2 (I-cos B)
=1+2 cos B
hence r=a (1+2 cos B).
This is the equation· of the locus of A in polar co-ordinates, B being the pole and BD being the polar axis.
Plotting the locus for values of B from 0° to 180° we have the curve DNOP (Fig. 5). For values of B from 180° to 360° the resulting curve will be symmetrical to DNOP with respect to OD. To trisect an angle make at the point P in the axis OD an angle say NPD equal the given angle. Join N with the pole 0 then the angle ONP equals one-third of the given angle.
Mathematics Teachers' Bulletin
The equation of the curve in rectangular co-ordinates reduces to x2+y2-2ax=ayx2+y2•
This, of course, is a higher plane curve. In plane elementary geometry the only curves used are the. straight line and circle so our problem is outside the domain of the geometry of the point, straight line and circle.
PRACTICAL SOLUTION
The following is an easy practical solution of the problem of trisecting an angle. Iiet ABC (Fig. 6) be the angle to be trisected. With B as
center and any radius BC de~cribe a semicircle CAD. Place a set square along CDG and. mark from the end of a ruler a distance equal the radius BC. Slide the ruler along the set s<1uare keeping the edge ahrnys on the point A. When the mark on the rnler coincides with a point on the circle dra1;· .AFE the trace of the ruler in this position, then angle FBD equals one-third of angle ABC
vell use it in .a better understanding of algebra. And again, there are some teachers who teach the graph throughout the two years of high school work.
To do the most effective work in high school algebra, plotting should by all means be introduced in the eighth grade, after the student has had some work in simple equations.
Knowing that the line or curve is the geometric representation of an equation in the book is more illuminating to the eighth grade boy or girl than one might think.
A class should never leave simultaneous equations without knowing how to solve them by the g·raphic method. Knowing how to check the so1utions of simultaneous equations both linear and quadratic by the graph bring to the student ideas that are very helpful means to a good understanding of the subject. Do not wait until the last week in the second year of algebra to introduce this important subject, but give bits of it
Mathematics Teachers' Bulletin
throughout the two years, and you "·ill see many good results and an added interest on the part of your class.
After the pupils have learned something about the quadratic equation, introduce the graph to explain what is meant by the roots of an equation. Dy means of the graph a teacher can represent geoln etrically real and imaginary roots and thus give the pupil a clearer conception of solutions of quadratics.
D. D. B.\TIX, Grubbs Vocational College.
THE BROWN MATHEMATICAL· PRIZES
Of interest to high school teachers of mathematics are the· Brown :Mathematical Entrance Prizes offered to freshmen in the fall. In case some do not know about these ~rizes, I quote· from the catalogue. ''Out of gratitude and respect to his Alma Mater, an alumnus of Brown University offers the fol1owing prizes, known as the Brown University Mathematical Prizes, to be awarded annually by the staff of the Department of Pure Mathematics on the basis of competitive examinations. "Entrance Prizes. Three prizes are offered of fifteen, ten, and five dollars respectively, to the freshmen making the best grades on a special examination to be helid during the third week of October. The examination will cover the minimum entrance requirements in mathematics, elementary algebra and plane geometry.''
The following is the examination given on October 14, 1922: Time: One hour.
1.
Through a given point draw a line intersecting a given circle so that the distance from the points of intersection to a given line have a given sum.
2.
Describe a circ1e touching two given parallel lines .and passing through a given point.
3.
A man travels fifty miles by the train A and then after a wait of five minutes returns by the train B, which runs five miles an hour faster than the train A. 'l'he entire journey occupies two hours twenty-six and two-third minutes. What are the rates of the two trains1
4.
The circumference of the hind wheel of a wagon exceeds that of a fore wheel hy 8 inches, and in traveling one mile this wheel makes 88 less revolutions than a fore wheel. Find the circumference of each wheel.
Note: In the geometrical problems no discussion of the circumstance under which the desired construction is possible or impossible or the number of so1utions obtainable in the possible case is demanded. A description of the steps in the construction and a proof that the suggested construction does indeed solve the problem, .are all that is wanted.
Mathemaitics Teachers' Bulletin
Prize Winners :
1st. Mildred Taylor, Weatherford H. S., \Yeatherford, Texas. 2nd. Eugenia Rountree, Paris H. S., Paris, Texas. 3rd. Sophie Lubben, Francitas H. S., Francitas, Texas.
\\-HAT IS THE ~\IATTER \VITH HIGH SCHOOL ::\IATHK\IATICS?
It is a fact that more students in our uninrsities are failing
in mathematics than in any other one branch. It is now· time
for the high school teachers to ask themselYes the question,
"\Vhat is the matter ·with our world" The trouble lies with
us: the foundations we han laid are of sand.
\Vhen I heard a bright young college student sa:·, ."I don't know an:· Plane Geometry, I memorized mine for that was the onl:· '"ay I could 'get by,' " I thought, "some teacher has not done his duty."
Begin the subject by telling the child the history of geometry -let him realize that the people of Egypt discoYered and deYeloped this wonderful science because of its use to them in replacing their boundaries after the annual oyerfio,Ys of the Xile. Let the child know that geometry, like all other mathematics, is to deYelop the power of reasoning, that no memory 'rnrk is required ,y}th the probable exception of the axioms and eYen these should be show·n to be true. r se some simple deYice as: T''"o boys each ''"ith two pieces of crayon and giYe each of them t'Yo more ; then the class "·ill readil:· see that axiom one is true. In the same ''"ay, or by correlating ''"ith algebra, explain eYery axiom.
I find that a great many high school students cannot read; that is, they do not get the full meaning of the "·ords; therefore, it is necessary not only to explain definitions but also to demonstrate where possible with the figure, as in the circle, the angle, etc., and sho''" the pupil that the figure and definition agree.
\Ve are now ready to begin the proofs of the theorems. I find it a goo,i idea to state all propositions with an pdYCrb clause introduced by if or il'hen. If the proposition is not stated this ·way in the text, 'Ye restate it. The pupil is taught · that there are fiye steps in eYery theorem, execpt in the cases of a few in proportion, where no figure is 11eeded. Fir.~t, state or restate the theorem; second, draw the iigure by making it agree with the English in the statement uf the theorem; third, let the adYerb clause always tell :what is giYPn; fourth, haYe
Mathematics Teachers' Bulletin
the noun clause tell what is to be proved. Now only the fifth step, the proof proper, remains. Teach the pupil to recall the conditions he Ju.s had that ·wilh aid in the 11roof, for all uf the pronf depends on axioms, definitions, and prt!v10us propositions. Use the same five steps and same method of solving in originals that you do in theorems-have every ste;1 a11li every reason just as complete.
I find that it often helps the beginner to understand the conditions of the proposition if I make the 6gures of paper. Let us take the theorem. If two triangle have three sides of one equal respectively to three sides of the ;itlier, they al'e congruent. I cut from paper .two triangles with their three sides respectively equal. I explain to the class by doing it,-that my second triangle is placed beneath the first with their bases coinciding but vertices opposite. The line that com:ects these two opposite vertices is a crease in the paper. I then show the class onl1y the left half of the figure and t:rnt in an ereet po<>ition so they can easily see the triangle. I thm show them the right half in the same way, and finally I show thrm the whole figure at once; and they thoroughly understand the angles and sides we have discussed.
Of course, all this requires effort on the part of the teacher, hut if we are not enthusiastic about our work, if we do not :;ee the beauty hidden thoughts in our subjects, if >Ve are not anxious to have our pupils enjoy this same knowledge ancl beauty, if we .are too lazy to put forward every effo1·t in our power, remembering to make demands of the pupils a~ to study and perfect order without which no teacher can be successful, we are not true teachers of mathematics, and we can expect our pupils to be failures in college.
LILLIAN E. TUTTLE,
Palestine High School.
THE STRAIGHT EDGE
\Yhat has become of the old-fashioned boy who used to object to having the teacher show him how to do originals?
A GOOD :\IAXY high schools are putting the soft pedal on mathematics. \Yonder if this has any relation to how :MANY GOOD high schools there are.
Have you ever noticed how likely it is that a student who is doing well in mathematics is doing well in his other subjects 1 Is the converse true?
Did you eYer see anybody try to prove a mathematical theorem by raising his Yoice, pounding a desk, and rumpling up his hair? Xo, one who is sure of his facts does not need fireworks.
Don't you think some training in a field where the facts are non-controversial and where cold reason holds sway may be of some use to a citizen ?