MOTIVATION IN THE TEACHING OF HIGH SCHOOL MATHEMATICS by Selwyn Archibald M i l l e r MOTIVATION IN THE TEACHING OP HIGH SCHOOL MATHEMATICS toy Selwyn Archibald M i l l e r do A Thesis submitted f o r the Degree of MASTER OP ARTS invthe Department of PHILOSOPHY oo The University of B r i t i s h Columbia A p r i l , 1936 TABLE OF CONTENTS Chapter I. Importance of Motivation i n the Teaching of High School Mathematics Page Chapter II. Types of Motivation Desirable i n the Teaching of High School Mathematics^) 10' Chapter III. Motivation i n the Teaching of Grade IX Geometry 22 Chapter IV. Motivation i n the Teaching of Grade XII Geometry 40 Chapter V. Motivation i n the Teaching of Grade IX Algebra 69 Chapter VI. Motivation i n the Teaching of Grade XII Algebra 87 Chapter VII.. Chapter VIII. Appendix Some Experimental Evidence to Show the Effects of Motivation General Conclusions Series of Questions 5 Answers 101 107 113 Bibliography 139 i MOTIVATION IN THE TEACHING OF HIGH SCHOOL MATHEMATICS Table of Contents with Subheadings Chapter I »Importance of Motivation i n the Teaching of High School Mathematics^ Meaning of motivation; l i t e r a l meaning; comparison of dictionary meanings; meaning as applied to teaching; meaning as applied p a r t i c u l a r l y to the teaching of mathematics. Mathematics i s an e s s e n t i a l part of a person's education; statements by authorities. S k i l l i n , and a l i k i n g f o r mathematics i s a great asset to a student studying any branch of science. In B r i t i s h Columbia mathematics i s a compulsory subject u n t i l the second year of the university; an early d i s l i k e for the subject may r u i n a student's educational l i f e . Where mathematics i s an optional subject i t i s very frequently avoided; evidence. A great number of students select subjects which they do not want, solely to avoid mathematics. A d i s l i k e for mathematics keeps students away from such subjects as physics and chemistry. Mathematics i s a comparatively d i f f i c u l t subject, and when It i s not motivated only the better students w i l l grasp i t . A great deal of review work i s necessary i n mathematics, and unless some form of motivation i s adop-ted, t h i s review.may become very monotonous and uninteresting - constant practice i n mechanical processes and continual reference to fundamentals are essential -motivation i s necessary f o r t h i s . Many pupils, e s p e c i a l l y g i r l s , get preconceived ideas that they can not do mathematics - t h i s could be avoided to a large extent by proper motivation. The large percentage of f a i l u r e s i n matriculation examinations could be reduced considerably by introduc-t i o n of motivated forms of teaching i n place of the o l d mechanical methods. Motivation i s necessary i n order to open a new f i e l d of inte r e s t to many students - a new and d e l i g h t f u l experience i s i n store for any student who develops a l i k i n g for mathematics - t h i s l i k i n g may be developed by s k i l f u l l y motivated teaching. i i Summary - m o t i v a t i o n i n t e a c h i n g mathematics w i l l s o l v e many d i f f i c u l t i e s and g i v e new l i f e t o the s u b j e c t . Chapter I I . Types o f M o t i v a t i o n D e s i r a b l e i n the T e a c h i n g o f High S c h o o l M a t h e m a t i c s ^ 1. I n t e r e s t i n t h i n g s new - n o v e l t y o f b e g i n n i n g a s u b j e c t not t a k e n b e f o r e . 2. I n t e r e s t i n m a t e r i a l s t u d i e d - s i t u a t i o n s c o n n e c t e d w i t h p l e a s u r a b l e e x p e r i e n c e s . 3. D e s i r e f o r s e c u r i n g p r a i s e and a v o i d i n g shame - p r a i s e o f t e a c h e r , p a r e n t s , p u p i l s . 4. D e s i r e t o a v o i d d i s g r a c e . 5. D e s i r e f o r good marks. 6. D e s i r e f o r p r o m o t i o n . 7. I n t e r e s t i n c o m p e t i t i o n s - a g a i n s t each o t h e r and a g a i n s t t i m e . 8. D e s i r e f o r a c t i v i t y - p h y s i c a l . 9. I n t e r e s t i n games. 10. I n t e r e s t i n humor. 11. D e s i r e f o r v a r i e t y - change. 12. I n t e r e s t i n c o n s t r u c t i n g - b u i l d i n g . 13. The t h r i l l o f d i s c o v e r y . 14. E f f e c t o f s p e c i a l p r i v i l e g e s . 15. I n t e r e s t i n s u b j e c t f o r i t s own sake - a new and p l e a s u r a b l e e x p e r i e n c e . 16. D e s i r e t o answer a c h a l l e n g e . 17. D e s i r e f o r e f f i c i e n c y i n l i f e ' s work - i m p o r t a n c e o f t e a c h i n g f o r t r a n s f e r . 18. S a t i s f a c t i o n t h r o u g h mastery. 19. D e s i r e f o r p e r f e c t i o n . 20. S a t i s f a c t i o n t h r o u g h h e l p i n g . 21. D e s i r e to be c o n s i d e r e d mature - advanced. 22. D e s i r e t o complete c o u r s e chosen. 23. D e s i r e t o pass m a t r i c u l a t i o n e x a m i n a t i o n s . 24. E f f e c t o f p r i z e s and awards - s c h o l a r s h i p s . These forms o f m o t i v a t i o n v a r y I n t h e i r s u i t a b i l i t y t o the d i f f e r e n t grades - how g r a d u a t e d - how s t r e s s e d . C h apter I I I . M o t i v a t i o n i n the T e a c h i n g o f Grade IX Geometry^) P o s s i b i l i t y o f c a p i t a l i z i n g on the n o v e l t y o f t h e s u b j e c t t o arouse i n t e r e s t - a b r i e f o u t l i n e o f the p l a c e geometry has h e l d i n v a r i o u s c i v i l i z a t i o n s - a s u b j e c t f u l l o f i n t e r e s t . N e c e s s i t y f o r k e e p i n g s u b j e c t e x p e r i m e n t a l a t f i r s t - p l e n t y o f a c t u a l c o n s t r u c t i o n work by p u p i l -t e a c h i n g o f sound g e o m e t r i c a l p r i n c i p l e s i n c o n n e c t i o n w i t h t h i s e x p e r i m e n t a l work. N e c e s s i t y f o r a l l o w i n g p u p i l s o f grade I X p l e n t y i l l of opportunity for physical a c t i v i t y - methods of marking - comparing r e s u l t s . In beginning the study of straight l i n e s , practice i n drawing and measuring can be made int e r e s t i n g by pupils guessing the length of a l i n e already drawn, or by drawing a l i n e of s p e c i f i e d length without measuring -competitions i n t h i s f i e l d - employment of millimetres and centimetres as well as inches. Motivating the study of angles - stress s p e l l i n g -competitions i n naming angles. Practice with angles by drawing from guess and checking by measuring and vice versa. - Teacher draws angles ©n board and pupils guess sizes. Use of mariner's compass - pupils t e l l number of degrees between various points on the compass - boy scouts and g i r l guides should be encouraged to show what they know about the compass. Practice with l i n e s and angles by following com-p l i c a t e d directions - ship s a i l i n g to desert i s l a n d -finding the hidden treasure from dir e c t i o n s . Treatment of p a r a l l e l s may be motivated by keeping i t very l a r g e l y experimental - strange d e f i n i t i o n s f o r p a r a l l e l l i n e s - drawing"parallels by observation; t e s t -ing by drawing transversal and measuring angles -introduce o p t i c a l i l l u s i o n s with regard to p a r a l l e l s . U t i l i z a t i o n of the p r i n c i p l e of s a t i s f a c t i o n through discovery i n the treatment of triangles - pupils try to construct triangles with sides 3, 5, 7 i n * ; 3, 5, 8 i n * ; 4, 6, 11 c » m ' ete. Pupils t r y to construct /S^kBG having AB * 3"; /_B s 82°; £ A = 98°. Teaching of ..standard constructions can be motivated by examining numerous methods suggested by the pupils and selecting the best. e.g. drawing perpendicular to a given s t r a i g h t l i n e from a given point outside i t . Interest i n geometrical language may be created by having pupils t r y to explain i n words what they have done with the instruments - compare constructing geometrical figures to b u i l d i n g . Construction of more d i f f i c u l t figures can be motivated by getting pupils to consider them equivalent to performing some very i n t r i c a t e b i t of handiwork, e.g. drawing inscribed, circumscribed and excribed c i r c l e s of a given t r i a n g l e ; constructing a quadril a t e r a l using i t s diagonals i Employ p r a c t i c a l problems such as drawing crests for sweaters, i n l a y work f o r trays, etc. Very c a r e f u l l y prepared tests should be welcomed by the pupil and should make him eager to go on and learn more about the wonders of geometry. Mathematical recreations suitable for grade IX geometry. i v Chapter IV. Motivation i n the Teaching of Grade XII Geometry /f; Development of proper attitudes towards grade XII geometry 1. Certain pupils should be encouraged to regard them-selves as "budding mathematicians - higher regard of teacher for pupil encourages more earnest e f f o r t . 2. D e s i r a b i l i t y of looking ahead - lay i n g foundation for higher mathematics - frequent reference to advanced work. 3. Pupils should become more independent - less explana-tion by teacher - more time f o r the pupil for thinking - develop sense of r e s p o n s i b i l i t y . 4. Development of sense of s a t i s f a c t i o n through mastery -very essential - extreme - s a t i s f a c t i o n derived from obtaining a solution a f t e r hours of tr y i n g . 5. Development of habit of v i s u a l i z i n g geometrical figures - seeing solutions to exercises while one i s walking along street, s i t t i n g i n street car, or waiting to meet someone. 6. Satisfaction' derived from working with most complica-ted looking diagrams - pride i n one's a b i l i t y (example). 7. Desire for absolute perfection - development of pride i n perfect solutions (reasoning and form). f$ ; Motivation In methods of presentation of grade XII geometry 1. Less formal treatment of theorems - Teach theorem (often by analysis); - b u i l d i t up together - test following day - develop idea that once key i s given to a theorem (or exercise) i t i s solved once and for a l l time. 2. Careful treatment of exercises - a c a r e f u l l y selected exercise to be written out and handed i n each day -marked - returned - :marks recorded - monthly t o t a l s read - excellent review d a i l y . 3. Keep ideals of good form before pupils always - pass extra good solutions around class - good effects on writer and observer. (Example) 4. Value of mimeographed sheet of exercises done during year - pupils r e a l i z e t h e i r accomplishments. 5. Use of questions from previous examination papers -t e l l pupils year and grade. Special inte r e s t i n c e r t a i n questions. (Example) 6. Value of short exercises at end of theorem - taken i n d i v i d u a l l y on board - race for solution - bright pupils give h i n t s . . 7. Value of occasional objective tests - pupils enjoy them - marks encouraging - examples. 8. Use of special exercises for bright pupils, examples; isosceles to equivalent e q u i l a t e r a l bisect by l i n e through point outside ( i n base produced) etc. 9. After-school discussions - informal - encouraged. 10. Development of pupil-teacher idea - grade XII pupils as tutors to.grade X friends - b e n e f i c i a l e f fects on both pupils - li m i t a t i o n s of this system. 11. Mathematical recreations i n geometry. Chapter V. Motivation i n the Teaching of Grade IX Algebra 1. The novelty of algebra (using l e t t e r s f o r quantities as well as numbers) gives the pupi l an i n i t i a l i n t e r e s t i n the subject - i f properly motivated, i n t e r e s t can be maintained. .2. In f i r s t l i t t l e problems involving symbols, make the problems " r e a l " , e.g. problems on baseball, swimming, running, etc. 3. Some of the early algebra problems seem exceedingly advanced to the beginner - plenty of praise f o r solutions of such problems i s a great motivating force. 4. The teaching of substitution - care must be taken hot to make t h i s topic too complicated - r e f e r to i t again l a t e r - pupils get tangled up very e a s i l y i n t h i s type of question - make the most of the peculiar r e s u l t s obtained by substitution of unity and zero-. 5. The teaching of addition i n algebra can be motivated so as to make i t an exceptionally i n t e r e s t i n g section of the work. - Addition of like"terms, not unlike (apples oranges s ? (grapefruit?) ) - r e a l , l i v i n g problems can be given In addition, e.g. See Thorndike - Pupils enjoy working big-looking questions (examples). 6. Subtraction the source.of many errors i n algebra. Motivate the learning of the rul e -for changing lower signs; {e.g. f i l l i n g in/holes, wiping off, debts). Subtraction r e a l l y addition. What do you add to 5 to make i t four? 7. Motivating the teaching of rules f o r m u l t i p l i c a t i o n -m u l t i p l i c a t i o n i s addition (of indices) - use 3a 3 x 2a 2 etc. Rule of signs (two wrongs make a right). Explanation -of -3ax-3a a 9a 2 - taking away a debt 3 times - f i l l i n g 3 holes - l i v i n g problems involving m u l t i p l i c a t i o n - examples. 8. Motivating the teaching of rules for d i v i s i o n - reverse of m u l t i p l i c a t i o n - use 9a 9 - 3a 3 etc. - competition by rows i n long d i v i s i o n . Allow pupils to work long , long d i v i s i o n questions to f i l l whole page, e.g. a + b ) a 1 2 - b 1 2 - races with long d i v i s i o n . 9. Removal of brackets - explanation of rule ( r e a l l y addition or subtraction) - t h r i l l of getting complica-ted expressions down to a or o - competition by rows,-v i point ahead to equations. 10. Fundamentals i n algebra must be thoroughly understood, as a l l future work i n algebra involves the use of these fundamentals - at the beginning of each period a review of the previous lesson i s essential - e.g. f i v e short questions on the board, - how many r i g h t out of f i v e - pupils stand (relaxation). 11. Ample allowance for pupils going on ahead In algebra -extra books, special questions - review sets - old examination papers. 12. Mathematical Recreations suitable for grade IX algebra. Chapter VI» Motivation i n the Teaching of Grade XII Algebra General methods of motivation outlined i n Chapter II - s p e c i f i c means by which these methods of motivation may be brought into operation given i n present chapter. Development of intere s t i n the subject for i t s own sake - how this may be achieved - Three-unit cycle; unit for teaching, unit f o r review, uni t for coordination. 1 E f f e c t of matriculation examinations «- how matric-u l a t i o n examinations should be regarded - how they may be made a desirable motivating force. Minor methods of motivation: (a) Special treatment of algebraical problems - introduc-tion of puzzles. (b) Use of mathematical recreations - examples. (c) Friendly r i v a l r y between classes. (d) Emulation of past achievements. (e) E f f e c t of scholarships and prizes . Chapter VII* Experimental Evidence to Show the Results of ITS Increasing the Motivation bC the Teaching of High School Mathematics 1. Outline of experiment car r i e d out by writer. 2. Table of marks obtained i n Junior Matriculation Examinations. 3. Table showing the r e l a t i v e amounts of enjoyment derived from the study of the various subjects on the high school curriculum. 4. Table showing estimates of e f f i c i e n c y . Chapter VlII»Conclusions regarding the effects of increasing motivating in-the teaching of high school mathematics^ Appendix. Question l i s t s . Answers. Bibliography. v i i L i s t of Figures Figure I. Figure I I . Figure I I I . Figure IV. Figure V. Figure VI. Figure VII. Page Example of Complicated Diagram 44A Example of Complicated Diagram 44B Example of Attempt for Perfection . 44C Example of Attempt for Perfection 44D Example Showing E f f e c t of Praise .51A Example Showing E f f e c t of Praise 51B i Graphical I l l u s t r a t i o n s of E f f e c t s of Motivation 111 L i s t of Tables Page Table I. Table I I . Table of marks for Home Exercises 50A Table of Matriculation Marks f o r Experimental class 104 Table I I I . Table Comparing Averages made by Experimental Class with Other Averages. 105 v i i i MOTIVATION IN THE TEACHING OP HIGH SCHOOL MATHEMATICS CHAPTER I/a THE IMPORTANCE OP MOTIVATION IN THE TEACHING OP MATHEMATICS/.^ What i s motivation? In i t s l i t e r a l sense i t i s "the process of inducing movement;" and i n i t s broader sense i t i s "the process of producing stimuli which i n i t i a t e , d i r e c t and sustain a c t i v i t y (2). Motivation i n connection with teaching might he considered as the introduction of cer t a i n factors into teaching, which i n i t i a t e , d i r e c t , and sustain the a c t i v i t y of the in d i v i d u a l taught. When con-sidering motivation i n r e l a t i o n to the teaching of high school mathematics, we s h a l l consider the numerous ways by which a teacher may induce each pupil i n h i s class to put f o r t h the maximum amount of e f f o r t of which he i s capable, and to derive the greatest possible benefit from that e f f o r t . I t i s obvious » that, f o r various reasons, a great many pupils do not put fo r t h t h e i r best e f f o r t s i n the study of mathematics, and that they do not derive a l l the benefits which they might derive from the study of such a wonderful subject. (1) F. Goodenough - Developmental Psychology, P. 498. (2) A. Gates - Psychology for Students of Education, P. 182. 1 2 The p r i n c i p a l objects of motivation in. teaching are two i n number: 1. The creation of a maximum amount of int e r e s t on the part of the learner; f 1 ^ 2. The attainment of maximum e f f o r t by each p u p i l . - In the case of perfect r e a l i z a t i o n of this aim each pupil w i l l have reached an A. Q. of 100. I f these two objectives could be reached, the study of mathematics i n our high schools would take on a new lease of l i f e , and many of the c r i t i c i s m s l e v e l l e d at the teaching of mathematics would be made groundless indeed. In the following pages special reference w i l l be made to the forms of motivation suitable for grades IX and XII, with a comparison.of the types of motivation used i n these grades. The study of mathematics, when devoid of any motivating forces, becomes- a d u l l and uninteresting task, and yet mathematics i s an essential part of a person's education. There i s no substitute for i t , and no complete education without i t . ( 2) It therefore behoves the teacher of mathematics to bend every e f f o r t towards presenting the subject to hi s pupils i n such a way that they w i l l have motivating forces urging them on to th e i r greatest e f f o r t s , (1) The importance of creating i n t e r e s t i s discussed by M.J. Stormzand, and he arrives at the conclusion that "the pro-blem of int e r e s t plays such an important part i n education because success i n a l l teaching involves the arousing of s u f f i c i e n t i n t e r e s t . Progressive Methods i n Teaching, P. 129. (2) " A l l s c i e n t i f i c education, which does not commence with mathematics, i s , of necessity, defective at i t s founda-t i o n . " Comte. 3 and producing most b e n e f i c i a l r e s u l t s , That mathematics i s an es s e n t i a l part of ajperson 1s education i s almost axiomatical. "Our entire present c i v i l i z a t i o n , " says Professor Voss, "as far"as i t depends upon the i n t e l l e c t u a l penetration.!and investigation of nature, has i t s r e a l founda-ti o n i n the mathematical sciences." ( 2) The physical laws of the universe are so link e d up with mathematics that our innate desire to examine the laws of nature i n order to f i n d explanations for a l l the various natural phenomena, would be doomed to disappointment without a mathematical foundation on which to work. ( 3) W.A.Miilis, when discussing the value of mathematics as a high school subject, makes the statement that "for algebra there i s no substitute. The elimination of algebra as a pure science from the curriculum would cut the foundation from under a l l s c i e n t i f i c procedure." v ' I f mathematics, therefore, i s r e a l l y an essential part of a person's education, should we not bend every e f f o r t to so motivate the teaching of the subject, that each pupil would (1) "Eu c l i d has done more to develop the l o g i c a l f a c u l t y of the world than any book ever written. I t has been the i n s p i r i n g influence of s c i e n t i f i c thought f o r ages, and Is one of the cornerstones of modern c i v i l i z a t i o n . " -Brooks. S.I.Jones - Mathematical Wrinkles, P. 245-6. (2) A. Schultze - The Teaching of Mathematics i n Secondary Schools, P. 17. (3) riIt i s when we examine the r e l a t i o n of mathematics to science, both pure and applied, that we see most f o r c i b l y i t s i n d l s p e n s a b i l i t y as a propaedeutic" - Charles De Garmo - The Studies of the Secondary School, P. 65. (4) W.A.Miilis - The Teaching of High School Subjects, P. 240. 4 derive the greatest possible value from i t s study, and would apply himself to his work with an eagerness that would bring him to h i s highest l e v e l of e f f i c i e n c y . Another reason why i t Is so important that every teacher should make the most of every opportunity to motivate the teaching of mathematics i s that i n many provinces mathematics i s one of the compulsory subjects of the curriculum u n t i l the student has completed two years of un i v e r s i t y work. Whether th i s regulation i s a wise one or not i s not for discussion here, but as long as the requirements remain as they are, i t i s imperative that a student be so instructed i n mathematics i n high school that he w i l l develop a l i k i n g f o r the subject and not f i n d i t a millstone about h i s neck year after year. Many a student has had the joy taken out of h i s educational l i f e simply by developing an early hatred for mathematics, and had the subject been properly motivated for those pupils, that abhorrence i n many cases may have been e n t i r e l y eliminated. Motivation In mathematics i s p a r t i c u l a r l y important when dealing with students who are favorably Inclined towards s c i e n t i f i c study. I f a boy or a g i r l develops a l i k i n g f o r physics, chemistry, geology, astronomy or, i n i f a c t , any of the sciences, i t i s very important that he or she also develop a l i k i n g f o r mathematics. I f such a student i s given a bad impression of mathematics by lack of motivation i n i t s presentation, then he may abandon the study of a science i n which he i s r e a l l y interested simply because of the 5 mathematical calculations involved i n i t . W.A.Miilis i n "The Teaching of High School Subjects," points out that mathematics i s unpopular with a great many high school students and enumerates reasons for i t s unpopularity. One of his chief reasons i s that the subject i s more d i f f i c u l t than most other subjects, and i f pupils are given the option of another subject i n place of mathematics, many students select another (1) subject Instead of mathematics. v 7 Not only do pupils abandonsthee study of some of the sciences on account of the mathematics involved, but they select subjects In which they have no p a r t i c u l a r Interest, and for which they are not suitably adapted, for the sole purpose of avoiding the only other alternative - mathematics. Un-doubtedly a number of these students, at some l a t e r date, regret t h e i r actions when they suddenly r e a l i z e to. what extent they are handicapped by lacking a fundamental knowledge of mathematics. E.L.Thorndike made an investigation into the popularity of the various subjects on the high school curriculum. The voting was done by grade XII pupils i n the High Schools of New York City. In the f i n a l ranking of subjects, algebra ranked 13th. out of 22 with boys and 25th. out of 27 with g i r l s . Geometry ranked 16th. out of 22 with boys, and 26th. (1) W.A.Miilis, Opcit. Pp. 231-3. 6 out of 27 with g i r l s . t^1^ The obvious conclusion., to be drawn from the re s u l t s of these tests conducted by Dr. Thorn-dike i s that In the past the 'teaching of mathematics has not (o) been c a r r i e d out i n a wholly s a t i s f a c t o r y manner. v ' Mathematics i s apparently a subject which can become very froresome and d i s t a s t e f u l to students i f presented In a cold, pedantic manner; but on the other hand, i t may become a subject of r e a l and l i v i n g i n t e r e s t i f the teacher uses s k i l l f u l l y motivated methods of presentation. A further reason for the necessity of motivating the teaching of mathematics i n high school i s that mathematics i s r e a l l y one of the more d i f f i c u l t subjects of the curriculum.-A p u p i l of In f e r i o r i n t e l l i g e n c e can very often reach the required standard i n ce r t a i n subjects such as French, s o c i a l studies, grammar, geography and similar subjects by means of frequent r e p e t i t i o n and laborious memorization; but such a student has much more d i f f i c u l t y In reaching the required standard i n mathematics. The very nature of the subject demands that i t s teaching be motivated to the f u l l e s t possible extent, i n order to encourage those who are not espe c i a l l y bright to develop their mathematical s k i l l to the extreme (1) E.L.Thorndike, Psychology of Algebra, P. 386. (2) "One reason f o r the unsatisfactory status of the subject (mathematics) i s poor teaching." - W.A.Miilis, Opcit. P. 233. (3) "The extension of the elective system has revealed that pupils do not l i k e these subjects (Algebra and Geometry)"-Ibid., P. 232. 7 l i m i t of t h e i r a b i l i t y . Without such motivation mathematics Is to these pupils an exceedingly d i f f i c u l t subject which they w i l l drop at the f i r s t opportunity. Since the usual regula-tions for promotion, however, demand that every student, d u l l or bright, must reach a c e r t a i n s p e c i f i e d standard, i t becomes incumbent upon the teacher of mathematics to motivate h i s teaching i n such a way that the d u l l e r pupils as well as the brighter ones Y/111 develop a s u f f i c i e n t i n t e r e s t i n the subject to encourage each one to reach his highest l e v e l of e f f i c i e n c y . Another reason why i t i s so essential for the teacher of mathematics to motivate h i s teaching to the f u l l e s t extent, i s that i n t h i s subject a great deal of review work i s necessary and unless some form of motivation i s adopted, t h i s review work may become very monotonous and uninteresting. Constant practice i n the mechanical processes and continual reference to fundamentals are e s s e n t i a l , and yet constant r e p e t i t i o n i n "cut and dried fashion"may take the l i f e out of the subject for pupil and teacher a l i k e . Both i n algebra and i n geometry, every succeeding topic i s b u i l t up on the r e s u l t s of what has gone before. There i s no p o s s i b i l i t y of a student making a new s t a r t i n mathematics beginning at a cert a i n chapter unless he i s w i l l i n g ,to go back and master a l l the fundamentals upon which that chapter i s based. This state of a f f a i r s makes i t imperative that even from"the very f i r s t day when the study of algebra and geometry i s begun, the material should be 8 presented i n such a way as to create an earnest desire on the part of the pupil to master the subject i n every d e t a i l . Every teacher of mathematics finds at some time or other pupils i n h i s class who d i s l i k e mathematics because they have a preconceived idea that they are not mathematically i n c l i n e d . When such a pup i l i s asked why he d i s l i k e s the subject the usual response i s such as, "Oh, I can't do mathematics. I never was any good at i t . " Upon hearing such a statement, one cannot help but wonder what i t was that turned that pu p i l against mathematics; and the l o g i c a l sequence of thought i s to consider whether or not the reason for such d i s l i k e l a y i n the pupil's f i r s t introduction to the subject, which was very l i k e l y void of any form of motivation whatsoever. Cases such as these are a l l too numerous, and the u t i l i z a t i o n of the various forms .of motivation to a f u l l e r extent would undoubted-l y reduce the number of such cases considerably. Probably the most important reason why the teaching of mathematics should be motivated to i t s f u l l e s t extent, i s that i t i s the means of opening up a new f i e l d of inte r e s t to many students. A new and d e l i g h t f u l experience i s i n store for any pupi l who develops a l i k i n g for mathematics; an experience which may become the chief i n t e r e s t i n that student's educa-t i o n a l career. The opening to that new and d e l i g h t f u l experience i s made by the teacher who so motivates his teaching as to create a desire i n the student to know more about this wonderful science of mathematics. Even the study 9 of very elementary mathematics, when accompanied by the proper motivating forces, becomes the source of abundant pleasure, a type of pleasure which i s d i f f e r e n t from a l l other pleasurable experiences. The study of mathematics for i t s own sake - for the s a t i s f a c t i o n and enjoyment derived from i t s pursuit - i s one of the prime reasons for i t s i n c l u s i o n i n the high school curriculum and a student may be l e d to this new source of pleasure by s k i l l f u l l y motivated forms of presentation on the part of the teacher. For these several reasons enumerated above i t i s obvious-l y imperative for the o l d pedantic methods of i n s t r u c t i o n to be replaced by highly motivated forms of teaching, i f the subject of mathematics i s to hold the place It should hold and f u l f i l the objects i t should f u l f i l i n the educational l i v e s of i n d i v i d u a l s . (1) "We study music because i t gives us pleasure So i t i s with geometry. We study i t because we derive pleasure from contact with a great and ancient body of learning that has occupied the attention of master minds during the thousands of years i n whioh i t has been perfected and we are u p l i f t e d by i t . " D.E.Smith - The Teaching of Geometry. 10 CHAPTER I I . TYPES OF MOTIVATION APPLICABLE TO THE TEACHING OF HIGH SCHOOL MATHEMATICS Having observed the necessity f o r u t i l i z i n g motivating forces to t h e i r f u l l e s t extent i n the teaching of high school mathematics, It i s now necessary to consider what are the various forms of motivation which are applicable to the teaching of high school mathematics. It i s impossible to compile a l i s t and'say, "These are a l l the forms of motivation which may be used i n teaching high school mathematics;" because every p u p i l r i s d i f f e r e n t , and that which i s a motivating force to one student may have no eff e c t upon another student whatsoever. Moreover, the motives urging a pu p i l to do h i s best are d i f f e r e n t at the various stages of that ind i v i d u a l ' s school l i f e . A very powerful motivating force to a pupil when i n grade IX may have no influence whatsoever upon that same pupil when he reaches grade XII. However, there are certain forms of motivation which may be u t i l i z e d i n the various grades of high school, and the u t i l i z a t i o n of which may bring new l i f e and int e r e s t to the study of mathematics. ^ ) This present chapter (1) H.B.Wilson and G.M.Wilson, The Motivation of School Work, P. 47. In t h i s book there are enumerated eleven d i f f e r e n t types of motivation. A l l but two of these are applicable to the teaching of mathematics; - v i z . Earning money, and the a c q u i s i t i o n of a c o l l e c t i o n . 11 contains merely the enumeration of the various types of motivation which may be used i n teaching high school mathema-t i c s , and the s p e c i f i c methods by which they may be put into e f f e c t w i l l be given i n succeeding chapters. 1. A natural Interest i n new experiences. - When a pupil enters grade IX of high school, the novelty of the new subjects which he has not studied before has a great appeal to him. The wise teacher w i l l c a p i t a l i z e on th i s novelty, and i n his teaching of algebra and geometry w i l l t r y to give the pup i l a proper outlook towards these subjects at a time when the pupil i s eager to hear about the wonders of these new spheres of knowledge. 2. An Interest i n the in d i v i d u a l topics studies. - After an appropriate introduction to the subject, and a proper attitude towards i t has been created i n the mind of the pu p i l , i t i s essential that the in d i v i d u a l topics studied be s u f f i c i e n t l y v i t a l to the pupil to maintain the interest which has been aroused. I f the material studied i s closely associated with the pleasurable aspects of the pupil's exper-ience, then there i s a desirable motivating force acting upon the pupil at a l l times. 3. Desire for Praise. - This i s a very powerful motivating force which i s p a r t i c u l a r l y strong i n grade IX and continues to a large extent throughout a pupil's high school career. P r a c t i c a l l y every p u p i l i s encouraged to better e f f o r t s i f he knows that he w i l l receive the praise of his teacher, his 12 parents or his fellow pupils by making that extra e f f o r t . The s k i l f u l use of praise by the teacher can be made a very e f f e c t i v e motivating force,.not only for the b r i l l i a n t student but also f o r the d u l l a r d who i s trying to do h i s best. 4. Desire to avoid disgrace. - This motivating force i s a l l i e d very cl o s e l y to the one immediately preceding, but there are a great many pupils, e s p e c i a l l y i n the group of average i n t e l l i g e n c e , who are encouraged to better e f f o r t s by the fear of being u t t e r l y disgraced by f a i l u r e to reach a certai n standard or to gain promotion. I t i s not so much a desire for praise that urges these pupils on as i t i s the desire to avoid the shame which would come upon them should they f a i l to measure up to the standard which they believe they should be able to reach. 5. Desire for good marks. - This i s a motivating force which i s active throughout the grades of high school, but which can be made espe c i a l l y useful In securing greater e f f o r t from a pu p i l i n the e a r l i e r grades. Pupils i n grades IX and X prize their marks very highly, and a teacher who marks ju d i c i o u s l y can make a great deal of the motivating power of marks. Pupils l i k e to receive marks even f o r the smallest set of questions, and i f these marks are recorded and made the basis of careful comparison, the pupils f i n d an added interest even In d a i l y tests. 6. Desire for promotion. - This form of motivation i s present 13 i n a l l the grades of high school, hut i t i s more active toward the end of each school year than at the heginning. At the commencement of a term promotion time seems somewhat distant, and desire for promotion i s not a very strong motivating force, hut there are ways and means by which the teacher, may increase i t s power even from the beginning of the term. 7. Interest i n competitions. - The use of competitions of various kinds i s a very useful form of motivation, especial-l y i n grades IX and X. The majority of pupils i n these grades are very keen on competitions, both against each other and against time. There are numerous ways by which thi s form of motivation may be used to good advantage. 8. Desire for a c t i v i t y . - One of the dangers In a subject l i k e mathematics i s for i t to become too inactive, ^ i t t i n g at one's desk for a considerable time working a long series of questions does not appeal to the ordinary high school student. There i s a tendency to boredom which should be overcome by the introduction of more physical a c t i v i t y into the mathematics lesson. The s p e c i f i c methods by which thi s a c t i v i t y may be introduced w i l l be discussed i n a l a t e r chapter. 9. Interest i n games. - Pupils i n a l l grades of high school derive a great deal of pleasure from games. There are a large number of mathematical recreations which are admirably suited to high school students, and I f these are 14 c a r e f u l l y arranged and properly placed i n the mathematics lesson they w i l l have a wonderful motivating e f f e c t on the whole subject of mathematics. 10. Interest i n humor. - A mathematics teacher, and especially one who has been teaching the subject f o r a number of years, i s very often i n c l i n e d to become mechanical i n h i s methods of presentation, and overlook some of the p o s s i b i l i t i e s which exist for making a mathematics lesson r e a l l y enjoy-able. Admittedly, the number of opportunities for humorous i l l u s t r a t i o n s and analogies i s not as great i n a mathematics lesson as i n lessons i n many of the other sub-r j e c t s . nevertheless the teacher keenly Interested i n motivating h i s teaching to the f u l l e s t extent should make the most of every l i t t l e opportunity that arises f o r introducing even a s l i g h t touch of humor into the mathematics lesson. 11. Desire for change, - vari e t y . - Any experienced mathematics teacher knows that there i s a decided tendency on h i s part to present lessons of a similar nature i n almost i d e n t i c a l manners. In the teaching of geometry th i s tendency i s p a r t i c u l a r l y noticeable. The treatment of one theorem after another, or one exercise after another i n the same manner day afte r day i s certain!to become tedious to pupils. Pupils l i k e v a r i e t y . An essential form of motivation, therefore, e s p e c i a l l y i n the teaching of geometry, i s a variety of methods of presentation for lessons of a similar 15 nature. 12. Interest In constructing. - In gradeIX to a great extent, and i n the other grades to a smaller extent, pupils f i n d a great deal of pleasure In doing actual constructive work. They enjoy b u i l d i n g . There i s a very close analogy between constructive geometry and building, and i f t h i s f a c t i s f u l l y appreciated by the teacher, a great deal of the work In geometry can be motivated very highly by u t i l i z i n g the pupil's keen inter e s t In b u i l d i n g . 13. The t h r i l l of discovery. - "To discover t h r i l l s them," says R.W.Pringle when r e f e r r i n g to the nature of adolescents. This t r a i t of the adolescent i s active i n a l l grades of high school. When a pupil i n the f i r s t week of his study of geometry discovers for himself some new (to him) fac t about t r i a n g l e s , or when a matriculation student by h i s own e f f o r t s discovers the fundamental nature of an e l l i p s e , the t h r i l l of discovery i s s u f f i c i e n t to give that student an ardent desire to delve more deeply into the mysteries of mathematics. The part played by the teacher i s to start the pupil on the road to discovery and guide him at d i f f i c u l t crossings. 14. E f f e c t of special p r i v i l e g e s . - A pupi l i n any grade of high school l i k e s to think that he i s a s p e c i a l l y p r i v i l e g e d person. By this i t i s not meant that he l i k e s (1) R.W.Pringle, Methods with Adolescents, P. 126. 16 to be "teacher's pet," but that as a reward for some special e f f o r t on hi s part he i s allowed to emjoy some special p r i v i l e g e not enjoyed by the remainder of the clas s . There are a great many ways by which good work i n mathematics may be rewarded by special p r i v i l e g e s , and this type of motivation i s very e f f e c t i v e with certain types of students. 15. Interest i n mathematics for i t s own sake. - I f the teaching of mathematics has been properly notivated by various means during the early part of a pupil's high school career, there should come a time, probably at the end of grade X or the beginning of grade XI, when that pupil discovers that he r e a l l y enjoys the study of mathematics just f or i t s own sake. He may leave some • other things undone, but he w i l l not neglect h i s mathematics because of the enjoyment he derives simply by Its pursuit. When t h i s stage has been reached, and i t i s reached by a great many high school students, there i s added to the motivating forces already at work a new and extremely powerful one; one which may cause the pupil to devote h i s l i f e to the pursuit of mathematical knowledge. 16. E f f e c t of a challenge. - Most human beings, and especially boys and g i r l s with the red blood of youth i n the i r veins, respond very r e a d i l y to a challenge and do not rest content u n t i l that challenge has been answered. This characteris-t i c of human psychology makes i t possible for the teacher 17 of mathematics to encourage the pupils to put f o r t h t h e i r best e f f o r t s by making mathematical problems appear as d e f i n i t e challenges to t h e i r i n t e l l i g e n c e and ingenuity. In the advanced grades of high school the motivating force of a challenge issued by a cer t a i n algebraical or geometrical problem spurs the students on to e f f o r t s f a r beyond t h e i r customary l e v e l s . 17. Desire for e f f i c i e n c y i n l i f e ' s work. - Some students i n high school are encouraged to better e f f o r t s i n mathematical study because of t h e i r desire to go out into the world better f i t t e d for t h e i r l i f e ' s work by reason of t h e i r study of mathematics. I f the teacher points out to these students the various p o s s i b i l i t i e s of transfer from th e i r study of mathematics to t h e i r intended occupations or professions, then t h e i r desire for e f f i c i e n c y may become a r e a l motivating force i n the study of mathematics. 18. S a t i s f a c t i o n through mastery. - This form of motivation i s one of the most Important, i f not the most important, of a l l the forms which can be applied to the teaching of mathematics i n high school. I f , i n the early stages of his study of mathematics, a pup i l has the teaching of mathematics motivated for him so as to give him an early l i k i n g f o r the subject, he w i l l i n turn put f o r t h h i s best e f f o r t i n that subject, and most l i k e l y f i n d that he has succeeded i n mastering the early part of the work. The s a t i s f a c t i o n which that pup i l derives from the mastery of 18 hi s early work i s a compelling influence to further e f f o r t . A pup i l enjoys a subject which he understands well, and i f a teacher can u t i l i z e the various minor methods of motivation i n order to encourage a pupil to master each section of the work as he goes along, then the very f a c t of h i s mastery over a preceding section i s s u f f i c i e n t motivation to create i n him a desire to proceed to the following section. The motivating force of t h i s s a t i s -f a c t i o n gained through mastery Is indeed a powerful Influence In the teaching of high school mathematics. 19. Eagerness f o r perfection. - The object of a great many forms of motivation i n teaching i s to encourage the poorer students to put forth greater e f f o r t and a t t a i n a higher standard of e f f i c i e n c y . While t r y i n g to accomplish th i s aim, the fac t must not be overlooked that those pupils who have already attained a very high degree of e f f i c i e n c y might be encouraged to do even better work than they have done. One of the methods of motivation which applies p a r t i c u l a r l y to thi s top-ranking class of pupils, i s the creation of an eagerness for perfection. Mathema-t i c s i s one subject In which perfection can be reached i n a great many cases, and i f the teacher can create an eagerness on the part of the pupil to a t t a i n absolute perfection i n h i s work, then he w i l l be urging that pup i l to extend himself to the l i m i t of h i s a b i l i t y and a new Interest w i l l be added to that pupil's work i n mathematics. 20. S a t i s f a c t i o n through helping. - When a student reaches grade XI or XII he has gained s u f f i c i e n t knowledge of elementary mathematics to enable him to o f f e r some assistance to pupils i n grades IX and X. I f opportunities are provided by the teacher f o r the o f f e r i n g of t h i s assistance, then the senior grade p u p i l w i l l s t r i v e to understand his work more thoroughly i n order to be able to teach h i s friends i n the lower grades more s k i l f u l l y . A student gets a great deal or s a t i s f a c t i o n out of helping other students, and at the same time he i s strengthening h i s own grasp of the subject by constant review of fundamentals. 21. Desire to be considered mature. - In the upper grades of high school most pupils l i k e to believe that they are getting on into advanced mathematics, and that they w i l l soon be blossoming into mature mathematicians. A c e r t a i n amount of encouragement to the adoption of this a ttitude can be given by the teacher to very good e f f e c t . I f a teacher treats his matriculation students as mature persons of whom much more i s expected than Is expected of pupils i n the lower grades, then those matriculation; students w i l l respond and endeavour to show that they are indeed advanced mathematicians capable of concentrated e f f o r t . 22. Desire to complete the course selected. - One or the motivating forces urging students i n the upper grades of 20 school to work d i l i g e n t l y i s the desire on the part of the p u p i l to complete the course which he has selected. Most pupils i n these grades look forward eagerly to the time when they w i l l be graduated from high school, and be able to go out into the world with a successfully completed high school course behind them. The a n t i c i p a t i o n of t h i s future pleasure has a valuable motivating e f f e c t on the student's present e f f o r t s . 2 3 • Eagerness to pass matriculation examination. - For pupils who are i n the f i n a l year of high school and who are taking the matriculation course, there i s one very powerful motivating force at work; namely, the desire to pass the matriculation examinations. The motivating force of matriculation examinations i s not always the most desirable form of motivation, but, as w i l l be. seen i n a l a t e r chapter, a proper attitude toward matriculation examinations may r i d them or most or t h e i r undesirable e f f e c t s , and transform them Into admirable motivating forces, not; only for pupils i n the matriculation grade, but also for pupils working up towards i t . 24. E f f e c t of prizes and scholarships. - The motivating power of prizes and scholarships i s obviously very l i m i t e d i n i t s scope. I t i s only those students who are exceptionally good at t h e i r work who have any i n t e r e s t i n scholarships, and usually these rewards are offered only to matriculation students. On account of t h e i r narrow scope, and also on 21 account of the fact that the working for the reward may rob a subject of much of i t s r e a l value, the o f f e r i n g of prizes and scholarships can not be considered one of the valuable forms of motivation for the high school student. To t h i s l i s t of the various types of motivation a p p l i -cable to the teaching of high school mathematics, there might be added a great many other forms of motivation which are more li m i t e d i n t h e i r scope or which are applicable only to special types of students. The foregoing l i s t , however, contains most of the more general forms of motivation which are applicable to students attending high school. In the following chapters the s p e c i f i c means by which these types or motivation may be applied to the teaching or algebra and geometry i n grades IX and XII w i l l be discussed. 22 CHAPTER III MOTIVATION IN THE TEACHING OP GRADE IX GEOMETRY To the teacher of geometry i n grade IX a wonderful opportunity i s presented; an opportunity for opening up to h i s pupils a new f i e l d of knowledge which i s f u l l of i n t e r e s t and enjoyment. It becomes incumbent upon the teacher to make the most of t h i s opportunity Dy so motivating h i s teaching that a l l the intere s t and enjoyment latent In the subject of geometry i s discovered by the pupils under h i s care. But what are the various methods of motivating the teaching of grade IX geometry i n order to achieve t h i s object? In the f i r s t place, the teacher can c a p i t a l i z e upon the novelty of the subject. People are inherently interested i n things new, so while the pupils are In the proper frame of mind the teacher can inform them of some of the wonders of th i s science of geometry, and give the pupils a proper outlook towards the subject. An introduction might include reference to some of the i n t e r e s t i n g features about aefew?.of the world's great mathematicians such as Pythagoras, E u c l i d and E i n s t e i n , and also give some i n d i c a t i o n as to the extent to which modern c i v i l i z a t i o n i s b u i l t up on a mathematical basis. After the pupil's inte r e s t has been aroused i n the subject, i t i s essential that t h i s i n t e r e s t be maintained by s k i l f u l l y motivated teaching. There i s a danger i n grade IX 23 that the novelty of geometry may wear o f f before the pup i l has developed a r e a l i n t e r e s t In the subject f o r i t s own sake. One of the means by which t h i s l a s t i n g i n t e r e s t may be aroused even at a very early stage i s by keeping the f i r s t part of the work mostly experimental. A pup i l i n t h i s grade does not l i k e to s i t s t i l l i n his seat and watch constructions being done on the blackboard by the teacher; but he wants to do the con-struc t i v e work himself. The blackboard explanations by the teacher should be as short and concise as i s conveniently possible, and then the pup i l should be allowed to experiment for himself. Although the early part of the study of geometry should be l a r g e l y experimental and of a constructive nature, never-theless i t i s important that sound geometrical p r i n c i p l e s be taught i n conjunction with these constructive exercises. I f t h i s i s not done, the pup i l w i l l f i n d out i n a very short time that his knowledge of geometry has been b u i l t up on a rather feeble foundation; and consequently he w i l l lose that early i n t e r e s t which he had i n the subject. He w i l l miss the r e a l enjoyment which would have been i n store for him had he b u i l t his geometrical knowledge on sound mathematical p r i n c i p l e s . In order to l a y a good foundation i t i s not necessary for the pupi l to memorize l i s t s of d e f i n i t i o n s , axioms and postulates; but rather he should be directed towards gathering accurate information about fundamental geometrical f a c t s . For example, i t i s e s s e n t i a l that a pup i l know exactly what i s meant by 24 " v e r t i c a l l y opposite angles 1 1 and what r e l a t i o n they bear to each other; but what p r o f i t would there be i n demanding that he should be able to r e c i t e the d e f i n i t i o n for v e r t i c a l l y opposite angles? One very important f a c t to remember i n motivating the teaching of grade IX geometry or algebra i s the f a c t that the pupils i n this grade require a c e r t a i n amount of physical a c t i v i t y ; - opportunity for actual movement of body and limbs. They are not capable of sustained e f f o r t i n concentration over a very long period of time, but they require opportunity for p e r i o d i c a l physical a c t i v i t y . This need i s supplied i n part by the introduction of constructive exercises i n which the pupils a c t u a l l y do the work. I t i s s a t i s f i e d to some extent also by extensive use of the blackboard by the pupils; but there Is s t i l l another very valuable means of supplying physical relaxation for the pupils, and at the same time s a t i s f y i n g t h e i r desire for competitive forms of a c t i v i t y . A series of short questions (ten for example) i s placed on the blackboard and the pupils are instructed to work the questions and on completion to turn t h e i r books face down on the desk. The f i r s t twenty pupils f i n i s h e d write t h e i r names on the blackboard i n order as they f i n i s h . When a l l pupils have f i n i s h e d the questions, books are exchanged and marks are assigned to the questions. Extra marks are a l l o t t e d to stu-dents who answer a l l questions c o r r e c t l y and also f i n i s h i n time to get their names on the board. These bonus marks are graduated according to the order of the names on the board. 25 Books are returned and r e s u l t s are compared i n the following manner: A l l the pupils stand, and as the teacher says, "One right 1'; "two r i g h t " ; "three r i g h t " , etc., pupils with the corresponding number of correct solutions take t h e i r seats. When "ten r i g h t " i s about to be reached, only those with perfect solutions are l e f t standing, and these can be ranked i n the order i n which t h e i r names appear on the blackboard. This at f i r s t glance appears to be a very ordinary b i t of teaching procedure, but on closer analysis i t w i l l be found to contain several valuable features which are very e f f e c t i v e forms of motivation. In the f i r s t place, the pupils who succeed i n getting most of the questions correct f e e l a cer t a i n sense of mastery over the work that has been covered and they w i l l attack new work with confidence. Also, the very best students w i l l be s t r i v i n g for perfect solutions In order to be able to continue standing u n t i l the l a s t . Besides t h i s , i f the questions are c a r e f u l l y graded, some of them being comparatively easy, then even the poorest pupils w i l l f i n d that they have developed a c e r t a i n amount of s k i l l , and on the next occasion they w i l l endeavour to remain standing for a longer time. No p u p i l wants to be the f i r s t to have to take his seat, so even the very poorest pupil i n the class has a very strong motive for trying to improve h i s work. The very fact that a l l the pupils i n the classi;have been standing for a few minutes, and that twenty of them have made t r i p s to the blackboard, provides an opportunity for relaxa-26 t i o n of muscles which become t i r e d from maintaining a s i t t i n g posture. This simple method of procedure, therefore, i s extremely useful as a motivating force i n the teaching of grade IX geometry. The foregoing methods for motivating grade IX geometry are a l l of a general nature, but the follow-ing are a 'number of s p e c i f i c methods by which c e r t a i n topics may be made much more i n t e r e s t i n g and valuable to the p u p i l . At the very beginning of the course i n geometry, when commencing the study of the straight l i n e , practice i n drawing and measuring can be made extremely i n t e r e s t i n g by u t i l i z a t i o n of the pupil's i n t e r e s t i n estimating or guessing. Questions such as the following prove very i n t e r e s t i n g to grade IX students just commencing the study of geometry: 1. By using the back of your r u l e r , draw a l i n e which you believe to be 7 i n . long. Turn the r u l e r over and measure. How many are within 1/16 of an inch? how many within 1/81'? within 1/4"? within 1/2"? etc. 2. Using the longest side of your set square, draw a l i n e which you believe to be 4|r i n . long. Measure. How many are within 1/16 of an Inch? etc. 3. Draw two columns, one for the estimated length and the other f o r the measured length i n each of the following. Use the longest side of your set square for drawing, and your r u l e r for measuring. (a) Draw a l i n e of any length. Estimate i t s length i n inches ( i n proper column). Measure. Record measure-ment. (b) Draw a l i n e which you believe to be 5 i n . long. Measure. (c) Draw a l i n e of any length diagonally on the page. Estimate i t s length i n Inches and In millimetres. Measure. (d) Estimate the length of your geometry book i n c e n t i -metres and millimetres. Measure. (e) Estimate the width of your desk i n inches. Measure. Mark each aftswer correct which i s within % i n . or 5 m.m. of the measured length. Series of questions s i m i l a r to the above may be arranged In competitive form and done from the 27 "blackboard. After the study of the straight l i n e the study of angles i s commenced, and there are numerous ways by which the teaching of this topic can be very e f f e c t i v e l y motivated. During the introduction to the topic, the teacher may ask the pupils to write down the following sentence: "Geometry teaches us to b i s e c t angles," and then see how many have the words "geometry11, "bisect" and "angles" spelled c o r r e c t l y . Mention might be made of the l i t t l e boy who wrote "Geometry teaches us to bisex angels." After the method f o r naming angles has been explained, s k i l l i n naming angles correctly can be developed i n an i n t e r e s t i n g manner by seeing who can name a l l the angles (not r e f l e x ) on a fig u r e such as the accompanying one. Each angle must be named only once. Practice i n drawing angles of various sizes, measuring angles already drawn, and developing a good idea of angular sizes can be given i n a rather i n t e r e s t i n g manner by using a method similar to that f o r giving practice with straight l i n e s . After explaining the use of the protractor, a series of questions such as the following may be given and the pupils instructed to keep a record of the ones which they get correct: 1. Draw any angle ABC. Estimate i t s size In degrees. Measure. 28 2. Draw an angle which you believe to be 65 . Measure. 3. Of the two angles on the blackboard which i s -the larger? How many degrees larger? (Angles such as the following may be drawn; - the smaller angle having the longer arms.) 4. Draw any acute angle on your paper. Draw another angle which you believe i s exactly twice the f i r s t . Measure both angles. Multiply size of f i r s t angle by two and compare with the size of the second. 5. Draw a figure on your paper similar to the one on the blackboard. (Accompanying). Estimate the size of each of the following angles: t AOB - /BOG -"BOD s /BOE -"DOE s Ty°A = "AOE z 7"AOC S Ch~eck by measurement". In marking the above questions, three marks are allowed i f the answer i s correct to within 1°; two marks i f within 2°; and one mark i f within 3°. Marks are t o t a l l e d and compared by the standing method. In connection with the study of angles, use of the mariner's compass can be made to good advantage. After a short discussion of the mariner's compass, i t s structure and use, the teacher may enquire i f there are any Boy Scouts or G i r l Guides i n the c l a s s . (Members of these organizations are supposed to know the thirty-two points of the compass.) If there are any present they may be allowed to display t h e i r knowledge by drawing a diagram of the mariner's compass on the blackboard. Prom one of these diagrams much practice can be 29 given i n angular sizes by asking questions such as the following: 1. A boat Is s a i l i n g due north, and then changes i t s course to N N VV. Through how many degrees does the keel of the boat turn? 2. Two boats are approaching the same port from d i f f e r e n t d i r e c t i o n s . One i s coming from a N by N E d i r e c t i o n and the other from a N by N W d i r e c t i o n . How many angular degrees are there between the l i n e s i n d i c a t i n g t h e i r routes? Many other similar questions on angles may be asked using the mariner's compass as a basis, and many in t e r e s t i n g facts about navigation may be brought into the discussion of t h i s topic. • This a l l adds i n t e r e s t to a lesson on angles, and i t i s a t y p i c a l way of motivating what might otherwise be a rather abstract and uninteresting lesson. Another p r a c t i c a l method by which in t e r e s t may be aroused i n the study of l i n e s and angles Is by making use of the pupils' sense of s a t i s f a c t i o n at being able to follow a number of rather complicated directions and arrive at the proper destination. Finding the hidden treasure or catching the t h i e f are forms of t h i s type of motivation. A question such as the following Is f u l l of in t e r e s t to a grade IX pu p i l just beginning the study of geometry: A jewel t h i e f stole a diamond r i n g and was caught by the police after pursuing him over the following course. Can you follow them? How f a r from the scene of the crime was the t h i e f when caught? (1 mile - 1 In.) From the scene of the crime he travels 1^ miles N.E.; then 1 mile due N.; from there § mile N,W.; he then turns and goes 1^ miles S.W.; then 3 miles due E. He turns again and goes 7/8 of a mile S.S.W.; and f i n a l l y he turns due E. and goes 1 mile before being caught. Questions such as t h i s can e a s i l y be made competitive by 30 seeing who can catch the t h i e f f i r s t , or who can f i n d the hidden treasure. The introduction to the study of p a r a l l e l l i n e s i s very often a rather d i f f i c u l t lesson i n which to arouse the int e r e s t of the pupils to any great extent. The app l i c a t i o n of care-f u l l y planned methods of motivation, however, may change t h i s s i t u a t i o n considerably and convert the lesson into one of exceptional i n t e r e s t to the p u p i l . After a discussion of the word " p a r a l l e l " - i t s s p e l l i n g , meaning and application to straight l i n e s , common examples of p a r a l l e l l i n e s may be taken, such as the two edges of the desk; the two edges of the blackboard, or the two edges of a book. D e f i n i t i o n s f o r p a r a l l e l l i n e s may be suggested by the pupils, and In t h i s connection the teacher may repeat some rather humorous de f i n i t i o n s received on examination papers at some time or another. The following are examples of such d e f i n i t i o n s : " P a r a l l e l l i n e s are l i n e s which run along together side by side l i k e the car tracks, sometimes for miles, and never converse." " P a r a l l e l l i n e s never meet unless you bend (2) one dr. both of them." "A p a r a l l e l l i n e Is one that when produced to meet i t s e l f does not meet." When the meaning of " p a r a l l e l l i n e s " i s p e r f e c t l y under-stood by the pupils, they may be given questions such as the (1) Received on Grade IX examination paper i n 1934, Britannia High School. (2) ( 3 j Alexander Abingdon, The Omnibus Boners. P. 69. 31 following: 1. On unruled paper draw two straight l i n e s across the page about two Inches apart, and which you believe to be p a r a l l e l . Test by measuring their distance apart at several places. 2. Draw two l i n e s diagonally across the page which you believe to be p a r a l l e l . Test as before. 3. Draw a l i n e across the page. Draw two other l i n e s which you believe to be p a r a l l e l to the f i r s t one - one about an inch above and the other about an inch below the given l i n e . Test the two new l i n e s to see i f they are p a r a l l e l . To questions such as these the teacher may add exercises from the blackboard, such as getting the pupils to say whether l i n e s drawn on the blackboard are p a r a l l e l or not, and i f not p a r a l l e l which way they converge. Interest may be added to the lesson by the introduction of c e r t a i n o p t i c a l i l l u s i o n s , involving p a r a l l e l l i n e s . The following are some examples of the type-or o p t i c a l i l l u s i o n s which are suitable at t h i s stage: 1. Are the two l i n e s i n the following diagram p a r a l l e l ? 3. In the following diagram, which i s longest, AB or BC or 2. Which or two l i n e s below i s the longer? CD? (1) Morgan, Foberg, Breckenridge> Plane Geometry. P. 13. 32 After the essential facts regarding p a r a l l e l s and transversals have been studied, the inaccuracy of the former method of te s t i n g p a r a l l e l s may be pointed out and a more s c i e n t i f i c method substituted; namely, the measuring of a pair of alternate or corresponding angles. The old method of measuring the distance between the l i n e s w i l l then be abandoned and the new method adopted. As a l i t t l e recreation at the end of a lesson on p a r a l l e l s , the following sentence, suggested by the word "p a r a l l e l 1 ' , may be given to the pupils as a s p e l l i n g t e s t . "In a cemetery an embarrassed cobbler and an harassed peddler were gauging the symmetry of a lady's tomb-stone with unparalleled ecstasy. 1 1 When approaching the study or tr i a n g l e s , the teacher i s offered a great opportunity for allowing the pupils to experience the " t h r i l l of discovery", and In t h i s way increase th e i r i n t e r e s t and s a t i s f a c t i o n i n the study of geometry. There are a great many facts about tr i a n g l e s with which the pupil i n grade IX i s not f a m i l i a r ; and i f the material i s presented i n such a way as to allow the pupil to discover these facts for himself, then he w i l l derive a great deal of s a t i s f a c t i o n from so-doing. Some of these facts which a pupil may be l e d to discover for himself i n an experimental 33 way are as follows: 1. The sum of the angles of a t r i a n g l e Is equal to 180°. 2. Any two sides of a t r i a n g l e are together greater than the t h i r d side. 3. I f the sides of a t r i a n g l e are produced i n order, the sum or the exterior angles so formed i s 180°. 4. I f one side of a t r i a n g l e i s produced, the exterior angle so formed i s equal to the sum of the i n t e r i o r opposite angles. 5. The bisectors of the angles of a t r i a n g l e meet at a point. 6. The perpendicular bisectors of the sides of a t r i a n g l e meet at a point. 7. The three medians of a t r i a n g l e meet at a point. (The pupils may experiment with paper tria n g l e s to see I f the point at which the medians meet i s tne centre of gravity.) 8. In a right-angled t r i a n g l e , the mid-point of the hypotenuse Is equidistant from the three v e r t i c e s . 9. In a 30°, 60<>, 90° t r i a n g l e the longest side i s double the shortest. 10. The area of a t r i a n g l e i s less than the area of a square with the same perimeter. The means of motivating the teaching of the standard constructions i n grade IX geometry are many and varied. The main types of motivation which are applicable to t h i s phase of the work are (1) The pupil's i n t e r e s t i n b u i l d i n g and construction work; (2) The pupil's desire for physical a c t i v i -ty; and (3) The pupil's i n t e r e s t i n things new. His i n t e r e s t i n b u i l d i n g can be used to advantage here by attacking each problem experimentally - a c e r t a i n figure has to be put together," can the p u p i l f i n d a method for doing i t ? A straight l i n e has to be erected perpendicular to another straight l i n e from a given point i n that l i n e . How can i t be done? Can the pup i l discover a method? Vi/hen experimentation i s over and r e s u l t s are compared,1 the comparative values of the various methods adopted by the pupils may be considered, and the best ones studied more c a r e f u l l y . 33.a The physical energy expended by the p u p i l i n working these constructions i s a very important part of the grade IX pupil's a c t i v i t y . Without periodic opportunities f o r physical a c t i v i t y , the pupi l Is l i k e l y to become r e s t l e s s and uneasy. Most youths f i n d i t d i f f i c u l t to keep s t i l l f or any great length of time and opportunities for muscular movement such as the one just mentioned allow the youth to s a t i s f y h i s desire for physical action. In the teaching of standard constructions i t i s well f o r the teacher to remember that these are new revelations to the grade IX student. The very novelty of such work i s an incentive to the pupil to learn. In the manner of expressing i n words the various methods of construction, there i s a certain amount of.novelty also; and i n t e r e s t i n t h i s phase of the work may be increased by seeing which pupils can express i n the clearest andmo>st concise form the actual work c a r r i e d but i n the process of construction. The pupils may be encouraged i n developing an eagerness to be able to express themselves i n correct geometrical language. They may be t o l d that a c e r t a i n professor wired from Vancouver to New York for the publishers to hold up the publication of h i s new book u n t i l i a c e r t a i n change was made. He wished one of the questions which began "P Is a point In the l i n e AB" to be changed to "The point P i s i n the l i n e AB" because i t i s bad form to begin a l i n e with a symbol. This shows the exactness of (1) Information obtained from Professor himself. 34 the science of mathematical expression, and the development of s k i l l i n t h i s science sometimes fascinates the young people beginning the study of geometry. When the e s s e n t i a l constructions have been mastered, pupils may be allowed to construct from s p e c i f i c a t i o n s a few of the more d i f f i c u l t figures involving combinations of the basic constructions. In t h i s part of the work the p u p i l should be given the impression that he i s now t a c k l i n g an I n t r i c a t e b i t of handiwork, and i f he i s successful he has developed a certain amount of geometrical constructive a b i l i t y Problems suitable for t h i s purpose are ones such as drawing the inscribed, circumscribed,and escribed c i r c l e s to triangles constructing quadrilaterals necessitating the use of diagonals and the development of geometrical designs suitable for crests Inlay work, linoleum and t i l i n g patterns. As mathematical recreations In connection with t h i s phase of the work, the following puzzles are very suitable: 1. A farmer had h i s prize sheep i n six pens, constructed of t h i r t e e n sections of fencing, as follows: Somebody stole one section of the fence, so the farmer rearranged the remaining 12 sections so as to s t i l l have six pens, a l l the same size and shape. How did he do i t ? 2. Given a piece of cardboard 15 Inches long and 3 inches wide how i s i t possible to cut i t so that the pieces when re-arranged s h a l l form a perfect square? I f tests are given at various stages of the work i n grade 35 IX geometry, i t i s esse n t i a l that the pupils develop a correct attitude toward them. The pupil should regard them as opportunities for showing h i s constructive and reasoning a b i l i t y . In each test he has the p r i v i l e g e of performing more new and i n t e r e s t i n g constructions, as well as the opportunity for reasoning out certai n mathematical problems. I f the teaching of the work has been motivated so as to arouse the pupil's i n t e r e s t i n the subject, and i f the tests a r e . s k i l -f u l l y arranged so as to follow up that motivated form of teaching, then the pupil w i l l indeed regard these tests i n the proper l i g h t and look forward with eagerness toward them. The following are some examples of tests designed to allow the pupil an opportunity of showing h i s constructive and reasoning a b i l i t y , and from the working of which a pu p i l may derive a great deal of s a t i s f a c t i o n and enjoyment. 1. Give d e f i n i t i o n s of the following terms. I f you cannot give d e f i n i t i o n s , explain each term c l e a r l y . (a) Acute angle (b) Obtuse angle. (c) Adjacent angles. (d) Supplementary angles. (e) P a r a l l e l l i n e s . ' " 2. Name as many angles as you can from the following diagram. Name only angles containing less than 180°. Measure to the nearest degree the size of each angle. \C 36 3. (a) Draw / AOB - 42°; at 0 make /J30C - 53°; at 0 make / COD »,37 G; at 0 make / DOE ~ 48°, making each angle adjacent to the one immediately preceding, (b) Without protractor, how could you test quickly to see whether the f i n a l r e s u l t i s accurate or not? 4. (a) Draw a straight horizontal l i n e AB 3" i n length. Mark X the mid-point of AB. At X, make / BXY • 50°, drawing t h i s angle on the upper side of AB. Make XY s 1" i n length. At Y, make / XYZ alternate to / YXB; and equal to 50°. Make YZ = 2^~"in length. At poinTT X, make /BXH s 70°, drawing th i s angle on the lower side of AB. Make XH s 1" i n length. At H, make / XHK, alternate t© /BXH, and equal to 70°. Make HK • 2? In length. (b) wEat r e l a t i o n exists between the straight l i n e s ZY and AB? Give reason f o r your answer. (c) What r e l a t i o n exists between s t . l i n e s AB and KH? Give reason f o r your answer. (d) What r e l a t i o n exists between s t . l i n e s ZY and KH? Give reason for your answer. (e) By using r u l e r and protractor only, how could you tes t t h i s r e l a t i o n s h i p between ZY and KH? 5. 6. In the diagram above, AB and CD are two p a r a l l e l l i n e s . (a) What name Is given to st. l i n e XY? (b) Name a pair of alternate angles. (e) Name a pair of corresponding angles. (d) Name a pair of v e r t i c a l l y opposite angles. (e) Name a pai r of adjacent angles. Name a pair of supplementary' angles. I f / a - 112, what Is size of / f? Do not obtain your answer by measuring. ' (f) (g) In the above diagram CX i s p a r a l l e l to BA. (a) Name any pairs of angles which you know to be equal, giving the reason i n each case. 37 (b) I f / A s 85° and /CD - 48 , without measuring f i n d the size of / ACB. Show your method of c a l c u l a t i o n . 7. 10° HP "B N In the accompanying diagram. AD i s perpendicular to BC. I f / DAC - 45 and / B - 30°, f i n d without measuring the size of each of the angles ACE, BAC, FCE. Show your method of c a l c u l a t i o n i n each case. 8. (a) Representing 1 mile by h a l f an inch, draw a diagram to I l l u s t r a t e the following journey, using directions as indicated by the accompanying arrows: A man starts from a place A and walks to a place B which i s 3 miles due east of A. He then walks from B to a place 1 >e c W h i c h i s 4 miles north-west. of B. Prom G he goes to D which i s 2 miles north-east of C. From D he goes 3^ miles due south to E. From E he walks 5 to place F, which i s 5 miles south-west of E. From F he goes 2^ miles due east to a place G. (b) Measure the d i r e c t distance from A to G, and t e l l how many miles G i s distant from A. , An inherent inte r e s t i n games i s one of the motivating forces which may be u t i l i z e d extensively throughout the entire high school course. The following are a few mathematical recreations connected with the work of grade IX geometry i n some way. The judicious use of these recreational problems may add much intere s t to the study of geometry for many of the pupils. 38 1. A man who owned a piece of land i n the form of a square, died and l e f t the property to h i s wife and four sons i n the following way: The wife was to receive the quarter i n the corner of the square where the house stood, and the remaining three-quarters was to be divided evenly among the four sons and a l l the four parts received by the sons must be the same size and shape. How was the property divided? 2. Why does i t take no more pickets to b u i l d a fence down a h i l l and up another than i n a straight l i n e from top to top, no matter how deep the gully? 3. Given a plank 12 inches square, required to cover a hole i n a f l o o r 9 Inches by 16 inches, cutting the plank into only two pieces. 4. A farmer has six pieces of chain, each piece containing f i v e l i n k s . I f i t costs 2^ for each cut and 3$zf for each weld, what w i l l i t cost him to have them made into an endless chain? ( i . e . chain i n the form of a c i r c l e . ) 5. Three missionaries t r a v e l -l i n g i n cannibal country, came to a r i v e r and could not get across. There were three cannibals on the bank of the r i v e r and a boat t i e d at the shore. The mission-aries were offered the use of the boat, but the boat could carry no more than two at a time, and i t was unsafe to leave more cannibals than missionaries i n any one place at any one time, f o r the cannibals outnumbering the missionaries would devour them. A l l the missionaries could row, but only one cannibal (Marked^) ) had learned to row. How d i d a l l the :r: ^ missionaries and cannibals get across the r i v e r , and what i s the l e a s t number of times the boat need cross the river? 6. A man having a fox, a goose, and a peck of corn i s desirous of crossing a r i v e r . He can take but one at a time. The fox w i l l k i l l the goose and the goose w i l l eat the corn i f they are l e f t together. How can he get them safely across? 7. A room i s 30 f t . long, 12 f t . wide, and 12 f t . high. On the middle l i n e of one of the smaller side walls and one foot from: the c e i l i n g i s a spider. On (the middle l i n e of the opposite wall and eleven feet from the c e i l i n g , i s a f l y . The f l y , being paralyzed by fear, remains s t i l l u n t i l the spider reaches I t by crawling the shortest route. How f a r d i d the spider crawl? 3 0 ©00 39 8. 4L .ft 1^ ) A g i r l had a piece of cl o t h of the shape shown i n the diagram. She wished to cut i t into three pieces, a l l of the same size and shape. How could she do It? 3f4. 9. A man and hi s wife, each weighing 150 pounds, with two sons each weighing 75 pounds, have to cross a r i v e r i n a boat which Is capable of carrying only 150 pounds' weight. How did they get across? 10. Take two pennies face upward on a table and edges i n contact. Suppose one i s f i x e d and the other r o l l s on i t without s l i p p i n g , making one complete revolution about i t and returning to i t s o r i g i n a l p o s i t i o n . How1many revolu-tions about i t s own centre has the moving coin made? 40 CHAPTER IV MOTIVATION IN THE TEACHING OP GRADE XII GEOMETRY Part A. Development of Proper Attitudes Towards Grade XII Geometry ¥i/hether a student i n grade XII i s influenced by s u f f i c i e n t motivating forces to encourage him to reach his highest l e v e l of e f f i c i e n c y i n the study of geometry depends very l a r g e l y upon the attitude which he adopts towards the subject. I f a teacher can lead a pupil to develop a proper attitude toward the subject of geometry, then that pu p i l w i l l have made the f i r s t step toward deriving the numerous benefits and pleasures latent i n the study of thi s most fas c i n a t i n g subject. But what i s the proper attitude toward geometry i n grade XII? In the f i r s t place cert a i n students i n their f i n a l year of high school may be encouraged to regard themselves as potent i a l mathematicians. I f the teacher regards a pupil as an advanced student launching out into the depths of higher mathematics, then that pupil w i l l endeavour to respond to this attitude, and w i l l put forth every e f f o r t to show•that he Is worthy of such recognition. A higher regard f o r the pupil (1) "What science can there be more noble, more excellent, more useful to men, more admirably high and demonstrative, than this of mathematics?" - Benjamin Franklin. S.I.Jones, Mathematical Wrinkles, P. 257. 41 undoubtedly encourages more earnest e f f o r t on his part. Another means of encouraging a proper attitude towards geometry i n grade XII, i s the development of the forward-looking att i t u d e . The teacher may f i n d many opportunities for making s l i g h t reference to some advanced work i n mathematics, and thereby show the pupils that they are r e a l l y laying the foundations for the study of higher mathematics; for example, i n the study of graphs the pupils' i n t e r e s t may be aroused to a considerable extent i n the study of a n a l y t i c a l geometry. Also, In connection with factoring, e s p e c i a l l y by the grouping method, there Is an opportunity for mentioning permutations and combinations. This topic could be mentioned also i n con-nection with evolution and invol u t i o n . When a pu p i l i s asked to write down the square of an expression such as 3x + 4y -t- z-2w, he has to use the p r i n c i p l e of combinations i n order to be able to write the terms i n the answer Involving twice the product of each p a i r . Longer expressions might be taken, and the number of groups of two calculated without actually p a i r i n g them o f f . By this means a student's i n t e r e s t and c u r i o s i t y is aroused In the topics which l i e just ahead of him, and he i s motivated to put f o r t h h i s best e f f o r t s and continue h i s mathematical studies. Besides developing the forward-looking attitude i n grade XII, i t i s Important that the student become more independent, and less r e l i a n t upon the teacher. I f the pupil can be l e d to develop the idea that the teacher i s there to guide and not 42 simply to Instruct, then that pupil w i l l develop a sense of r e s p o n s i b i l i t y and w i l l f e e l that he himself Is the one to do the thinking and the reasoning; and he w i l l f i n d much more s a t i s f a c t i o n i n doing h i s work In geometry than I f t h i s attitude of independence were not developed. Probably the most Important attitude towards grade XII geometry Is the development of the sense of s a t i s f a c t i o n through mastery. There i s no greater enjoyment i n any phase of school work than the t h r i l l derived from obtaining a solution to a d i f f i c u l t geometrical problem which has required a great deal of concentration. I f the teacher has a select group of problems at h i s disposal which are of just the r i g h t d i f f i c u l t y for the pupils at each stage of t h e i r development, he can use these to wonderful advantage by giving the pupils an opportunity for experiencing that sense of s a t i s f a c t i o n through mastery which i s such a strong motivating force to further e f f o r t . "Success begets success," and every exper-ience of t h i s nature acts as a "stepping-stone to higher things." In connection with the solution of geometrical exercises, the students i n grade XII may be encouraged to develop the power of v i s u a l i z i n g geometrical solutions. I f a pu p i l has been working at a problem for some time he may develop a very accurate mental picture of the diagram with yfoich he i s working. Even when he ceases to work at the problem he may s t i l l v i s u a l i z e the diagram very c l e a r l y , and i f a student i s encouraged to continue working at the problem from his mental 43 picture of the diagram he might discover the solution even while walking along the street, s i t t i n g i n a street car, or waiting for a f r i e n d to keep an appointment. The extreme enjoyment derived from being able to solve exercises i n such a manner i s a very strong incentive towards further development of that s k i l l . S t i l l another phase of the development of the proper attitude towards grade XII geometry i s the arousal of int e r e s t i n complicated geometrical diagrams. I f the teacher has a selection of exercises which are not very d i f f i c u l t i n them-selves but which produce a rather complicated looking diagram, the pupils may be l e d to develop a keen in t e r e s t i n such figures and derive much pleasure both from th e i r construction and from th e i r analysis. There i s a great fascination about complicated geometrical diagrams, and when a student finds that he can not only draw the figure from s p e c i f i c a t i o n s but also analyse i t and prove a certa i n required fact about i t , then h i s f a i t h i n hi s own a b i l i t y i s greatly strengthened, and this i n turn Is an incentive towards continued a c t i v i t y along these l i n e s . The following exercises are examples of the foregoing type; they produce rather complicated looking diagrams but their proofs are not p a r t i c u l a r l y d i f f i c u l t : 1. I f a triangle i s inscribed In a c i r c l e and perpendiculars are drawn from any point on the cir cumference of the c i r c l e to the sides of the t r i a n g l e , the faet of .the three per-pendiculars are i n one straight l i n e . (Simpson's l i n e . ) 2. To construct a tr i a n g l e having a base equal to a given straight l i n e , a v e r t i c a l angle equal to a given angle, and 44 an area equal to the area of a given parallelogram. The following exercises handed i n by grade XII pupils indicate that s:ome pupils are indeed interested In drawing complicated looking figures, and that t h i s i s one form of motivation In grade XII geometry. (Figures I and II overleaf.) Another attitude towards grade XII geometry i s the development on the part of the pupils of a desire for absolute perfection. There i s nothing quite as stimulating to a student's enthusiasm as to be t o l d that h i s solution of a problem i s perfect. In the solution of geometrical exercises i t Is cuite possible for a pupil to reach perfection - perfect-l y l o g i c a l reasoning expressed In accurate geometrical terms. When a pupil knows that such an attainment Is quite within h i s reach, he i s encouraged to put f o r t h special e f f o r t i n order to derive the s a t i s f a c t i o n of producing a perfect solution. That students are interested i n making the i r solutions perfect i s evident from the accompanying solutions to exercises given to a grade XII class simply as optional home exercises. (Figures III and IV overleaf.) I 6 j f ^ feojjix. r3eD ^ ^ ^ v . . . U r j ISA If" BC, 6 D , 0 * W i « * . a * . E . X a* t * * c«.xo a_ ( W r f a A / erf t*vju CD n e> e. D . ***** < c o e> =. ^ e R (.t rJr.e.f (c~fa~tni) . L A.H C .o^ncl L h7> -£ . L MM-. , ^.T / / ^ .4J7 : y 4 a ) ' A C ,5//7CP Z.4 7ZT —fcUda) *,M~~. - 1 i L ! U 51A . x y l j a W Jno tee.. " = .', <3_ . e _ y = < E_. — : • { D F E . 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ZFs COK iF zrs: 3 52 10. Construct a square three times as large as a given square. 11. 0 i s a point inside a rectangle ABCD. Prove that 0A 2+0C 2 e 0B2-f-0D2. 12. BE, CP are al t i t u d e s of an acute-angled t r i a n g l e ABC. Prove that AE.AC = AF.AB. 13. The sum of the squares on the sides of a parallelogram i s equal to the sum of the swuares on the diagonals. 14. In a tr i a n g l e , three times the sum of the squares on the sides i s equal to four times the sum of the squares on the medians. 15. The shortest chord that can be drawn through a'point inside a c i r c l e i s that which i s perpendicular to the diameter through the point. 16. Show how to draw a chord of a c i r c l e , equal to a given chord and p a r a l l e l to a given straight l i n e . 17. Show how to draw two tangents to a c i r c l e making a given angle with each other. 18. The bisectors of the three angles of a tr i a n g l e meet at a point. 19. Show how to draw three equal c i r c l e s each touching the other two, and how to circumscribe a fourth c i r c l e around the other three. 20. Find the distance between the centres of two c i r c l e s , t h e i r r a d i i being 5 cm. and 7 cm. and t h e i r common chord 8 cm. (Two cases.) 21. ABCD i s a quadrila t e r a l i n s c r i b e d i n a c i r c l e . DA and CB are produced to meet at E; AB, DC to meet at F. Prove that i f a c i r c l e can be drawn through the points A, E, F, C, then EF i s the diameter of t h i s e i r c l e ; and BD i s the diameter of the c i r c l e ABCD. 22. The straight l i n e b i s e c t i n g the angles of any convex quad r i l a t e r a l form a c y c l i c q u a d r i l a t e r a l . 23. Through a point 2 i n . outside a c i r c l e of radius 2 i n . draw a l i n e to pass 1 i n . from the centre of the c i r c l e . Calculate the part inside the c i r c l e . , 24. Circumscribe about a c i r c l e of radius 5 cm. a t r i a n g l e having i t s sides p a r a l l e l to three given l i n e s . 25. Show how to construct a t r i a n g l e on a given base with a given v e r t i c a l angle and a given median. When i s t h i s impossible? 26. ABC i s an e q u i l a t e r a l triangle inseelbed i n a c i r c l e ; prove that PA - PB + PC. 27. AOB, COD are two chords of a c i r c l e , i n t e r s e c t i n g at r i g h t angles. Prove that arc AC f arc BD s arc CD -+ arc DA. 28. A, B, C are three points on a c i r c l e . The bisector of /ABC meets the c i r c l e again at D. DE i s drawn p a r a l l e l to AB and meets the c i r c l e again at E. Prove that DE = BC. 29. A, C are two fi x e d points, one upon each of two c i r c l e s which intersect at B, B. Through B i s drawn a variable chord PBQ, cutting the two c i r c l e s i n P, Q. PA, QC (produced i f necessary) meet at R. Prove that the locus 53 of R i s a c i r c l e . 30. BE, CF are two a l t i t u d e s of a triangle-ABC V-O and they i n t e r -sect at H. BE produced meets the circumcircle at K. Prove that E i s the mid-point of HK. Another means of motivating the teaching of geometry i n grade XII i s hy selecting exercises of special s i g n i f i c a n c e . One type of exercise which i s of special significance to the pupils i s an exercise which has appeared on a matriculation examination paper i n a recent year. Matriculation pupils are usually keen to see i f they are able to obtain a solution to an exercise which was worthy of a place on a matriculation examination paper. I f a teacher compiles a l i s t of such exercises, he can give them to the pupils one at a time during the year as s u f f i c i e n t work i s covered to enable the pupils to obtain a solution. I f the pupils are informed as to the year i n which the question appeared, and the number of marks assigned to i t , then a s t i l l greater i n t e r e s t i s created i n the question. The following i s a l i s t of questions, compiled from matriculation papers, which would be suitable for use as indicated herein: .1. In the isosceles t r i a n g l e ABC, having AB e AC, X i a any point i n AB, and Y i s taken i n AC.so that XY i s p a r a l l e l to BC. Prove: BY 2-CY 2 BC.XY-. 2. Two c i r c l e s touch i n t e r n a l l y at A. A chord BC of the l a r -ger c i r c l e touches the smaller c i r c l e at D. Prove that AB:AC s BD:DC. 3. The base and v e r t i c a l angle of a tri a n g l e are given. Find the locus of the i n t e r s e c t i o n of l i n e s drawn from the ends of the base perpendicular to the opposite sides of the t r i a n g l e . 4. With a c i r c l e of radius r, draw two c i r c l e s of r a d i i rg and r j which touch each other externally and both of which 'touch the c i r c l e of radius r, i n t e r n a l l y . 54 5. DEF i s a t r i a n g l e inscribed i n a c i r c l e with centre 0. The diameter perpendicular to EP cuts DE at P and FD produced at Q. Prove that CE i s the mean proportional between OP and OQ. 6. AB and CD are two diameters of the c i r c l e ADBC at r i g h t angles to each other. EP i s a chord such that the straight l i n e s CE and CP, when produced, cut AB produced i n G and H respectively. Prove that the rectangle contained by CE and Gil i s equal to the rectangle contained by EF and CH. 7. ABC i s an e q u i l a t e r a l triangle and D i s any point i n the base, BC. On the base produced (both ways) points E and F are taken such that the angles EAD and DAF are bisected i n t e r n a l l y by AB and AC respectively. Show that the » .'triangles ABE and ACF are similar and that BE.CF - 3C . 8. Two c i r c l e s touch one another externally at A; BA and AC are diameters of the c i r c l e s . BD i s a chord of the f i r s t c i r c l e which, when produced, touches the second at X, and CE i s a chord of the second c i r c l e which, when produced, touches the f i r s t at Y. Prove that BD.CE a 4DX.EY. 9. I f two tangents at the ends of one diagonal of a c y c l i c q u a d r i l a t e r a l intersect on the other diagonal produced, the rectangle contained by one pair of opposite sides of the quad r i l a t e r a l Is equal to that contained by the other four. 10. D, E, F are the mid-points of the sides BC, CA, AB of a trian g l e ABC. AL i s an a l t i t u d e . Prove D, E, F, L are concyclic. 11. AB i s a fix e d chord of a c i r c l e ; CD i s .a diameter perpen-di c u l a r to AB. P i s a variable point on the c i r c l e ; AP, BP.cut CD (produced i f necessary) i n X, Y respectively. I f 0 i s the centre of the c i r c l e , prove OX.0IY i s constant. 12. ABCD i s a qua d r i l a t e r a l inscribed i n a c i r c l e ; i t s diagonals AC and BD int e r s e c t at E. The l i n e j oining E to the c i r -cumcentre of the tr i a n g l e AEB cuts CD i n F. I f CR i s a ciameter of the circumscribing circle, of the qu a d r i l a t e r a l ABCD, prove that the rectangle contained by DE and BC i s equal to the rectangle contained by EF and CR. 13. A and B are f i x e d points on a c i r c l e , and PQ i s any chord of constant length. Find the locus of the point of i n t e r -section of AP and BQ. Give a proof. 14. C i r c l e s are described, on the sides of a right-angled triangle, as diameters. Through the ri g h t angle at A, between the arms AB and AC, a straight l i n e APQR i s drawn cutting the three c i r c l e s i n P, Q, R respectively. Prove that AP Is equal to QR. -.. •-15. Show how to bisect the t r i a n g l e whose sides are 2, 2% and 3 inches long by a straight l i n e p a r a l l e l to the longest . side. 16. The sides BA, CD of a c y c l i c q u a d r i l a t e r a l ABCD are pro-duced and intersect at an angle of 30° , and the- sides BA, CB when produced intersect at an angle of 40°. Calculate a l l the angles of the q u a d r i l a t e r a l . 17. Two c i r c l e s intersect at P and Q. Through P a straight 55 l i n e DPE i s drawn terminated by the circumferences at D and E. The bisector of the angle DQE meets DE at P. Prove: (a) The angle DQE i s constant. (b) The locus of F i s a c i r c l e . 18.Let ABC be a t r i a n g l e . Draw ,AD, BE perpendicular to BC,CA respectively, meeting at P. Join CP and produce i t to cut AB at F. ,Prove:-(a) Angle DPC = angle DEC = angle FBD. (b) CF i s perpendicular to AB. The motivating power of tests i s another factor which must be taken into consideration when discussing motivation i n the teaching of geometry. I f the teaching of theorems and exercises has been motivated s u f f i c i e n t l y to bring each pupil up to h i s maximum l e v e l of e f f i c i e n c y , the pupils w i l l regard the working of geometry tests simply as opportunities for showing t h e i r s k i l l In geometrical reasoning. However, i t i s important that the t e s t be such as to encourage the pupils i n their good work, and In no way give them a f e e l i n g of discouragement. Tests could be set which would be too d i f f i c u l t for even the best In the class, and such tests as these would undermine the confidence of the pupils and have a detrimental e f f e c t upon t h e i r study of geometry. The i d e a l test i s one which i s d i f f i c u l t enough to give the best p u p i l an opportunity to show hi s mathematical a b i l i t y , and at the same time easy enough to allow a l l pupils i n the class to experience a c e r t a i n amount of s a t i s f a c t i o n through mastery. A c a r e f u l l y graded test of a somewhat objective nature appears to s a t i s f y these conditions most suitably. The following are examples of tests which might be given towards the end of the 56 school year, or parts of which might be given at the conclusion of c e r t a i n sections of the work: Test I - (May be given at the end of the work i n c i r c l e s . ) Completion Test ^ 1. The greatest chord i n a c i r c l e i s the . 2. The largest central angle has degrees. 3. The l i n e that touches a c i r c l e at only one point, however far produced, i s a . 4. I f AB and CD are two chords of a c i r c l e each 10" from the centre, they, are • . 5. A q u a d r i l a t e r a l ABCD i s inscribed i n a c i r c l e . The angle A s 80°, and the angle B = 90°. The angle C = degrees and the angle D = degrees. 6. I f A i s any point within a c i r c l e , the shortest chord through A i s r___ t o fc^e diameter through A. 7. AB and CD are two diameters of a c i r c l e , then ABCD i s a • 8. I f the c e n t r a l angle AOB i s 60°, and the radius OA i s 20", the chord AB Is inches. 9. An inscribed angle and a central angle intercept the same arc. The central angle Is the inscribed angle. 10. A chord AB meets a tangent AT at an angle of 50°. The angle i n the minor segment cut o f f by AB i s • degrees. Test II - (At the end of c i r c l e s ) True - False Test ( 2) 1. An angle i s inscribed i n an arc of 60°. The angle contains 2. I f two c i r c l e s are equal or unequal, angles i n s c r i b e d i n arcs of the same number of degrees are equal. 3. An angle inscribed i n an arc of 200° Is acute. 4. An angle inscribed i n an arc of 100° i s obtuse. _______ 5. I f one of two arcs Intercepted by two p a r a l l e l l i n e s i s 25°, the other i s 25°. 6. A central angle has the same number of angular degrees as i t s arc has of are degrees. 7. If. two c i r c l e s touch externally, the l i n e of centres Is : equal to the sum of the r a d i i . 8. I f -twoccircles are concentric, a l l tangents to the smaller c i r c l e , cut o f f by the larger c i r c l e are equal chords of (1) Morgan, Foberg, Breckenridge, Plane Geometry. (2) Morgan, Foberg, Breckenridge, Opcit. 57 the larger c i r c l e . 0 9. I f a c i r c l e circumscribes a t r i a n g l e , the tr i a n g l e Ts e q u i l a t e r a l . 10. A diameter which bisects a chord i s perpendicular to the chord. 11. A qua d r i l a t e r a l inscribed i n a c i r c l e has i t s opposite angles supplementary. 12. A trapezium i n s c r i b e d i n a c i r c l e i s iso s c e l e s . 13. I f two chords of a c i r c l e b i s e c t each other they are perpendicular. 14. Two chords which intersect In a c i r c l e are equal. 15. An angle inscribed i n a semicircle contains 89°60' 16. A parallelogram Inscribed i n a c i r e l e i s a rectangle. 17. A c i r c l e can be circumscribed about a rectangle. 18. Two concentric c i r c l e s have r a d i i 4 i n . and 6 In. res-pec t i v e l y . A c i r c l e of radius 5 i n . may be drawn to touch both c i r c l e s . 19. I f two c i r c l e s touch externally, they have but two common tangents. 20. I f a c i r c l e can be inscribed i n a qu a d r i l a t e r a l , the qua d r i l a t e r a l i s a parallelogram. 21. Through a point within a c i r c l e , i t i s always possible to draw a chord which i s bisected by the point. 22. A c i r c l e can always be constructed to touch each of three given l i n e s . 23. I f the angle between two tangents i s 60°, the lengtn" of the chord joining the points of contact i s equal to the length of each tangent. 24. I f two c i r c l e s touch a t h i r d c i r c l e they also touch .each other. 25. Two c i r c l e s which touch the same straight l i n e at the same point touch each other. 26. A c i r c l e of radius 4" has twice the area of a c i r c l e with radius 2". 27. I f two tangents are drawn to a c i r c l e from an external point, the angle subtended at the centre by the two points of contact i s supplementary to the angle between the tangents. 28. The sum of the squares on two tangents drawn from an external point i s equal to the square on the l i n e joining the external point to the centre. 29. If two c i r c l e s touch Internally they have but one common tangent. 30. In a c i r c l e of radius 3 i n . , a chord 2 i n . long i s twice as far from the centre as a chord 4 In. long. Test I I I . (At end of c i r c l e s . ) Multiple-Choice Test, t 1 ) (1) Morgan, Foberg, Breckenridge, Opcit. 58 In each of the following, t e l l which answer you believe to be correct and why. 1. I f an angle i s inscribed i n an arc of 150°, the angle con-tains 150°, 75°, 300°. 2. I f an inscribed angle intercepts an arc less than a semi-c i r c l e , the angle i s acute, obtuse, r i g h t , s t r a i g h t . 3. One angle of a c y c l i c q u a d r i l a t e r a l i s 70°. The opposite angle i s 102°, 70°, 20°, 110°, cannot t e l l . 4. Two tangents are drawn to a c i r c l e from a point p. They include between t h e i r points of contact an arc of 100°. The angle between the tangents i s 100°, 200°, 80°, 70°, cannot t e l l . 5. Two chords are equal i f they are p a r a l l e l , i f they are the same distance from the centre, i f they bis e c t each other, cannot t e l l . » • • 6. Two chords which int e r s e c t i n a c i r c l e are equal, i f they make equal angles with the diameter through t h e i r point of i n t e r s e c t i o n , i f the figure formed by joining t h e i r ends Is an isosceles trapezium, i f they b i s e c t each other. 7. A perpendicular to a diameter at i t s extremity Is a secant, a chord, a-sector, a tangent, a segment. 8. Two secants drawn from a point P form an angle of 30°. The larger intercepted arc i s 80°. The smaller arc i s 40°, 60°, 20°, 30°. 9. An angle formed by a tangent and a chord which passes through the point of contact has as many degrees as the intercepted arc, h a l f the sum of the intercepted arcs, h a l f the intercepted arc, h a l f the difference of the intercepted arcs, cannot t e l l . 10. A parallelogram inscribed i n a c i r c l e i s a rhombus, a square, a rectangle, a rhomboid. MIDDLE SCHOOL GEOMETRY t1' Time 2| hours NOTE: Please read c a r e f u l l y the instructions given before attempting the paper. (1) Do the questions i n order. (2) Do not spend too long at any one question. Pass on to the next question and return to the unsolved questions aft e r completing the paper. (3) At the close of the examination, hand t h i s paper to the presiding examiner. A. CONSTRUCTIONS REQUIRED TO SOLVE PROBLEMS: (1) Example of good geometry test as published i n findings of a committee appointed to Investigate types of examinations suitable for High Schools i n Ontario. 59 NOTE: (1) Make the constructions indicated on the diagrams below each question. (2) The figures should he neat and approsimately correct; absolute accuracy i s not required. (3) Ruler and compasses are the only instruments to be used. (4) A l l construction l i n e s should be c l e a r l y shown. (5) No written statement i s necessary i n t h i s part of the paper. (6) In drawing p a r a l l e l l i n e s use your eye and the ru l e r ; i n other constructions show a l l construction l i n e s . 1. Bisect this angle. 2. Draw the r i g h t bisector of this l i n e . 3. Construct a rectangle equal 4;Construct a triangle equal i n i n area to thi s t r i a n g l e . area to t h i s polygon. 5. Find the centre of the c i r c l e of which, the arc below i s a part. 6. Draw tangents from P to t h i s cir c i e . 7. Cut off 3/7 of thi s l i n e . 8. Find a l i n e which w i l l be a mean proportional between these two l i n e s . 60 9. Draw a l i n e p a r a l l e l to BC so that the part be-tween AB and AC may equal BP. 10. Draw a perpendicular to AB at A without producing AB. 11. From the qu a d r i l a t e r a l , cut o f f a part s i m i l a r to i t and - 4/9 of i t s area. 12. Draw a transverse common tangent to these c i r c l e s . 13. The radius of t h i s c i r c l e 14. Draw a straight l i n e \/Q i s If- Inches. Place i n i t inches long, a eiiord 2 inches long and calculate i t s distance from the centre. B. CONSTRUCTIONS REQUIRED TO PROVE THEOREMS: See the Instructions under section A. Lines or angles made equal should be so marked on the figure. No proof i s required. 1. Make the construction that 2. I f BC i s greater than AC then w i l l prove angle ABD the angle A i s greater than greater than angle BDC. the angle B.-61 3. ML-t MN i s greater than LN. 4. The c i r c l e s touch at P to prove AD p a r a l l e l to EG. C. PROOFS OF THEOREMS: NOTE: Pupil w i l l make on the figure any construction necessary and w i l l write the proof only, i n as concise a form as possible. Example: In the figure below BC = DC and BE = DE, prove CE the r i g h t bisector of BD. Proof In / _ 3 BCE and CDE y BC s CD BE - ED CE s CE In/_a BCF and DCF BC - CD CF r CF /JBCF = DCF BF s FD and /BFC = DFC = 1 Rt. /__, C . l . AE and BE bisec t the angles A and B of the parallelogram. Prove angle AEB = r i g h t angle. 62 ~D c 2. Prove that angle AOB Is double the angle APE-. Given 0 the centre and P on the circumference. R 63 4. AP i s a tangent to the c i r c l e ABQ and AQ i s a tangent to the c i r c l e ABP# Prove angle ABP s angle ABQ. D. NUMERICAL CALCULATIONS: 1. How many degrees are there i n one of the i n t e r i o r angles of a regular eight sided figure? 2. The exterior angle made by producing one side of a regular polygon i s 20°. How many sides has i t ? 3. I f you were asked to draw triangles (a) 3", 6", 9" (b) 4", 5", 6" (c) 5", 6", 8" (d) 15", 36", 39" which of these would be (Place a, b, c, or d;in (1) A r i g h t angled t r i a n g l e brackets) , ( ) (2) An acute angled t r i a n g l e ( ) (3) An obtuse angled triangle ( ) (4) Not a triangle ( ) 4. In t r i a n g l e ABC side CB - CA and AD bisects the angle CAE. Also AC s 3 and AB r 5. Find CD. 5. In tr i a n g l e ABC, AB = 5, AC = 6, and BC * 7. AD i s at ri g h t angles to BC. Find BD and CD. 64 6. In these similar triangles the perpendiculars are as 7:5. What is the r a t i o of the areas? 7. Two straight l i n e s CAB and OCD cut a c i r c l e . Given that CA - 3 1/3, OC - 4 1/3, CD a 3 2/3 and BD - 6 2/5, f i n d OB and AC. One of the motivating forces mentioned i n connection with grade XII work i n geometry i s the s a t i s f a c t i o n derived through helping others. By what means can t h i s form of motivation be brought into operation? The answer i s , by i n s t i t u t i n g a p u p i l -teacher arrangement between the students i n grade XII and those i n the lower grades. This arrangement consists of providing an opportunity f o r pupils i n the lower grades to secure help from those i n the matriculation year. A teacher who teaches both i n grade X and i n Grade XII has an excellent opportunity i n t h i s regard, because he can r e f e r certain grade X pupils to certain students i n grade XII for assistance. I f the p u p i l -teacher idea i s encouraged i t i s found that grade X pupils who have friends i n grade XII re f e r to these friends f o r assistance. 65 w i t h o u t a c t u a l l y b e i n g i n s t r u c t e d t o do so b y t h e t e a c h e r . T h i s h e l p o b t a i n e d f r o m p u p i l s i n a h i g h e r g r a d e i s by no means meant t o r e p l a c e t h e h e l p o f f e r e d b y t h e t e a c h e r b u t m e r e l y t o s u p p l e m e n t i t . I n t h e c a s e where an e x e r c i s e i s a s s i g n e d f o r home s o l u t i o n , and a p u p i l i s u n a b l e t o s o l v e i t e v e n a f t e r r e p e a t e d a t t e m p t s , t h e n i n s t e a d o f s i m p l y w a i t i n g u n t i l t h e n e x t g e o m e t r y l e s s o n and s e e i n g t h e e x e r c i s e done i n c l a s s , he c a n go t o h i s g r a d e X I I f r i e n d a nd r e c e i v e enough a s s i s t a n c e t o e n a b l e h i m t o c o m p l e t e t h e e x e r c i s e . The b e n e f i c i a l e f f e c t s o f t h e a r r a n g e m e n t a r e s h a r e d b y b o t h p u p i l s . The s e n i o r s t u d e n t f i n d s t h a t he d e r i v e s a g r e a t d e a l o f s a t i s f a c t i o n f r o m b e i n g a b l e t o be o f a s s i s t a n c e t o h i s l e s s m a t u r e f r i e n d , a n d t h e j u n i o r p u p i l r e c e i v e s e x t r a a s s i s t a n c e i n h i s work w h i c h he w o u l d o t h e r w i s e n o t r e c e i v e , a n d he i s a b l e t o come t o c l a s s p r e p a r e d t o g i v e a s o l u t i o n t o t h e home e x e r c i s e i n s t e a d o f s i m p l y r e l y i n g on some o t h e r member o f t h e c l a s s t o do s o . I n t h e e x p e r i e n c e o f t h e w r i t e r , who h a s s e e n s u c h a p u p i l - t e a c h e r p l a n i n o p e r a t i o n , i t h a s b e e n d i s t i n c t l y n o t i c e a b l e t h a t p u p i l s who t a k e p a r t i n s u c h an a r r a n g e m e n t a p p e a r t o t a k e an add e d i n t e r e s t I n t h e s u b j e c t o f g e o m e t r y . I f t h e t e a c h i n g o f g e o m e t r y i n g r a d e X I I i s m o t i v a t e d p r o -p e r l y , i t w i l l be f o u n d t h a t t h e s t u d e n t s i n t h i s g r a d e t a k e a r e a l I n t e r e s t i n d i s c u s s i n g g e o m e t r i c a l p r o b l e m s . T h i s i n i t -s e l f c a n b y made an a d d i t i o n a l m o t i v a t i n g f o r c e b y t h e i n s t i t u t i o n o f a f t e r s c h o o l d i s c u s s i o n g r o u p s . T h i s c a n be 66 c a r r i e d out i n a very informal way, and the teacher may take a very small part or no part at a l l i n the discussion. The problem for discussion may be one that has been assigned for homework; one which a pupil discovered on an old examination paper; .one which was given to a bright pupil as a special p r i v i l e g e ; or one from some outside source altogether. The diagram i s placed on the board and the students cooperate Informally i n obtaining a solution. The writer has seen such discussions as these c a r r i e d on so long after school that the caretaker has had to t e l l the students either to go home or be locked i n the school a l l night. The number taking part In these discussions may be only a small f r a c t i o n of the class, but nevertheless the motivating Influence upon that few i s indeed valuable. The motivating force of games i s one which i s extremely powerful i n a l l grades of high school. The types of games may vary considerably i n the d i f f e r e n t grades, but the i n t e r e s t i n games i s just as much a l i v e i n the grade XII student as i n the p u p i l of grade IX. The method of u t i l i z i n g this motiva-tin g force i n the teaching of grade XII geometry i s c h i e f l y by the judicious use of suitable mathematical recreations. I t Is, of course, inadvisable to make these recreations a main, feature of the course, but they are of immense value i n arousing the Interest i n things of a mathematical nature i n those pupils who are less responsive to the other forms of motivation. A good selection of these mathematical recreations 67 i n g e o m e t r y i s a v a l u a b l e a s s e t t o t h e t e a c h e r o f m a t h e m a t i c s , an d i f t h e y a r e p r e s e n t e d a t s u i t a b l e t i m e s t h e y may p r o d u c e most d e s i r a b l e e f f e c t s upon t h e p u p i l s . The f o l l o w i n g a r e a number o f m a t h e m a t i c a l r e c r e a t i o n s s u i t a b l e f o r u s e i n t h e t e a c h i n g o f g r a d e X I I g e o m e t r y : M a t h e m a t i c a l R e c r e a t i o n s r- ^ - - 2 5 03-— i 1. A h o u s e and b a r n a r e 25 r o d s a p a r t . The h o u s e i s 12 r o d s a n d t h e b a r n 5 r o d s f r o m a b r o o k r u n n i n g i n a s t r a i g h t l i n e . What i s t h e s h o r t e s t d i s t a n c e one must w a l k f r o m t h e h o u s e t o g e t a p a i l o f — w a t e r f r o m t h e b r o o k and t a k e i t t o t h e b a r n ? 2. To c o n s t r u c t t h e famous N i n e - P o i n t C i r c l e . i . e . I f a c i r c l e be d e s c r i b e d a b o u t t h e p e d a l t r i a n g l e o f any g i v e n t r i a n g l e , i t . w i l l p a s s t h r o u g h t h e m i d d l e . p o i n t s o f t h e l i n e s drawn f r o m t h e o r t h o c e n t r e t o t h e v e r t i c e s o f t h e t r i a n g l e , and t h r o u g h t h e m i d d l e p o i n t s o f t h e s i d e s o f t h e t r i a n g l e , ' i n a l l , t h r o u g h n i n e p o i n t s . 3. To p r o v e t h a t i t i s p o s s i b l e to l e t f a l l two p e r p e n d i c u l a r s t o a l i n e f r o m an e x t e r n a l p o i n t . Take two i n t e r s e c t i n g c i r c l e s w i t h c e n t r e s 0, 0 . L e t one p o i n t o f i n t e r s e c t i o n be P, a n d draw t h e d i a m e t e r s PM, FN. Draw MN, c u t t i n g t h e c i r c u m f e r e n c e s a t A, B. J o i n PA, PB. P r o o f : S i n c e / PBM i s i n s c r i b e d i n a s e m i c i r c l e i t I s a r i g h t a n g l e . A l s o , s i n c e / PAN i s i n -s c r i b e d i n a s e m i c i r c l e , i t i s a r i g h t a n g l e . . . PA and PB a r e b o t h j_ t o MN. Where I s t h e f a l l a c y ? 4. To p r o v e t h a t p a r t o f an a n g l e e q u a l s t h e whole a n g l e . Take a s q u a r e ABCD and draw MM'P, t h e p e r p e n d i c u l a r b i s e c t o r o f CD. Then MM'P I s a l s o t h e p e r p e n d i c u l a r b i s e c t o r o f AB. From B draw any l i n e BX e q u a l t o AB. Draw DX, and b i s e c t i t b y t h e p e r p e n d i c u l a r NP. S i n c e D X - - i n t e r -68 s e c t s CD, t h e p e r p e n d i c u l a r s c a n n o t be p a r a l l e l a n d must meet a t P. Draw PA, PD, PG, PX, PB. P r o o f : s i n c e MP i s t h e _ j _ b i s e c t o r o f CD, PD = PC S i m i l a r l y PA = PB and PD = PX . \ PX B PD m PC• Bu t BX - BC b y c o n s t r u c t i o n and PB i s common t o t r i a n g l e s PBX, PBC .". /\PBX 5 A P B C (3 s i d e s s 3 s i d e s ) .'. 2_XBP = / C B P . The whole / XBP e q u a l s i t s p a r t t h e / CBP. F i n d t h e f a l l a c y . G i v e n a p i e c e o f c a r d b o a r d i n t h e f o r m o f an e q u i l a t e r a l t r i a n g l e . R e q u i r e d t o c u t i t i n t o f o u r p i e c e s t h a t may be p u t t o g e t h e r t o f o r m a p e r f e c t s q u a r e . G i v e n , f i v e s q u a r e s o f c a r d b o a r d a l i k e i n s i z e . R e q u i r e d , t o c u t them so t h a t b y r e a r r a n g e m e n t o f t h e p i e c e s y o u c a n f o r m one l a r g e s q u a r e . . 68A Theory without practice i s as f a i t h without works -dead," says James Strachan when writ i n g about the study of geometry i n Dr. John Adam's book e n t i t l e d "The Hew -Tea'chlrag'*, and the statement c e r t a i n l y contains much truth. A boy or g i r l who never sees any p r a c t i c a l appl i c a t i o n of the facts learned i n h i s t h e o r e t i c a l study of geometry w i l l never appreciate the f u l l value of h i s mathematical education. Moreover, the very f a c t of being able to apply the t h e o r e t i c a l knowledge to r e a l situations adds to the int e r e s t of the subject immensely and becomes a genuine motivating force. There are three very appropriate ways by which th i s motivating force may be applied to the teaching of geometry; namely, (1) by u t i l i z i n g the geometry of the manual t r a i n i n g shops; (2) by u t i l i z i n g the geometry of outdoor measurements; and (3) by u t i l i z i n g the geometry of a r c h i t e c t u r a l forms. The following l i s t of examples w i l l indicate c e r t a i n s p e c i f i c methods by which these three forms of motivation might be brought into operation: 1. Bisecting an angle by the carpenter's square. To bisec t the angle A, take AC - AD. Place the R square so that BC = BD. Prove AB i s the bis e c t o r of angle A. (1) Adams, J . , The New Teaching. P. 108. 68B 2. Explain how the p a r a l l e l r u l e r may be used to draw li n e s p a r a l l e l to a given l i n e . The r u l e r moves f r e e l y about the points A, B, C, and D. 3. To f i n d the centre of a c i r c l e the carpenter's square may be used as i n the figure and the l i n e AB draWn. Move the square and draw another diameter intersecting AB at the centre. 4. To f i n d the distance between two points on opposite sides of a r i v e r . To f i n d AB, lay o f f any convenient distance AC perpen-dicula r to AB, and CD perpendi-cular to BC to meet BA"produced at D. Prove AB = AC 2 - AD. B (1) Exercises from Morgan, Foberg, Breckenridge, Opcit. P.413. For more complete l i s t see Morgan, Foberg, Breckenridge, Opcit. Appendix. 68C I f A i s the centre of the arc BC, B the centre of the arc AC and t r i a n g l e ABC i s e q u i l a t e r a l , the figure thus formed by AB, arc AC and arc BC i s an eq u i l a t e r a l Gothic arch. AB i s i t s span. I f a window has the form of an eq u i l a t e r a l Gothic arch with a span of 4 f t . , f i n d the area of the glass i n the window. 69 CHAPTER V MOTIVATION IN THE TEACHING OF GRADE IX ALGEBRA The various forms of motivation suitable for use i n the early grades of high school have already been discussed i n chapter I I . The present chapter w i l l outline s p e c i f i c methods by which the d i f f e r e n t topics i n grade IX algebra may be pre-sented so as to u t i l i z e c e r t a i n motivating forces to t h e i r f u l l e s t extent. When we know what motivating forces are at our disposal, the question is", "By what means can we bring these motivating forces into f u l l operation?" When commencing the study of algebra i n grade IX, the pupil i s n a t u r a l l y interested i n i t on account of i t s novelty. It i s something e n t i r e l y new to him to f i n d out that i n this peculiar subject he i s going to use l e t t e r s as well as numbers to represent quantities. He i s amazed when he i s t o l d that i n a very short time he w i l l be able to add, subtract, multiply and divide by using l e t t e r s of the alphabet throughout. The teacher can c a p i t a l i z e upon th i s novelty of the s i t u a t i o n , and arouse an i n i t i a l i n t e r e s t by allowing the pupils to r e a l i z e that they are commencing a r e a l l y strange and i n t e r e s t i n g study. In the Introduction to the use of symbols, the motivating force of the novelty of the subject may gradually be supple-mented by the arousal of the pupils' i n t e r e s t i n the actual 70 situations studied. For the i l l u s t r a t i o n of the use of symbols i t i s necessary to use various types of l i t t l e problems i n which l e t t e r s , a r e employed instead of numbers. I f these problems are made " r e a l " and " l i v i n g " to the p u p i l , he w i l l ., f i n d himself intensely interested i n the actual material at hand because of i t s association with the more pleasurable experiences of h i s l i f e . Problems involving baseball, swimming, running, skating, b u i l d i n g and numerous other pleasurable a c t i v i t i e s , make an appeal to the p u p i l which more abstract problems can never make. The following i s a compari-son between some " l i v i n g " problems and some of t h e , " l i f e l e s s " or abstract types: 1. (a) Divide 84 into three parts so that two of the parts are equal and the. other part f i v e times as great as either of the equal parts.(1) (b) Three boys went fo r a drive i n an automobile a distance of 84 miles. George and Jack each drove the :same distance, but Harold drove f i v e times as far as either of the other two. How many miles did each boy drive? 2. (a) How many square feet are there i n a rectangle which has adjacent sides measuring 2p-t- q, and 3p-4q feet res-pectively? (2) (b) A boy l i n e d out a f o o t b a l l f i e l d which was 2p + q feet long, and 3p-4q feet wide. What i s the area of the f i e l d ? How many feet of sawdust did he lay making the outside lines? It i s a great incentive to industry, when a pu p i l f e e l s that he i s mastering something which i s r e a l l y somewhat d i f f i c u l t to master. The early problems i n algebra are n a t u r a l l y extremely elementary, but to the beginner some of them appear to be very i n t r i c a t e indeed. I f the teacher i s (1) H.S.Hall, Elementary Algebra, P. 20. (2) Ibid, P. 43. 71 i s observant as to which problems seem to the beginner to be of a rather advanced nature, then he can be l i b e r a l with h i s praise when solutions to these problems are reached by the pup i l ; and the combined ef f e c t of the praise of the teacher and the r e a l i z a t i o n that he has r e a l l y made a d e f i n i t e accomplish-ment, i s an exceedingly strong motivating force inducing further e f f o r t on the part of the p u p i l . The following are examples of problems which might appe ar to the beginner to be rather complicated, and for the solution of v/hich l i b e r a l praise might be given: 1. A man who balances h i s accounts at the end of every quarter finds that he has three gains followed by a l o s s . The t h i r d gain i s 4 times the second, and the second i s three times the f i r s t . The loss i s twice the f i r s t gain. I f c : v oh^the whole he gains $1,120, f i n d the amount of the l o s s . 2. A boy begins to play marbles with x marbles. He wins y more and then loses x. He takes h i s marbles home and gives h i s l i t t l e brother one-third of them. How many marbles has he now? 3. A c i r c u l a r racetrack i s m yards around. One boy rides around i t on his b i c y c l e n times and another boy rides around i t n more times than the f i r s t . How many yards did both boys together ride? 4. I f Jack can run k miles per hour, and George twice as fas t , while Henry can run only h a l f as fast as Jack, how many miles w i l l the three boys together cover i n x hours? When the study of the topic of substitution i s reached, there i s a grave tendency f o r the i n t e r e s t of the pu p i l to wane. The novelty of beginning a new subject has worn o f f to a large extent, and the topic of substitution does not lend I t s e l f very r e a d i l y to the use of problems which may be made " r e a l " to the p u p i l . Moreover, there are ce r t a i n types of substitution questions which may quite e a s i l y cause even advanced pupils to become "tangled up". 72 For these reasons It i s advisable to give the beginner the general Idea behind the process of substitution, but not cause him to become confused by confronting him v/ith the more technical points involved In substitution. He can proceed to work i n the fundamental processes of addition, subtraction, m u l t i p l i c a t i o n and d i v i s i o n without having wrestled with the t e c h n i c a l i t i e s of substitution, and then when these fundamen-t a l s have been mastered, he w i l l be able to come back to substitution and)understand i t much more r e a d i l y . For example, on page 12 of Hall's School Algebra, there i s a whole page of questions on substitution, thirty-three i n a l l , some of which are as follows: 1. I f a = 2, b = 1, c = 3, x = 4, y = 6, z = 0, f i n d the value 9** 27 * x2 2. With same values as above, f i n d the value of a^b^ (b -t c - z ) 2 Questions such as these could very conveniently be omitted u n t i l the study of the fundamentals ha"Se been completed, and the p o s s i b i l i t y of a l i f e l e s s , uninteresting section of the work k i l l i n g the enthusiasm of the pupils w i l l be eliminated. Even i n the development of the general idea of substitution, s p e c i f i c forms of motivation might be employed i n order to maintain the int e r e s t of the p u p i l . Races may be held to f i n d the value of a certa i n expression, such as: a 2+ 2ab + 4b-2a +• b 2 when a - 2 and b = 3. Also the use of unity and zero may be made rather e f f e c t i v e , as 73 a pupil i s somewhat amazed to f i n d that some complicated looking expression reduces r i g h t down to unity or zero. In the teaching of addition i n algebra, many of the general types of motivation suitable f o r grade IX may be used, such as int e r e s t i n competitions, desire for good marks, desire for praise; and these may be supplemented by other methods p a r t i c u l a r l y suited to the teaching of thi s section of the subject. In the teaching of addition there i s an excellent opportunity for the use of problems which are of r e a l i n t e r e s t to the p u p i l . Here i t i s possible to introduce questions which are connected with the most i n t e r e s t i n g phases of the pupil's l i f e . A few examples of questions i n addition which are admirably suited to pupils of grade IX are as follows: 1. A man on a motor t r i p covers x miles the f i r s t day, three times as many the second day, y miles the t h i r d day and z more miles the fourth day than on the t h i r d . Express the number of miles covered i n the four days. 2. The hockey team on which Jack plays scored k goals i n t h e i r f i r s t game, two less i n th e i r second game, three more i n thei r t h i r d game than i n th e i r second, and twice as many i n the i r fourth game as i n th e i r f i r s t . Express the t o t a l number of goals scored. 3. A boy, f l y i n g h is model airplane, found that i t flew m feet on i t s f i r s t f l i g h t and 57 feet further on i t s second f l i g h t . On i t s t h i r d f l i g h t i t flew 10 feet less than on i t s second, and on i t s fourth f l i g h t three times as f a r as on i t s f i r s t . Express the t o t a l distance flown. The working of the more mechanical types of addition questions can also be motivated somewhat by making use of the fact that a pupil derives great s a t i s f a c t i o n from being able to work "big-looking" questions c o r r e c t l y . Some questions i n addition occupy a large amount of space, and appear very 74 formidable indeed, but t h e i r actual working i s comparatively simple. A pupil's confidence i n his own a b i l i t y i s greatly strengthened when he discovers that he has obtained the correct answer to such an enormous looking question as the following: Add together the following expressions: 7 x 6 y 4 z - 6 x 5 y 3 z 2 - H 5 x 4 y 4 z 3 ; 5 x 6y 4z-8x 4y 4z 3+ 2 x 3 y 3 z 2 ; -12x 6y 4z + 5 x 5 y 3 z 2 i - 3 x 4 y 4 z 3 - 3 x 3 y 3 z 2 . The topic of subtraction i s one of the most d i f f i c u l t sections of grade IX algebra to handle s a t i s f a c t o r i l y . The rule for subtraction i s indeed very simple, but the explanation of the rule i s a d i f f e r e n t matter. The method of working subtraction questions can be learned very r a p i d l y by the pupils, and yet i t Is the source of numerous errors even i n advanced classes. For these reasons i t i s necessary f o r the teacher to use to t h e i r f u l l e s t extent any means of motivating t h i s section of the work. The d i f f i c u l t y which pupils experience i n understanding the rule for subtraction may be overcome to some extent by the use of f i t t i n g I l l u s t r a t i o n s such as the follow-ing: (1) Taking away holes i n the ground - by adding s o i l . (2) Taking away debts - by adding money. (3) Taking away the c h i l l from water. (4) Taking away the need for food. As a means of impressing the rule for subtraction more firmly upon the minds of the pupils, they may be t o l d to go home and t e l l t h e i r parents that they have discovered that there i s no such thing as subtraction; and when the parents 75 question the truth of such a statement, i t can he explained that i t i s simply addition with the sign changed. In connec-ti o n with this the pupils may be asked what must be added to ten to make i t nine. After the answer "minus one" has been obtained the teacher may argue that i t should be "one", giving his reason thus: Add I to X and you obtain IX. As a means of overcoming the d i f f i c u l t y presented by the fact that the mechanical method of working subtraction ques-tions seems so very easy to learn and at the same time very l i k e l y to cause mistakes, i t i s necessary to motivate the mechanical process of subtraction so as to induce the pupils to do s u f f i c i e n t practice i n i t without that practice becoming tiresome. One means of accomplishing t h i s i s for the teacher to place a cert a i n number of subtraction questions on the board and have the pupils compete against one another to see who can obtain the largest number of correct answers i n a certa i n s p e c i f i e d time. This procedure may be varied somewhat by having the pupils place their names on the blackboard i n order as they f i n i s h , and then i t can be ascertained who was the f i r s t one f i n i s h e d with a l l answers oorrect. Such methods as these develop the necessary mechanical s k i l l i n subtraction and at the same time prevent the topic from becoming d u l l and l i f e l e s s . In the teaching o f ' m u l t i p l i c a t i o n i n algebra, the d i f f i c u l t i e s are very similar to those encountered i n the teaching of subtraction, and they can be overcome very l a r g e l y '76 by the same methods of motivation as were outlined i n the preceding section. In m u l t i p l i c a t i o n , however, there i s more opportunity for developing the pupil's i n t e r e s t i n the material at hand, because i n this phase of the work there i s an excellent opportunity for introducing problems based on i n t e r e s t i n g l i f e experiences. Problems such as the following give the necessary practice i n m u l t i p l i c a t i o n and at the same time arouse the Interest of the pup i l immediately: 1. I f an airplane can f l y 3x-t- 2y miles per hour, how f a r w i l l i t t r a v e l i n 3x-4y hours? 2. I f the distance around a skating rink i s 4a-7b feet, how far w i l l a boy skate i n going around 5a-8b times? 3. The circumference of an automobile t i r e i s 7p-8q fe e t . How far w i l l the automobile t r a v e l while the wheel i s making 6p -t- 5q revolutions? 4. In one baseball there are 2a-3b+4c yards of thread. How many yards of thread would there be i n 3a + b-2c baseballs? In the teaching of d i v i s i o n the usual forms of motivation can be employed, but here there Is an excellent opportunity to add i n t e r e s t to the topic by u t i l i z i n g the motivating e f f e c t of competition. V\Jhen the p r i n c i p l e s of long d i v i s i o n have been taught, and a certain amount of practice given i n i t s application, the teacher may select a c e r t a i n f a i r l y long question and require the f i r s t person i n each row to go to the board, write the question on the board and perform the f i r s t set of operations (I.e. writing the f i r s t term i n the quotient, multiplying and subtracting.) When the f i r s t pupil f i n i s h e s his part, the next pupil i n the row goes to the board and performs the second step, providing he thinks the f i r s t step i s correct. I f he has noticed ai y mistake i n the f i r s t step, 77 he must correct i t before proceeding. This procedure i s continued u n t i l each of the questions i s f i n i s h e d c o r r e c t l y . The row which obtains the correct answer f i r s t i s declared the winning row, and the others i n order as they f i n i s h . This form of competition adds very greatly to the i n t e r e s t which pupils take i n long d i v i s i o n , and i t encourages each one to t r y to develop h i s s k i l l i n this type of question so as to make a good showing when his turn comes to go to the board. The fa c t that each pupi l going to the board has to correct any mistake i n the question, keeps pupils following the working very c a r e f u l l y r i g h t from the beginning. Since long d i v i s i o n involves short d i v i s i o n , m u l t i p l i c a t i o n and subtraction, t h i s type of contest may be used as a review of these other operations as well as practice i n long d i v i s i o n . A suitable question for such a competition would be one l i k e any of the following: 1. Divide 6x 5-x 4+ 4x 3-5x 2-x-15 by 2x 2-x + 3. 2. Divide a 1 2 4 - 2 a 6 b 6 H b 1 2 by a 4 + 2 a 2 b 2 + b 4 . 3. Divide x 7-2y 1 4-7x 5y 4-7xy l 2+ 14x 3y 8 by x-2y 2. The motivating e f f e c t of special p r i v i l e g e s may be u t i l i z e d to a certain extent i n the teaching of long d i v i s i o n . The teacher may as a special p r i v i l e g e for work well done, allow certain pupils to try t h e i r s k i l l at working a question which w i l l occupy p r a c t i c a l l y a whole page. A question such as, Divide x 7-2y 1 4-7x 5y 4-7xy l 2W- 14x 3y 8 by X-2V 8 might be used for t h i s purpose, and the pupils w i l l derive considerable 78 enjoyment from securing the answer to a question of such length. When dealing with the removal of brackets, after the process has been explained as r e a l l y a form of addition and subtraction, the rule f o r removal of brackets may be impressed upon the minds of the pupils by comparing the removal of brackets to the removal of the wrappings from a Christmas parcel. In the case of brackets, the innermost ones come o f f f i r s t , but i n the case of the parcel the outer Wrappings are removed before the innermost. The removal of brackets lends i t s e l f very conveniently to the employment of competition by rows such as was c a r r i e d out i n long d i v i s i o n . In t h i s case, however, each person going to the board removes one set of brackets, and then he i s followed by the next person i n the row. This i s a very e f f e c t i v e method for increasing inte r e s t In the longer types of questions which might e a s i l y become very mechanical and monotonous. A question such as the following would be very suitable as the subject of a competition by rows: l _ a - ( l - a + a 2 ) - ^L-(a-a 2+ a3/} - [ l - ^a-(a 2-a3 a 4) There i s a certain f a s c i n a t i o n about removing brackets from a long and apparently complicated expression, and f i n d i n g that i s a l l reduces down to a very simple answer such as a, 1, or zero. I f questions such as these are selected by the teacher, the pupils w i l l derive that extra b i t of enjoyment which comes from the working of such questions. The following 79 question i s an example of the type which might be used f o r this purpose: Simplify: 2c 2-d(3c+ d)- £ c 2 - d ( 4 c - d ) j + { 2d 2-e(c + d ) J (Answer i s 0.) In the treatment of brackets, the teacher may develop a forward-looking attitude i n the pupils by showing them how knowledge of thi s process i s a prerequisite to the study of equations. Pupils may be allowed to glance ahead to grade X work i n equations to see how f u l l of brackets some of those equations are. The various i n c i d e n t a l b i t s of motivation as outlined immediately above a l l help to stimulate Interest i n the study of algebra, and encourage the pu p i l to develop h i s a b i l i t y to i t s highest degree along these l i n e s . Nevertheless, although a l l these forms of motivation' be applied to the teaching of grade IX algebra, and although the pupil be encouraged by these means to develop a keen in t e r e s t i n the subject, t h i s i n t e r e s t w i l l not endure unless i t i s coupled with the s a t i s f a c t i o n that the material covered has been thoroughly mastered. There i s no motivating force more conducive to the development of l a s t i n g i n t e r e s t i n a subject than the sense of s a t i s f a c t i o n derived from a thorough knowledge of the subject matter studied. An opportunity for experiencing t h i s sense of s a t i s f a c t i o n can by given to the pupils by making use of special review questions at the beginning of each period. (1) Cf. Pp. 74, 76, Hall' s School Algebra. 80 A short series of c a r e f u l l y selected questions, bearing on topics previously studied, i s placed on the board. The questions should be c a r e f u l l y graded, the f i r s t -ones being comparatively simple and the following ones of increasing d i f f i c u l t y . The l a s t one or two might be r e a l l y d i f f i c u l t ones In t h e i r c l a s s . The following i s an example of a set of questions which would be suitable for this purpose a f t e r the topics of m u l t i p l i c a t i o n and d i v i s i o n have been studied: 1. Multiply 3a 3b 2-5a 2b+ ab 4 by 4a 2b 3. 2. Divide 12x 6y 2-8x 4y 3+ 20x 3y 8 by 4x 3 y 2 . 3. Multiply a 2b-3ac 2+ 7 b 2 c 3 by -3a 3b 4c. A ™ 4 j , c 8 6 o c 12 9 „ K 4 12 , _ 4 3 4. Divide 15x y -25x y -35x y by -5x y . 5. Multiply 3a-r4b by 5a-2b. 6. Divide 6x 2-7x-3 by 2x-3. 7. Multiply -ax 2 +• 3axy 2-9ay 4 by -ax-3ay 2. 8. Divide 15+-2m4-31m + 9m2 + 4m3+ m5 by 3-2m-m2. By working a set of questions such as these, each pupi l has the opportunity of determining the degree of mastery which he has developed up to th i s stage. I f marks are assigned to these questions, and the marks are compared by the method outlined i n chapter I I I , then the pupils become motivated to greater e f f o r t s i n the i r study of algebra at the beginning of each period; and at the same time they are laying for them-selves a strong foundation upon which, to b u i l d t h e i r mathema-t i c a l superstructure. There are usually a few pupils i n each grade IX class who 81 master the work very quickly and are eager to go ahead to new work. Such pupils derive a great deal of enjoyment from knowing that they are considerably ahead of the remainder of the class, and by allowing these pupils to proceed i n t h i s manner, the teacher i s motivating the work for them to a very considerable degree. In a large cl a s s , however, i t would obviously be very d i f f i c u l t to put t h i s p r i n c i p l e into opera-tion to any great extent, because of the i m p o s s i b i l i t y of one teacher giving i n d i v i d u a l i n s t r u c t i o n to a large number of pupils working at d i f f e r e n t places i n the test; nevertheless, the same motivating effect can be obtained i f the teacher has at h i s disposal a large amount of extra material to supplement the work of the text. This material may be i n the form of a series of more d i f f i c u l t types of questions on the various topics studied; a number of supplementary texts i n which certain questions are marked; mimeographed sheets of review questions; or a supply of old examination papers. When a bright p u p i l f i n i s h e s a c e r t a i n assignment s a t i s f a c t o r i l y , he may be given the p r i v i l e g e of selecting one of these supple-mentary groups of questions. The following are a few examples of supplementary questions which are very valuable i n bringing into operation the type of motivation referred to here: 1. Colle c t terms: 5a-6 +9-8a+ a +8 2. Collect terms: 2x-5y+ 3y-3x+x-y 3. Colle c t terms: 2a 2b-5ab 2-ab 2+ a^b 4. Add: 2x+3; 4-x; -3x *• 1; -5 + 4x 5. Add: 2c-3d; -2d*»c; 5d-2c 6. Subtract: 3x-2 from 3-2x 7. From ~2a 3b take 4b-3a A 8. Take 3x 2-5x * 1 from 2x2-f6x~5 82 9. Subtract 2a 2b-3ab 2 from 4ab 2 + 3a 2b ' 10. Find the sum of 3x 2+ 2xy-y 2j ~6x2-»- 3y 2; 5xy-4y 2 11. Multiply (~6x2y) by (4x) 12. Multiply (~4cd2) by (^3c 2d 3) 15. Find the product of -5xy and 1 2 14. Give the square of -4a 15. Give the cube of 3x 16. Give the cube of -2a 4 17. Simplify (-5x 3) 2 18. Simplify (-4xy 2) 3 19. Multiply (-3a+4) by 4a 20. Multiply (2xy-3) by -4x 21. Multiply (3a 2b-2) by (-4ab2) Simplify: 22. -3a(-4a + 6b) 23. (6x - f 3 ) (6x-3 ) 24. (7x-2y)(ox+y) 25. (3a-2b) 2. 26. ( l - 3 x 3 ) 2 27. (9a 2b 2) ~ (-9b) 2 8 « "5x 2y 3 29. C-3xy+6y) - (3y) -5y 3 30. x 2-6x+ 8 x 31. -5a +6a J -3a 32. 3a 3-2a 2b 33. 50x2~5x-6 ' " a 5x + 2 34. 8a 3-27 2a-3 35. The number 78 i s divided into two parts so that one i s f i v e times the other. What i s the smaller part? 36. I f 90 cents be divided among A, B, and C so that A gets three times as much as G and B twice as much as C, what i s A's share? 37. I f x s 3 and y = 4, f i n d value of 2y-xy 2 38. I f x = -2 and y - 1, f i n d value of ( x - y ) 2 39. I f a : 2, b a 3, o s 0, f i n d value of -3(b 2-ac) 40. I f x = -1 and y = 2, f i n d value of 2x 3-4y 2 41. Find the sum of (2x-3) 2 and (3x + l ) 2 42. Add (2a-3b)(a + b) to ( 2 a + b ) 2 43. Take 3x-2y from zero 83 44. Subtract 2a 2-3b 2 from 1 45. I f a boy i s 10 years old now, i n how many years from now w i l l he be y years old? 46. I f a boy w i l l be x years o l d 4 years hence, how old was he y years ago?. 47. How many inches altogether i n x yards y feet? 48. At the rate of x miles per hour, how many miles are t r a -v e l l e d i n b minutes? 49. At the rate of c miles per minute how many minutes are taken to go d miles? 50. What i s the t o t a l cost i n d o l l a r s of x books at y cents e a ch. Wha-t- -ia- -ay- -to-t-aJU -lo-as-?-51. I buy k books at b cents each and s e l l a l l of them at y cents each. -What i s my t o t a l loss? 52. Find the sum of three consecutive numbers, the smallest of which i s x. 53. Find the sum of two consecutive even numbers i f the larger one i s y. 54. What must be added to 3x-2y to give 2x-*-5y? 55. What must be subtracted from 4x-3y to leave 5x+ 2y? 56. From what must -3a +b be taken to leave 4af b? 57. What.must be subtracted from zero, to leave 3x? 58. A rectangle i s x feet long and y inches wide. Give the number of Inches i n i t s perimeter. 59. The perimeter of a square i s y inches. Give i t s area i n square inches. 60. Express the area i n square feet of a rectangle b feet long and 4 inches wide. 61. In a school with 212 students, 3x t 5 are boys and 4x-3 are g i r l s . How many g i r l s are there? Simplify: 62. (la-3b) + (a-lb) 63. (x-ly) t (lx 1 y) t (x-ly) 64. ( 2 a 2 - ^ ) - ( ^ a 2 - l ) 65. fy2)(-^) 66. (|a-2b) 2 67. (Jx-3y)(gx-t y) 68. (3a 2-2gb) - (§) 69. (2x 2-r2x-§) - (x-1) 70. g(5a-6) -t g(2a-3) 71. 3x(lx-1 )-4x(ix-1) 72. 9(a * b ) 2 vs. 2x- (3-(-x+2) (3-2x)} 3(a t b) 74. 2 |5x-2(-x+ y)J 75. Add 5(a + b)-2(a-b) to 4(a + b) f 3(a-b) 76. Subtract 3(x-2)f 5(y+ 1) from 2(x-2)-4(yt1) 77. I f x - 1 and y • JL f i n d value of l x 2 y 3 3 2 2 ' 84 78. % e n a s 1 give value of l-3a +- 3 a 2 - a 3 2 79. When b a -| give value of b 3-5b 2+ 1 80. Expand (x-2) 3 81. Simplify - 1=5 82. Simplify 3£lZ + 7 x ^ 3 y 83. P i l l i n brackets xa-3ya-a - a( ) 84. P i l l i n brackets ax-2bx-x s -x( ) 85. I f A = 3x-2 and B = 2x +3 f i n d value of A -AB 86. Simplify (a-2b) 2-3(2a-b)(a+b) 87. I bought an a r t i c l e for x d o l l a r s and sold i t at a gain of 5% of cost. Find the s e l l i n g p r i c e . 88. I sold an a r t i c l e f o r y d o l l a r s which cost me x d o l l a r s . Find the gain % reckoned on cost. 89. I f It takes y seconds to t r a v e l c feet, express the rate i n yards per minute. 90. I f a a -3, b a 1, f i n d value a 2-5b 2 91. I f x s -3, y a 3, f i n d value of 3 (3x 2-2xy + 3) 92. Find value of (2a + l ) a when a - 2 93. Find the sum of the numerical c o e f f i c i e n t s i n the expan-sion of ( 2 x 2 + xy-y2)2 94. A dealer buys x books at c d o l l a r s each. He s e l l s a l l of them at a uniform price, making a t o t a l gain of b d o l l a r s . For what does he s e l l each book? Pupils i n grade IX are ever ready to attempt the solution of a puzzle, or to j o i n i n some form of mathematical recrea-t i o n . The use of such material i n grade IX, i f c a r e f u l l y selected and presented at suitable times, has a very valuable motivating e f f e c t upon the pupils. It gives them an oppor-tunity of experiencing that " t h r i l l of discovery" which comes from the solution of mathematical problems, and this leads to the development of that inexplicable f e e l i n g of s a t i s f a c t i o n which comes from the working of questions of a mathematical nature. The following are some of the simpler forms of 85 mathematical recreations, which are suitable for introduction into grade IX work i n algebra: Mathematical Recreations 1. I f you drive from Vancouver to San Francisco at an average speed of 30 miles per hour, and return at an average speed of 00 miles per hour, what i s your average speed for the whole journey? 2. Pat and Mike, two painters, were given the contract to paint the lamp-posts on a c e r t a i n s t r e e t . Pat was to paint the posts on the ftorth side and Mike the posts on the south side. There were the same number of posts on each side. Pat ar r i v e d early and began to paint. ?fnen Mike arrived he found that Pat was painting and was just completing the t h i r d post on the south side. Mike pointed out the error to Pat and sent him over to h i s own side where he again commenced to paint at post number one. Mike completed h i s posts f i r s t , so i n order to help Pat he went over and painted six posts for Pat. Now, Pat painted three posts for Mike i n the morning, and Mike painted six f o r Pat i n the evening; who painted the larger number of posts, and ronhow many more than the other. 3. A farmer died and l e f t h i s stock to h i s three sons. The oldest son was to receive one-half of the stock, the second son was to receive one-third of the stock; and the youngest son one-ninth of the stock. When the stock was counted there were seventeen head of c a t t l e . How were they divided, none being k i l l e d ? 4. Express a l l the numbers from one to twenty-one, each time using four 4's and arithmetical signs, e.g. 5 4 x 4 -4-4 4 5. I f a f i s h weighs 13 l b s . and h a l f Its own weight, what i s the.weight of the fish? 6. Write down any f o u r - d i g i t number. Reverse the order of the d i g i t s . Subtract the smaller of these two numbers from the larger. Stroke out one of the d i g i t s i n the answer. T e l l me the remaining d i g i t s and I s h a l l t e l l you which one you struck out. 7. Which would you prefer: a h a l f ton of sovereigns or a ton of half-sovereigns? 8. A man went into a shoe store and bought a pair of shoes. He gave the shoe merchant a ten d o l l a r b i l l , but the shoes cost only $5.00. The merchant could not change the b i l l , so he went over to the drug store and changed i t . He gave the customer $5.00 change and the shoes. After the customer had gone, the drug clerk came over and said that the $10.00 b i l l that the shoe merchant had given him was a counterfeit; and i t was. The shoe merchant then had to give the drug clerk a good ten d o l l a r s f o r the counterfeit b i l l . How much money as well as the shoes did the shoe merchant lose? 86 9. I f 6 cats eat 6 rats i n 6 minutes, how many cats w i l l i t take to eat 100 rats i n 100 minutes. 10. The head of a f i s h i s 9 i n . long. The t a i l i s as long as the head and one-half of the "body, and the body i s as long as the head and t a i l . What i s the length of the fish? 11. I f a log starts from the source of a r i v e r on Friday and f l o a t s 80 miles down the stream during the day, but comes back 40 miles during the night with the return t i d e . On what day of the week w i l l i t reach the mouth of the r i v e r which i s 300 miles long? 12. A king has a horse shod and agrees to pay 1 cent for the f i r s t n a i l , 2 cents for the second, 4 cents for the t h i r d , doubling each time. What w i l l the shoeing with 32 n a i l s cost? 13. A hare i s 10 rods i n front of a hound, and the hound can run 10 rods while the hare runs 1 rod. Prove that the hound w i l l never catch the hare. 14. A and B have an 8-gallon can of milk and wish to divide the milk into two equal parts. The only measures they have are a 5-gallon can and a 3-gallon can. How can they divide the milk? 15. I bought a horse for $90 and sold i t for $100, and soon repurchased i t f o r $80. How much did I gain by trading? 16. • Three books were placed on a shelf i n proper order as shown In the diagram. Each book was three Vol. Vol. Vol. inches thick including the covers, each of which was 1/8 of an inch thick. A bookworm bored a hole 1- Jl Ill from the f i r s t page of volume I, straight through to the l a s t page of volume I I I . How f a r did he travel? 87 CHAPTER VI MOTIVATION IN THE TEACHING OP GRADE XII ALGEBRA In chapter I I , the various forms of motivation used i n the higher grades of high school were outlined. In'."the present chapter those, types which are e s p e c i a l l y applicable to the teaching of grade XII algebra w i l l be discussed more f u l l y , and s p e c i f i c methods w i l l be outlined by which these forms of motivation may be u t i l i z e d to their f u l l e s t extent. In the f i r s t place, we must r e a l i z e that i n t h i s grade there i s ever present that great motivating force of matricu-l a t i o n examinations. The very f a c t that a pupil knows that at the end of the school year he w i l l be required to write departmental examinations i s sometimes s u f f i c i e n t to cause him to put f o r t h a maximum amount of e f f o r t , and reach h i s l i m i t of mastery. But the presence of matriculation examinations i s by no means an i d e a l form of motivation. A p u p i l may work at the highest l e v e l of h i s a b i l i t y In order to avoid f a i l u r e , but at the same time he may have very l i t t l e Interest i n the work which he i s doing. The f i r s t step towards motivation i n grade XII, therefore, should be the creation of a proper attitude towards matriculation examinations.. The teacher should i n s t i l i n the minds of the pupils the idea that examinations are c a r r i e d on not f o r the express purpose of torturing pupils, but rather because they are a desirable means 88 of allowing a pupil the opportunity of s a t i s f y i n g himself that he has reached a high l e v e l of mastery i n the work which he has been doing. Any pupil who has taken a keen i n t e r e s t i n h i s school work, and has discovered the p o s s i b i l i t y of securing a great amount of pleasure from atta i n i n g mastery over successive units of work, w i l l look on the matriculation examinations as another opportunity f o r securing that stimulating sensation of s a t i s f a c t i o n through mastery. From t h i s It must not be i n f e r r e d that the present system of examinations for grade XII pupils i s a good one, because i t i s obvious that matriculation examinations as at present constituted have a great many very undesirable features. Their psychological e f f e c t upon high school students i s often f a r from b e n e f i c i a l and sometimes decidedly harmful. However, as long as the present system i s i n vogue, i t i s imperative that high school teachers make the best of the s i t u a t i o n as i t exists and encourage the pupils to regard examinations i n such a way that they may be affected by whatever b e n e f i c i a l influences are contained i n them. Having created a proper attitude towards examinations, i t remains for the teacher to cause the pupils to develop an int e r e s t i n th e i r work, simply for the work's sake. This can be done very conveniently i n algebra by a spe c i a l adaptation of the unit system. In grade XII algebra the topics studied are of such a nature that there i s very l i t t l e i n common between them. In any one section there i s but s l i g h t reference 89 to the work of sections immediately preceding; and, consequent-l y , there i s p r a c t i c a l l y no opportunity for reviewing a previous unit while studying a new one. Quadratics, indices, surds, r a t i o and graphs are almost complete units i n themselves, and for this reason i t i s convenient to operate three separate unit systems during the course of the year. During the f i r s t part of the year, the various topics are studied from the text book. It would require far too much time, and be too labor-ious a task f o r a pupil to work every question i n every set, but c a r e f u l l y selected and well-graded examples are chosen to comprise each unit of the text. When the tests at the end of each unit indicate that the majority of the pupils have attained a s u f f i c i e n t degree of mastery In that topic, the following unit i s treated i n a sim i l a r manner. By this means a l l the units can be covered i n approximately two-thirds of the school year. Since each unit i n grade XII algebra i s studied as a separate entity, when a l l have been completed i t i s necessary to have some form of review and also some means of coordina-tin g the various u n i t s . These' two objectives - review and coordination - can be reached by making use of two separate sets of unit sheets. The f i r s t set consists of a series of c a r e f u l l y graded questions arranged i n the same order as they were studied. In thi s way a pupil can check up on his knowledge of the various units, and thereby ascertain the exact spot where concentrated review i s necessary, .in the 90 appendix a series of questions i s given which were drawn up for this purpose. The pupils are allowed to work the questions as quickly as they wish, with a minimum number to be done each week. If the series i s begun twelve weeks before the end of the school year, the teacher may divide the l i s t into about ten equal parts and make one part a minimum for each week. At the end of each week the answers are given to the questions i n the section just completed, and special attention i s paid to those questions with which the pupils experienced d i f f i c u l t y . The above-mentioned series of questions, the series f o r review, i s supplemented by a eeries for coordination. This consists of ten separate sheets designed for the purpose of r e l a t i n g the various topics and providing a general survey of the whole subject. Each sheet contains ten questions dealing with d i f f e r e n t sections of the work,?..and the various types are arranged i n a d i f f e r e n t order on each sheet. These questions vary from comparatively easy ones to those of considerable d i f f i c u l t y . A good many of these questions are chosen from previous examination papers, and i f the pupils are informed of this fact they show an added in t e r e s t i n the questions on that account. When working on one of these sheets, i f a pupil finds that he i s unable to master one of the questions, he refers to h i s review sheets and checks up on h i s knowledge of (1) See pJH3 91 that p a r t i c u l a r type of question. The series for coordination i s also given i n the appendix, and these questions may be given during the SET' l a s t ten weeks of the school year at the rate of one^per week. The algebra period on one p a r t i c u l a r day, Friday for example, may be set aside for the discussion of these papers. On the f i r s t Friday, sheet number one i s d i s t r i b u t e d . The pupils work these questions during the week, and hand i n t h e i r solutions on the following Friday. At the end of this period they receive sheet number two. On Friday of the next week the teacher hands back the solutions to sheet number one, marked, c o l l e c t s the solutions to sheet number two, and hands out sheet number three. On each succeeding Friday the same procedure i s followed, the teacher c o l l e c t i n g one set of answers; handing back another set, marked; and d i s t r i b u t i n g a new set for the following week. Most of the algebra period each Friday i s devoted to discussing the papers which have just been returned, and i n examining the questions which gave the pupils most trouble. The adoption of t h i s three-unit method - the u n i t for learning, the unit for review, and the unit for coordination -has a great motivating e f f e c t upon the pupils, and coupled with the development of a proper attitude toward matriculation examinations, i t i s almost s u f f i c i e n t motivation i n i t s e l f for (1) See p. 124 92 making the grade XII course i n algebra extremely i n t e r e s t i n g and s a t i s f y i n g . It brings into operation most e f f e c t i v e l y the motivating forces of s a t i s f a c t i o n through mastery, and of the development of an Interest i n algebraical calculations as a new and d e l i g h t f u l experience. The three-unit system of teaching algebra i n grade XII can be made even more successful by overcoming one weakness which i s common to most unitary methods of i n s t r u c t i o n ; namely, that when one unit i s f i n i s h e d and a new one commenced, the knowledge of the e a r l i e r u n i t gradually fades unless some means i s adopted for keeping In constant contact with i t . This weakness can be overcome by making use of a set of review questions selected to cover a l l the s a l i e n t points i n the .various units without Involving very lengthy c a l c u l a t i o n s . A few of these questions can be assigned for homework each day, handed i n by the pupils, marked by the teacher (probably out of ten) and the marks t o t a l l e d at the end of each, month. The • f i r s t part of each lesson (about ten minutes) may be devoted to going over the questions which the teacher has marked and Is returning to the pupils. Such a system keeps a pup i l constantly i n touch with the material contained i n previous units; and when the time comes to commence the hundred review questions, he finds them much more f a m i l i a r than he would have done had he not had this d a i l y review. The following l i s t of questions w i l l give an idea of the type of question suitable f o r this purpose: 93 1. Solve c / 2x-l ,") „ x + 2 ., „ 5 x - ( — + 3 X + ~ 2 ~ + 7 2. A man bought a number of a r t i c l e s for $200. He kept f i v e and sold the remainder for $180, gaining $2 on each a r t i c l e sold. How many did he buy? 3. Solve 5x2-9x-4 a 0. 4. The hypotenuse of a right-angled t r i a n g l e i s 25 i n . and the perimeter i s 56 i n . Find the remaining sides. p 5. Solve x -xy = 6 y 2-3xy =10 6. Factor (a) 4m4-21m2n2+ n 4 . (b) a 3 f b3-*- a + b (c) 12x 4-27a 2x 2. (d) 4(2a-3n) 2-(3a-7b) 2 7. By use of factors f i n d the product of: fx 2-2(Xrl)/«n-^+ 2(x-l)J 8. I f x + y s 1, prove x 3 ( y + 1)-y 3(x + 1)-x ry = 0 9. Solve x 2-3x = 0. 10. Solve 4-9x = 13x 2. 11. Solve 6y-4x _ 5z - x _ y - 2z 3x-4y ~ 2y-3z ~ 3y-2z ~ : 12. Solve 4x 2 r x-1. 13. Solve 42x 2-28c 2 - 25cx. 4 2 14. Solve x + 2x s 3x . 15. Simplify 3 x 2 a - 4 x 2a~2 94 17. Solve: + -5 » 4 I x^ y^ * 1 - 1 l i 18. Solve: 7 y 2 r 15xy s -68 x 2 + 2xy + 2y 2 = 17 19. Express i n i t s simplest form: 4( /~~5-f 1) _ - 2 •* /~5~ [5-1 2- fi 20. Rationalize the denominator of: 21. Find the square root of: tp48~ - "i/ii " " 22. Solve: -j/I+a t f x ^ b = a V5T a 23. The price of photographs is raised $3 per dozen; and cus-• tomers consequently receive seven less pictures than before for $21. What was the o r i g i n a l price of the pictures per dozen? 24. Solve: x ^ l „ , f x V 1 — — — — — « O r — — — i f ^ - i 2 25. Write down the roots of: (a) (x-a+b)(x-a-b) = 0 (b) x 2+ 2x - 0 26. Simplify: 27. Simplify: 2 / § ) - J3(2 f2- fz) 39. Simplify by removing negative indices:, a ^ - f b"^" a *V b ^ 40. Find the value of: x 3 - f x 2+x-/-l when x s 1 41. Simplify: 42. A man has h hours at h i s disposal. How many miles can he ride out at r miles per hour i f he must walk back a t w miles per hour? 43. Find the mean proportional between /27^3 fz and ^ 7 - r 3«T¥ T a a+b-f c t d a-b-r- c-d a c 44. I f — ~ — prove T - ~} a ^ b - c - d a-b-c-fd. d. 45. Find the square root of | -t fb~ 46. Using the scale 1 u n i t = \ inch, f i n d the point of i n t e r -5 section of the graphs of 2x-3y s 24 and — _ X _ 12 3 2 -47. Find the equation for the straight l i n e passing fchfcough the points (3,4), (-2,5) 48. Solve for a and for n: S - ^(B^-f£) 49. Solve for r and for v: F - m v 2 gr 96 50. Solve for x: x(a-x) = c . Give the numerical value of the roots when a = 16 and c = 6. There are several other minor, but nevertheless important, means by which the teaching of grade XII algebra may be motivated to a f u l l e r extent, and one of these i s a careful treatment of algebraical problems. A number of students seem to develop somewhat of a d i s l i k e for problems, and become possessed with, the idea that grade XII problems are t r i c k y b i t s of mathematical reasoning almost beyond any ordinary student's a b i l i t y . The development of thi s erroneous impres-sion can be prevented by prefacing the work on problems with a few c a r e f u l l y selected puzzles which require considerable thought for t h e i r solution, but which are by no means out of the range of any average student. I f these puzzles are presented so as to give each, pupil the idea that they are a challenge to h i s ingenuity and resourcefulness, then he w i l l respond to the challenge and do h i s utmost to obtain solutions. The following are examples of rather simple puzzles which might very well be used as a preface to a lesson on problems: 1. A blacksmith had a stone weighing 40 pounds, and a s k i l l e d mason broke i t into 4 pieces whereby any number of pounds from l ' t o 40 could be weighed on scales. Find the weight of each of the four pieces. 2. Two automobiles 20 miles apart are approaching each other, each t r a v e l l i n g 10 miles per hour. A bee, which f l i e s at the rate of 15 miles per hour starts at the radiator of one automobile and f l i e s back and forth between t h e i r radiators u n t i l the automobiles meet. How f a r does the bee f l y ? 3. When I was born my s i s t e r was one-fourth mother's age, but she i s now one-third father's age. I am now one-fourth mother's age, but i n four years I s h a l l be one-fourth father's age. How old Is each of us? 97 4. How s h a l l we buy 12 eggs f o r eighty cents, i f hen eggs s e l l at 5^ each, duck eggs at 7^ each, and turkey eggs at Qtf each, and i f we buy some of each? Puzzles l i k e the ones given above are extremely i n t e r e s t -ing to the high school student, and he soon begins to r e a l i z e that r e a l enjoyment can be derived from attempting to solve problems of various kinds. Using such puzzles as these as an introduction, the teacher can lead the class on to the investi g a t i o n of problems which involve the use of equations i n t h e i r solution, and once a pupil experiences the " t h r i l l of discovery" derived from the solution of a mathematical problem, he i s eager to tackle more problems which w i l l enagle him to experience that t h r i l l more often. There are many ether forms of mathematical recreations, besides the s t r i c t l y "problem1' type, which can be used very conveniently as a means of motivating the work i n grade XII algebra. The following are some examples of such: Mathematical Recreations 1. A ship i s twice as old as i t s engine was when the ship was as old as Its engine i s now. Their combined ages are 42. ~ How old are they? 2. A farmer bought one hundred head of 3tock consisting of calves, sheep and lambs. The calves cost $10 each; the sheep $3 each; and the lambs, 50^ each. Altogether the hundred head cost him one hundred d o l l a r s . How many of each did he buy? 3. My father was born on 2 (l* J U E f l 2 J L (2.3.3 s.6 2)-3 J)' What was his age on August 10th, .1935? 4. I f Dr. Jones loses 3 patients out of 7; Dr. Smith, 4 out of 13; and Dr. Brown, 5 out of 19; what chance has a sick man for h i s l i f e who i s dosed by the three doctors for the same disease? 98 5. Mr. Dough, a business manager, wished to h i r e one of three men. In order to decide which was the smartest, he adopted the following system. He c a l l e d the three men to him and showed them f i v e pieces of paper on h i s desk. Two of the papers were black and three were white. He then t o l d the men to turn around, and he pinned a paper on each man's back, putting the two remaining papers i n h i s pocket. He to l d the men to look at the papers on each other's backs, and by so doing decide what color of paper was on t h e i r own backs. The f i r s t man to decide the color of the paper on his back was to come up to the desk and explain how he made the decision. I f his explanation were sound he would get the po s i t i o n . One of the applicants was successful i n determining the color of paper on his own back. Which one could i t have been, and how did he decide? 6. An eagle and a sparrow are i n the a i r , the eagle 100 f t . above the sparrow. If the sparrow f l i e s straight forward i n a horizontal l i n e and the eagle f l i e s twice as f a s t d i r e c t l y towards the sparrow, how f a r w i l l each f l y before the eagle reaches the sparrow"? 7. To prove that 1 s 2 Let x - 1 Then x" - x . . x 2 ~ l - x-1 Factoring (x-l)(x-t-l) = x-1 Dividing x + 1 = 1 but x = 1 . [ . 1 + 1 = 1 . . . S a l 8. To prove that -1 = 1 i I - a 3 _ a* ~? "~ "I? . '. ( - a i ) 2 - ( a * ) 2 m ?i 1 -a - a . . —1 s 1 Where i s the f a l l a c y ? 9. Find the keyword i n the following problem i n "Letter D i v i -sion": 0PN)A0UIERT(PCAAU CPN PIUI PUCM RRIE RHAH REER RHAH RIRT RCUN 99 A t e a c h e r who t e a c h e s more t h a n one c l a s s o f g r a d e X I I a l g e b r a h a s a t h i s d i s p o s a l a n o t h e r means o f m o t i v a t i n g h i s t e a c h i n g t o some e x t e n t ; n a mely, t h e s t i m u l a t i o n o f f r i e n d l y r i v a l r y b e t w e e n c l a s s e s . I f t h e same t e s t i s g i v e n t o b o t h c l a s s e s , a n d t h e p u p i l s know b e f o r e h a n d t h a t t h e r e s u l t s o b t a i n e d i n t h e two c l a s s e s a r e g o i n g t o be compared, t h e n e a c h p u p i l w i l l do h i s u t m o s t to a v o i d p u l l i n g down t h e a v e r -age o f t h e c l a s s . The same t y p e o f m o t i v a t i o n i s e m p l o y e d when a t e a c h e r h a s a r e c o r d o f t h e r e s u l t s o b t a i n e d b y p r e v i o u s m a t r i c u l a t i o n c l a s s e s , a n d e n c o u r a g e s h i s p r e s e n t c l a s s t o s t r i v e t o s u r p a s s e v e n t h e b e s t o f t h e p r e v i o u s r e s u l t s . I f c e r t a i n f o r m e r s t u d e n t s c a n b e named who o b t a i n e d 100$ i n t h e i r m a t r i c u l a t i o n e x a m i n a t i o n s , t h e good s t u d e n t s i n t h e c l a s s w i l l make up t h e i r minds t h a t t h e y , t o o , w i l l o b t a i n a p e r f e c t mark. C a r e must be t a k e n , o f c o u r s e , t o p r e v e n t t h e p u p i l s f r o m d e v e l o p i n g more I n t e r e s t I n e x a m i n a t i o n marks t h a n i n t h e s t u d y o f t h e s u b j e c t i t s e l f ; n e v e r t h e l e s s , t h e i n t r o d u c t i o n o f a s l i g h t e l e m e n t o f c o m p e t i t i o n i s a v a l u a b l e means o f m o t i v a t i n g t h e work i n g r a d e X I I a l g e b r a . The m o t i v a t i n g f o r c e o f s c h o l a r s h i p s and p r i z e s i s one w h i c h i s o p e r a t i v e t o a c e r t a i n e x t e n t i n t h e m a t r i c u l a t i o n g r a d e . T h i s f o r m o f m o t i v a t i o n a f f e c t s o n l y a v e r y s m a l l p e r c e n t a g e o f t h e c l a s s , b u t i t I s ' v a l u e i n e n c o u r a g i n g t h e few o u t s t a n d i n g s t u d e n t s t o do t h e i r v e r y b e s t . T h e s e p u p i l s s h o u l d be i n f o r m e d b y t h e t e a c h e r as t o what s c h o l a r s h i p s and 100 p r i z e s a r e o f f e r e d t o m a t r i c u l a t i o n p u p i l s , a n d i t s h o u l d be i m p r e s s e d u p o n them t h a t t h e y a r e a b l e t o o b t a i n one o f t h e s e awards j u s t as w e l l as any o t h e r s t u d e n t i n t h e p r o v i n c e . As i n t h e c a s e o f r i v a l r y b e t w e e n c l a s s e s , t h i s f o r m o f m o t i v a -t i o n s h o u l d n o t be a l l o w e d t o become t h e d o m i n a t i n g f o r c e b e h i n d a s t u d e n t ' s e f f o r t s , b u t s h o u l d o n l y s u p p l e m e n t t h e m a j o r f o r m s o f m o t i v a t i o n d i s c u s s e d i n t h e e a r l i e r p a r t o f t h i s c h a p t e r . 101 CHAPTER VII SOME EXPERIMENTAL EVIDENCE TO SHOW THE EFFECTS OF MOTIVATION Every teacher of mathematics has noticed that c e r t a i n of his lessons have seemed very d u l l and uninteresting both to himself and to h i s pupils, while other lessons have been'ex-ceedingly enjoyable. When.the f i r s t type of lesson i s complet-ed the teacher experiences a sense of r e l i e f , and the pupils lose no time i n getting away t h e i r mathematics books and preparing f o r the next lesson; but when the second type of lesson i s f i n i s h e d the teacher experiences a sense of great s a t i s f a c t i o n , and the pupils continue working even afte r the b e l l has rung, t r y i n g to complete as many questions as possible before the next lesson begins. The reason for the difference i s obvious. The f i r s t type of lesson was lacking In motiva-tin g value, while the second had some features about i t which aroused and held the interest of both pupils and teacher. The b e n e f i c i a l r e s u l t s derived from t h i s second lesson would un4 doubtedly outnumber those derived from the f i r s t ; and at the same time, the material studied i n the highly motivated lesson would be better understood and more e a s i l y remembered than that studied i n the d u l l mechanical lesson. The writer has recently completed a three-year experiment with methods of motivating the teaching of high school mathematics. Although the results of t h i s experiment cannot be accepted as conclusive because of the smallness of the o 102 and because of the fac t that a number of variable factors were not controlled, nevertheless they may be considered as evidence to show that e f f i c i e n c y and Interest can be developed i n the subjects of algebra and geometry to a marked degree by the use of a highly motivated form of teaching. The writer conducted the experiment between the years 1931 and 1934. He taught the experimental class f o r three years i n succession i n grades Ten, Eleven and Twelve. The class was an extremely heterogeneous one, being composed of pupils of many d i f f e r e n t types and of several d i f f e r e n t n a t i o n a l i t i e s . Some of the pupils had done t h e i r grade IX work i n junior high school and others had come d i r e c t l y to the senior high school from grade VIII i n the elementary school. There was a wide v a r i a t i o n i n the i r academic standing, some advancing to grade X with a very good scholastic record and others being promoted only on t r i a l . The class as a whole was not considered a good cla s s , the p r i n c i p a l ' s opinion being that i t was a very weak one. On being assigned as teacher to the above-mentioned class, the writer attempted to develop the pupils' i n t e r e s t i n mathematics and at the same time to bring them to a high l e v e l of e f f i c i e n c y . In attempting to do t h i s he adopted most of the methods mentioned i n thi s thesis. The re s u l t s of the experiment can best be judged from the tables which follow. In the week previous to the matriculation examina-tions, the pupils of the class were asked to write down the 103 names of the three subjects which they had enjoyed the most during t h e i r high school course - these names to be i n order of merit. The res u l t s of t h i s vote, giving Five for f i r s t choice, Three for second choice and One f o r t h i r d , are as follows* Algebra . . . . 54 Chemistry . . . 51 Physics . . . . 31 Geometry . . . 24 French . . . . 17 Literature . . 19 So c i a l Studies. 16 Composition . . 3 Physiology . . 1 Grammar . . . . 0. The above procedure had i t s l i m i t a t i o n s i n as much as the writer conducted the voting and the b a l l o t s were signed by the pupils. However, the pupils were Instructed very emphatically to overlook the personal element e n t i r e l y and give exact statements of the i r preferences. At the same time the pupils were asked to write down the names of the three subjects i n which they thought they had obtained the highest degree of e f f i c i e n c y , and i n which they were roost confident of making good marks i n the f o r t h -coming matriculation examinations. The re s u l t s were as follows: ; Algebra . . . . 54 Geometry . . . 44 Chemistry . . . 28 Literature . . 22 Grammar . . . . 20 Physics . . . . 19 Social Studies 13 French . . . . 10 Composition . . 4 Physiology . . 1 104 The marks actually obtained by the pupils of this class i n the Junior Matriculation'examinations are shown i n the table given below. TABLE II . • ~ ' Class marks - Junior Matriculation examinations Pupil Algebra Geometry A 92 92 B 75 71 C 19 17 D 74 50 E 95 81 P 83 79 G 84 61 H 83 70 I • 85 61 J 44 45 K 80 69 L 84 78 M 87 80 N 63 60 0 83 83 P 94 92 Q 68 67 R 58 74 S 94 83 'T 92 72 U 81 69 V 55 57 W 100 82 X 94 89 Y 62 67 Z 56 55 a 82 73 Notes: 1. Pu p i l C, whose marks, brought the class average down considerably, was only a conditioned student, not having passed grade XI; and during the spring term he was injured i n a t r a f f i c accident which caused him to be absent from school for s i x weeks. 2. There was one "repeater" i n the c l a s s . The following table compares^the averages of the other classes i n the same school and with the City (Vancouver) and 105 P r o v i n c i a l (B.C.) averages: TABLE III Comparison of Averages i n Subjects A l l Other Algebra Geometry Subjects Averages of Experimental Class 76.6 69.5 59.5 Averages of other Classes i n Same 57.2 66.8 60.2 School City averages (Vancouver) 63. 65.8 59.8 P r o v i n c i a l Averages (B.C.) 63.5 63.9 60.7 In the above table c e r t a i n points should be noticed when considering the r e s u l t s of using a s p e c i a l l y motivated form of teaching. In the f i r s t place, the table indicates very c l e a r l y that the class was c e r t a i n l y not a "picked" c l a s s . The average of the class In a l l subjects other than algebra and geometry i s 59.5$ which i s below the average of the three other grade XII classes i n .the same school; i t i s below the c i t y average and below the p r o v i n c i a l average. This compari-son indicates that the class as a whole was rather on the weak side/.- In contrast with t h i s comparison, the algebra and geometry averages for the experimental class are considerably above the averages for other classes i n the same school, as. as well as being much higher than both the c i t y and p r o v i n c i a l averages f o r these subjects. It should be noticed s t i l l further that the average In algebra f o r the experimental class i s exceptionally high, and when the voting was taken 106 before the matriculation examinations were held, the pupils expressed themselves as f e e l i n g better prepared for t h e i r algebra examination than for any other. They also indicated by t h e i r b a l l o t s that they had received more enjoyment from the study of algebra than from the study of any other subject i n t h e i r high school course. The evidence seems to indicate, therefore, that the teaching of mathematics can be made much more e f f e c t i v e by a wider application of the p r i n c i p l e s of motivation. It seems safe to i n f e r that, by a careful study of the various s p e c i f i c means of motivating the teaching of mathematics In high school, average and even weak classes may be brought up to a high l e v e l of e f f i c i e n c y i n both algebra and geometry; and also that the enjoyment of these subjects by the pupils may be increased very greatly. 107 CHAPTER VIII GENERAL CONCLUSIQNS In closing t h i s t r e a t i s e on Motivation i n the Teaching of High School Mathematics, i t i s apparent that the following conclusions might he reached; namely, (1) that mathematics i s a very essential part of an i n d i v i d u a l ^ education; (2) that the teaching of mathematics i n high school i s at present not of a s u f f i c i e n t l y high standard to enable students to derive the maximum amount of benefit from t h e i r mathematical studies; and (3) that the si t u a t i o n can be improved tremendously by a more extensive application of the p r i n c i p l e s of motivation. The statement that mathematics i s an es s e n t i a l part of an individual's education i s corroborated by the statements of numerous authorities on the subject, who commend i t s study both from a u t i l i t a r i a n and an aesthetic point of view. The assertion of Comte that " A l l s c i e n t i f i c education which does not commence with mathematics i s , of necessity, defective at i t s foundation" ^ i s emphatic indeed; while W. A. M i l l i s states that "For algebra there i s no substitute. The elimina-t i o n of algebra as a pure science from the curriculum would (2) nut the foundation from under a l l s c i e n t i f i c procedure." David Eugene Smith speaks of geometry i n metaphorical language as follows: "Geometry i s a mountain. Vigor i s needed for i t s (1) Jones, Mathematical Wrinkles. P.^s". (2) M i l l i s , Opcit. P. 240. 108 ascent. The views a l l along the road are magnificent. The e f f o r t of climbing i s stimulating. A guide who points out the grandeur and the special places of i n t e r e s t commands the admiration of h i s group of pilgrims." Numerous other quotations might be given emphasizing the same fac t - that the study of mathematics i s a most essential part of a high school student's education. In support of the conclusion that the teaching of mathematics i s at present not of a s u f f i c i e n t l y high standard to enable the students to derive the maximum amount of benefit from t h e i r mathematical studies, we have the statement of Wi A» l i i i i s - i - - t h a t "The reason for the present unsatisfac-tory status of mathematics i s poor teaching." There i s also the s t a t i s t i c a l evidence given by Professor Judd showing that the percentages of f a i l u r e s i n mathematics, and with-(3 drawals due to lack of i n t e r e s t i n mathematics, i s very high; To these might be added s t a t i s t i c a l evidence given by M i l l i s emphasizing the same point and outlined i n h i s "Teaching of (4) High School Subjects". E. R. Br i s l o c k states that (5) "Algebra i s a subject d i f f i c u l t to learn and to teach," and Professor Judd writes that "Mathematics must be recognized as one of the most d i f f i c u l t subjects i n the high school (1) Jones, Ibid. P. a&'v-(2) a i l l l s V e , Opcit. P. 233 (3) Judd, CH. Psychology of H.S.Subjects. P. 18. (4) M i l l i e , Opcit, P. 232 (5) B r e s l i c h , Problems i n Teaching Secondary School Mathema-t i c s . P. ToTI 109 course." Both, these men follow up t h a i * statements hy the conclusions that the teaching of mathematics Is at present not up to the standard necessary for producing best r e s u l t s i n the subject. The t h i r d of our general conclusions, that the teaching of mathematics can be improved tremendously by a more extensive application of the p r i n c i p l e s of motivation, i s supported to a large extent by the evidence presented i n preceding chapters of this thesis, but these may be supplemen-ted by the opinions of authorities on these questions. Professor Judd says that "We s h a l l c e r t a i n l y need to inquire, i n such negative cases, what i s the psychological character of pleasure, and what the p o s s i b i l i t y of so readjusting the si t u a t i o n as to produce pleasure through the study of (2) geometry." v F. W. Westaway also stresses the importance of increasing the in t e r e s t of students i n the subject of mathematics, and he suggests several ways by which t h i s may be brought about. He says that "A young boy's (13-14) natural fondness f o r puzzles of a l l kinds may often u s e f u l l y be (3) employed for furthering h i s i n t e r e s t i n geometry." v ; Westaway also states that "No boy can become a successful mathematician unless he gights hard b a t t l e s on h i s own behalf," and that "There i s no better means of giving a boy a permanent (1) Judd. Opcit. P. 17. (2) Judd. Opcit. P. 89. (3) Westaway, Craftsmanship i n the Teaching of Elementary Mathematics. P. 229 110 in t e r e s t i n mathematics than to help him to achieve a mastery of the commoner forms of mathematical puzzles and f a l l a c i e s . The effects of paying special attention to motivation i n t the teaching of high school mathematics may he i l l u s t r a t e d graphically as shown on p. 111. The graphical I l l u s t r a t i o n s given on p. I l l are not based upon the s t a t i s t i c a l r e s u l t s of experiments, but they give us a v i v i d comparison between the amounts of i n t e r e s t created. It w i l l be noticed that graph II indicates that even though a teacher of mathematics ignores the p o s s i b i l i t i e s of motivation, some students develop a considerable i n t e r e s t i n the subject. These students, however, do not reach, the same height of inte r e s t which they would reach under more favorable conditions. The majority of students under the conditions of graph II w i l l undoubtedly develop a d i s l i k e for mathematics, and i n a few cases t h i s d i s l i k e may be exceedingly great. In graph III i n d i c a t i n g the amount of inte r e s t created by a teacher using a highly motivated form of teaching, we notice that the majority of pupils under such conditions develop a r e a l i n t e r e s t In the subject, and some of these reach an exceptionally high l e v e l . Only a small percentage of the pupils develop an actual d i s l i k e for mathematics. Graph IV shows us the s i t u a t i o n which might be reached i f an exception-a l l y well developed system of motivated teaching were employed. (1) Westaway, Ibid. P. 11. 112 In an average class i t should be possible to have every member interested to some extent i n the subject of mathematics. I f the conditions indicated i n graph IV were reached,, then students would not only gain much po s i t i v e enjoyment from the study of mathematics, but the l e v e l of e f f i c i e n c y would follow to some extent the l e v e l of in t e r e s t , and the whole status of mathematics i n our high schools would be tremendously Improved. There i s , therefore, an urgent need for a greater atten-t i o n to motivation i n the teaching of high school mathematics, and i t has been pointed out i n this thesis that there are various s p e c i f i c means by which the p r i n c i p l e s of motivation may be applied to the teaching of algebra and geometry. By employing c e r t a i n general methods of motivation throughout a l l the grades, and by supplementing these by more d e f i n i t e methods espec i a l l y suitable for ce r t a i n grades, the subject of mathematics could be made Intensely i n t e r e s t i n g , both to the pupils and to the teacher. ^ I f t h i s procedure were f o l -lowed by more teachers of high school mathematics, then the increase i n in t e r e s t , followed by a natural increase i n e f f i c i e n c y , would regain for mathematics some of the prestige i t has l o s t due to the use of mechanical methods of teaching seriously d e f i c i e n t i n motivating value. (1) The importance of creating i n t e r e s t i s discussed by Stormzand, and he arrives at the conclusion that "the pro-blem of Interest plays such an important part i n education because success i n a l l teaching involves the arousing of s u f f i c i e n t i n t e r e s t . " Stormzand, Progressive Methods of Teaching. P. 129. 113 APPENDIX I GRADE XII ALGEBRA QUESTIONS SERIES FOR REVIEW 1. Solve: 2 _ 3 41 x 2y - 35 2| . 3j _ _73 2x y ~ "70 2. Solve: 2y - x - 4xy 4 _ 3 y x ° 9-3. Solve: |x gy + gZ = 8 1 1 1 c gx - §y -+ gz - 5 l x 4- l y - z - 7 4. Solve: x-f-z-1 a |(x=*-4z-8) = |(x-*-9z~27) = y 5. Solve: I -t- | = 1 P 2 3 6 1^-1 - 2 - 8 2 3 6 " 5+2 _ i o 3 2 -6. Solve: x-8 x 4 x-5 . x-7 1 _ ^ x-10 x - 6 x-7 x-9 7. s o l v e : 5x-64 _ 2 x - l l _ 4x-55 _ x-6 x-13 x-6 - x-14 x^V 8. Solve: x 9~x r » 1 8-x x-2 T^x - x - 1 B^x 9. Solve: x b . _ x-v-b-a x -t-b-c ~ 10. Solve: x-bc . x-ca Lx-ab 11. Solve: 5 _8 1 x-t-2a x-a ~~ a 114 12. Solve: _£,x- 4 ^ _ 4 c Sd I "~ cx die 13. Solve: jx 2-6ax+9a 2-b 2 s 0 14. Solve: m(x-t-y)4h n(x-y) s 2mn m(x-+ y)-n(x-y) - mn 15. Solve: l x - my § c x y 16. Solve:' y s *±JL + £. 2 + 3 y-+ b , a X a 2 + 3 17. Solve: 2x , 5x-l 5 x - l l x-1 x + 2 - x-2 18. Solve: (a) 4x 2-10x = -5 (b) x 2-f 6X-J-3 - c 19. Write the general equation for quadratics and solve i t for x. 20. Solve: (a) x2-»- ax-a 2 - 0 (b) x(a-x) = c 2 21. Find two values for x which w i l l make x(3x-l) equal to 0.362, giving each value to the nearest hundredth. 2 2 " 22. Solve: (a) x2-»- = a 2-hb 2 (b) (x 2+ 2 ) 2 s 29(x2-H2)-198 (c) x(x-2x) = g 8 & 4 +-7a2 x -Sax 23. Solve: (a) x 3-3x-2 - 0 (b) x 4-t- 2x - 3x 2 (c) x 3 + 6a 3 = 7a 2x 3 P 24. Solve: 4x -15X^ H- 1 = 0 having given one root as x s -115 25. Solve: 26. Solve: ( x - 1 ) 2 - g ( x - l H l O * (a) 3x-+ 3y : 4 2xy-+y2 - 7y-2 27. Solve: (a) 1 . 1 _ = 178 To) 0>) xy = 35 1' 1 12 x y = ZE x &. y' 39xy -1 2 _1 x 1 1 -x y = 4 1* 28. Solve: (a) x -2xy - 24 .xy-2y 2 = 4 (b) 4x2-+ xy -. 7 3xy+ y 2 = 18 29. Solve: 30. Solve: 31. Solve: 32. Solve: 33. Solve: 34. Solve: 35. Solve: (a) ^ 2 x y + 1 0 y 2 s 145( b) 2x 2-3xy+ 2y 2 - 2 2xy x7+ y 2 z 24 x 2-4xy+ y^ -H f = 0 (a) x 4-+x 2y 2-t y 4 - 21 (b) 4x 2-2xy+ y 2 3 31 o p x^-i- xy +-y = 7 8x3-t- y 3 s 217 (a) x 3 r y 3 = 243 xy(y-x) s 162 (b) x-+ 3xy s 35 y-t- 2xy =22 (a) 2J/X-1 - V/4X-11 (b) 2^/5x-35 *• 5^/.2x-7 (a) |/xT~a-)/x-a s l/a (a) \/x -fr 3 3j/x-5 \/x - 2 ~ 3[/x-13 ( &) 6 \/x-7 _ 7 t/x-26 t/x-1 (b) x/'2 2* (b) \^x+ 7+|/x + 2 = y/6x-r 13 (b) )/x -t- a + / x T b - — V'X-»- a 7 Vx-21 5a 36. Solve: (a) \/9 t 2x - )/2x * l/a*< x 2 5 |/9 -t 2x \A t x^-j/x-7 _ 1/1+ x- (/x-7 37. (a) Prove t h a t a 0 = 1 = 2 (b) Prove t h a t a " n s -± a 116 •38. Multiply: 3m 3-3m 2m by 5m -f 4 39. Divide: x x 4 - y -2y by x -x A A T 40. Find the sq. root of: (a) 9x-12xs-t-10-~ •+• = VX x Cb ) 12ax-t- 4-6a 3 x+ a 4 x + 5 a 2 x 41. Simplify: r , \JH \^ . Q~ '/j ) ^ (b) ( o j \ '/3 l / ^ W r ^ " ) 42. Simplify: ( ^ - j -r ( ^ ^ 43. Simplify: 3 (a) (a 2-b 2)2x(a^-b)^x(a-b)§ -2 (b) l/(a 3b 3-4 a 6 f / C b ^ a ^ 3 ) - 1 44. Simplify: (a) 3.2"-4.2^g (b) 2f_l£ „ 6^111. 2 n.2 n" 1 15~n~"^ 5n~"1 45. Simplify: 4 ^ 6 . 3 n " V l 46. Factor: (a) x«2 \/x~15 (b) x a-49 1 A (c) 3a s+5a 4-»-2 (d) x3m-+ 27 (e) x"2c-+ x~ c~2c 47. Express i n i t s simplest form free from_radical signsi - 1-t ( a) x ~ 7 x * _ ^ J L w l (b) aJ->-ab \/iT x«5 Vx-14 V* ab~b l/a-b 48. Find the value o f : (a) 3J/147-7\/U - i i 1/1 (b) |/l8p 3q 3-pj/8pq 3-qj/50p 3q 117 49. Find the value of the following to two decimal places: (a) 54 2 1/3 (b) 7 1/3 - 5 ]/2 7-4 ]/3 ]/48 + V18 50. Find the value of: 3cx v 4cx 3c v 8c 51. Find the value of : (a) \/2__ 2+ x\/2 (b) 22 ( y f 8 W 7 .) 3~\/2 .7 3\/2*|/7 52. Express with r a t i o n a l denominators: (a) V& -t- x 4 \/a~x (b) \/a -»• x- V^a-x 2+ |/3 -»|/7 1 \/3 + Vs" - v^ s" 53. Express i n i t s simplest form: (a) l/x2-y2-» x _ x2-» y2>-y (b) 4(^34-1) 2+-1/3 i / 2 2 , / 2 2 1/3»1 2~ V/3 |/x -+ y 4-y x- |/x -y v / 54. Find the square root of: (a) 49^20/6 (b) \/175 - ^147~ (c) 3x-l-r 2 \/2x2-3x-2 (d) 2m-t- 2 \/m2-9n2 55. Express i n i t s simplest form: (a) \/l9 -t 4 1/21 H j/7 - /12 « \/29-2 /28 (b) \/27« /8 4 1/17 -t 12 1/2 - 1/28-6 ^ 3 56. Show thatj/a+yb cannot be expressed i n the form |/x + |/y 2 unless a -b i s a perfect square. 57. (a) Find the r a t i o compounded of the duplicate r a t i o 3:7 and the r a t i o of 35:27. (b) Find the r a t i o of x:y from the equation d a x b y - 221 2ax *by ax 58. (a) I f m:n i s the duplicate r a t i o of m-f-x:n-Kx, prove that 2 x = mn. 119 58. (b) I f a and b are unequal, and ab(c 2-j- d 2) - b 2 c 2 + a 2 d 2 shew that the r a t i o of a to b i s the duplicate r a t i o of c to d. 5 9 . I f 2 a S B i prove that _ H ± q s u a 3 ^ 2 u 2 q s u 6 0 . I f |« c g = | prove that each r a t i o i s equal to 5/ 6 a 2 c 2 e - c 4 e f 4-7ac5 2 2 4 2 5 6b d f-d f + 7ad 61. i f 22 :• shew that ap+bq-cr = 0 and bz-cy "" cx-t-az ~ ay-Hbx xp-^yq-4-zr 3 0. 5 2 ' I f I f r l S = I 1 ? = o,Til Prove that each of these r a t i o s i s oz« y_L z—x ^y**ox equal to £; hence show that either x - y or z a x t y . .y. 63. (a) I f g = § prove that £ + £ b d b "~ d (b) I f a s S prove that & = £ b . d c d 64. '(a) Find the fourth proportional to 12x 3, 9ax 2, 8a 3x. (b) Find the t h i r d proportional to 5 1/15. (c) Find the mean proportional to 2 1/18; 3 1/128. 65. I f a, b, c are three proportionals, shew that (b 2+ be +c 2)(ac-bc t c 2 ) - b 4 + ac 3+-c 4. 3 3 3 66. I f a:b - c:d, prove that a ^ ~° ^ = -HI b^c -c 3d ad 2 • 67. I f a, b, c, d are i n continued proportion, prove that a:d s a 3 + b 3 + c 3 : b 3 + c 3 - r - d 3 . 68. I f (a + b-3c-3d)(2a-2b-c-+d) 3 (2a + 2b-c-d)(a-b-3c -f3d) prove that a, b, c, d are proportionals. 69. I f b+c i s the mean proportional between a+b and c + a show that b+c:c-+a = c-a:a-b. 70. I f 2 , & then * 2-*y+y 2 _ x 2 J a+b 2 . . , - 2 - ~T> ^ a ab+b a 71. I f 12:x = x:y - y:z s z:18, calculate the value of x to two decimal places, and show that x 4 + y 4 + z 4 - ( x 2 + y 2 v - z 2 ) ( x 2 - y 2 +z 2) 120 72. I f a+-x:a~x i s the duplicate r a t i o of a+b:a-b, then x-b:a-x 5 b(a+b):a(a-b). 73. Resolve into factors: (a) c 3 d 6 e 9 - l (b) a 4-3a 3~a 3b -+ 3a 2b (c) 0 p x^-y^-t- x H-y (d) 9(a-2b) 2-(4a-7b) 2 (e) 2cd-2xy+x2-»-y2-c2-d2 (f) a 6-6a 3-a 2-2ab-b 2-+ 9. (g) c 2 +-9(d 2~a 2) +6cd (h.) x 4 +- 25y 4-19x 2y 2 (I) 4m4-21m2n2 + n 4 ( j ) 30a2-*-37ab-84b2 (k) - x 8 - y 8 (1) 3(2b 2~l)-7b (m) (c -hd)3-»- ( c - d ) 3 (n) x 3-37x~84 (0) 6a 3-f a 2~19a -t-6 (P) a-7 V a -t-12 (q) x a-49 (r) 3a«-+ 5^ 4 - 2 (s) x- 2 c+x- C-20 (t) x 3 m - t 27 74. Simplify: (a) 1-t 8x 3 v 4x-x 3 _ (l-2x) 2-h 2x x2 * 2 • 2 (2-x) l-4x 2-5x-t-2x ( b) x 2 ( x ~ 4 ) 2 . ( x 2 ~ 4 x ) 3 y 64-x 3 16-x 2 (x + 4 r ~ 4 x (x-f4)' 75. Simplify: (a) a 3a (a-x); (h) 1 1 2 2 x -+• ax-2a 2x 2a+x 4X1" a-x a -v x 2 , 2 a + x ( x 2 - a 2 ) 2 76. Find the value of: t \ 2 2 (a) -m nr L n rm Px2JPlIX . (rrr»n)(m-r) (n-r)(n-m) (r-m)(r-n) (h) q -t- r + r-4- p 4 p-» q (x-y)(x-z) (y-z)Cy-x) (z-x)(z~y) 77. Find the value of: (a) 1 - x-» y x+y-121 77. (b) 2 1-2- - 3 -d 1-x 78. (a) Divide ^ I6x-27 . 13 x^ -16 2 n e by x-1 + — ± (b) Multiply x + 2 a - ^ by 2 x - a . ^ 79. (a) Simplify: 3 (. 1 jUx 8(l-x) 8 ( l + x ) 4 ( l - t x 2 ) 4 ( x 2 - l ) (b )Simplify: x+-2y 5x 2+63xy-4-70y 2 •tjrx-y 2x 2H- 3xy-35y 2 80. Find the value of f - t 2 * + ~ — • + p 4 a b „ when x » - ^ r -2b.-x 2b-+x 7Ld-^bd a-+-b 81. Draw qua d r i l a t e r a l ABCD,"the coordinates of i t s angular points being A(12,4)j B(-2,8); C ( - 8 , - l l ) ; D(9,-9). Cal-culate i t s area. I f the unit used i s 1/10", what i s the area of the quadrilateral i n square inches? • 82. Draw graphs for the following l i n e s : (a) x + 7 = 0 (b) y-9 = 0 (c) 3x a 4y (d) 2x-3y - 6 (by i n -tercept method) (e) -3y~4x-*5 - 0. 5x-+-17 83. Draw a graph of the function — and from the graph read the value of the function 2 when x - 5 and when x - 8 . 84. Draw a graph to show the variations of the functions 1.2x-3 and 3.5-3.8x between the values 0, 1, 2, 3, 4 of x. Hence f i n d the value of x which s a t i s f i e s the equation 1.2X-3 = 3.5-3.8X. 85. Solve graphically and prove a l g e b r a i c a l l y : (a) 2y-5x « 20) (b) 2x-5y * 16) 4x4-3y = 7 ) 4x+y = 10) 86. (a) Draw the t r i a n g l e whose sides are given by the equations 3y-x = 9; x + 7y a 11; 3x-fy - 13, and f i n d the coordinates of i t s v e r t i c e s . (b) Find the equation for the l i n e joining the points (4,5), (11,11). Shew that the point (-3,-1) also l i e s on this l i n e . (c) Prove that the points (2,4), (-3,8), (12,-4) l i e on a straight l i n e which cuts the axis of x at a distance of 7 units from the o r i g i n . 87. (a) Solve graphically: x 2-»-y 2 - 41 y - 2x-3 ~ (b) Plot the graph of the function y <* x 2x-4 and give the coordinates of i t s turning point. 122 87. (c) Find graphically the roots of the following equation to two decimal places: 4X2-16XH-9 a 0 88. Draw the graph^of: ('a) y = 2x- (To) x 2-y = 4~2x In each case give the maximum or minimum value for x or y. 89. (a) Find graphically the roots of the following equations: x 2 - 7 x 4 - l l =0 ^ (b) What i s the minimum value of the expression x -7x +11? 90. (a) Show graphically that the expression x -2x-8 i s negative for a l l values of x between -2 and 4, and pos i t i v e for a l l values f o r x outside these l i m i t s . o p (b) Solve graphically and test a l g e b r a i c a l l y : x y - 53) y-x = 5) 91. An income of $160 Is derived p a r t l y from money invested at 3\% and pa r t l y from money invested at 3%, I f the Investments were interchanged the income would be '$165. How much i s invested at each rate? 92. The p r o f i t s of a business were '|150 i n the f i r s t year, and h a l f as much i n the second year as i n the t h i r d . In the fourth year they were three times as much as i n the f i r s t two years together. The t o t a l p r o f i t i n a l l four years was h a l f as much again as i n the f i r s t and fourth years together. Find the t o t a l p r o f i t s . 93. A man has a number of coins which he t r i e s to arrange, i n the form of a s o l i d square. On the f i r s t attempt he has 116 over, and when he increases the side of the square by 3 coins he wants 25 to complete the square.. How many coins has he? 94. An o f f i c e r forms h i s men into a hollow square four deep. If he has 1392 men, f i n d how many there w i l l be i n front. 95. Two men,- A and B, tr a v e l i n opposite directions along a i?oad 180 miles long, s t a r t i n g simultaneously from the ends of the road. A travels 6 miles a day more ,than B, and the number of miles t r a v e l l e d each day by B i s equal to double the number of days before they meet. Find the number of miles which each travels i n a day. 96. Find two numbers such that their product m u l t i p l i e d by the i r sum i s 330, and t h e i r product m u l t i p l i e d by t h e i r difference i s 30. 97. The interest on a sum of money for 1 year i s £31 17s 6d., i f the rate of in t e r e s t were l s 3 s by \ per cent i t would be necessary to invest £100 more to produce the same' amount of i n t e r e s t . Find the sum Invested at f i r s t . 98. There i s a number consisting of two d i g i t s such that the difference of the cubes of the d i g i t s i s 109 times the difference of the d i g i t s . Also the number exceeds twice the product of Its d i g i t s by the d i g i t In the units place. Find the number. 123 (a) A boat goes up-stream 30 miles and then downstream 44 miles i n 10 hours, and i t also goes upstream 40 miles and downstream 55 miles i n 15 hours. Find the rate of the stream and of the boat. (b) How long w i l l i t take each of two pipes to f i l l a c i s t e r n I f one of them alone takes 27 minutes longer to f i l l i t than the other and 75 minutes longer than the two together? (a) A man buys 99 oranges at a c e r t a i n price; they would have cost him 1 s h i l l i n g less i f he had obtained for each s h i l l i n g spent 4 more oranges than he a c t u a l l y did. What price d i d he pay? (b) A rectangular plot of ground i s surrounded by a gravel walk 4 f t . wide. The area of the plot i s 1200 s q . f t . , and the area of the walk Is 624 sq. f t . Find the dimensions of the p l o t . (c) A man invests some money i n 3 per cent stook; i f the price were £10 more, he would receive 1 per cent less for h i s money; at what price d i d he buy the stock? 124 SERIES FOR COORDINATION Grade XII Algebra I 1. Simplify: ^ a * KTS / b 2 o 2 1 1 X I 1 2bc -0 a b + c 2. Solve: x + g 1 0_ x2 1 0 x-2 4-x 2 x 2~4 3. Solve: .32x + .045x .05 .125 ~ 13.52 4. Seven years ago a boy was h a l f as old as he w i l l be one year hence. How old Is he now? 5. A c o l l e c t i o n of five-cent pieces and quarters contains 80 coins. Their t o t a l value i s $16. How many are there of, each? 6. Solve: 5x-+ 2y~3z ~ 160 3x -+ 9y+8z s 115 2x-3y-5z a 45 7. Factor: (a) x 4+ y 4 - 7 x 2 y 2 . (b) x 3 64 / 3,—^ \'/ 3 3 3. Factor: (a) m n 729" (b) x°-8y -27z -18xyz (c) x 4-15x 2y 2+9y 4; 4. a. Solve: 22x 2 = 3mn+7m2. b. The sum of the rec i p r o c a l s of two consecutive numbers i s 5 / 6 . Find the numbers. 5. Find to three places of decimals the value of: 5 4- y l d 4l^5 + 1/18 6 . Solve graphically: g x + x f Q 4 C 7. Factor: (a) a 4 - a 2 - 9 - 2 a 2 b 2 - i - b 4 4 - 6 a (b) a34- 3a 2b -*-3ab2 + 2b 3. (c) a 4b~a 2b 3-a 3b 2 4 -ab 4. 8. Find as simply as possible the value of: „2 .54532 - .45672 .5433 - .4567 9. Divide: 10. Reduce: x 4 y x-y 2 2 x -y x + y x 2 - y 2 X 2/3 - ^ 3/ — 131 X 2 1. In-the formula F 3 ^I_> given m = 12.075, r = 3, g . 32.2, F B 200, f i n d v. S r 2. Show that 1 + 1 _ V16 .+ 2 \/63 Vl6^2 \/63 3. (a) Simplify: ^ g 1 + 7 ^ x ^ I ^ f (b) Solve: 6 y/^11 ^ g y ^ l 3 1/x 2 4. Solve: (b) x -xy -6 (a) 5x 2-15x-Ml s 0 y 2 ^ 3xy .10 5. Th.e length of a f i e l d exceeds i t s breadth by 30 yds. I f the f i e l d were square, but of the same perimeter, i t s area would be 1/24 greater. Find the sides. 6. Solve: 2*-27 ^ = Z=±2 + x-14 x-8 x-13 x-9 7. J. number has three d i g i t s , the units being J the tens and 1/3 of the hundreds. I f 396 be subtracted from the number, the d i g i t s are reversed. Find the number. 8. Prove that the points (3,2), (8,8), (-2,-4) l i e on a straight l i n e . Find i t s equation. Prove a l g e b r a i c a l l y and graphically that i t cuts the x axis at a distance of 1 1/3 from the o r i g i n . 9. Solve: (b) x^-xyt 2y* = 4 x 2-3xy c-2 2 1 + 4 5 — + — = 12 . 10. Given /5 - 2.23607 f i n d to 4.places of decimals the value of: 7 yg" t l 5 \/5~2 \/5-l 3 4 1/5 132 APPENDIX I I Answers to Review Questions 1. x - - 2. x a J 3. x - 12 4. x ~ 6 2 I 7 r 18 y = 11 V - - 2 y - - z - 6 Z = 6 J - 2 ^ - 6 5. x s y a z = 12 6. x = 8 7. x - 10 8. x - 4 9. x s 0 ( a " b ) 10. x - bc+ can ab 11. x ss 13a or -a a 12. x P -2c-d 13. x s 3a-*-b 14. x =. 3n+m or-ec-»-d or 3a-b » 4 T c ^ ab-bm - „ v „ 3n-m 15. x s — - — 7a+8b y — 7 — c l 16. x - * 9 y - al-bm 8a j-7b 17. x . 4 18. x - 1.81 cm J V 9 or 3/2 (a) or .64 18. x - -5.449 19. x = " b " A 2 - 4 a o 20. x »|(/5-l) ( b ) o r - 5 5 1 2 a o * - . f ( / 5 f l ) 20. x s |(a - ^ a 2-4c) 21. x = .55 22. x - - a o r " * 2 2 (a) or - b 22. x = *3 22. x - 4a (b) or 44 (c) or -2a 23. a x = 2 ; - l or a (lo) x - 0; 1; -2 (c) x = 2a; a; -3a 24. x - 3.73; .27 , c , 26. (a) x = 5; I 25. x . 4; -1; 3; -J y s -8, * 26. < b ) x = 5 ; 7 27. (a) x = ^ ' 27. (b) x - J; -2 y = 3J y^;- - 3 y = 2; - § 28. (a) x = %6 28. (b) x = t l ; 29. (a) x =^5; - 12 . d 3 _ 29.(b) x 3 t 2 ; t f y - 23; X12 y - t 3 ; X J y = *ib *3 30.(a) x - i l ; i 2 (b) x = 3; % 2 y s *2; H y = 1; 6 31. (a) x . 6; 3 31. (b) x - 5; -| 3 2 ( , m g y s -3; -6 y - 2; - I i 32. (b) x B 2§ 33.(a) x m 2* ' 3 S - ( b ) ( x X « J e x t . } 3 133 34. 36. 37. 38. 40. 41. 43. 46. 47. 49. 51. 58. 53. 55. 64. 73. a) x - 49 a) x - 8 a) 15m-3m -2m- -t i A a) 3x s-2 4-x~a 34.(b) x - -36.(b) x = 8 ab , a 4-b 35.(a) x - 64 35.(b) x - i£ " 3 8m o9. X 4 x y +> 2y 40. (b) a 2 x-3a x-2 a) b 41. (b) x ^ 42. c^" 43. (a) (a-b) : b) a b ( b 6 - a 6 ) / 3 a) (Vx + 3 ) (Vx-5 ) b) ( X * 44.(a) 4. 44.(b) 864. 45. 1 7)(x V*'-7) (d) (x m4-3)(x 2 m-3x m+ 9) (e) ( x ~ c t 5)(x-°-4) c) (3a^4-2)(a^-hl) a) 1. (b) 8s a) 117.8897 (b) .4524 48.(a) 19 VZ 7 (b) -4pq /2pq 50. j/2.e a) 1. (b) 2. a) aj-^a^-x*1 x a) -8 a) 1. a) 6a 4. (b) 2 \/5 +3- 1/21 ( c ) 2 [/3 + 3 |/2t[/30 3 :—r£2—~ (b) 1. 54.(a) 5-2^6 WffityT ~ Vl) 54. (c) \/2x +1 + Yx-2 54.(d)l/m + 3n 4 l/m-3n (b) 4. 57. (a) 5:21 ^ ? - t> — 3b (b) \/3 (c) 12 \/2" /•u \ X D OD ID J — — v oi* S?~ y a u x 2a a) ( c d 2 e 3 - l ) ( c 2 d 4 e 6 4- c d 2 e 3 + 1) b) a 2(a-3)(a-b) c) d) e) f ) g) h) i ) 4) k ) x4-y)(x-y +1) b-a)(7a-l3b) a 3-x-a-b)(a 3-3 + a + b) c 4-3d-3a)(c 4-3d 43a) x 2-5y 2-3xy)(x 2-5y 2+ 3xy) 2m2 4-n2-5mn)(2m24- n 2 -v 5mn) 6a-yb)(5a + I2b) x-y)(x4-y)(x 2-^y 2)(x 4+-y 4) 1) 3b -M)(2b-3) m) 2 c ( c 2 l-3d 2) n) (x 4 3)(x-4)(x + 7) 134 (0) (a 4-2)(3a-l)(2a-3) (P) (|/a-4)()'a-3) (q) ( x ^ 7 ) ( x S 7) (r) ( 3 a ^ + 2 ) ( a ^ + 1 ) (s) ( i " f i - 4 ) ( x " c f 5 ) (t) (x" 1^ 3)(2 2 m-3x m-t-9) 74. (a) x(2-+x) (b) x T 4 75. (a) x x ( x " 4 ) u ^ P 75. (b) 8 a * , 2^ 2 W 2 2,2 76.la) °-(a -t- x )(a -x ) 76. (b) -(y-z) + q(z-x) +r(x-y) 77. (a) i 2 t y g (b) 6(x-l) (y-z)(z-x)(x-y) 2 y 2 x-4 78. (a) x-3 (b) (2x +5a)(x-2) x=¥ 79. (a) 2H-X +-3xg (b) 16x 4 8 0 < Q t 2(1-x 4) x 8-256 81. 2.525 sq.ln. 83. 21:28.5 84. 1.3. 85. (a) x ± -2 (b) x c 3 y = 5 y - -2 86. (a) (-3,2); (4,1); (3,4) (b) 7y-6x = 11 87. (a) x s 4, -1.6 (b) (-1,-5) (c) x - 3.32 y s 5, -6.2 or .68 88. (a) max. = 4 (b) min. - -5 89. (a) x = 2.38 or 4.62 (b) min. - -1.25 90. (b) x - 2, -7 91. |3,000 @ Z% 92. |l,800 y - 7, -2 $2,000 % 3 J # 93. 600 coins. 94. 91 men. 95. A, 18 miles; B, 12 miles 96. 6, 5 97. £750 98. 75 99. (a) 3 m-P* h« and 8 m«P» h» (b) 108 min.; 135 min. 100. (a) 5s. 6d. (b) 30 f t . x 40 f t . (c) £50. 135 ANSWERS TO ALGEBRA QUESTIONS FOR COORDINATION Paper I 1. 8a- 2-7a- 1 + 6 2- (a) ( 2 x +3)(3x-4y) (b) ( 2 a - b ) ( 2 a ^ b ) ( a 2 + b 2 ) (c) ( 3 r x - 5 x 3 x ) ( 3 r x - t - 5 x 3 x ) (d) x ( l +2ay)(l-2ay 4-4a 2y 2) 3. 2=2. x-1 4. x s"'-2; y a -3; z a -1 2 II R 5. (a) fl « R (b) H - * ss s 6. — ^ hrs. c-a YJ « be a 8. x. 9. x - 1 10. x = |, 2 7 = 4, 1 Paper II 2 1. (a-t-b 4-c) 2. x - -6 3. x s 2 4. 15 yrs. 2bc 5. 20 @ 5^ 6. x s 20 60 % 2H y s 15 9 9 9 9 Z =-10 7. (a) (x- y^ + Sxy) /-x /X 4 w x 2 , 1 . 16\ ( b ) (8 - ^ ) ( 6 4 + ' 2 4 - ^ ) (c) (x-2)(x+-2)(x 2i-4)(x -}-l)(x 2-x i-1) 8. (c) ll-\/..5 9 > g 0 d a y s @ $1.80. 10. x - 2; 3; 1/3-t 3; )/3-3 y - 3; 2', ^3-3; - 1/3-3 136 Paper III 1. |a 4-t |a 3x-t-j|a 2x 2+ j l-iax 3 2. ah(h-b) 3. 133j l b s . @ 40^. 4. $6. 3 _ 5. x M -2; -1 6. x = 2; y o 4-|; z a 3^ y s i l ; *2 7. (a) 3l/7 + 2]/5 8. x*-y^ (b) 5 \/2 - J/5 10. 56. Paper IV 1. 3a-13b -tlOc 2. (a) k = 5 (b) x - 2; y s 1 3 3. (a) (x-2)(x-2)(x-2) 3. (b) (3x-4a)(4a-3x) ( c ) (y-3x) (x-y) (x -ty) 4. (a) 12mn. (b) 9. 5. lpa 34- a 2-2 6. _J>Z 7. x - -2 ( a * 2)(2a + l ) ( 4 a 2 + 2a +1) 115 8. p = 157i 9. $1,000. o 10. x s i 4 ; - UZ 2- (-3.299) 3 _ 7 = £1; ;1 4 / g (-1.885) Paper V 1. -x4(a-3b2-+ c-v7)-x(b +c 2) 2. 4 m.p.h. and. 1 m.p.h. 3. 8, 4. 4. 3g 6. x r 8; -5 8 y x 6; -8 7. (a) (3-t 4x)(4-5a) 7. (b) (3y-8)(9y 2 + 24y4 64) ( c ) x 3 n ( 3x ni-1) ( 3 x n - l ) 7. (d) (x-3-t-2y)(x-3-2y) 8. 4. 9. 4a ( . i ? | | 8 ) o Paper VI 1. x = 5 2. x = -4 3. 68 yrs. 4. 16 yds. 137 5. m - 2; n = 3; p = -6. 6. x - i§, -1 7. 12'; 5'; 13»-. 8. (x+y)\/xy 9. -35. 10. (a) No. (b) x a 2, -2 y = -3, 5 Paper VII 1. x - 5g 2. X £ A 3. x =8 4. (a) (0,0) y 4. (b) |/x2-»-y2~ 5. 21x 2-9x 3 6. x = 8 7. j | 8. (a) x - .61a; -.52a (b) x = 2.643; .7566. 9. x I' -5; i s y = -6; *3 10. 4 yds. Paper VIII 1. (a) 2(a-b)(x-4)(x +1) (b)|a-b+ c~3)(a-b-c +3) 1.' (c) (a-tbx)b-+ax) 2. x n "* n+x n~ 1 3. 2-1/3 4. x = 64. 5. (b) (-2,4) 6. 1_ x +1 7. x = 9. 8. x = 3a; y = -2b _ 9. 100 benches. 10. x s i ; 4 3 2 Y - 1; -1 * ~ 2' 3 Paper IX 1. (a) 24.64 sq.in. (b) i f f t . 2. „ / » ,mn n wm 2n 2 . -mn , , , a. (a) (-g--!H 8 1 -+--^-+l) (b) (x-2y-3z)(x 2+ 4y2-+-9z 4-2xy +3xz-6yz) A a a 13 I J U H M A ' ^ X H . X T4"¥ (c) (x 2-3y 2-3xy)(x 2-3y 2 + 3xy) 4. (a) x - ^E;. - - (b) 2;3 5. 2.236. ** 11 2 6. x = -8; 0 7. (a) (a 2-b 2-a+5)(a 2-b 2 4 - a-3) 7. (b) (a4-2b)(a 2+ab+b 2) (c) ab(a-b)(a-b)(a-+ b) 8. 1. 9. x. 10. x %-2 x 138 Paper X 1. v - 40 3. (a) 7. (b) x a9 4. (a) x a 1.7236; 1.2764 (b) x - i l ; ±3. 5. 60 yd. x 90 yd. y - t5; *1. 6. x - 11 7. 642 8. 6x-5y = 8 9. (a) x = 3 (b) x = -I; ±2 2-y =-2 + 3 + z 3 1 y = t | ; t l 10. 1.11803. 139 BIBLIOGRAPHY Adams, John: The New Teaching. Hodder and Stoughton, Toronto, (1918) Mathematical Recreations and Essays. Macmillan, Toronto, 1928. A Short Account of the History of Mathematics. Macmillan, Toronto, 1912. Problems i n Teaching Secondary School .Mathematics. University of Chicago Press, Chicago, 1931. An Introduction to the Study of Education. Houghton M i f f l i n , New York. 1925. New Geometry Papers. MacMillan, London, 1915. Modern Methods In High School Teaching. Houghton M i f f l i n , N.Y. Psychology for Students of Education. MacMillan, N.Y. 1932. Godfrey and SIddon3:Elementary Geometry. Cambridge University Press. 1924. Goodenough, F.: Developmental Psychology. 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