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UBC Theses and Dissertations

Motivation in the teaching of high school mathematics Miller, Selwyn Archibald 1936

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MOTIVATION IN THE TEACHING OF HIGH SCHOOL MATHEMATICS by Selwyn A r c h i b a l d  Miller  MOTIVATION IN THE TEACHING OP HIGH SCHOOL MATHEMATICS  toy  Selwyn A r c h i b a l d  Miller  do  A Thesis  submitted f o r the Degree MASTER  OP  of  ARTS  i n v t h e Department o f  PHILOSOPHY  oo  The U n i v e r s i t y o f B r i t i s h Columbia April,  1936  TABLE OF CONTENTS Page Chapter  I.  Importance of M o t i v a t i o n i n the Teaching o f High School Mathematics 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 Teaching o f High School Mathematics^)  10'  Chapter I I I .  M o t i v a t i o n i n the Teaching o f Grade IX Geometry  22  Chapter  M o t i v a t i o n i n the Teaching o f Grade XII Geometry  40  Chapter  II.  IV.  Chapter  V.  M o t i v a t i o n i n the Teaching o f Grade IX A l g e b r a  69  Chapter  VI.  M o t i v a t i o n i n the Teaching o f Grade X I I A l g e b r a  87  Chapter VII..  Some Experimental Evidence to Show the E f f e c t s o f M o t i v a t i o n  101  Chapter V I I I .  General Conclusions  107  Appendix  Series of Questions  Bibliography  5  Answers  113 139  i  MOTIVATION IN THE TEACHING OF HIGH SCHOOL MATHEMATICS Table of Contents w i t h Subheadings  Chapter  I »Importance of M o t i v a t i o n i n the Teaching of High  School Mathematics^ Meaning of m o t i v a t i o n ; l i t e r a l meaning; comparison of d i c t i o n a r y meanings; meaning as a p p l i e d to t e a c h i n g ; meaning as a p p l i e d p a r t i c u l a r l y to the t e a c h i n g of mathematics. Mathematics i s an e s s e n t i a l p a r t of a person's education; statements by a u t h o r i t i e s . 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 s t u d y i n g any branch of s c i e n c e . In B r i t i s h Columbia mathematics i s a compulsory s u b j e c t u n t i l the second year of the u n i v e r s i t y ; an e a r l y d i s l i k e f o r the s u b j e c t may r u i n a student's e d u c a t i o n a l life. Where mathematics i s an o p t i o n a l subject i t i s very f r e q u e n t l y avoided; evidence. A great number of students s e l e c t s u b j e c t s which they do not want, s o l e l y to a v o i d mathematics. A d i s l i k e f o r mathematics keeps students away from such s u b j e c t s as p h y s i c s and chemistry. Mathematics i s a comparatively d i f f i c u l t s u b j e c t , and when I t i s not motivated o n l y the b e t t e r students w i l l grasp i t . A g r e a t d e a l of review work i s necessary i n mathematics, and u n l e s s some form of m o t i v a t i o n i s adopted, t h i s review.may become very monotonous and u n i n t e r e s t i n g - constant p r a c t i c e i n mechanical processes and c o n t i n u a l r e f e r e n c e to fundamentals are e s s e n t i a l m o t i v a t i o n i s necessary f o r t h i s . Many p u p i l s , e s p e c i a l l y g i r l s , get p r e c o n c e i v e d ideas t h a t they can not do mathematics - t h i s c o u l d be avoided to a l a r g e extent by proper m o t i v a t i o n . The l a r g e percentage of f a i l u r e s i n m a t r i c u l a t i o n examinations c o u l d be reduced c o n s i d e r a b l y by i n t r o d u c t i o n of motivated forms of t e a c h i n g i n p l a c e of the o l d mechanical methods. M o t i v a t i o n i s necessary i n order to open a new f i e l d of i n t e 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 s t o r e f o r any student who develops a l i k i n g f o r 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 t e a c h i n g . ii  Summary - m o t i v a t i o n i n t e a c h i n g m a t h e m a t i c s 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 t h e subject. 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  High School  of  Mathematics^  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 taken 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 with pleasurable experiences. 3. D e s i r e f o r s e c u r i n g p r a i s e a n d a v o i d i n g shame - p r a i s e of teacher, parents, 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 g o o d m a r k s . 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 e a c h o t h e r and against time. 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 . 1 3 . 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 s a k e - a new and pleasurable experience. 16. D e s i r e t o a n s w e r 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 w o r k - i m p o r t a n c e o f teaching for transfer. 18. S a t i s f a c t i o n t h r o u g h m a s t e r y . 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 . 2 1 . D e s i r e t o be c o n s i d e r e d m a t u r e - a d v a n c e d . 22. D e s i r e t o c o m p l e t e c o u r s e c h o s e n . 23. D e s i r e t o p a s s 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 a w a r d s - 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 t h e d i f f e r e n t g r a d e s - how g r a d u a t e d - how s t r e s s e d . Chapter I I I .  M o t i v a t i o n i n t h e T e a c h i n g o f G r a d e I X 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 t h e n o v e l t y o f t h e s u b j e c t to arouse i n t e r e s t - a b r i e f o u t l i n e of 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 of i n t e r e s t . Necessity f o r keeping subject experimental at 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 w o r k by p u p i l t e a c h i n g of 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 of grade IX p l e n t y ill  of o p p o r t u n i t y f o r p h y s i c a l a c t i v i t y - methods of marking - comparing r e s u l t s . In b e g i n n i n g the study o f s t r a i g h t l i n e s , p r a c t i c e i n drawing and measuring can be made i n t e r e s t i n g by p u p i l s guessing the l e n g t h o f a l i n e a l r e a d y drawn, or by drawing a l i n e o f s p e c i f i e d l e n g t h without measuring competitions i n t h i s f i e l d - employment of m i l l i m e t r e s and centimetres as w e l l as i n c h e s . M o t i v a t i n g the study of angles - s t r e s s s p e l l i n g competitions i n naming angles. P r a c t i c e with angles by drawing from guess and checking by measuring and v i c e v e r s a . - Teacher draws angles ©n board and p u p i l s guess s i z e s . Use of mariner's compass - p u p i l s t e l l number of degrees between v a r i o u s p o i n t s on the compass - boy scouts and g i r l guides should be encouraged to show what they know about the compass. P r a c t i c e w i t h l i n e s and angles by f o l l o w i n g comp l i c a t e d d i r e c t i o n s - ship s a i l i n g to d e s e r t i s l a n d f i n d i n g the hidden t r e a s u r e from d i r 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 v e r y 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 - d r a w i n g " p a r a l l e l s by o b s e r v a t i o n ; t e s t ing by drawing t r a n s v e r s a l and measuring angles i n t r o d u c e o p t i c a l i l l u s i o n s with r e g a r d to p a r a l l e l s . U t i l i z a t i o n o f the p r i n c i p l e o f s a t i s f a c t i o n through d i s c o v e r y i n the treatment of t r i a n g l e s - p u p i l s t r y t o c o n s t r u c t t r i a n g l e s with s i d e s 3, 5, 7 * ; 3, 5, 8 * ; 4, 6, 11 » ' e t e . P u p i l s t r y to c o n s t r u c t /S^kBG h a v i n g AB * 3"; /_B 82°; £ A = 9 8 ° . Teaching of ..standard c o n s t r u c t i o n s can be motivated by examining numerous methods suggested by the p u p i l s and s e l e c t i n g the b e s t . e.g. drawing p e r p e n d i c u l a r t o a g i v e n s t r a i g h t l i n e from a g i v e n p o i n t o u t s i d e i t . I n t e r e s t i n g e o m e t r i c a l language may be c r e a t e d by having p u p i l s t r y to e x p l a i n i n words what they have done w i t h the instruments - compare c o n s t r u c t i n g g e o m e t r i c a l f i g u r e s to b u i l d i n g . C o n s t r u c t i o n o f more d i f f i c u l t f i g u r e s can be motivated by g e t t i n g p u p i l s to c o n s i d e r them e q u i v a l e n t to performing some very i n t r i c a t e b i t o f handiwork, e.g. drawing i n s c r i b e d , c i r c u m s c r i b e d and e x c r i b e d c i r c l e s of a given t r i a n g l e ; c o n s t r u c t i n g a q u a d r i l a t e r a l u s i n g i t s diagonalsi Employ p r a c t i c a l problems such as drawing c r e s t s f o r sweaters, i n l a y work f o r t r a y s , e t c . Very c a r e f u l l y prepared t e s t s should be welcomed by the p u p i l and should make him eager to go on and l e a r n more about the wonders o f geometry. Mathematical r e c r e a t i o n s s u i t a b l e f o r grade IX geometry. i n  i n  c  m  s  iv  Chapter  IV.  M o t i v a t i o n i n the Teaching of Grade XII Geometry  /f; Development o f proper a t t i t u d e s towards grade XII geometry 1. C e r t a i n p u p i l s should be encouraged to r e g a r d thems e l v e s as "budding mathematicians - h i g h e r r e g a r d of teacher f o r p u p i l encourages more earnest e f f o r t . 2. D e s i r a b i l i t y of l o o k i n g ahead - l a y i n g f o u n d a t i o n f o r h i g h e r mathematics - frequent r e f e r e n c e to advanced work. 3. P u p i l s should become more independent - l e s s explanat i o n by teacher - more time f o r the p u p i l f o r t h i n k i n g - 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 v e r y e s s e n t i a l - extreme - s a t i s f a c t i o n d e r i v e d from o b t a i n i n g a s o l u t i o n a f t e r hours of t r y i n g . 5. Development of h a b i t of v i s u a l i z i n g g e o m e t r i c a l f i g u r e s - s e e i n g s o l u t i o n s to e x e r c i s e s w h i l e one i s walking along s t r e e t , s i t t i n g i n s t r e e t c a r , or w a i t i n g to meet someone. 6. S a t i s f a c t i o n ' d e r i v e d from working w i t h most complicat e d l o o k i n g diagrams - p r i d e i n one's a b i l i t y (example). 7. D e s i r e f o r a b s o l u t e p e r f e c t i o n - development of p r i d e i n p e r f e c t s o l u t i o n s (reasoning and form). f$ ; M o t i v a t i o n In methods of p r e s e n t a t i o n of grade XII geometry 1. Less formal treatment of theorems - Teach theorem ( o f t e n by a n a l y s i s ) ; - b u i l d i t up together - t e s t f o l l o w i n g day - develop i d e a t h a t once key i s g i v e n to a theorem (or e x e r c i s e ) i t i s s o l v e d once and f o r a l l time. 2. C a r e f u l treatment o f e x e r c i s e s - a c a r e f u l l y s e l e c t e d e x e r c i s e to be w r i t t e n out and handed i n each day marked - r e t u r n e d - :marks recorded - monthly t o t a l s read - e x c e l l e n t review d a i l y . 3. Keep i d e a l s of good form b e f o r e p u p i l s always - pass e x t r a good s o l u t i o n s around c l a s s - good e f f e c t s on w r i t e r and observer. (Example) 4. Value of mimeographed sheet of e x e r c i s e s done d u r i n g year - p u p i l s r e a l i z e t h e i r accomplishments. 5. Use of questions from p r e v i o u s examination papers t e l l p u p i l s year and grade. Special interest i n c e r t a i n q u e s t i o n s . (Example) 6. Value of short e x e r c i s e s a t end of theorem - taken i n d i v i d u a l l y on board - r a c e f o r s o l u t i o n - b r i g h t pupils give h i n t s . . 7. Value of o c c a s i o n a l o b j e c t i v e t e s t s - p u p i l s enjoy them - marks encouraging - examples.  8. Use o f s p e c i a l e x e r c i s e s f o r b r i g h t p u p i l s , examples; isosceles to equivalent e q u i l a t e r a l bisect by l i n e through p o i n t o u t s i d e ( i n base produced) e t c . 9. A f t e r - s c h o o l d i s c u s s i o n s - i n f o r m a l - encouraged. 10. Development of p u p i l - t e a c h e r i d e a - grade XII p u p i l s as t u t o r s to.grade X f r i e n d s - b e n e f i c i a l e f f e c t s on both p u p i l s - l i m i t a t i o n s o f t h i s system. 11. Mathematical r e c r e a t i o n s i n geometry. Chapter V.  M o t i v a t i o n i n the Teaching  o f Grade IX A l g e b r a  1. The n o v e l t y o f a l g e b r a ( u s i n g l e t t e r s f o r q u a n t i t i e s as w e l l as numbers) g i v e s the p u p i l an i n i t i a l i n t e r e s t i n the s u b j e c t - i f p r o p e r l y motivated, i n t e r e s t can be maintained. .2. I n f i r s t l i t t l e problems i n v o l v i n g symbols, make the problems " r e a l " , e.g. problems on b a s e b a l l , swimming, running, e t c . 3. Some of the e a r l y a l g e b r a problems seem e x c e e d i n g l y advanced t o the beginner - p l e n t y o f p r a i s e f o r s o l u t i o n s o f such problems i s a great m o t i v a t i n g f o r c e . 4. The t e a c h i n g o f s u b s t i t u t i o n - care must be taken hot to make t h i s t o p i c too complicated - r e f e r to i t again l a t e r - p u p i l s get t a n g l e d up v e r y e a s i l y i n t h i s type of q u e s t i o n - make the most of the p e c u l i a r r e s u l t s obtained by s u b s t i t u t i o n o f u n i t y and zero-. 5. The teaching o f a d d i t i o n i n algebra can be motivated so as t o make i t an e x c e p t i o n a l l y i n t e r e s t i n g s e c t i o n o f the work. - A d d i t i o n o f l i k e " t e r m s , not u n l i k e (apples oranges ? (grapefruit?) ) - real, l i v i n g problems can be g i v e n I n a d d i t i o n , e.g. See Thorndike - P u p i l s enjoy working b i g - l o o k i n g questions (examples). 6. S u b t r a c t i o n the source.of many e r r o r s i n a l g e b r a . Motivate t h e l e a r n i n g o f the r u l e -for changing lower s i g n s ; {e.g. f i l l i n g i n / h o l e s , wiping o f f , d e b t s ) . S u b t r a c t i o n r e a l l y a d d i t i o n . What do you add to 5 to make i t four? 7. M o t i v a t i n g the t e a c h i n g o f r u l e s 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 a d d i t i o n ( o f i n d i c e s ) - use 3 a x 2 a e t c . Rule o f signs (two wrongs make a r i g h t ) . E x p l a n a t i o n -of -3ax-3a a 9 a - t a k i n g away a debt 3 times - f i l l i n g 3 h o l e s - l i v i n g problems i n v o l v i n g m u l t i p l i c a t i o n - examples. 8. M o t i v a t i n g the t e a c h i n g o f r u l e s f o r d i v i s i o n - r e v e r s e of m u l t i p l i c a t i o n - use 9 a - 3 a e t c . - competition by rows i n l o n g d i v i s i o n . Allow p u p i l s to work l o n g , l o n g d i v i s i o n questions t o f i l l whole page, e.g. s  3  2  2  9  3  a +b)a -b - r a c e s with l o n g d i v i s i o n . 9. Removal o f b r a c k e t s - e x p l a n a t i o n o f r u l e ( r e a l l y a d d i t i o n or s u b t r a c t i o n ) - t h r i l l o f g e t t i n g complicated expressions down to a o r o - competition by rows,1 2  1 2  vi  p o i n t ahead t o equations. 10. Fundamentals i n a l g e b r a must be thoroughly understood, as a l l f u t u r e work i n algebra i n v o l v e s the use o f these fundamentals - a t the beginning o f each p e r i o d a review o f the previous l e s s o n i s e s s e n t i a l - e.g. f i v e short questions on the board, - how many r i g h t out o f f i v e - p u p i l s stand ( r e l a x a t i o n ) . 1 1 . Ample allowance f o r p u p i l s going on ahead I n a l g e b r a e x t r a books, s p e c i a l questions - review s e t s - o l d examination papers. 1 2 . Mathematical R e c r e a t i o n s s u i t a b l e f o r grade IX a l g e b r a . Chapter V I » M o t i v a t i o n i n the Teaching  o f Grade XII Algebra  General methods o f m o t i v a t i o n o u t l i n e d i n Chapter I I - s p e c i f i c means by which these methods of m o t i v a t i o n may be brought i n t o o p e r a t i o n g i v e n i n present chapter. Development o f i n t e r e s t i n the subject f o r i t s own sake - how t h i s may be achieved - Three-unit c y c l e ; u n i t f o r t e a c h i n g , u n i t f o r review, u n i t f o r c o o r d i n a t i o n . E f f e c t o f m a t r i c u l a t i o n examinations «- how m a t r i c u l a t i o n examinations should be regarded - how they may be made a d e s i r a b l e m o t i v a t i n g f o r c e . Minor methods o f m o t i v a t i o n : (a) S p e c i a l treatment o f a l g e b r a i c a l problems - i n t r o d u c tion of puzzles. (b) Use o f mathematical r e c r e a t i o n s - examples. (c) F r i e n d l y r i v a l r y between c l a s s e s . (d) Emulation o f past achievements. (e) E f f e c t o f s c h o l a r s h i p s and p r i z e s . 1  Chapter V I I * Experimental  Evidence  to Show the R e s u l t s o f  ITS  I n c r e a s i n g the M o t i v a t i o n bC the Teaching  o f High School  Mathematics 1. O u t l i n e o f experiment c a r r i e d out by w r i t e r . 2. Table o f marks o b t a i n e d i n J u n i o r M a t r i c u l a t i o n Examinations. 3. Table showing the r e l a t i v e amounts o f enjoyment d e r i v e d from the study of the v a r i o u s s u b j e c t s on the h i g h school c u r r i c u l u m . 4. Table showing estimates o f e f f i c i e n c y . Chapter VlII»Conclusions r e g a r d i n g the e f f e c t s o f i n c r e a s i n g m o t i v a t i n g i n - t h e t e a c h i n g o f h i g h school m a t h e m a t i c s ^ Appendix.  Question  lists.  Answers.  Bibliography. vii  L i s t of F i g u r e s Page Figure  I.  Example o f Complicated  Diagram  44A  Figure  II.  Example of Complicated  Diagram  44B  Figure I I I .  Example of Attempt f o r P e r f e c t i o n . 44C  Figure  IV.  Example of Attempt f o r P e r f e c t i o n  Figure  V.  Example Showing E f f e c t of P r a i s e  .51A  Figure  VI.  Example Showing E f f e c t of P r a i s e  51B  Figure VII.  44D  i  G r a p h i c a l I l l u s t r a t i o n s of E f f e c t s of M o t i v a t i o n  111  L i s t o f Tables Page Table  I.  Table  II.  Table I I I .  Table of marks f o r Home E x e r c i s e s  50A  Table of M a t r i c u l a t i o n Marks f o r Experimental c l a s s  104  Table Comparing Averages made by Experimental C l a s s with Other Averages.  105  viii  MOTIVATION IN THE TEACHING OP HIGH SCHOOL MATHEMATICS  CHAPTER I/a THE  IMPORTANCE OP MOTIVATION IN THE TEACHING OP MATHEMATICS/.^  What i s m o t i v a t i o n ?  In i t s l i t e r a l  process o f i n d u c i n g movement;" it  sense i t i s "the  and i n i t s broader  sense  i s "the process o f producing s t i m u l i which i n i t i a t e ,  d i r e c t and s u s t a i n a c t i v i t y  ( 2 ) . Motivation i n connection  w i t h t e a c h i n g might he c o n s i d e r e d as the i n t r o d u c t i o n of c e r t a i n f a c t o r s i n t o t e a c h i n g , which i n i t i a t e , d i r e c t , and s u s t a i n the a c t i v i t y o f the i n d i v i d u a l taught.  When con-  s i d e r i n g m o t i v a t i o n i n r e l a t i o n to the t e a c h i n g o f h i g h s c h o o l mathematics, we s h a l l c o n s i d e r the numerous ways by which a teacher may induce each p u p i l i n h i s c l a s s to put f o r t h the maximum amount o f e f f o r t o f which he i s capable, and t o d e r i v e the g r e a t e s t p o s s i b l e b e n e f i t from t h a t e f f o r t . »  I t i s obvious  t h a t , f o r v a r i o u s reasons, a g r e a t many p u p i l s do not put f o r t h t h e i r b e s t e f f o r t s i n the study o f mathematics, and that they do not d e r i v e a l l the b e n e f i t s which they might d e r i v e from the study o f such a wonderful  subject.  (1) F. Goodenough - Developmental Psychology, (2) A. Gates - Psychology  P. 498.  f o r Students o f E d u c a t i o n , P. 182. 1  2 The p r i n c i p a l o b j e c t s o f m o t i v a t i o n in. t e a c h i n g a r e two i n number: 1. The c r e a t i o n o f a maximum amount o f i n t e r e s t on the p a r t o f the l e a r n e r ;  f ^  each p u p i l .  I n the case of p e r f e c t r e a l i z a t i o n o f t h i s aim  -  1  2. The attainment o f maximum e f f o r t by  each p u p i l w i l l have reached an A. Q. o f 100. o b j e c t i v e s c o u l d be reached,  I f these two  the study of mathematics i n our  h i g h schools would take on a new l e a s e 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 t e a c h i n g o f mathematics would be made groundless indeed.  I n the f o l l o w i n g pages s p e c i a l  r e f e r e n c e w i l l be made to the forms o f m o t i v a t i o n s u i t a b l e f o r grades IX and X I I , with a comparison.of m o t i v a t i o n used i n these The  the types of  grades.  study o f mathematics, when devoid of any m o t i v a t i n g  f o r c e s , becomes- a d u l l and u n i n t e r e s t i n g task, and y e t mathematics i s an e s s e n t i a l p a r t o f a person's  education.  There i s no s u b s t i t u t e f o r i t ,  education  without i t .  ( ) 2  and no complete  I t t h e r e f o r e behoves the teacher of  mathematics to bend every e f f o r t towards p r e s e n t i n g the s u b j e c t to h i s p u p i l s i n such a way t h a t they w i l l have m o t i v a t i n g f o r c e s u r g i n g them on t o t h e i r g r e a t e s t e f f o r t s ,  (1) The importance o f c r e a t i n g i n t e r e s t i s d i s c u s s e d by M.J. Stormzand, and he a r r i v e s a t the c o n c l u s i o n that "the p r o blem o f i n t e r e s t p l a y s such an important p a r t i n e d u c a t i o n because success i n a l l t e a c h i n g i n v o l v e s the a r o u s i n g o f sufficient interest. P r o g r e s s i v e Methods i n Teaching, P. 129. (2) " A l l s c i e n t i f i c education, which does n o t commence w i t h mathematics, i s , o f n e c e s s i t y , d e f e c t i v e at i t s foundation." 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 e s s e n t i a l p a r t of a j p e r s o n s education i s almost 1  axiomatical.  "Our  P r o f e s s o r Voss,  e n t i r e present c i v i l i z a t i o n , "  "as f a r " a s i t depends upon the  says  intellectual  penetration.!and i n v e s t i g a t i o n of nature, has i t s r e a l t i o n i n the mathematical  sciences." ( ) 2  founda-  The p h y s i c a l laws of  the u n i v e r s e are so l i n k e d up with mathematics t h a t our i n n a t e d e s i r e to examine the laws of nature i n order to f i n d e x p l a n a t i o n s f o r a l l the v a r i o u s n a t u r a l phenomena, would be doomed to disappointment  without  a mathematical  foundation on  which to work. ( ) W . A . M i i l i s , when d i s c u s s i n g the value of 3  mathematics as a h i g h s c h o o l s u b j e c t , makes the statement " f o r a l g e b r a there i s no s u b s t i t u t e .  The  that  e l i m i n a t i o n of  a l g e b r a as a pure s c i e n c e from the c u r r i c u l u m would cut the f o u n d a t i o n from under a l l s c i e n t i f i c procedure."  v  '  If  mathematics, t h e r e f o r e , i s r e a l l y an e s s e n t i a l p a r t of a person's  e d u c a t i o n , should we not bend every e f f o r t to so  motivate the t e a c h i n g of the s u b j e c t , t h a t each p u p i l would  (1) " E u c l i d has done more to develop the l o g i c a l f a c u l t y o f the world than any book ever w r i t t e n . I t has been the i n s p i r i n g i n f l u e n c e 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 W r i n k l e s , P. 245-6. (2) A. S c h u l t z e - The Teaching of Mathematics i n Secondary S c h o o l s , P. 17. (3) It i s when we examine the r e l a t i o n of mathematics to s c i e n c e , both pure and a p p l i e d , 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 p r o p a e d e u t i c " - Charles De Garmo - The S t u d i e s o f the Secondary School, P. 65. (4) W . A . M i i l i s - The Teaching of High School S u b j e c t s , P. 240. ri  4 d e r i v e the g r e a t e s t p o s s i b l e value from i t s study, and would apply h i m s e l f to h i s work w i t h an eagerness him  that would b r i n g  to h i s h i g h e s t l e v e l o f e f f i c i e n c y . Another reason why i t I s so important that every teacher  should make the most o f every o p p o r t u n i t y t o motivate the t e a c h i n g o f mathematics i s that i n many p r o v i n c e s mathematics i s one o f the compulsory student has completed  s u b j e c t s o f the c u r r i c u l u m u n t i l the  two years o f u n i v e r s i t y work.  Whether  t h i s r e g u l a t i o n i s a wise one or not i s not f o r d i s c u s s i o n here, but as l o n g as the requirements remain  as they are, i t  i s i m p e r a t i v e that a student be so i n s t r u c t e d i n mathematics i n h i g h s c h o o l that he w i l l develop a l i k i n g f o r the s u b j e c t and not f i n d i t a m i l l s t o n e about h i s neck year a f t e r y e a r . Many a student has had the j o y taken out o f h i s e d u c a t i o n a l life  simply by d e v e l o p i n g an e a r l y h a t r e d f o r mathematics, and  had the s u b j e c t been p r o p e r l y motivated f o r those p u p i l s , that abhorrence  i n many cases may have been e n t i r e l y e l i m i n a t e d .  M o t i v a t i o n I n mathematics i s p a r t i c u l a r l y important when d e a l i n g w i t h students who are f a v o r a b l y I n c l i n e d towards scientific  study.  I f a boy or a g i r l develops a l i k i n g f o r  p h y s i c s , chemistry, geology, astronomy o r , i n i f a c t , any o f the s c i e n c e s , i t i s very important t h a t he or she a l s o develop a l i k i n g f o r mathematics.  I f such a student i s g i v e n a bad  impression o f mathematics by l a c k o f m o t i v a t i o n i n i t s p r e s e n t a t i o n , then he may abandon the study o f a s c i e n c e i n which he i s r e a l l y i n t e r e s t e d simply because o f the  5 mathematical c a l c u l a t i o n s i n v o l v e d i n i t . Teaching  W.A.Miilis  in  "The  of High School S u b j e c t s , " p o i n t s out t h a t mathematics  i s unpopular with a g r e a t many h i g h school students enumerates reasons f o r i t s u n p o p u l a r i t y .  One  reasons i s that the s u b j e c t i s more d i f f i c u l t  and  of h i s c h i e f than most other  s u b j e c t s , and i f p u p i l s are g i v e n the o p t i o n of another s u b j e c t i n p l a c e of mathematics, many students  s e l e c t another  (1) s u b j e c t I n s t e a d of mathematics.  v  7  Not o n l y do p u p i l s abandonsthee study  of some o f  the  s c i e n c e s on account of the mathematics i n v o l v e d , but  they  s e l e c t s u b j e c t s In which they have no p a r t i c u l a r I n t e r e s t , and for  which they are not  s u i t a b l y adapted, f o r the s o l e purpose  of a v o i d i n g the only other a l t e r n a t i v e - mathematics. doubtedly  a number of these  students, at some l a t e r  r e g r e t t h e i r a c t i o n s when they they  Un-  date,  suddenly r e a l i z e to. what extent  are handicapped by l a c k i n g a fundamental knowledge of  mathematics. E.L.Thorndike made an i n v e s t i g a t i o n i n t o the p o p u l a r i t y of the v a r i o u s s u b j e c t s on the h i g h school c u r r i c u l u m .  The  v o t i n g was  of  New  done by grade XII p u p i l s i n the High Schools  York C i t y .  In the f i n a l  ranking of subjects,  algebra  ranked 13th. out of 22 w i t h boys and 25th. out of 27 girls.  with  Geometry ranked 16th. out o f 22 w i t h boys, and  (1) W . A . M i i l i s ,  Opcit.  Pp.  231-3.  26th.  6 out o f 27 w i t h g i r l s . ^ ^ t  1  The obvious conclusion., to be  drawn from the r e s u l t s of these t e s t s conducted by Dr. Thorndike i s t h a t In the past the 'teaching of mathematics has not (o)  been c a r r i e d out i n a wholly Mathematics i s a p p a r e n t l y  s a t i s f a c t o r y manner.  subject of r e a l  '  a s u b j e c t which can become very  froresome and d i s t a s t e f u l to students pedantic manner;  v  i f presented  In a c o l d ,  but on the other hand, i t may become a 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 A f u r t h e r reason  methods of p r e s e n t a t i o n . f o r the n e c e s s i t y of m o t i v a t i n g the  t e a c h i n g of mathematics i n h i g h school i s t h a t mathematics i s r e a l l y one of the more d i f f i c u l t  s u b j e c t s o f the curriculum.-  A p u p i l o f I n f e r i o r i n t e l l i g e n c e can very o f t e n reach the r e q u i r e d standard  i n c e r t a i n s u b j e c t s such as French,  social  s t u d i e s , grammar, geography and s i m i l a r s u b j e c t s by means o f frequent r e p e t i t i o n and l a b o r i o u s memorization; but such a student has much more d i f f i c u l t y I n r e a c h i n g standard  i n mathematics.  The very nature  demands t h a t i t s t e a c h i n g be motivated extent,  the r e q u i r e d  o f the s u b j e c t  to the f u l l e s t p o s s i b l e  i n order t o encourage those who a r e not e s p e c i a l l y  b r i g h t t o develop t h e i r mathematical s k i l l  to the extreme  (1) E.L.Thorndike, Psychology o f Algebra, P. 386. (2) "One reason f o r the u n s a t i s f a c t o r y s t a t u s o f the s u b j e c t (mathematics) i s poor t e a c h i n g . " - W . A . M i i l i s , O p c i t . P. 233. (3) "The e x t e n s i o n o f the e l e c t i v e system has r e v e a l e d t h a t p u p i l s do n o t l i k e these s u b j e c t s (Algebra and Geometry)"I b i d . , P. 232.  7 l i m i t of t h e i r a b i l i t y . Is  Without such m o t i v a t i o n mathematics  to these p u p i l s an e x c e e d i n g l y d i f f i c u l t  w i l l drop a t the f i r s t o p p o r t u n i t y . t i o n s f o r promotion, or  s u b j e c t which they  Since the u s u a l r e g u l a -  however, demand t h a t every student,  dull  b r i g h t , 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 t e a c h e r o f mathematics to motivate h i s t e a c h i n g i n such a way  t h a t the d u l l e r p u p i l s as w e l l as the  b r i g h t e r 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 s u b j e c t to encourage each one to reach h i s h i g h e s t l e v e l of efficiency. Another reason why  i t i s so e s s e n t i a l f o r the teacher of  mathematics to motivate h i s t e a c h i n g to the f u l l e s t extent, i s that i n t h i s s u b j e c t a great d e a l o f review work i s necessary and u n l e s s some form of m o t i v a t i o n i s adopted, work may  become very monotonous and u n i n t e r e s t i n g .  p r a c t i c e i n the mechanical to "cut  this  review Constant  processes and c o n t i n u a l r e f e r e n c e  fundamentals are e s s e n t i a l , and y e t constant r e p e t i t i o n i n and d r i e d fashion"may take the l i f e out of the s u b j e c t f o r  p u p i l and teacher a l i k e .  Both i n a l g e b r a and i n geometry,  every succeeding t o p i c i s b u i l t up on the r e s u l t s of what has gone b e f o r e . new  There i s no p o s s i b i l i t y of a student making a  s t a r t i n mathematics b e g i n n i n g at a c e r t a i n chapter u n l e s s  he i s w i l l i n g ,to go back and master a l l the fundamentals upon which t h a t chapter i s based.  T h i s s t a t e of a f f a i r s makes i t  imperative t h a t even from"the v e r y f i r s t day when the of  study  a l g e b r a and geometry i s begun, the m a t e r i a l should be  8 p r e s e n t e d i n such a way  as to c r e a t e an earnest d e s i r e on the  p a r t of the p u p i l to master the s u b j e c t i n every  detail.  Every teacher o f mathematics f i n d s a t some time or other p u p i l s i n h i s c l a s s who  d i s l i k e mathematics because they have  a p r e c o n c e i v e d i d e a t h a t they are not mathematically When such a p u p i l i s asked why u s u a l response i s such as, "Oh, never was  any good a t i t . "  he d i s l i k e s the s u b j e c t the I can't do mathematics.  I  Upon h e a r i n g such a statement,  cannot h e l p but wonder what i t was a g a i n s t mathematics; and  inclined.  one  t h a t turned that p u p i l  the l o g i c a l sequence o f thought i s to  c o n s i d e r whether or not the reason f o r such d i s l i k e l a y i n the p u p i l ' s f i r s t i n t r o d u c t i o n to the s u b j e c t , which was l i k e l y v o i d o f any form o f m o t i v a t i o n whatsoever.  very  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 v a r i o u s forms .of m o t i v a t i o n to a f u l l e r extent would undoubtedl y reduce the number o f such cases c o n s i d e r a b l y . Probably the most important reason why  the t e a c h i n g o f  mathematics should be m o t i v a t e d to i t s f u l l e s t extent, i s that it  i s the means of opening up a new  students.  A new  f i e l d o f i n t e r e s t to many  and d e l i g h t f u l experience i s i n s t o r e f o r any  p u p i l who  develops a l i k i n g f o r mathematics; an experience  which may  become the c h i e f i n t e r e s t i n t h a t student's educa-  t i o n a l career.  The opening to t h a t new  experience i s made by the teacher who  and  delightful  so motivates h i s  teaching as to c r e a t e a d e s i r e i n the student to know more about  t h i s wonderful  s c i e n c e of mathematics.  Even the study  9 of  v e r y elementary mathematics, when accompanied by the  proper  m o t i v a t i n g f o r c e s , becomes the source of abundant p l e a s u r e , a type o f p l e a s u r e which i s d i f f e r e n t from a l l other p l e a s u r a b l e experiences.  The  study of mathematics f o r i t s own  sake - f o r  the s a t i s f a c t i o n and enjoyment d e r i v e d from i t s p u r s u i t - i s one  of the prime reasons  curriculum  f o r i t s i n c l u s i o n i n the h i g h  and a student may  p l e a s u r e by s k i l l f u l l y motivated p a r t of the  source  forms of p r e s e n t a t i o n on  of  the  teacher.  For these s e v e r a l reasons ly  be l e d to t h i s new  school  enumerated above i t i s obvious-  imperative f o r the o l d pedantic methods of i n s t r u c t i o n to  be r e p l a c e d by h i g h l y motivated  forms of t e a c h i n g , i f the  s u b j e c t of mathematics i s to h o l d the p l a c e I t should h o l d and f u l f i l l i v e s of  the o b j e c t s i t should f u l f i l  i n the e d u c a t i o n a l  individuals.  (1) "We study music because i t g i v e s us p l e a s u r e So i t i s w i t h geometry. We study i t because we d e r i v e p l e a s u r e from c o n t a c t with a great and a n c i e n t body of l e a r n i n g t h a t has occupied the a t t e n t i o n of master minds d u r i n g the thousands o f y e a r s i n whioh i t has been p e r f e c t e d and we are u p l i f t e d by i t . " D.E.Smith - The Teaching o f Geometry.  10 CHAPTER I I . TYPES OF MOTIVATION APPLICABLE TO THE TEACHING OF HIGH SCHOOL MATHEMATICS Having observed  the n e c e s s i t y f o r u t i l i z i n g  motivating  f o r c e s t o t h e i r f u l l e s t extent i n the t e a c h i n g of h i g h mathematics, I t i s now necessary  school  to c o n s i d e r what are the  v a r i o u s forms o f m o t i v a t i o n which are a p p l i c a b l e to the t e a c h i n g o f h i g h s c h o o l mathematics. It all  i s i m p o s s i b l e to compile  a list  and'say, "These a r e  the forms o f m o t i v a t i o n which may be used i n t e a c h i n g h i g h  s c h o o l 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 m o t i v a t i n g f o r c e to one student may have no e f f e c t upon another  student whatsoever.  Moreover, the motives  u r g i n g a p u p i l to do h i s best are d i f f e r e n t a t the v a r i o u s stages o f t h a t i n d i v i d u a l ' s s c h o o l l i f e .  A very  powerful  m o t i v a t i n g f o r c e to a p u p i l when i n grade IX may have no i n f l u e n c e whatsoever upon t h a t same p u p i l when he reaches grade X I I .  However, there a r e c e r t a i n forms o f m o t i v a t i o n  which may be u t i l i z e d i n the v a r i o u s grades o f h i g h s c h o o l , and the u t i l i z a t i o n o f which may b r i n g new l i f e  and i n t e r e s t  to  chapter  the study o f mathematics. ^ )  T h i s present  (1) H.B.Wilson and G.M.Wilson, The M o t i v a t i o n o f School Work, P. 47. I n t h i s book there a r e enumerated e l e v e n d i f f e r e n t types o f m o t i v a t i o n . A l l but two o f these a r e a p p l i c a b l e to the t e a c h i n g o f mathematics; - v i z . E a r n i n g money, and the a c q u i s i t i o n o f a c o l l e c t i o n .  11 c o n t a i n s merely the enumeration o f the v a r i o u s types o f m o t i v a t i o n which may be used i n t e a c h i n g h i g h school mathemat i c s , and the s p e c i f i c methods by which they may be put i n t o e f f e c t w i l l be g i v e n i n succeeding  chapters.  1. A n a t u r a l I n t e r e s t i n new experiences.  - When a p u p i l  enters  grade IX o f h i g h s c h o o l , the n o v e l t y o f the new s u b j e c t s which he has not s t u d i e d b e f o r e has a g r e a t appeal to him. The wise teacher w i l l c a p i t a l i z e on t h i s n o v e l t y , and i n his  t e a c h i n g o f a l g e b r a and geometry w i l l  p u p i l a proper  t r y to g i v e the  outlook towards these s u b j e c t s at a time when  the p u p i l i s eager to hear about the wonders o f these new spheres o f knowledge. 2. An I n t e r e s t i n the i n d i v i d u a l t o p i c s s t u d i e s . -  A f t e r an  a p p r o p r i a t e i n t r o d u c t i o n to the s u b j e c t , and a proper a t t i t u d e towards i t has been c r e a t e d i n the mind o f the p u p i l , i t i s e s s e n t i a l t h a t the i n d i v i d u a l t o p i c s s t u d i e d be  s u f f i c i e n t l y v i t a l to the p u p i l t o m a i n t a i n  which has been aroused.  the i n t e r e s t  I f the m a t e r i a l s t u d i e d i s c l o s e l y  a s s o c i a t e d w i t h the p l e a s u r a b l e aspects o f the p u p i l ' s experi e n c e , then there i s a d e s i r a b l e m o t i v a t i n g f o r c e a c t i n g upon the p u p i l a t a l l times. 3. D e s i r e f o r P r a i s e . -  T h i s i s a v e r y powerful  motivating  f o r c e which i s p a r t i c u l a r l y strong i n grade IX and continues to  a l a r g e extent throughout a p u p i l ' s h i g h school c a r e e r .  P r a c t i c a l l y every p u p i l i s encouraged to b e t t e r e f f o r t s i f he knows that he w i l l r e c e i v e the p r a i s e o f h i s teacher, h i s  12 parents or h i s f e l l o w p u p i l s by making t h a t e x t r a e f f o r t . The  s k i l f u l use o f p r a i s e by the teacher can be made a very  e f f e c t i v e m o t i v a t i n g f o r c e , . n o t o n l y f o r the b r i l l i a n t student b u t a l s o f o r the d u l l a r d who i s t r y i n g to do h i s best. 4. D e s i r e to a v o i d d i s g r a c e . - T h i s m o t i v a t i n g f o r c e i s a l l i e d v e r y c l o s e l y to the one immediately  p r e c e d i n g , b u t there  are a great many p u p i l s , e s p e c i a l l y i n the group o f average i n t e l l i g e n c e , who are encouraged to b e t t e r e f f o r t s by the f e a r o f being u t t e r l y d i s g r a c e d by f a i l u r e to reach a c e r t a i n standard or to g a i n promotion. a d e s i r e f o r p r a i s e t h a t urges  I t i s n o t so much  these p u p i l s on as i t i s the  d e s i r e to a v o i d the shame which would come upon them should they f a i l  to measure up to the standard which they b e l i e v e  they should be able to r e a c h . 5. D e s i r e f o r good marks. i s a c t i v e throughout  T h i s i s a m o t i v a t i n g f o r c e which  the grades o f h i g h s c h o o l , b u t which  can be made e s p e c i a l l y u s e f u l I n s e c u r i n g g r e a t e r e f f o r t from a p u p i l i n the e a r l i e r grades.  P u p i l s i n grades IX  and X p r i z e t h e i r marks very h i g h l y , and a teacher who marks j u d i c i o u s l y can make a great d e a l o f the m o t i v a t i n g power o f marks.  P u p i l s l i k e to r e c e i v e marks even f o r the  s m a l l e s t s e t o f q u e s t i o n s , and i f these marks are recorded and made the b a s i s o f c a r e f u l comparison, the p u p i l s f i n d an added i n t e r e s t even I n d a i l y 6. D e s i r e f o r promotion. -  tests.  T h i s form o f m o t i v a t i o n i s present  13 i n a l l the grades o f h i g h s c h o o l , hut i t i s more a c t i v e toward the end o f each school year than at the h e g i n n i n g . At the commencement o f a term promotion time seems somewhat d i s t a n t , and d e s i r e f o r promotion i s not a very strong m o t i v a t i n g f o r c e , hut there are ways and means by which the teacher, may i n c r e a s e i t s power even from the b e g i n n i n g o f the term. 7. I n t e r e s t i n c o m p e t i t i o n s . -  The use o f competitions o f  v a r i o u s kinds i s a very u s e f u l form o f m o t i v a t i o n , l y i n grades IX and X.  especial-  The m a j o r i t y o f p u p i l s i n these  grades are very keen on competitions, both a g a i n s t each other and a g a i n s t time.  There are numerous ways by which  t h i s form o f m o t i v a t i o n may be used to good advantage. 8. D e s i r e f o r a c t i v i t y .  - One o f the dangers In a s u b j e c t l i k e  mathematics i s f o r i t to become too i n a c t i v e ,  ^ i t t i n g at  one's desk f o r a c o n s i d e r a b l e time working a l o n g s e r i e s o f questions does not appeal to the o r d i n a r y h i g h school student.  There i s a tendency to boredom which should be  overcome by the i n t r o d u c t i o n o f more p h y s i c a l a c t i v i t y i n t o the mathematics l e s s o n .  The s p e c i f i c methods by which  t h i s a c t i v i t y may be i n t r o d u c e d w i l l be d i s c u s s e d i n a later  chapter.  9. I n t e r e s t i n games. -  P u p i l s i n a l l grades o f h i g h school  d e r i v e a great d e a l o f p l e a s u r e from games.  There are a  l a r g e number of mathematical r e c r e a t i o n s which are admirably  s u i t e d to high school students, and I f these are  14 c a r e f u l l y arranged  and p r o p e r l y p l a c e d i n the mathematics  l e s s o n they w i l l have a wonderful m o t i v a t i n g e f f e c t on the whole s u b j e c t o f mathematics. 10. I n t e r e s t i n humor. -  A mathematics teacher, and e s p e c i a l l y  one who has been t e a c h i n g the s u b j e c t f o r a number o f y e a r s , i s very o f t e n i n c l i n e d to become mechanical i n h i s methods of p r e s e n t a t i o n , and overlook  some o f the p o s s i b i l i t i e s  which e x i s t f o r making a mathematics l e s s o n r e a l l y able.  Admittedly,  enjoy-  the number o f o p p o r t u n i t i e s f o r  humorous i l l u s t r a t i o n s and a n a l o g i e s i s not as great i n a mathematics l e s s o n as i n l e s s o n s i n many of the other jects.  sub-r  n e v e r t h e l e s s the teacher k e e n l y I n t e r e s t e d i n  m o t i v a t i n g h i s t e a c h i n g to the f u l l e s t extent should make the most o f every l i t t l e  opportunity that arises f o r  i n t r o d u c i n g even a s l i g h t touch o f humor i n t o the mathematics l e s s o n . 11. D e s i r e f o r change, - v a r i e t y . - Any experienced mathematics teacher knows t h a t there i s a decided tendency on h i s p a r t to present l e s s o n s of a s i m i l a r nature manners.  i n almost  identical  In the t e a c h i n g o f geometry t h i s tendency i s  particularly noticeable. a f t e r another,  The treatment  o f one theorem  or one e x e r c i s e a f t e r another  i n the same  manner day a f t e r day i s c e r t a i n ! t o become t e d i o u s to p u p i l s . Pupils l i k e variety.  An e s s e n t i a l form o f m o t i v a t i o n ,  t h e r e f o r e , e s p e c i a l l y i n the t e a c h i n g o f geometry, i s a v a r i e t y of methods o f p r e s e n t a t i o n f o r l e s s o n s of a s i m i l a r  15 nature. 12.  I n t e r e s t In c o n s t r u c t i n g . - I n gradeIX to a g r e a t  extent,  and i n the other grades to a s m a l l e r extent, p u p i l s f i n d a g r e a t d e a l o f p l e a s u r e I n doing a c t u a l c o n s t r u c t i v e work. They enjoy b u i l d i n g .  There i s a very c l o s e analogy  between c o n s t r u c t i v e geometry and b u i l d i n g , and i f t h i s f a c t i s f u l l y a p p r e c i a t e d by the teacher, a great d e a l o f the work In geometry c a n be motivated  very h i g h l y by  u t i l i z i n g the p u p i l ' s keen i n t e r e s t In 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 . -  "To d i s c o v e r 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 o f adolescents.  T h i s t r a i t o f the adolescent  i n a l l grades o f h i g h s c h o o l .  i s active  When a p u p i l i n the f i r s t  week of h i s study o f geometry d i s c o v e r s f o r h i m s e l f some new ( t o him) f a c t about t r i a n g l e s , or when a m a t r i c u l a t i o n student by h i s own e f f o r t s d i s c o v e r s the fundamental nature  o f an e l l i p s e , the t h r i l l o f d i s c o v e r y i s  s u f f i c i e n t t o g i v e that student more deeply  i n t o the mys t e r i e s  p l a y e d by the teacher to  an ardent  d e s i r e to delve  of mathematics.  The p a r t  i s t o s t a r t the p u p i l on the road  d i s c o v e r y and guide him at d i f f i c u l t c r o s s i n g s .  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 . -  A p u p i l i n any grade o f  h i g h s c h o o l l i k e s to t h i n k 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 t h i s i t i s not meant t h a t he l i k e s  (1) R.W.Pringle, Methods w i t h Adolescents,  P. 126.  16 to be "teacher's p e t , " b u t t h a t as a reward f o r some s p e c i a l e f f o r t on h i s p a r t he i s allowed to emjoy some s p e c i a l p r i v i l e g e n o t enjoyed by the remainder of the class.  There are a great many ways by which good work i n  mathematics may be rewarded by s p e c i a l p r i v i l e g e s , and t h i s type of m o t i v a t i o n i s very e f f e c t i v e with c e r t a i n types of students. 15. I n t e r e s t i n mathematics f o r i t s own sake. -  I f the  t e a c h i n g o f mathematics has been p r o p e r l y n o t i v a t e d by v a r i o u s means d u r i n g the e a r l y p a r t o f a p u p i l ' s h i g h s c h o o l c a r e e r , there should come a time, probably a t the end of grade X o r the beginning o f grade X I , when t h a t p u p i l d i s c o v e r s that he r e a l l y enjoys the study of mathematics j u s t f o r i t s own sake.  He may leave some  • other t h i n g s undone, but he w i l l not n e g l e c t h i s mathematics because o f t h e enjoyment he d e r i v e s simply by Its pursuit.  When t h i s stage has been reached,  and i t i s  reached by a great many h i g h school students, there i s added to the m o t i v a t i n g f o r c e s a l r e a d y a t work a new and extremely powerful one; one which may cause the p u p i l to devote h i s l i f e t o the p u r s u i t o f mathematical knowledge. 16. E f f e c t o f a c h a l l e n g e . -  Most human b e i n g s , and e s p e c i a l l y  boys and g i r l s w i t h the r e d b l o o d o f youth i n t h e i r v e i n s , respond v e r y r e a d i l y to a c h a l l e n g e and do not r e s t until  that c h a l l e n g e has been answered.  content  This characteris-  t i c o f human psychology makes i t p o s s i b l e f o r the teacher  17 of mathematics to encourage the p u p i l s to put f o r t h  their  best e f f o r t s by making mathematical problems appear as d e f i n i t e c h a l l e n g e s to t h e i r i n t e l l i g e n c e and i n g e n u i t y . In the advanced grades of h i g h school the m o t i v a t i n g f o r c e of  a c h a l l e n g e i s s u e d by a c e r t a i n a l g e b r a i c a l or  g e o m e t r i c a l 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. 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. -  Some students i n  h i g h school are encouraged to b e t t e r e f f o r t s i n mathematical study because of t h e i r d e s i r e to go out  into  the world b e t t e r f i t t e d f o r t h e i r l i f e ' s work by reason t h e i r study o f mathematics. these  students  of  I f the teacher p o i n t s out to  the v a r i o u s p o s s i b i l i t i e s of t r a n s f e r from  t h e i r study of mathematics to t h e i r i n t e n d e d  occupations  or p r o f e s s i o n s , then t h e i r d e s i r e f o r e f f i c i e n c y  may  become a r e a l m o t i v a t i n g f o r c e i n the study of mathematics. 18. S a t i s f a c t i o n through mastery. -  T h i s form of m o t i v a t i o n  i s one o f the most Important, i f not the most of  a l l the forms which can be a p p l i e d to the t e a c h i n g o f  mathematics i n h i g h s c h o o l . his  important,  I f , i n the e a r l y stages  study o f mathematics, a p u p i l has  mathematics motivated  f o r him  of  the t e a c h i n g of  so as to g i v e him  an e a r l y  l i k i n g f o r the s u b j e c t , he w i l l i n t u r n put f o r t h h i s b e s t e f f o r t i n t h a t s u b j e c t , and most l i k e l y f i n d t h a t he succeeded i n mastering  the e a r l y p a r t o f the work.  has The  s a t i s f a c t i o n which t h a t p u p i l d e r i v e s from the mastery o f  18 h i s e a r l y work i s a compelling i n f l u e n c e to f u r t h e r e f f o r t . A p u p i l enjoys a subject which he understands  w e l l , and i f  a teacher can u t i l i z e the v a r i o u s minor methods o f m o t i v a t i o n i n order to encourage a p u p i l t o master each s e c t i o n o f the work as he goes along, then the very f a c t of h i s mastery over a p r e c e d i n g s e c t i o n i s s u f f i c i e n t m o t i v a t i o n to c r e a t e i n him a d e s i r e to proceed following section.  The m o t i v a t i n g f o r c e o f t h i s  to the satis-  f a c t i o n gained through mastery I s indeed a powerful I n f l u e n c e I n the t e a c h i n g o f h i g h s c h o o l mathematics. 19. Eagerness f o r p e r f e c t i o n . - The o b j e c t o f a great many forms o f m o t i v a t i o n i n t e a c h i n g i s to encourage the poorer students to put f o r t h g r e a t e r e f f o r t and a t t a i n a h i g h e r standard o f e f f i c i e n c y .  While t r y i n g t o accomplish  t h i s aim, the f a c t must not be o v e r l o o k e d t h a t those p u p i l s who have a l r e a d y a t t a i n e d a v e r y h i g h degree o f e f f i c i e n c y might be encouraged to do even b e t t e r work than they have done.  One o f the methods o f m o t i v a t i o n which  a p p l i e s p a r t i c u l a r l y to t h i s top-ranking c l a s s o f p u p i l s , i s the c r e a t i o n o f an eagerness f o r p e r f e c t i o n .  Mathema-  t i c s i s one s u b j e c t I n which p e r f e c t i o n can be reached i n a great many cases, and i f the teacher can c r e a t e an eagerness on the p a r t o f the p u p i l to a t t a i n a b s o l u t e p e r f e c t i o n i n h i s work, then he w i l l be u r g i n g t h a t p u p i l to extend h i m s e l f to the l i m i t o f h i s a b i l i t y and a new I n t e r e s t w i l l be added to t h a t p u p i l ' s work i n mathematics.  20. S a t i s f a c t i o n through h e l p i n g . - When a student reaches grade XI or XII he has gained s u f f i c i e n t knowledge o f elementary mathematics to enable him to o f f e r some a s s i s t a n c e to p u p i l s i n grades IX and X. are  I f opportunities  p r o v i d e d by the teacher f o r the o f f e r i n g o f t h i s  a s s i s t a n c e , then the s e n i o r grade p u p i l w i l l  s t r i v e to  understand h i s work more thoroughly i n o r d e r to be a b l e to  t e a c h h i s f r i e n d s i n the lower grades more s k i l f u l l y .  A student gets a g r e a t d e a l o r s a t i s f a c t i o n out o f h e l p i n g other s t u d e n t s , and a t the same time he i s s t r e n g t h e n i n g his  own grasp o f the s u b j e c t by constant review o f  fundamentals. 21. D e s i r e t o be c o n s i d e r e d mature. h i g h s c h o o l most p u p i l s l i k e  In the upper grades o f  to b e l i e v e t h a t they a r e  g e t t i n g on i n t o advanced mathematics,  and that they w i l l  soon be blossoming i n t o mature mathematicians.  A certain  amount o f encouragement to the adoption o f t h i s  attitude  can be g i v e n by the teacher to v e r y good e f f e c t .  If a  teacher t r e a t s h i s m a t r i c u l a t i o n students as mature persons o f whom much more i s expected than I s expected o f p u p i l s i n the lower grades, then those m a t r i c u l a t i o n ; students w i l l respond and endeavour to show t h a t they a r e indeed advanced mathematicians  capable o f c o n c e n t r a t e d  effort. 22. D e s i r e to complete  the course s e l e c t e d . - One o r the  m o t i v a t i n g f o r c e s u r g i n g students i n the upper grades o f  20 school to work d i l i g e n t l y i s the d e s i r e on the p a r t of p u p i l to complete the course which he has  selected.  p u p i l s i n these grades l o o k forward e a g e r l y when they w i l l be graduated from h i g h  to the  school,  and  the  Most time  be  able  to go out i n t o the world w i t h a s u c c e s s f u l l y completed high  school  course behind them.  future pleasure  has  student's present 2 3  •  The  a n t i c i p a t i o n of t h i s  a valuable motivating  e f f e c t on  efforts.  Eagerness to pass m a t r i c u l a t i o n examination. who  are i n the  f i n a l year of h i g h  school  t a k i n g the m a t r i c u l a t i o n course, there motivating  the  For  and who  i s one  pupils  are  v e r y powerful  f o r c e a t work; namely, the d e s i r e to pass  m a t r i c u l a t i o n examinations.  The m o t i v a t i n g  force  the  of  m a t r i c u l a t i o n examinations i s not always the most d e s i r a b l e form of m o t i v a t i o n ,  but,  as w i l l be. seen i n a  l a t e r chapter, a proper a t t i t u d e toward m a t r i c u l a t i o n examinations may e f f e c t s , and  r i d them or most or t h e i r  t r a n s f o r m them Into  undesirable  admirable  motivating  f o r c e s , not; only f o r p u p i l s i n the m a t r i c u l a t i o n grade, but a l s o f o r p u p i l s working up 24.  E f f e c t of p r i z e s and o f p r i z e s and scope.  towards i t .  scholarships. -  scholarships  i s obviously  I t i s o n l y those students who  good a t t h e i r work who  The  have any  motivating  very l i m i t e d i n i t s  are  exceptionally  interest in  scholarships,  and u s u a l l y these rewards are o f f e r e d o n l y to students.  power  matriculation  On account of t h e i r narrow scope, and  also  on  21 account of the f a c t t h a t the working f o r the reward may rob a s u b j e c t o f much o f i t s r e a l v a l u e , the o f f e r i n g o f p r i z e s and s c h o l a r s h i p s can not be c o n s i d e r e d one o f the v a l u a b l e forms o f m o t i v a t i o n f o r the h i g h s c h o o l To t h i s l i s t  student.  o f the v a r i o u s types o f m o t i v a t i o n  appli-  c a b l e t o the t e a c h i n g o f h i g h s c h o o l mathematics, there might be added a great many other forms o f m o t i v a t i o n which a r e more limited  i n their  scope or which are a p p l i c a b l e o n l y to s p e c i a l  types o f students.  The f o r e g o i n g l i s t , however, c o n t a i n s most  of the more g e n e r a l forms o f m o t i v a t i o n which a r e a p p l i c a b l e to  students a t t e n d i n g h i g h s c h o o l .  I n the f o l l o w i n g chapters  the s p e c i f i c means by which these types o r m o t i v a t i o n may be a p p l i e d to the t e a c h i n g o r a l g e b r a and geometry i n grades IX and X I I w i l l be d i s c u s s e d .  22 CHAPTER I I I MOTIVATION IN THE TEACHING OP GRADE IX GEOMETRY To  the teacher  o f geometry i n grade IX a wonderful  o p p o r t u n i t y i s presented;  an o p p o r t u n i t y f o r opening up to h i s  p u p i l s a new f i e l d o f knowledge which i s f u l l of i n t e r e s t and enjoyment.  I t becomes incumbent upon the teacher t o make the  most o f t h i s o p p o r t u n i t y all  Dy so m o t i v a t i n g h i s t e a c h i n g t h a t  the i n t e r e s t and enjoyment l a t e n t I n the s u b j e c t o f  geometry i s d i s c o v e r e d by the p u p i l s under h i s care.  But  what are the v a r i o u s methods of m o t i v a t i n g  the t e a c h i n g o f  grade IX geometry i n order to achieve  object?  this  In the f i r s t p l a c e , the teacher can c a p i t a l i z e upon the n o v e l t y of the s u b j e c t . t h i n g s new, so w h i l e  People are i n h e r e n t l y i n t e r e s t e d i n  the p u p i l s a r e I n the proper  frame o f  mind the teacher can i n f o r m them o f some o f the wonders o f t h i s s c i e n c e of geometry, and g i v e the p u p i l s a proper towards the s u b j e c t . to  outlook  An i n t r o d u c t i o n might i n c l u d e r e f e r e n c e  some o f the i n t e r e s t i n g f e a t u r e s about aefew?.of the world's  g r e a t mathematicians such as Pythagoras, E u c l i d and E i n s t e i n , and a l s o g i v e 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 b a s i s . A f t e r the p u p i l ' s i n t e r e s t has been aroused i n the s u b j e c t , i t i s e s s e n t i a l t h a t t h i s i n t e r e s t be maintained s k i l f u l l y motivated  teaching.  by  There i s a danger i n grade IX  23 that the n o v e l t y of geometry may developed One  wear o f f b e f o r e the p u p i l  a r e a l i n t e r e s t In the s u b j e c t f o r i t s own  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  even at a very e a r l y stage i s by keeping work mostly to  experimental.  sit still  has  sake. be  aroused  the f i r s t p a r t o f the  A p u p i l i n t h i s grade does not  like  i n h i s seat and watch c o n s t r u c t i o n s b e i n g done on  the b l a c k b o a r d by the teacher; but he wants to do the cons t r u c t i v e work h i m s e l f .  The b l a c k b o a r d e x p l a n a t i o n s by  the  teacher should be as short and c o n c i s e as i s c o n v e n i e n t l y p o s s i b l e , and then the p u p i l should be allowed to experiment for  himself. Although  the e a r l y p a r t of the study of geometry should  be l a r g e l y experimental  and of a c o n s t r u c t i v e nature,  never-  t h e l e s s i t i s important  t h a t sound g e o m e t r i c a l p r i n c i p l e s  be  taught i n c o n j u n c t i o n w i t h these c o n s t r u c t i v e e x e r c i s e s .  If  t h i s i s not done, the p u p i l w i l l f i n d out i n a v e r y short  time  that h i s knowledge of geometry has been b u i l t up on a r a t h e r f e e b l e f o u n d a t i o n ; and consequently he w i l l l o s e t h a t e a r l y i n t e r e s t which he had i n the s u b j e c t .  He w i l l miss the r e a l  enjoyment which would have been i n s t o r e f o r him had he his  built  g e o m e t r i c a l knowledge on sound mathematical p r i n c i p l e s .  In order to l a y a good f o u n d a t i o n i t i s not necessary f o r the p u p i l to memorize l i s t s  o f d e f i n i t i o n s , axioms and p o s t u l a t e s ;  but r a t h e r he should be d i r e c t e d towards g a t h e r i n g a c c u r a t e i n f o r m a t i o n about fundamental g e o m e t r i c a l f a c t s . it  For example,  i s e s s e n t i a l t h a t a p u p i l know e x a c t l y what i s meant by  24 " v e r t i c a l l y opposite angles  and what r e l a t i o n they bear  11  to  each o t h e r ; but what p r o f i t would there be i n demanding t h a t he should be a b l e to r e c i t e the d e f i n i t i o n f o r v e r t i c a l l y opposite One  angles? very important  f a c t to remember i n m o t i v a t i n g the  t e a c h i n g of grade IX geometry o r a l g e b r a i s the f a c t t h a t the p u p i l s i n t h i s grade r e q u i r e a c e r t a i n amount of p h y s i c a l a c t i v i t y ; - o p p o r t u n i t y f o r a c t u a l movement of body and They are not  capable of s u s t a i n e d e f f o r t i n c o n c e n t r a t i o n over  a v e r y l o n g p e r i o d of time, but periodical physical activity. by  limbs.  they r e q u i r e o p p o r t u n i t y f o r T h i s need i s s u p p l i e d i n p a r t  the i n t r o d u c t i o n of c o n s t r u c t i v e e x e r c i s e s i n which the  p u p i l s 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  a l s o by e x t e n s i v e use o f the b l a c k b o a r d by the p u p i l s ; there I s s t i l l  but  another very v a l u a b l e means o f s u p p l y i n g  p h y s i c a l r e l a x a t i o n f o r the p u p i l s , and a t the  same time  s a t i s f y i n g t h e i r d e s i r e f o r c o m p e t i t i v e forms of a c t i v i t y . A s e r i e s of short questions (ten f o r example) i s p l a c e d on the b l a c k b o a r d and  the p u p i l s are i n s t r u c t e d to work the  questions and on completion the desk.  to t u r n t h e i r books f a c e down on  The f i r s t twenty p u p i l s f i n i s h e d w r i t e t h e i r names  on the blackboard i n order as they f i n i s h .  When a l l p u p i l s  have f i n i s h e d the q u e s t i o n s , books are exchanged and marks are a s s i g n e d to the q u e s t i o n s . dents who  E x t r a marks are a l l o t t e d to s t u -  answer a l l questions c o r r e c t l y and a l s o f i n i s h i n  time to get t h e i r names on the board. graduated  These bonus marks are  a c c o r d i n g t o the order of the names on the  board.  25 Books are r e t u r n e d and r e s u l t s are compared i n the f o l l o w i n g manner:  A l l the p u p i l s stand, and as the teacher says,  r i g h t ' ; "two 1  right";  "One  "three r i g h t " , e t c . , p u p i l s w i t h the  corresponding number of c o r r e c t s o l u t i o n s take t h e i r s e a t s . When "ten r i g h t " i s about to be reached, p e r f e c t s o l u t i o n s are l e f t in  s t a n d i n g , and these can be  ranked  the order i n which t h e i r names appear on the b l a c k b o a r d . T h i s at f i r s t glance appears  of  o n l y those w i t h  t e a c h i n g procedure,  to be a v e r y o r d i n a r y b i t  but on c l o s e r a n a l y s i s i t w i l l  be  found to c o n t a i n s e v e r a l v a l u a b l e f e a t u r e s which are v e r y e f f e c t i v e forms of m o t i v a t i o n . In  the f i r s t  p l a c e , the p u p i l s who  succeed i n g e t t i n g  most of the questions c o r r e c t f e e l a c e r t a i n sense of mastery over the work t h a t has been covered and they w i l l a t t a c k  new  work w i t h c o n f i d e n c e .  be  A l s o , the v e r y b e s t students w i l l  s t r i v i n g f o r p e r f e c t s o l u t i o n s In order to be s t a n d i n g u n t i l the l a s t . c a r e f u l l y graded,  able to continue  Besides t h i s , i f the questions are  some of them b e i n g comparatively easy,  even the poorest p u p i l s w i l l f i n d t h a t they have  developed  a c e r t a i n amount o f s k i l l , and on the next o c c a s i o n they endeavour to remain s t a n d i n g f o r a l o n g e r time. to be the f i r s t  then  will  No p u p i l wants  to have to take h i s seat, so even the very  p o o r e s t p u p i l i n the c l a s s has a very s t r o n g motive f o r t r y i n g to  improve h i s work. The v e r y f a c t t h a t a l l the p u p i l s i n the classi;have been  standing f o r a few minutes,  and that twenty of them have made  t r i p s to the b l a c k b o a r d , p r o v i d e s an o p p o r t u n i t y f o r r e l a x a -  26 t i o n of muscles which become t i r e d from m a i n t a i n i n g posture.  a  sitting  T h i s simple method of procedure, t h e r e f o r e , i s  extremely u s e f u l as a m o t i v a t i n g grade IX geometry.  The  f o r c e i n the t e a c h i n g  of  f o r e g o i n g methods f o r m o t i v a t i n g  grade IX geometry are a l l o f a g e n e r a l nature, but  the f o l l o w -  i n g are a 'number of s p e c i f i c methods by which c e r t a i n t o p i c s may  be made much more i n t e r e s t i n g and v a l u a b l e to the At the v e r y beginning  of the course  pupil.  i n geometry, when  commencing the study of the s t r a i g h t l i n e , p r a c t i c e i n drawing and measuring can be made extremely i n t e r e s t i n g by  utilization  of the p u p i l ' s i n t e r e s t i n e s t i m a t i n g or guessing.  Questions  such as the f o l l o w i n g prove very i n t e r e s t i n g to grade IX students  j u s t commencing the study o f geometry:  1. By u s i n g the back of your r u l e r , draw a l i n e which you b e l i e v e to be 7 i n . l o n g . Turn the r u l e r over and measure. How many are w i t h i n 1/16 of an inch? how many w i t h i n 1/8 '? w i t h i n 1/4"? w i t h i n 1/2"? e t c . 2. Using the l o n g e s t s i d e o f your set square, draw a l i n e which you b e l i e v e to be 4|r i n . l o n g . Measure. How many are w i t h i n 1/16 of an Inch? e t c . 3. Draw two columns, one f o r the estimated l e n g t h and the other f o r the measured l e n g t h i n each o f the f o l l o w i n g . Use the l o n g e s t s i d e of your set square f o r drawing, and your r u l e r f o r measuring. (a) Draw a l i n e of any l e n g t h . Estimate i t s l e n g t h i n inches ( i n proper column). Measure. Record measurement. (b) Draw a l i n e which you b e l i e v e to be 5 i n . l o n g . Measure. (c) Draw a l i n e of any l e n g t h d i a g o n a l l y on the page. Estimate i t s l e n g t h i n Inches and In m i l l i m e t r e s . Measure. (d) Estimate the l e n g t h of your geometry book i n c e n t i metres and m i l l i m e t r e s . Measure. (e) Estimate the width of your desk i n i n c h e s . Measure. Mark each aftswer c o r r e c t which i s w i t h i n % i n . or 5 m.m. of the measured l e n g t h . S e r i e s o f questions s i m i l a r to the above may be arranged In c o m p e t i t i v e form and done from the 1  27 "blackboard. A f t e r the study of the s t r a i g h t l i n e the study o f angles is  commenced, and there are numerous ways by which the  t e a c h i n g o f t h i s t o p i c can be v e r y e f f e c t i v e l y m o t i v a t e d . D u r i n g the i n t r o d u c t i o n to the t o p i c , the t e a c h e r may p u p i l s to w r i t e down the f o l l o w i n g sentence:  "Geometry  teaches us to b i s e c t a n g l e s , " and then see how words "geometry , " b i s e c t " and "angles" s p e l l e d 11  Mention might be made of the l i t t l e boy who  ask the  many have the correctly.  wrote "Geometry  teaches us to b i s e x a n g e l s . " A f t e r the method f o r naming angles has been e x p l a i n e d , skill  i n naming angles c o r r e c t l y can be developed i n an  i n t e r e s t i n g manner by s e e i n g who  can  name a l l the angles (not r e f l e x ) on a f i g u r e such as the accompanying one.  Each angle must be named o n l y  once. P r a c t i c e i n drawing angles o f v a r i o u s s i z e s , measuring  angles  a l r e a d y drawn, and d e v e l o p i n g a good i d e a of angular  sizes  can be g i v e n i n a r a t h e r i n t e r e s t i n g manner by u s i n g a method s i m i l a r to t h a t f o r g i v i n g p r a c t i c e w i t h s t r a i g h t  lines.  A f t e r e x p l a i n i n g the use o f the p r o t r a c t o r , a s e r i e s of q u e s t i o n s such as the f o l l o w i n g may  be g i v e n and the p u p i l s  i n s t r u c t e d to keep a r e c o r d o f the ones which they get c o r r e c t : 1. Draw any angle ABC.  Estimate i t s s i z e In degrees. Measure.  28 2. Draw an angle which you b e l i e v e to be 65 . Measure. 3. Of the two angles on the b l a c k b o a r d which i s -the l a r g e r ? How many degrees l a r g e r ? (Angles such as the f o l l o w i n g may be drawn; - the s m a l l e r angle h a v i n g the l o n g e r arms.)  4. Draw any acute angle on your paper. Draw another angle which you b e l i e v e i s e x a c t l y twice the f i r s t . Measure both a n g l e s . M u l t i p l y s i z e of f i r s t angle by two and compare w i t h the s i z e of the second. 5. Draw a f i g u r e on your paper s i m i l a r to the one on the b l a c k b o a r d . (Accompanying). Estimate the s i z e of each of the f o l l o w i n g angles: AOB /BOG "BOD s /BOE -  t  "DOE s "AOE  Ty° = A  7"AOC  z  S  Ch~eck by measurement".  In marking the above q u e s t i o n s , t h r e e marks are a l l o w e d if  the answer i s c o r r e c t to w i t h i n 1°; two marks i f w i t h i n 2°;  and one mark i f w i t h i n 3 ° .  Marks are t o t a l l e d and  compared  by the s t a n d i n g method. In  c o n n e c t i o n w i t h the study of angles, use of the  mariner's compass can be made to good advantage.  After a  short d i s c u s s i o n of the mariner's compass, i t s s t r u c t u r e use, the teacher may G i r l Guides  and  enquire i f t h e r e are any Boy Scouts or  i n the c l a s s .  (Members of these o r g a n i z a t i o n s  are supposed t o know the t h i r t y - t w o p o i n t s of the compass.) I f t h e r e are any present they may knowledge by drawing a diagram blackboard.  be allowed to d i s p l a y  their  o f the mariner's compass on the  Prom one of these diagrams much p r a c t i c e can be  29 g i v e n i n angular s i z e s by asking q u e s t i o n s such as the following: 1. A boat I s s a i l i n g due n o r t h , and then changes i t s course to N N VV. Through how many degrees does the k e e l o f the boat turn? 2. Two boats are approaching the same p o r t from d i f f e r e n t directions. 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 t h e l i n e s i n d i c a t i n g t h e i r routes? Many other s i m i l a r questions on angles may be asked u s i n g the mariner's  compass as a b a s i s , and many i n t e r e s t i n g  n a v i g a t i o n may be brought i n t o  facts  about  the d i s c u s s i o n o f t h i s t o p i c . •  T h i s a l l adds i n t e r e s t t o a l e s s o n on angles, and i t i s a t y p i c a l way o f m o t i v a t i n g what might otherwise be a r a t h e r a b s t r a c t and u n i n t e r e s t i n g l e s s o n . Another p r a c t i c a l method by which i n t e r e s t may be aroused i n the study o f l i n e s and angles I s by making use of the pupils'  sense o f s a t i s f a c t i o n a t b e i n g a b l e to f o l l o w a number  of r a t h e r complicated d i r e c t i o n s  and a r r i v e a t the proper  destination.  t r e a s u r e or c a t c h i n g the  F i n d i n g the hidden  t h i e f a r e forms o f t h i s type o f m o t i v a t i o n .  A question  as the f o l l o w i n g Is f u l l of i n t e r e s t to a grade IX p u p i l  such just  b e g i n n i n g the study o f geometry: A jewel t h i e f s t o l e a diamond r i n g and was caught by the p o l i c e a f t e r p u r s u i n g him over the f o l l o w i n g course. Can you f o l l o w them? How f a r from the scene o f the crime was the t h i e f when caught? (1 m i l e - 1 In.) From the scene o f the crime he t r a v e l s 1^ m i l e s N.E.; then 1 m i l e due N.; from there § m i l e N,W.; he then turns and goes 1^ m i l e s S.W.; then 3 m i l e s due E. He turns again and goes 7/8 of a m i l e S.S.W.; and f i n a l l y he turns due E. and goes 1 m i l e b e f o r e b e i n g caught. Questions  such as t h i s can e a s i l y be made c o m p e t i t i v e by  30 seeing who can c a t c h the t h i e f f i r s t , o r who can f i n d the hidden t r e a s u r e . The  i n t r o d u c t i o n to the study o f p a r a l l e l l i n e s i s v e r y  often a rather d i f f i c u l t  l e s s o n i n which to arouse  of the p u p i l s t o any g r e a t extent.  the i n t e r e s t  The a p p l i c a t i o n o f c a r e -  f u l l y planned methods o f m o t i v a t i o n , however, may change t h i s situation  c o n s i d e r a b l y and convert the l e s s o n i n t o one o f  exceptional interest  to the p u p i l .  A f t e r a d i s c u s s i o n o f the  word " p a r a l l e l " - i t s s p e l l i n g , meaning and a p p l i c a t i o n t o straight taken,  l i n e s , common examples o f p a r a l l e l l i n e s may be  such as the two edges of the desk; the two edges o f the  blackboard,  or the two edges o f a book.  Definitions f o r  p a r a l l e l l i n e s may be suggested by the p u p i l s , and I n t h i s connection the teacher may repeat some r a t h e r humorous definitions another.  r e c e i v e d on examination  papers a t some time or  The f o l l o w i n g are examples o f such  definitions:  " P a r a l l e l l i n e s a r e l i n e s which r u n along together s i d e by s i d e l i k e the c a r t r a c k s , sometimes f o r m i l e s , and never converse."  " P a r a l l e l l i n e s never meet u n l e s s you bend (2)  one dr. both of them."  "A p a r a l l e l l i n e I s one t h a t when  produced to meet i t s e l f does n o t meet." When the meaning o f " p a r a l l e l l i n e s " i s p e r f e c t l y understood by the p u p i l s , they may be g i v e n questions such as the (1) Received on Grade IX examination paper i n 1934, B r i t a n n i a High S c h o o l . (2) ( j Alexander Abingdon, The Omnibus Boners. P. 69. 3  31 following: 1. On u n r u l e d paper draw two s t r a i g h t l i n e s a c r o s s the page about two Inches a p a r t , and which you b e l i e v e to be p a r a l l e l . Test by measuring t h e i r d i s t a n c e a p a r t a t s e v e r a l p l a c e s . 2. Draw two l i n e s d i a g o n a l l y across the page which you b e l i e v e to be p a r a l l e l . T e s t as b e f o r e . 3. Draw a l i n e a c r o s s the page. Draw two other l i n e s which you b e l i e v e t o be p a r a l l e l to the f i r s t one - one about an i n c h above and the other about an i n c h below the g i v e n l i n e . T e s t the two new l i n e s to see i f they a r e p a r a l l e l . To questions such as these the teacher may add e x e r c i s e s from the blackboard,  such as g e t t i n g the p u p i l s to say whether  l i n e s drawn on the b l a c k b o a r d a r e p a r a l l e l o r not, and i f not p a r a l l e l which way they converge. I n t e r e s t may be added t o the l e s s o n by the i n t r o d u c t i o n 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 lines.  f o l l o w i n g are some examples o f the type-or o p t i c a l which are s u i t a b l e a t t h i s 1. Are the two l i n e s  The  illusions  stage:  i n the f o l l o w i n g diagram  parallel?  2. Which or two l i n e s below i s the longer?  3. I n the f o l l o w i n g diagram, which i s l o n g e s t , AB o r BC o r CD?  (1) Morgan, Foberg,  Breckenridge>  Plane Geometry.  P. 13.  32  A f t e r the e s s e n t i a l f a c t s r e g a r d i n g p a r a l l e l s and t r a n s v e r s a l s have been s t u d i e d , the i n a c c u r a c y o f the former method o f t e s t i n g p a r a l l e l s may be p o i n t e d out and a more s c i e n t i f i c method s u b s t i t u t e d ; namely, the measuring of a p a i r o f a l t e r n a t e o r corresponding  angles.  The o l d method  of measuring the d i s t a n c e 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 r e c r e a t i o n a t the end o f a l e s s o n on p a r a l l e l s , the f o l l o w i n g sentence, "parallel ',  suggested  by the word  may be g i v e n t o the p u p i l s as a s p e l l i n g  1  test.  "In a cemetery an embarrassed c o b b l e r and an h a r a s s e d  peddler  were gauging the symmetry o f a l a d y ' s tomb-stone w i t h unparalleled  ecstasy.  11  When approaching  the study o r t r i a n g l e s , the teacher i s  o f f e r e d a great o p p o r t u n i t y f o r a l l o w i n g the p u p i l s t o experience  the " t h r i l l  o f d i s c o v e r y " , and I n t h i s way i n c r e a s e  t h 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 o f geometry. There a r e a great many f a c t s about t r i a n g l e s with which the p u p i l i n grade IX i s not f a m i l i a r ; presented  and i f the m a t e r i a l i s  i n such a way as t o allow the p u p i l t o d i s c o v e r  these f a c t s f o r h i m s e l f , then he w i l l d e r i v e a g r e a t deal o f s a t i s f a c t i o n from so-doing.  Some o f these f a c t s which a  p u p i l may be l e d t o d i s c o v e r f o r h i m s e l f 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 s i d e s of a t r i a n g l e are together g r e a t e r than the t h i r d side. 3. I f the s i d e s of a t r i a n g l e are produced i n order, the sum or the e x t e r i o r 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 e x t e r i o r angle so formed i s equal to the sum of the i n t e r i o r o p p o s i t e angles. 5. The b i s e c t o r s of the angles of a t r i a n g l e meet at a p o i n t . 6. The p e r p e n d i c u l a r b i s e c t o r s o f the s i d e s of a t r i a n g l e meet at a p o i n t . 7. The three medians of a t r i a n g l e meet at a p o i n t . (The p u p i l s may experiment w i t h paper t r i a n g l e s to see I f the p o i n t at which the medians meet i s tne c e n t r e of g r a v i t y . ) 8. In a r i g h t - a n g l e d t r i a n g l e , the mid-point of the hypotenuse Is e q u i d i s t a n t 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 l o n g e s t s i d e i s double the shortest. 10. The area of a t r i a n g l e i s l e s s than the area of a square w i t h the same p e r i m e t e r . The  means o f m o t i v a t i n g  constructions  of the  standard  i n grade IX geometry are many and v a r i e d .  main types o f m o t i v a t i o n of the work are  (1) The  (3) The  The  which are a p p l i c a b l e to t h i s phase pupil's interest i n building  c o n s t r u c t i o n work; (2) The t y ; and  the t e a c h i n g  and  pupil's desire for physical  p u p i l ' s i n t e r e s t i n t h i n g s new.  activi-  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 a t t a c k i n g each problem e x p e r i m e n t a l l y together," can l i n e has  to be  - a c e r t a i n f i g u r e has  to be  put  the p u p i l f i n d a method f o r doing i t ? erected perpendicular  from a g i v e n p o i n t i n t h a t l i n e . p u p i l d i s c o v e r a method?  How  A straight  to another s t r a i g h t l i n e can i t be  done?  Can  Vi/hen experimentation i s over  r e s u l t s are compared, the comparative values 1  methods adopted by the p u p i l s may ones s t u d i e d more c a r e f u l l y .  be  o f the  considered,  and  the  and  various the  best  33.a The  p h y s i c a l energy expended by the p u p i l i n working  these c o n s t r u c t i o n s i s a very important p a r t o f the grade IX pupil's activity.  Without p e r i o d i c o p p o r t u n i t i e s f o r p h y s i c a l  a c t i v i t y , the p u p i l I s l i k e l y t o 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 o r any g r e a t  l e n g t h o f time and o p p o r t u n i t i e s f o r muscular movement such as the one j u s t mentioned allow  the youth to s a t i s f y h i s d e s i r e  for physical action. In the t e a c h i n g o f standard the teacher  constructions i t i s well f o r  to remember t h a t these are new r e v e l a t i o n s to the  grade IX student.  The v e r y n o v e l t y o f such work i s an  i n c e n t i v e to the p u p i l t o l e a r n .  In the manner of e x p r e s s i n g  i n words the v a r i o u s methods o f c o n s t r u c t i o n , there i s a c e r t a i n amount o f . n o v e l t y a l s o ; and i n t e r e s t i n t h i s phase o f the work may be i n c r e a s e d by seeing which p u p i l s can express i n the c l e a r e s t andmo>st c o n c i s e form the a c t u a l work c a r r i e d but  i n the process  of construction.  encouraged i n d e v e l o p i n g  The p u p i l s may be  an eagerness to be able t o express  themselves i n c o r r e c t g e o m e t r i c a l that a c e r t a i n p r o f e s s o r wired  language.  They may be t o l d  from Vancouver to New York f o r  the p u b l i s h e r s to h o l d up the p u b l i c a t i o n o f 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 o f the questions  which began "P Is a p o i n t I n the l i n e AB" to be changed t o "The  p o i n t P i s i n the l i n e AB" because i t i s bad form to  b e g i n a l i n e w i t h a symbol. (1) Information  obtained  T h i s shows the exactness o f  from P r o f e s s o r  himself.  34 the s c i e n c e of mathematical e x p r e s s i o n , of s k i l l i n t h i s science b e g i n n i n g the  and  the development  sometimes f a s c i n a t e s the young people  study of geometry.  When the e s s e n t i a l c o n s t r u c t i o n s have been mastered, p u p i l s may  be  allowed to c o n s t r u c t from s p e c i f i c a t i o n s a  of the more d i f f i c u l t f i g u r e s i n v o l v i n g combinations o f basic constructions. should be g i v e n  few the  In t h i s p a r t o f the work the p u p i l  the impression  I n t r i c a t e b i t of handiwork, and  that he i f he  i s now  t a c k l i n g an  i s s u c c e s s f u l he  developed a c e r t a i n amount of g e o m e t r i c a l  has  constructive  ability  Problems s u i t a b l e f o r t h i s purpose are ones such as drawing the i n s c r i b e d , c i r c u m s c r i b e d , a n d e s c r i b e d c i r c l e s to t r i a n g l e s c o n s t r u c t i n g q u a d r i l a t e r a l s n e c e s s i t a t i n g the use and the development o f g e o m e t r i c a l I n l a y work, l i n o l e u m  and  tiling  of  diagonals  designs s u i t a b l e f o r c r e s t s  patterns.  As mathematical r e c r e a t i o n s In connection phase o f the work, the f o l l o w i n g p u z z l e s  with t h i s  are very s u i t a b l e :  1. A farmer had h i s p r i z e sheep i n s i x pens, c o n s t r u c t e d t h i r t e e n s e c t i o n s of f e n c i n g , as f o l l o w s :  of  Somebody s t o l e one s e c t i o n o f the fence, so the farmer r e a r r a n g e d the remaining 12 s e c t i o n s so as to s t i l l have s i x pens, a l l the same s i z e and shape. How d i d he do i t ? 2. Given a p i e c e o f cardboard 15 Inches l o n g and 3 inches wide how i s i t p o s s i b l e to cut i t so t h a t the p i e c e s when r e arranged s h a l l form a p e r f e c t square? I f t e s t s are g i v e n a t v a r i o u s  stages o f the work i n grade  35 IX geometry,  i t i s e s s e n t i a l t h a t the p u p i l s develop a c o r r e c t  a t t i t u d e toward them.  The p u p i l s h o u l d r e g a r d them as  o p p o r t u n i t i e s f o r showing h i s c o n s t r u c t i v e and r e a s o n i n g ability. new  I n each t e s t he has the p r i v i l e g e o f p e r f o r m i n g more  and i n t e r e s t i n g c o n s t r u c t i o n s , as w e l l as the o p p o r t u n i t y  f o r r e a s o n i n g out c e r t a i n mathematical problems.  I f the  t e a c h i n g o f the work has been m o t i v a t e d so as t o arouse the p u p i l ' s i n t e r e s t i n the s u b j e c t , and i f the t e s t s  are.skil-  f u l l y arranged so as t o f o l l o w up t h a t m o t i v a t e d form o f t e a c h i n g , then the p u p i l w i l l i n d e e d r e g a r d these t e s t s i n the proper l i g h t and l o o k forward w i t h eagerness toward them. The f o l l o w i n g a r e some examples  o f t e s t s designed t o  a l l o w the p u p i l an o p p o r t u n i t y o f showing h i s c o n s t r u c t i v e and r e a s o n i n g a b i l i t y , and from the working o f which a p u p i l may  d e r i v e a g r e a t d e a l o f s a t i s f a c t i o n and enjoyment.  1. G i v e d e f i n i t i o n s o f t h e f o l l o w i n g terms. I f you cannot g i v e d e f i n i t i o n s , e x p l a i n each term c l e a r l y . (a) Acute angle (b) Obtuse a n g l e . (c) Adjacent a n g l e s . (d) Supplementary a n g l e s . (e) P a r a l l e l l i n e s . ' " 2. Name as many angles as you can from the f o l l o w i n g diagram. Name o n l y angles c o n t a i n i n g l e s s than 180°. Measure t o the n e a r e s t degree the s i z e o f each angle. \C  36 3. (a) Draw / AOB - 42°; a t 0 make /J30C - 53°; a t 0 make / COD » , 3 7 ; a t 0 make / DOE ~ 48°, making each angle adjacent t o the one immediately p r e c e d i n g , (b) Without p r o t r a c t o r , how c o u l d you t e s t q u i c k l y t o see whether the f i n a l r e s u l t i s accurate or not? 4. (a) Draw a s t r a i g h t h o r i z o n t a l l i n e AB 3" i n l e n g t h . Mark X t h e mid-point o f AB. At X, make / BXY • 50°, drawing t h i s angle on the upper s i d e o f AB. Make XY s 1" i n l e n g t h . At Y, make / XYZ a l t e r n a t e t o / YXB; and equal to 50°. Make YZ = 2^~"in l e n g t h . A t poinTT X, make /BXH 70°, drawing t h i s angle on the lower s i d e o f AB. Make XH 1" i n l e n g t h . At H, make / XHK, a l t e r n a t e t© /BXH, and equal t o 70°. Make HK • 2? I n l e n g t h . (b) wEat r e l a t i o n e x i s t s between the s t r a i g h t 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 e x i s t s 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 e x i s t s between s t . l i n e s ZY and KH? G i v e reason f o r your answer. (e) By u s i n g r u l e r and p r o t r a c t o r o n l y , how c o u l d you t e s t t h i s r e l a t i o n s h i p between ZY and KH? 5. G  s  s  In t h e diagram above, AB and CD are two p a r a l l e l l i n e s . (a) What name I s g i v e n t o s t . l i n e XY? (b) Name a p a i r o f a l t e r n a t e a n g l e s . (e) Name a p a i r o f corresponding angles. (d) Name a p a i r o f v e r t i c a l l y o p p o s i t e a n g l e s . (e) Name a p a i r o f adjacent a n g l e s . ( f ) Name a p a i r of supplementary' a n g l e s . (g) I f / a - 112, what Is s i z e o f / f ? Do n o t o b t a i n your answer by measuring. ' ' 6.  In the above diagram CX i s p a r a l l e l to BA. (a) Name any p a i r s o f angles which you know t o be equal, g i v i n g the reason i n each case.  37 (b) I f / A s 85° and / C D - 48 , without measuring f i n d the s i z e o f / ACB. Show your method o f c a l c u l a t i o n . 7.  10°  HP  "B  I n the accompanying diagram. AD i s p e r p e n d i c u l a r to BC. I f / DAC - 45 and / B - 30°, f i n d without measuring the s i z e of each o f 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) R e p r e s e n t i n g 1 m i l e by h a l f an i n c h , draw a diagram to I l l u s t r a t e the f o l l o w i n g journey, u s i n g d i r e c t i o n s as i n d i c a t e d by t h e N accompanying arrows: A man s t a r t s from a p l a c e A and walks to a p l a c e B which i s 3 m i l e s due east o f A. He then walks from B to a p l a c e 1 >e h i c h i s 4 m i l e s north-west. o f B. Prom G he goes t o D which i s 2 m i l e s n o r t h - e a s t o f C. From D he goes 3^ m i l e s due south to E . From E he walks 5 to p l a c e F, which i s 5 m i l e s south-west of E . From F he goes 2^ m i l e s due east to a p l a c e G. (b) Measure the d i r e c t d i s t a n c e from A to G, and t e l l how many m i l e s G i s d i s t a n t from A. , c  W  An i n h e r e n t i n t e r e s t i n games i s one o f the m o t i v a t i n g f o r c e s which may be u t i l i z e d e x t e n s i v e l y throughout the e n t i r e h i g h school c o u r s e .  The f o l l o w i n g are a few mathematical  r e c r e a t i o n s connected w i t h the work of grade IX geometry i n some way.  The j u d i c i o u s use o f these r e c r e a t i o n a l  may add much i n t e r e s t t o the study o f geometry pupils.  problems  f o r many o f the  38 1.  A man who owned a p i e c e of l a n d i n the form of a square, d i e d and l e f t the p r o p e r t y to h i s w i f e and f o u r sons i n the f o l l o w i n g way: The wife was to r e c e i v e the q u a r t e r i n the corner o f the square where the house stood, and the remaining t h r e e - q u a r t e r s was to be d i v i d e d evenly among the f o u r sons and a l l the f o u r p a r t s r e c e i v e d by the sons must be the same s i z e and shape. How was the p r o p e r t y d i v i d e d ? 2. Why does i t take no more p i c k e t s to b u i l d a fence down a h i l l and up another than i n a s t r a i g h t l i n e from top to top, no matter how deep the g u l l y ? 3. Given a plank 12 inches square, r e q u i r e d to cover a h o l e i n a f l o o r 9 Inches by 16 i n c h e s , c u t t i n g the plank i n t o o n l y two p i e c e s . 4. A farmer has s i x p i e c e s of chain, each p i e c e c o n t a i n i n g five links. I f i t c o s t s 2^ f o r each cut and 3$zf f o r each weld, what w i l l i t c o s t him to have them made i n t o an endless chain? ( i . e . c h a i n i n the form of a c i r c l e . ) 5. Three m i s s i o n a r i e s t r a v e l l i n g i n c a n n i b a l country, came to a r i v e r and c o u l d not get a c r o s s . There were three c a n n i b a l s on the bank of the r i v e r and a boat t i e d at the shore. The m i s s i o n a r i e s were o f f e r e d the use of the boat, but the boat c o u l d c a r r y no more than two at a time, and i t was unsafe to l e a v e more 3 0 c a n n i b a l s than m i s s i o n a r i e s i n any one p l a c e at any one time, f o r the c a n n i b a l s outnumbering the m i s s i o n a r i e s would devour them. A l l the m i s s i o n a r i e s c o u l d row, but only one c a n n i b a l (Marked^) ) had l e a r n e d to row. How d i d a l l the :r: ^ m i s s i o n a r i e s and c a n n i b a l s get across the r i v e r , and what i s the l e a s t number o f times the boat need cross the r i v e r ? 6. A man h a v i n g a fox, a goose, and a peck of corn i s d e s i r o u s o f c r o s s i n g 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 t o g e t h e r . How can he get them s a f e l y across? 7. A room i s 30 f t . l o n g , 12 f t . wide, and 12 f t . h i g h . On the middle l i n e of one of the s m a l l e r s i d e w a l l s and one f o o t from: the c e i l i n g i s a s p i d e r . On (the middle l i n e o f the o p p o s i t e w a l l and e l e v e n f e e t from the c e i l i n g , i s a f l y . The f l y , b e i n g p a r a l y z e d by f e a r , remains s t i l l u n t i l the s p i d e r reaches I t by c r a w l i n g the s h o r t e s t r o u t e . How f a r d i d the s p i d e r crawl?  ©00  39 8.  4L .ft  1^)  A g i r l had a p i e c e o f c l o t h of the shape shown i n the diagram. She wished to c u t i t i n t o three p i e c e s , a l l o f the same s i z e and shape. How c o u l d she do I t ?  3f4. 9. A man and h i s w i f e , each weighing 150 pounds, w i t h two sons each weighing 75 pounds, have t o c r o s s a r i v e r i n a boat which Is capable o f c a r r y i n g o n l y 150 pounds' weight. How d i d they get a c r o s s ? 10. Take two pennies face upward on a t a b l e and edges i n c o n t a c t . Suppose one i s f i x e d and the o t h e r r o l l s on i t without s l i p p i n g , making one complete r e v o l u t i o n about i t and r e t u r n i n g to i t s o r i g i n a l p o s i t i o n . How many r e v o l u t i o n s about i t s own c e n t r e has the moving c o i n made? 1  40 CHAPTER  IV  MOTIVATION IN THE TEACHING OP GRADE X I I GEOMETRY P a r t A.  Development o f Proper A t t i t u d e s  Towards Grade XII Geometry ¥i/hether a student i n grade XII i s i n f l u e n c e d by s u f f i c i e n t m o t i v a t i n g f o r c e s to encourage him t o r e a c h h i s h i g h e s t l e v e l of  e f f i c i e n c y i n the study of geometry depends v e r y l a r g e l y  upon the a t t i t u d e which he adopts towards the s u b j e c t . teacher can l e a d a p u p i l to develop a proper a t t i t u d e  Ifa toward  the s u b j e c t o f geometry, then t h a t p u p i l w i l l have made the first  step toward d e r i v i n g the numerous b e n e f i t s and p l e a s u r e s  l a t e n t i n the study o f t h i s most f a s c i n a t i n g s u b j e c t . But what i s the proper a t t i t u d e toward XII?  geometry i n grade  I n the f i r s t p l a c e c e r t a i n students i n t h e i r f i n a l  of h i g h s c h o o l may be encouraged p o t e n t i a l mathematicians.  year  to r e g a r d themselves as  I f the teacher regards a p u p i l as  an advanced student l a u n c h i n g out i n t o the depths o f h i g h e r mathematics, then t h a t p u p i l w i l l endeavour to respond t o t h i s a t t i t u d e , and w i l l put f o r t h every e f f o r t to show•that he I s worthy of such r e c o g n i t i o n .  A h i g h e r r e g a r d f o r the p u p i l  (1) "What s c i e n c e can there be more noble, more e x c e l l e n t , more u s e f u l to men, more admirably h i g h and demonstrative, than t h i s of mathematics?" - Benjamin F r a n k l i n . S.I.Jones, Mathematical W r i n k l e s , P. 257.  41 undoubtedly  encourages more earnest e f f o r t on h i s p a r t .  Another means of encouraging  a proper a t t i t u d e towards  geometry i n grade X I I , i s the development o f the forwardlooking attitude.  The teacher may f i n d many o p p o r t u n i t i e s f o r  making s l i g h t r e f e r e n c e to some advanced work i n mathematics, and thereby show the p u p i l s t h a t they are r e a l l y l a y i n g the foundations f o r the study of h i g h e r mathematics; f o r example, i n the study of graphs the p u p i l s ' i n t e r e s t may be aroused to a c o n s i d e r a b l e extent i n the study o f a n a l y t i c a l geometry. A l s o , I n c o n n e c t i o n w i t h f a c t o r i n g , e s p e c i a l l y by the grouping method, there Is an o p p o r t u n i t y f o r mentioning and combinations.  permutations  T h i s t o p i c c o u l d be mentioned a l s o i n con-  n e c t i o n w i t h e v o l u t i o n and i n v o l u t i o n .  When a p u p i l i s asked  to w r i t e down the square of an e x p r e s s i o n 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 w r i t e the terms i n the answer I n v o l v i n g twice the product of each p a i r .  Longer e x p r e s s i o n s might be taken, and  the number of groups o f two c a l c u l a t e d without them o f f .  actually pairing  By t h i s means a student's i n t e r e s t and c u r i o s i t y i s  aroused In the t o p i c s which l i e j u s t ahead of him, and he i s motivated to put f o r t h h i s b e s t e f f o r t s and continue h i s mathematical  studies.  Besides d e v e l o p i n g the f o r w a r d - l o o k i n g a t t i t u d e i n grade XII, i t i s Important  t h a t the student become more  and l e s s r e l i a n t upon the teacher. develop  independent,  I f the p u p i l can be l e d t o  the i d e a t h a t the teacher i s there t o guide and not  42 simply to I n s t r u c t , then t h a t p u p i l w i l l develop a sense o f r e s p o n s i b i l i t y and w i l l f e e l t h a t he h i m s e l f I s the one to do the  t h i n k i n g and the r e a s o n i n g ; 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 I n geometry than I f t h i s a t t i t u d e o f independence were not developed. Probably the most Important a t t i t u d e towards grade XII geometry Is the development through mastery. of  o f the sense o f s a t i s f a c t i o n  There i s no g r e a t e r enjoyment i n any phase  school work than the t h r i l l d e r i v e d from o b t a i n i n g a  s o l u t i o n to a d i f f i c u l t g e o m e t r i c a l problem which has r e q u i r e d a great d e a l o f c o n c e n t r a t i o n .  I f the t e a c h e r has a s e l e c t  group of problems a t h i s d i s p o s a l which are of j u s t the r i g h t difficulty  f o r the p u p i l s a t each stage o f t h e i r  development,  he can use these to wonderful advantage by g i v i n g the p u p i l s an o p p o r t u n i t y f o r e x p e r i e n c i n g that sense o f s a t i s f a c t i o n through mastery which i s such a s t r o n g m o t i v a t i n g f o r c e to further e f f o r t .  "Success begets s u c c e s s , " and every exper-  i e n c e of t h i s nature a c t s as a " s t e p p i n g - s t o n e to h i g h e r things." In the  c o n n e c t i o n w i t h the s o l u t i o n o f g e o m e t r i c a l e x e r c i s e s ,  students i n grade X I I may be encouraged to develop the  power o f v i s u a l i z i n g g e o m e t r i c a l s o l u t i o n s .  I f a p u p i l has  been working a t a problem f o r some time he may develop a very a c c u r a t e mental p i c t u r e of the diagram w i t h 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 h i s mental  43 p i c t u r e o f the diagram he might d i s c o v e r the s o l u t i o n even while walking along the s t r e e t , s i t t i n g i n a s t r e e t c a r , or w a i t i n g f o r a f r i e n d to keep an appointment.  The extreme  enjoyment d e r i v e d from being able to s o l v e e x e r c i s e s i n such a manner i s a very strong i n c e n t i v e towards f u r t h e r development of t h a t  skill.  Still  another  phase o f the development o f the proper  a t t i t u d e towards grade X I I geometry i s the a r o u s a l of i n t e r e s t i n complicated g e o m e t r i c a l diagrams.  I f the teacher has a  s e l e c t i o n o f e x e r c i s e s which are n o t very d i f f i c u l t  i n them-  s e l v e s but which produce a r a t h e r complicated l o o k i n g diagram, the p u p i l s may be l e d to develop  a keen i n t e r e s t i n such  f i g u r e s and d e r i v e much p l e a s u r e both from t h e i r c o n s t r u c t i o n and from t h e i r a n a l y s i s .  There i s a great f a s c i n a t i o n about  complicated g e o m e t r i c a l diagrams, and when a student  finds  that he can not only draw the f i g u r e from s p e c i f i c a t i o n s but a l s o analyse i t and prove a c e r t a i n r e q u i r e d f a c t about i t , then h i s f a i t h i n h i s own a b i l i t y i s g r e a t l y strengthened, and t h i s i n t u r n I s an i n c e n t i v e towards c o n t i n u e d a c t i v i t y these  along  lines. The  f o l l o w i n g e x e r c i s e s are examples of the f o r e g o i n g  type; they produce r a t h e r complicated l o o k i n g diagrams but t h e i r p r o o f s are not p a r t i c u l a r l y  difficult:  1. I f a t r i a n g l e i s i n s c r i b e d In a c i r c l e and p e r p e n d i c u l a r s are drawn from any p o i n t on the cir cumference o f the c i r c l e to the s i d e s of the t r i a n g l e , the f a e t of .the three perp e n d i c u l a r s are i n one s t r a i g h t l i n e . (Simpson's l i n e . ) 2. To c o n s t r u c t a t r i a n g l e h a v i n g a base equal to a g i v e n s t r a i g h t l i n e , a v e r t i c a l angle equal to a g i v e n angle, and  44 an area equal t o the area of a g i v e n The  parallelogram.  f o l l o w i n g e x e r c i s e s handed i n by grade XII p u p i l s  i n d i c a t e that s:ome p u p i l s are indeed i n t e r e s t e d In drawing complicated l o o k i n g f i g u r e s , and that t h i s i s one form of motivation  I n grade XII geometry.  ( F i g u r e s I and I I o v e r l e a f . )  Another a t t i t u d e towards grade X I I geometry i s the development on the p a r t o f the p u p i l s o f a d e s i r e f o r perfection.  absolute  There i s nothing q u i t e as s t i m u l a t i n g to a  student's enthusiasm as to be t o l d that h i s s o l u t i o n o f a problem i s p e r f e c t .  I n the s o l u t i o n o f g e o m e t r i c a l  it  f o r a p u p i l to r e a c h p e r f e c t i o n - p e r f e c t -  Is c u i t e p o s s i b l e  exercises  l y l o g i c a l r e a s o n i n g expressed In a c c u r a t e g e o m e t r i c a l  terms.  When a p u p i l knows t h a t such an attainment Is q u i t e w i t h i n h i s reach, he i s encouraged to put f o r t h s p e c i a l e f f o r t i n order to d e r i v e the s a t i s f a c t i o n of producing a p e r f e c t  solution.  That students a r e i n t e r e s t e d i n making t h e i r  solutions  p e r f e c t i s evident given  from the accompanying s o l u t i o n s to e x e r c i s e s  to a grade XII c l a s s simply as o p t i o n a l home e x e r c i s e s .  ( F i g u r e s I I I and IV o v e r l e a f . )  6jf  ^  I  feojjix.  r3eD ^ ^  ^v...Urj  ISA  If" BC,  c«.xo  6D  ,0*Wi«*.  a*. E  .  X a*  a_ ( W r f a A / erf t*vju CD  n e> e.  ***** < c o e> =.  ^eR  .'.  V. E S J-o  <  D F E .  D .  Tu 1  <SE6 - < a o % .  <M^-t  t**  <X  --  J^t  <.  <•  _a 3 C Sfnce  AC.  tS A . (>(.t r f 7f>J .e.f r  < 7  (c~fa~tni)  L A.H C .o^ncl L h7> <L = / >-£ . L  .  .  ,  ^.T  /  /  ^  .4J7  MM-  y 4 a)'  :  A C  ,5// CP Z.4 7ZT 7  —fcUda) *, <S'LAIC*(At- fi-o<;e<{. M  44D  45 CHAPTER IV P a r t B.  M o t i v a t i o n i n the Methods o f  P r e s e n t a t i o n of Grade XII Geometry In P a r t A of t h i s chapter I t was  p o i n t e d out that b e f o r e  a grade XII student can d e r i v e the maximum amount of  satis-  f a c t i o n and enjoyment from the study o f geometry he must develop the proper a t t i t u d e toward the s u b j e c t , and t h a t t h i s d e s i r a b l e a t t i t u d e i s composed of s e v e r a l d i f f e r e n t The present s e c t i o n w i l l o u t l i n e how s t u d i e d i n grade XII geometry may flevelop  factors.  the m a t e r i a l to be  be p r e s e n t e d so as to  i n the p u p i l s a d e s i r a b l e a t t i t u d e toward the s u b j e c t ,  and at the same time enable them to r e a c h a v e r y h i g h l e v e l of e f f i c i e n c y . In grade XII the method of t e a c h i n g the p r e s c r i b e d theorems can be done i n such a way  as to motivate  of the work to a c o n s i d e r a b l e e x t e n t . theorems may  this  section  In t h i s grade the  be t r e a t e d i n a much l e s s f o r m a l manner than i n  the p r e v i o u s grades.  When a new  commenced, the diagram may  theorem i s about to be  b e . p l a c e d on the b l a c k b o a r d ,  and  then thoroughly d i s c u s s e d i n order to make sure what f a c t s are g i v e n about i t and what new  t r u t h i s to be d e r i v e d .  then analyse the problem j u s t as i n the case of an  The  pupils  original  e x e r c i s e , and w i t h a few g u i d i n g suggestions from the t e a c h e r , I f neoessary,  the d e s i r e d r e s u l t i s e v e n t u a l l y reached.  c l a s s as a whole b u i l d s up a s u i t a b l e statement  for this  The new  46 t r u t h , and then i t heoomes to them a theorem or e s t a b l i s h e d f a c t which may truths.  be used i n the d i s c o v e r y of other  In the l e s s o n which f o l l o w s the one  theorem i s developed,  geometrical  i n which a  the teacher t e s t s the c l a s s on the f a c t  l e a r n e d i n the p r e v i o u s l e s s o n , and the p u p i l s are r e q u i r e d to show, f i r s t of  of a l l , t h a t they understand  r e a s o n i n g steps necessary  the l o g i c a l  to prove the f a c t ; and  sequence secondly,  that they are able to express t h i s l o g i c i n c o n v e n i e n t l y arranged g e o m e t r i c a l terms. i s adopted, i t prevents  I f t h i s a t t i t u d e towards theorems  students from r e g a r d i n g them as  ated g e o m e t r i c a l f a c t s which have to be remembered, and p r o o f s f o r which have to be expressed The  students  isolthe  i n s t e r e o t y p e d form.  should c o n s i d e r each theorem as a newly d i s c o v e r e d  t r u t h , which, when e s t a b l i s h e d , becomes an important stone i n the pyramidal  foundation  s t r u c t u r e of geometry.  In grade XII the study o f the r e q u i r e d theorems i s but a small f r a c t i o n o f the work.  The  l a r g e r p o r t i o n o f the  time  i n t h i s grade i s spent i n the s o l u t i o n of o r i g i n a l e x e r c i s e s of v a r i o u s k i n d s , i n order to develop skill  In the p u p i l a c e r t a i n  In g e o m e t r i c a l r e a s o n i n g both of the i n d u c t i v e and  deductive type.  I t i s i n the treatment  of these  original  e x e r c i s e s t h a t there Is the g r e a t e s t need f o r u t i l i z i n g suitable"means the work may  of m o t i v a t i o n .  every  A l l i n t e r e s t i n t h i s phase of  q u i t e e a s i l y be k i l l e d i f the p u p i l s are  simply  r e q u i r e d to attempt the e x e r c i s e s i n order as they occur i n the t e x t book, and then see these e x e r c i s e s gone over on  the  47 blackboard solutions.  a f t e r a c e r t a i n number o f p u p i l s have  obtained  This mechanical way of t r e a t i n g e x e r c i s e s robs the  s u b j e c t o f geometry o f much o f the i n t e r e s t o f which i t i s so full,  and causes p u p i l s to miss a m u l t i t u d e  experiences  of pleasurable  which might have been t h e i r s had a m o t i v a t e d  method o f treatment been adopted. How', then, can the treatment o f o r i g i n a l e x e r c i s e s i n geometry be motivated?  I n the f i r s t  place,  at the end of  almost every theorem there a r e a number o f short and comparat i v e l y easy e x e r c i s e s .  These can very c o n v e n i e n t l y be  made the s u b j e c t o f r a p i d s o l u t i o n c o n t e s t s , where each e x e r c i s e I s taken s e p a r a t e l y and p u p i l s compete to see who can o b t a i n the s o l u t i o n f i r s t . he  When a p u p i l sees a s o l u t i o n  turns h i s book f a c e downward on the desk and w r i t e s h i s  name on the b l a c k b o a r d . the board the wlnnerr-Is his  explanation  his  opportunity.  A f t e r a l a r g e number o f names are on asked t o e x p l a i n h i s s o l u t i o n . I f  i s not c o r r e c t , the p u p i l next i n order has Each p u p i l keeps t r a c k of h i s c o r r e c t  answers, and a f t e r s e v e r a l e x e r c i s e s have been given i n t h i s manner, r e s u l t s are compared to see which p u p i l s o b t a i n e d the l a r g e s t number o f c o r r e c t s o l u t i o n s . When d e a l i n g w i t h e x e r c i s e s of a l i t t l e  (1)  See  greater  (214, ex. 1119-1126) pages (236, ex. 1222-1231) Godfrey and Siddons (325, ex. 1699-1704) Elementary Geometry.  48 d i f f i c u l t y than those mentioned  above  d i f f e r e n t procedure i s d e s i r a b l e .  a slightly  I n t h i s case, p u p i l s are  allowed a few minutes i n v/hich to r e a d the e x e r c i s e and make a mental note as t o what f a c t s are g i v e n and what i s r e q u i r e d to be proved.  At the end of the a l l o t t e d time, a l l books are  turned f a c e down, and one p u p i l i s r e q u i r e d to draw a s u i t a b l e diagram on the b l a c k b o a r d without the a i d o f the text.  I f the diagram i s not c o r r e c t , another p u p i l i s  s e l e c t e d and s o on u n t i l a s u i t a b l e f i g u r e i s drawn.  Pupils  then examine the diagram on the b o a r d and r a c e f o r a s o l u t i o n . Upon seeing a s o l u t i o n , a p u p i l r a i s e s h i s hand.  When a  number o f hands have been r a i s e d , the teacher asks  certain  p u p i l s , who a p p a r e n t l y have o b t a i n e d a s o l u t i o n , t o g i v e some h i n t which w i l l h e l p those not y e t s u c c e s s f u l .  In t h i s way  the b r i g h t e r p u p i l s f e e l that they a r e h e l p i n g the slower ones to some extent, and t h e i r i n t e r e s t i n the e x e r c i s e i s maintained even a f t e r a s o l u t i o n has been o b t a i n e d .  The  f o l l o w i n g are some examples of e x e r c i s e s which are s u i t a b l e f o r t h i s type o f treatment: 1. AD i s j _ to the base BC o f AABC; AE i s a diameter o f the circumscribing c i r c l e . Prove that /j\ ABD, AEC are equiangular. ( I n t h i s case h e l p might be g i v e n by a p u p i l who sees the s o l u t i o n naming the theorems upon which the solution rests.) 2. I f two c i r c l e s touch e x t e r n a l l y at A, and touched a t P, Q by a l i n e PQ, then PQ, subtends a r i g h t angle a t A. (The s  (1)  (261, ex. 1351-1353) See pages (277, ex. 1425-1438) Godfrey and Siddons (316, ex. 1678-1681) Elemtary Geometry.  49 s u g g e s t i o n i n t h i s case might be f o r the teacher to ask a p u p i l who sees the s o l u t i o n what i s u s u a l l y the l i n k between two c i r c l e s which touch e x t e r n a l l y . ) When d e a l i n g w i t h e x e r c i s e s  of greater  difficulty  e i t h e r o f the types mentioned above, i t i s s t i l l  than  desirable  to a v o i d mechanical treatment of one e x e r c i s e a f t e r another taken c o n s e c u t i v e l y  from the t e x t book.  The thorough a n a l y s i s  o f one good e x e r c i s e , from the s o l u t i o n of which the b r i g h t e r p u p i l s can d e r i v e a great  d e a l o f s a t i s f a c t i o n , and from the  a n a l y s i s o f which the poorer p u p i l s may g a i n much in. l e a r n i n g how such e x e r c i s e s  are attacked,  i s much b e t t e r  than an incomplete survey of s e v e r a l e x e r c i s e s which a thorough a n a l y s i s i s g i v e n . out  f o r none o f  One means of c a r r y i n g  t h i s p l a n i s f o r the teacher to a s s i g n one s u i t a b l e  exercise it  assistance  each day f o r home s o l u t i o n .  I f a s o l u t i o n i s obtained,  i s w r i t t e n out and handed i n a t the b e g i n n i n g o f the next  geometry l e s s o n .  The work handed i n i s examined by the  teacher and marked (probably  out of 10), and the marks are  r e a d p u t when the papers are r e t u r n e d  during  the f o l l o w i n g  l e s s o n , a t which time a thorough a n a l y s i s o f the problem i s c a r r i e d out.  The marked e x e r c i s e s  are r e t a i n e d by the p u p i l s ,  a f t e r the marks have been r e c o r d e d by the teacher.  At the end  of each month the marks f o r each p u p i l are t o t a l l e d and r e a d to the c l a s s . T h i s method o f procedure i s f u l l o f m o t i v a t i n g and  forces,  i t encourages each p u p i l to do h i s utmost to develop h i s  skill  at s o l v i n g geometrical  problems.  The teacher should by  50 no means make the p u p i l f e e l t h a t t h i s i s compulsory homweork which must be done i n order t o a v o i d some p e n a l t y , but  each  e x e r c i s e should be presented as a new  o p p o r t u n i t y f o r the  p u p i l to t r y h i s s k i l l  reasoning.  at g e o m e t r i c a l  That p u p i l s w i l l respond  f a v o r a b l y to such a method i s  evidenced by the r e s u l t s of an experiment along these conducted  lines  by the w r i t e r w i t h a grade XII c l a s s i n the school  year 1933-4.  The  accompanying form i n d i c a t e s t h a t  a b s o l u t e l y no pressure was  e x e r t e d upon the p u p i l s i n order to  get them to hand i n s o l u t i o n s , the matter b e i n g optional,, ©ith each one,  although  entirely  they were eager to o b t a i n s o l u t i o n s  and hand them i n to be marked:  (Table  I.)  Another m o t i v a t i n g f o r c e can be added to the s o l u t i o n of e x e r c i s e s a c c o r d i n g to the p l a n o u t l i n e d above by  keeping  b e f o r e the c l a s s models of p e r f e c t i o n i n r e a s o n i n g and T h i s may  style.  be done by p i c k i n g out examples which have been  e x c e p t i o n a l l y w e l l done and p a s s i n g them around the c l a s s that each p u p i l may solution.  have an o p p o r t u n i t y o f examining a model  A p u p i l d e r i v e s immense s a t i s f a c t i o n from h a v i n g  h i s work passed i n c e n t i v e to him  around the c l a s s as a model, and i t i s a great to keep up h i s good work, and to the o t h e r s  to t r y to emulate him. obvious  so  That t h i s i s a c t u a l l y the case  i n the experiment r e f e r r e d to immediately  was  above,  and  the accompanying s o l u t i o n s are examples of work which undoubtedly i s the r e s u l t of such m o t i v a t i o n .  Both of the  f o l l o w i n g examples are home e x e r c i s e s handed i n by p u p i l s a f t e r  JLfLLLY  50A "  /VJ«KK  51 p r e v i o u s e x e r c i s e s of t h e i r s had been h e l d up to the c l a s s as examples of work w e l l done.  ( F i g u r e s V and VI, o v e r l e a f )  In the above method of t e a c h i n g the s o l u t i o n of g e o m e t r i c a l e x e r c i s e s , the teacher  should encourage each  p u p i l to f i l e h i s e x e r c i s e s as they are r e t u r n e d to him. the end o f the year, then, that he has  accomplished  the p u p i l i s r e a l l y amazed to t h i n k  such an amount of work.  source of great s a t i s f a c t i o n to him to  It is a  to t h i n k t h a t he was  solve such a great number of a p p a r e n t l y d i f f i c u l t  and i t g i v e s him  At  confidence  able  exercises,  to go forward and accomplish  even  greater things. The for  following i s a l i s t  treatment  of e x e r c i s e s which are s u i t a b l e  by the method i n d i c a t e d above.  The order of  the e x e r c i s e s f o l l o w s the/order o f the t o p i c s s t u d i e d i n grade X I I , so that when any p a r t i c u l a r q u e s t i o n i s reached c i e n t m a t e r i a l has been covered  to enable  the students  suffito  obtain a solution: 1. A and B are two p o i n t s on o p p o s i t e s i d e s o f a s t . l i n e CD; show how to f i n d a p o i n t P i n CD such t h a t / APC = / BPC. 2. A and B are two p o i n t s on the same side o f a s t . l i n e CD; show how to f i n d the p o i n t P i n CD f o r which AP+PB i s least. 3. Show how to draw a s t r a i g h t l i n e equal and p a r a l l e l to a g i v e n s t . l i n e and having i t s ends oh two g i v e n s t . l i n e s . 4. Transform a g i v e n t r i a n g l e i n t o an e q u i v a l e n t i s o s c e l e s t r i a n g l e with base equal to a g i v e n l i n e . 5. Draw a l i n e through a g i v e n p o i n t In a s i d e of a t r i a n g l e to b i s e c t t h e t r i a n g l e . 6. Construct a t r i a n g l e equal to the sum of two g i v e n triangles. 7. C o n s t r u c t a t r i a n g l e equal to the d i f f e r e n c e of two g i v e n triangles. 8. Construct a square equal to the sum of three g i v e n squares. 9. C o n s t r u c t a square equal to the d i f f e r e n c e of two g i v e n squares»  ck,"  *  3**3 ***Jf'  dA'aqcT^tx^  f^^a!j5Ste,0  J^&AAA**,  <x*r & .  ,F • R  I  Tki. CD R & £>.  - 1 i ! L  U  51A  . xylja  W Jno tee.. "  .', < 3 _ . e _ y = <  • { v W t  E_.  E.  D F  = —  :  O J A J ^ , ov/wc  < R.  -TJO' /WV_  ul. I0  VKJU V=- C> E_  <  D F F _  • '. <, D  £^/3*> Ci F=  <t  R  <  B  R  I2> <^  R  12,  • F =. < R.  <2L-  B cm  .oJL 'JC  D E. • B  -  F^ F . (L R .  -3.  Vage. 38i  a.  ATT  Air  2> •ZWa.--  /V  is a. f^^lklojra,^  R£CZ>  parallel / RJ)  Jmun  . From any paint E  +o rneei 3>C  0  H  F  *nd  }  f« We F  c/i<zjonal RC  FR  }  F>C  is <Ftawn p & Y a . l U Fo  is  meef 1FC a.-l F. T f c ^ / W -h~Houe\- A RID  is s ihn, 1st  Conjiruc-hi CI^ :• J~oin f> B  A R&C  Vrcc-f.'- XnFh< .'. W£  RE  _  Rc  HF-  )  5/«ce  t>F RB  ii p^h*-  Z f =  f o  F  ZF  ------  II -to  i s  . _-. zr &  - ' ~  p ro t/e  /?D  FC  HC  FC  HFi*  H  *  ?J>'-  ~  » IF^C__—  mm  * LFHF  JMhL  ZFSZ a..  ' - -  s-Ov)  - - - - - -  ZTsG't)  ~  .UUXL-mJL-  ~  &J£C  .\ UFM. - L&LE LMF~  - LE££. t  • '. IMH ML  Si ^ hc  ^  .'.  * A'S  l&jj)  IMFJZ,  -  '  t  ^  6  , .  }  3.X/0M,.  LEEC-  L££J± "SLUEHF-  RED  L&£>B_ = /Bflt)  " -  LME-  LEFLFL  — ^  -  lS'fl$1> is ..\  ^oi)  LUECL.  IME  In  22  _  £F  FC  ?u-t  darts.. Jl  ..  B  .'.  F/F is  is p^a-IU / Fc lie/  FF  .\  AEFH  FC  FF  3ince  }  Fd  - -  LEFJL L  ^  J  "  ZFs  COK  £  efuia.Kjv.fa-*'  ff-'BD'ts froporFivnz-l  +°  -I* A  ^  F~ f-f F  EhFF  r. P.  zrs:  3  iF  52 10. C o n s t r u c t a square t h r e e times as l a r g e as a g i v e n square. 11. 0 i s a p o i n t i n s i d e a r e c t a n g l e ABCD. Prove t h a t 0A +0C e 0B -f-0D . 12. BE, CP are a l t i t u d e s o f an acute-angled t r i a n g l e ABC. Prove that AE.AC = AF.AB. 13. The sum o f t h e squares on the s i d e s of a p a r a l l e l o g r a m i s equal to the sum of the swuares on the d i a g o n a l s . 14. I n a t r i a n g l e , three times the sum o f the squares on the s i d e s i s equal to f o u r times the sum o f the squares on the medians. 15. The s h o r t e s t chord t h a t can be drawn through a ' p o i n t i n s i d e a c i r c l e i s t h a t which i s p e r p e n d i c u l a r to the diameter through the p o i n t . 16. Show how to draw a chord o f a c i r c l e , equal to a g i v e n chord and p a r a l l e l to a g i v e n s t r a i g h t l i n e . 17. Show how to draw two tangents to a c i r c l e making a g i v e n angle with each o t h e r . 18. The b i s e c t o r s o f the three angles of a t r i a n g l e meet a t a point. 19. Show how to draw three equal c i r c l e s each touching the other two, and how to c i r c u m s c r i b e a f o u r t h c i r c l e around the other t h r e e . 20. F i n d the d i s t a n c e between the c e n t r e s o f two c i r c l e s , t h e i r r a d i i b e i n g 5 c m . and 7 c m . and t h e i r common chord 8 c m . (Two cases.) 21. ABCD i s a q u a d r i l a 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 a t E; AB, DC to meet a t F. Prove t h a t i f a c i r c l e can be drawn through the p o i n t s A, E, F, C, then EF i s the diameter o f t h i s e i r c l e ; and BD i s t h e diameter o f the c i r c l e ABCD. 22. The s t r a i g h t l i n e b i s e c t i n g the angles o f any convex q u a d 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 p o i n t 2 i n . o u t s i d e a c i r c l e of r a d i u s 2 i n . draw a l i n e to pass 1 i n . from the c e n t r e of the c i r c l e . C a l c u l a t e the p a r t i n s i d e the c i r c l e . , 24. C i r c u m s c r i b e about a c i r c l e o f radius 5 c m . a t r i a n g l e h a v i n g i t s s i d e s p a r a l l e l t o three g i v e n l i n e s . 25. Show how t o c o n s t r u c t a t r i a n g l e on a g i v e n base w i t h a g i v e n v e r t i c a l angle and a g i v e n 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 t r i a n g l e i n s e e l b e d i n a c i r c l e ; prove t h a t PA - PB + PC. 27. AOB, COD are two chords o f a c i r c l e , i n t e r s e c t i n g a t r i g h t a n g l e s . Prove that arc AC f a r c BD a r c CD -+ a r c DA. 28. A, B, C are t h r e e p o i n t s on a c i r c l e . The b i s e c t o r o f / A B C 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 f i x e d p o i n t s , one upon each o f two c i r c l e s which i n t e r s e c t a t B, B . Through B i s drawn a v a r i a b l e chord PBQ, c u t t i n g the two c i r c l e s i n P, Q. PA, QC (produced i f necessary) meet a t R. Prove t h a t the l o c u s 2  2  2  s  2  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 t r i a n g l e - A B C V-O and they i n t e r s e c t at H. BE produced meets the c i r c u m c i r c l e at K. Prove t h a t E i s the mid-point of HK. Another means of m o t i v a t i n g the t e a c h i n g of geometry i n grade XII i s hy s e l e c t i n g e x e r c i s e s of s p e c i a l One  significance.  type of e x e r c i s e which i s of s p e c i a l s i g n i f i c a n c e to the  p u p i l s i s an e x e r c i s e which has appeared on a m a t r i c u l a t i o n examination paper i n a r e c e n t y e a r .  M a t r i c u l a t i o n p u p i l s are  u s u a l l y keen to see i f they are able to o b t a i n a s o l u t i o n t o an e x e r c i s e which was examination paper.  worthy o f a p l a c e on a m a t r i c u l a t i o n  I f a teacher compiles a l i s t  of such  e x e r c i s e s , he can g i v e them t o the p u p i l s one a t a time d u r i n g the year as s u f f i c i e n t work i s covered to enable the p u p i l s to obtain a solution.  I f the p u p i l s are informed as to the year  i n which the q u e s t i o n appeared, a s s i g n e d to i t , the q u e s t i o n .  then a s t i l l  and the number of marks  greater i n t e r e s t i s created i n  The f o l l o w i n g i s a l i s t  of q u e s t i o n s , compiled  from m a t r i c u l a t i o n papers, which would be s u i t a b l e f o r use as indicated herein: .1. In the i s o s c e l e s t r i a n g l e ABC, h a v i n g AB e AC, X i a any p o i n t 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 -CY BC.XY-. 2. Two c i r c l e s touch i n t e r n a l l y a t A. A chord BC of the l a r ger c i r c l e touches the s m a l l e r c i r c l e at D. Prove t h a t AB:AC s BD:DC. 3. The base and v e r t i c a l angle o f a t r i a n g l e are g i v e n . F i n d the l o c u s o f 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 p e r p e n d i c u l a r to the o p p o s i t e s i d e s of the triangle. 4. With a c i r c l e of r a d i u s r , draw two c i r c l e s of r a d i i r g and 2  2  r j which touch each other e x t e r n a l l y and both of which 'touch the c i r c l e of r a d i u s 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 i n s c r i b e d i n a c i r c l e w i t h c e n t r e 0. The diameter p e r p e n d i c u l a r to EP cuts DE a t P and FD produced a t Q. Prove that CE i s the mean p r o p o r t i o n a l between OP and OQ. 6. AB and CD are two diameters o f the c i r c l e ADBC a t r i g h t angles to each other. EP i s a chord such t h a t the s t r a i g h t l i n e s CE and CP, when produced, c u t AB produced i n G and H respectively. Prove t h a t the r e c t a n g l e c o n t a i n e d by CE and Gil i s equal to the r e c t a n g l e c o n t a i n e d by EF and CH. 7. ABC i s an e q u i l a t e r a l t r i a n g l e and D i s any p o i n t i n the base, BC. On the base produced (both ways) p o i n t s E and F are taken such t h a t the angles EAD and DAF are b i s e c t e d i n t e r n a l l y by AB and AC r e s p e c t i v e l y . Show that t h e » . ' t r i a n g l e s ABE and ACF are s i m i l a r and t h a t BE.CF - 3C . 8. Two c i r c l e s touch one another e x t e r n a l l y a t A; BA and AC are diameters o f the c i r c l e s . BD i s a chord o f 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 o f t h e second c i r c l e which, when produced, touches the f i r s t a t Y. Prove t h a t BD.CE a 4DX.EY. 9. I f two tangents a t the ends o f one d i a g o n a l o f a c y c l i c q u a d r i l a t e r a l i n t e r s e c t on the other d i a g o n a l produced, the r e c t a n g l e c o n t a i n e d by one p a i r o f o p p o s i t e s i d e s o f the q u a d r i l a t e r a l I s equal to t h a t c o n t a i n e d by the other f o u r . 10. D, E, F a r e the mid-points o f t h e s i d e s BC, CA, AB o f a t r i a n g l e ABC. AL i s an a l t i t u d e . Prove D, E, F, L a r e concyclic. 11. AB i s a f i x e d chord o f a c i r c l e ; CD i s .a diameter perpend i c u l a r to AB. P i s a v a r i a b l e p o i n t on the c i r c l e ; AP, BP.cut CD (produced i f necessary) i n X, Y r e s p e c t i v e l y . I f 0 i s the centre o f the c i r c l e , prove OX.0IY i s constant. 12. ABCD i s a q u a d r i l a t e r a l i n s c r i b e d i n a c i r c l e ; i t s diagonals AC and BD i n t e r s e c t a t E. The l i n e j o i n i n g E to the c i r cumcentre of t h e t r i a n g l e AEB cuts CD i n F. I f CR i s a ciameter o f the c i r c u m s c r i b i n g c i r c l e , o f the q u a d r i l a t e r a l ABCD, prove that the r e c t a n g l e c o n t a i n e d by DE and BC i s equal t o the r e c t a n g l e c o n t a i n e d by EF and CR. 13. A and B are f i x e d p o i n t s on a c i r c l e , and PQ i s any chord of constant l e n g t h . F i n d the l o c u s o f t h e p o i n t o f i n t e r s e c t i o n o f AP and BQ. Give a p r o o f . 14. C i r c l e s a r e described, on t h e s i d e s o f a r i g h t - a n g l e d t r i a n g l e , as diameters. Through the r i g h t angle at A, between the arms AB and AC, a s t r a i g h t l i n e APQR i s drawn c u t t i n g the three c i r c l e s i n P, Q, R r e s p e c t i v e l y . Prove t h a t AP I s equal to QR. -.. •15. Show how to b i s e c t the t r i a n g l e whose s i d e s a r e 2, 2% and 3 inches long by a s t r a i g h t l i n e p a r a l l e l to the l o n g e s t . side. 16. The s i d e s 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 produced and i n t e r s e c t a t an angle o f 3 0 ° , and the- s i d e s BA, CB when produced i n t e r s e c t 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 i n t e r s e c t at P and Q. Through P a s t r a i g h t  55 l i n e DPE i s drawn terminated by the circumferences at D and E. The b i s e c t o r of the angle DQE meets DE at P. Prove: (a) The angle DQE i s constant. (b) The l o c u s of F i s a circle. 18.Let ABC be a t r i a n g l e . Draw ,AD, BE p e r p e n d i c u l a r to BC,CA r e s p e c t i v e l y , meeting at P. J o i n CP and produce i t to cut AB at F. ,Prove:(a) Angle DPC = angle DEC = angle FBD. (b) CF i s p e r p e n d i c u l a r to AB.  The m o t i v a t i n g power o f t e s t s i s another  f a c t o r which  must be taken i n t o c o n s i d e r a t i o n when d i s c u s s i n g m o t i v a t i o n i n the t e a c h i n g of geometry. e x e r c i s e s has been m o t i v a t e d  I f the t e a c h i n g of theorems  and  s u f f i c i e n t l y to b r i n g each p u p i l  up to h i s maximum l e v e l of e f f i c i e n c y , the p u p i l s w i l l  regard  the working of geometry t e s t s simply as o p p o r t u n i t i e s f o r showing t h e i r s k i l l I n g e o m e t r i c a l r e a s o n i n g . important  t h a t the t e s t be such as to encourage the p u p i l s  i n t h e i r good work, and In no way discouragement. difficult  However, i t i s  g i v e them a f e e l i n g of  T e s t s c o u l d be s e t which would be  f o r even the b e s t In the c l a s s ,  too  and such t e s t s as  these would undermine the confidence of the p u p i l s and have a d e t r i m e n t a l e f f e c t upon t h e i r study of geometry.  The  t e s t i s one which i s d i f f i c u l t enough to g i v e the b e s t  ideal pupil  an o p p o r t u n i t y to show h i s mathematical a b i l i t y , and a t the same time easy enough to a l l o w a l l p u p i l s i n the c l a s s to experience a c e r t a i n amount o f s a t i s f a c t i o n through A c a r e f u l l y graded  mastery.  t e s t of a somewhat o b j e c t i v e nature  to s a t i s f y these c o n d i t i o n s most s u i t a b l y .  The  appears  f o l l o w i n g are  examples of t e s t s which might be g i v e n towards the end of the  56 school year, or p a r t s of which might be g i v e n at the c o n c l u s i o n of  c e r t a i n s e c t i o n s of the work: Test I - (May be g i v e n at the end of the work in circles.) Completion  Test  ^  1. The g r e a t e s t chord i n a c i r c l e i s the . 2. The l a r g e s t c e n t r a l angle has degrees. 3. The l i n e t h a t touches a c i r c l e at o n l y one p o i n t , however f a r produced, i s a . 4. I f AB and CD are two chords o f a c i r c l e each 10" from the c e n t r e , they, are • . 5. A q u a d r i l a t e r a l ABCD i s i n s c r i b e d 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 p o i n t w i t h i n a c i r c l e , the s h o r t e s t chord through A i s ___ ^ diameter through A. 7. AB and CD are two diameters of a c i r c l e , then ABCD i s a t o  fc  e  r  •  8. I f the c e n t r a l angle AOB i s 60°, and the r a d i u s OA i s 20", the chord AB Is inches. 9. An i n s c r i b e d angle and a c e n t r a l angle i n t e r c e p t the same arc. The c e n t r a l angle Is the i n s c r i b e d angle. 10. A chord AB meets a tangent AT at an angle of 5 0 ° . The angle i n the minor segment cut o f f by AB i s • degrees. Test I I - (At the end of  circles)  True - F a l s e T e s t  ( ) 2  1. An angle i s i n s c r i b e d i n an arc of 6 0 ° .  The angle c o n t a i n s  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 a r c s of the same number of degrees are e q u a l . 3. An angle i n s c r i b e d i n an arc of 200° I s a c u t e . 4. An angle i n s c r i b e d i n an arc of 100° i s obtuse. _______ 5. I f one of two a r c s I n t e r c e p t e d 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 c e n t r a l angle has the same number of angular degrees as i t s a r c has of are degrees. 7. If. two c i r c l e s touch e x t e r n a l l y , the l i n e o f c e n t r e s Is equal to the sum of the r a d i i . 8. I f -twoccircles are c o n c e n t r i c , a l l tangents to the s m a l l e r c i r c l e , cut o f f by the l a r g e r c i r c l e are equal chords o f :  (1) Morgan, Foberg, Breckenridge, Plane Geometry. (2) Morgan, Foberg, Breckenridge, O p c i t .  57 the l a r g e r c i r c l e . 9. I f a c i r c l e c i r c u m s c r i b e s a t r i a n g l e , t h e t r i a n g l e T s equilateral. 10. A diameter which b i s e c t s a chord i s p e r p e n d i c u l a r to the chord. 11. A q u a d r i l a t e r a l i n s c r i b e d i n a c i r c l e has i t s o p p o s i t e 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 i s o s c e l e s . 13. I f two chords o f a c i r c l e b i s e c t each other they a r e perpendicular. 14. Two chords which i n t e r s e c t I n a c i r c l e a r e e q u a l . 15. An angle i n s c r i b e d i n a s e m i c i r c l e c o n t a i n s 89°60' 16. A p a r a l l e l o g r a m I n s c r i b e d i n a c i r e l e i s a r e c t a n g l e . 17. A c i r c l e can be c i r c u m s c r i b e d about a r e c t a n g l e . 18. Two c o n c e n t r i c c i r c l e s have r a d i i 4 i n . and 6 I n . r e s p e c t i v e l y . A c i r c l e o f r a d i u s 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 e x t e r n a l l y , they have but two common tangents. 20. I f a c i r c l e can be i n s c r i b e d i n a q u a d r i l a t e r a l , the quadrilateral i s a parallelogram. 21. Through a p o i n t w i t h i n a c i r c l e , i t i s always p o s s i b l e to draw a chord which i s b i s e c t e d by the p o i n t . 22. A c i r c l e can always be c o n s t r u c t e d to touch each o f three g i v e n l i n e s . 23. I f the angle between two tangents i s 6 0 ° , the lengtn" of the chord j o i n i n g the p o i n t s o f contact i s equal to the l e n g t h o f 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 a l s o touch .each o t h e r . 25. Two c i r c l e s which touch the same s t r a i g h t l i n e a t the same p o i n t touch each o t h e r . 26. A c i r c l e o f r a d i u s 4" has twice the area of a c i r c l e w i t h r a d i u s 2". 27. I f two tangents are drawn to a c i r c l e from an e x t e r n a l p o i n t , the angle subtended a t the c e n t r e by the two p o i n t s o f contact i s supplementary to the angle between the tangents. 28. The sum o f the squares on two tangents drawn from an e x t e r n a l p o i n t i s equal to the square on the l i n e j o i n i n g the e x t e r n a l p o i n t to the c e n t r e . 29. I f two c i r c l e s touch I n t e r n a l l y they have but one common tangent. 30. I n a c i r c l e o f r a d i u s 3 i n . , a chord 2 i n . l o n g i s twice as f a r from the centre as a chord 4 I n . l o n g . 0  Test I I I .  (At end o f c i r c l e s . )  Multiple-Choice Test, t ) 1  (1) Morgan, Foberg, Breckenridge,  Opcit.  58 In each of the f o l l o w i n g , t e l l which answer you b e l i e v e to be c o r r e c t and why. 1. I f an angle i s i n s c r i b e d i n an a r c o f 150°, the angle cont a i n s 150°, 7 5 ° , 300°. 2. I f an i n s c r i b e d angle i n t e r c e p t s an a r c l e s s than a semic 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 o f a c y c l i c q u a d r i l a t e r a l i s 70°. The o p p o s i t e angle i s 102°, 7 0 ° , 20°, 110°, cannot t e l l . 4. Two tangents are drawn to a c i r c l e from a p o i n t p. They i n c l u d e between t h e i r p o i n t s o f c o n t a c t an a r c of 1 0 0 ° . The angle between the tangents i s 100°, 200°, 8 0 ° , 7 0 ° , 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 d i s t a n c e from the c e n t r e , i f they b i s e c t each o t h e r , cannot t e l l . » • • 6. Two chords which i n t e r s e c t i n a c i r c l e are e q u a l , i f they make equal angles w i t h the diameter through t h e i r p o i n t of i n t e r s e c t i o n , i f the f i g u r e formed by j o i n i n g t h e i r ends Is an i s o s c e l e s trapezium, i f they b i s e c t each o t h e r . 7. A p e r p e n d i c u l a r to a diameter at i t s e x t r e m i t y I s a secant, a chord, a - s e c t o r , a tangent, a segment. 8. Two secants drawn from a p o i n t P form an angle o f 30°. The l a r g e r i n t e r c e p t e d a r c i s 80°. The s m a l l e r a r c i s 40°, 60°, 20°, 3 0 ° . 9. An angle formed by a tangent and a chord which passes through the p o i n t o f contact has as many degrees as the i n t e r c e p t e d a r c , h a l f the sum o f the i n t e r c e p t e d a r c s , h a l f the i n t e r c e p t e d a r c , h a l f the d i f f e r e n c e of the i n t e r c e p t e d a r c s , cannot t e l l . 10. A p a r a l l e l o g r a m i n s c r i b e d i n a c i r c l e i s a rhombus, a square, a r e c t a n g l e , a rhomboid. MIDDLE SCHOOL GEOMETRY  t' 1  Time 2| hours NOTE:  P l e a s e read c a r e f u l l y the i n s t r u c t i o n s g i v e n b e f o r e attempting the paper. (1) Do the q u e s t i o n s i n o r d e r . (2) Do n o t spend too l o n g a t any one q u e s t i o n . Pass on to the next q u e s t i o n and r e t u r n to the u n s o l v e d questions a f t e r completing the paper. (3) At the c l o s e o f the examination, hand t h i s paper to the p r e s i d i n g examiner.  A. CONSTRUCTIONS REQUIRED TO SOLVE PROBLEMS:  (1) Example o f good geometry t e s t as p u b l i s h e d i n f i n d i n g s o f a committee a p p o i n t e d to I n v e s t i g a t e types o f examinations s u i t a b l e f o r High Schools i n O n t a r i o .  59  NOTE: (1) Make the c o n s t r u c t i o n s i n d i c a t e d on the diagrams below each q u e s t i o n . ( 2 ) The f i g u r e s should he neat and approsimately c o r r e c t ; a b s o l u t e accuracy i s not r e q u i r e d . (3) R u l e r and compasses are the o n l y instruments to be used. ( 4 ) A l l c o n s t r u c t i o n l i n e s should be c l e a r l y shown. ( 5 ) No w r i t t e n statement i s necessary i n t h i s p a r t o f the paper. (6) In drawing p a r a l l e l l i n e s use your eye and the r u l e r ; i n other c o n s t r u c t i o n s show a l l c o n s t r u c t i o n l i n e s . 1. B i s e c t t h i s a n g l e . 2. Draw the r i g h t b i s e c t o r o f this line.  3. Construct a r e c t a n g l e equal 4;Construct a t r i a n g l e equal i n i n area t o t h i s t r i a n g l e . area t o t h i s polygon.  5. F i n d the centre o f the c i r c l e o f which, the a r c below i s a p a r t .  6. Draw tangents from P to t h i s cir c i e .  7. Cut o f f 3/7 o f t h i s  8. F i n d a l i n e which w i l l be a mean p r o p o r t i o n a l between these two l i n e s .  line.  60 9. Draw a l i n e p a r a l l e l to BC so that the p a r t between AB and AC may equal BP.  11. From the q u a d r i l a t e r a l , cut o f f a p a r t s i m i l a r to i t and - 4/9 o f i t s area.  10. Draw a p e r p e n d i c u l a r t o AB at A without producing AB.  12. Draw a t r a n s v e r s e common tangent to these c i r c l e s .  13. The r a d i u s o f t h i s c i r c l e 14. Draw a s t r a i g h t l i n e \/Q i s If- Inches. Place i n i t inches l o n g , a eiiord 2 inches l o n g and c a l c u l a t e i t s d i s t a n c e from the c e n t r e .  B. CONSTRUCTIONS REQUIRED TO PROVE THEOREMS: See the I n s t r u c t i o n s under s e c t i o n A. L i n e s o r angles made equal should be so marked on the f i g u r e . No proof i s required. 1. Make the c o n s t r u c t i o n t h a t 2. I f BC i s g r e a t e r than AC then w i l l prove angle ABD the angle A i s g r e a t e r than g r e a t e r than angle BDC. the angle B.-  61  3. ML-t MN i s g r e a t e r than LN.  4. The c i r c l e s touch a t P t o prove AD p a r a l l e l to EG.  C. PROOFS OF THEOREMS: NOTE:  P u p i l w i l l make on the f i g u r e any c o n s t r u c t i o n necessary and w i l l w r i t e the proof o n l y , i n as c o n c i s e a form as p o s s i b l e . Example: I n the f i g u r e below BC DC and BE DE, prove CE the r i g h t b i s e c t o r o f BD. Proof In / _ BCE and CDE y BC CD BE - ED CE s CE =  =  3  s  In/_  a  BCF BC CF /JBCF BF  and DCF - CD r CF = DCF s FD and / B F C = DFC = 1 R t . /__,  C . l . AE and BE b i s e c t the angles A and B o f the p a r a l l e l o g r a m . Prove angle AEB = r i g h t a n g l e .  62  ~D  c  2. Prove t h a t angle AOB Is double the angle APE-. centre and P on the circumference.  R  Given 0 the  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 o f the i n t e r i o r angles o f a regular eight sided figure? 2. The e x t e r i o r angle made by producing one s i d e o f a r e g u l a r polygon i s 20°. How many s i d e s has i t ? 3. I f you were asked t o draw t r i a n g l e s (a) 3", 6", 9" (b) 4", 5", 6" (c) 5", 6", 8" (d) 15", 36", 39" which o f these would be (Place a, b, c, or d ; i n (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 t r i a n g l e ( ) (4) Not a t r i a n g l e ( ) 4. I n t r i a n g l e ABC s i d e CB - CA and AD b i s e c t s the angle CAE. Also AC 3 and AB r 5. F i n d CD. s  5. I n t r i a n g l e ABC, AB = 5, AC = 6, and BC * 7. r i g h t angles to BC. F i n d BD and CD.  AD i s a t  64 6. In these s i m i l a r t r i a n g l e s the p e r p e n d i c u l a r s are as What i s the r a t i o of the areas?  7:5.  7. Two s t r a i g h t l i n e s CAB and OCD cut a c i r c l e . Given t h a t 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 m o t i v a t i n g f o r c e s mentioned i n c o n n e c t i o n w i t h  grade XII work i n geometry i s the s a t i s f a c t i o n d e r i v e d helping others.  through  By what means can t h i s form of m o t i v a t i o n be  brought i n t o 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 i n the lower grades.  those  T h i s arrangement c o n s i s t s of p r o v i d i n g  an o p p o r t u n i t y f o r p u p i l s i n the lower grades to secure h e l p from those i n the m a t r i c u l a t i o n y e a r .  A t e a c h e r who  teaches  both i n grade X and i n Grade XII has an e x c e l l e n t o p p o r t u n i t y i n t h i s r e g a r d , because he  can r e f e r c e r t a i n grade X p u p i l s to  c e r t a i n students i n grade XII f o r a s s i s t a n c e .  I f the p u p i l -  teacher i d e a i s encouraged i t i s found that grade X p u p i l s  who  have f r i e n d s i n grade XII r e f e r to these f r i e n d s f o r a s s i s t a n c e .  65 without This  actually being  help  obtained  means meant merely  even  to replace  f o r home  after  until  from  t o supplement  assigned  The  pupils.  deal  of satisfaction  less  mature  friend,  the  home  such  distinctly  an  arrangement  of  geometry.  perly,  t o do s o .  appear  self  c a n b y made  and r e c e i v e  that  that  i n discussing  a  great  relying  plan  pupils  an added  a solution to  o n some  other  of the writer, i t has  who  i n such  take  part  interest In the subject  the students  discussion  extra  i n operation,  i n grade  geometrical  to h i s  not receive,  to give  XII i s motivated i n this  grade  problems.  This  an a d d i t i o n a l m o t i v a t i n g  of after school  enough  he d e r i v e s  In the experience  o f geometry  done i n  are shared by  otherwise  prepared  that  to take  be f o u n d  Interest  the exercise  to be of a s s i s t a n c e  which he would  noticeable  real  institution  able  o f simply  waiting  the exercise.  finds  a pupil-teacher  the teaching i tw i l l  being  but  to solve i t  o f simply  and the junior p u p i l receives  instead  of the class  who h a s s e e n  XII friend  t o come t o c l a s s  exercise  i s unable  and seeing  student  from  i s b y no  an e x e r c i s e i s  e f f e c t s o f the arrangement  i n h i s work  he i s a b l e  where  instead  him t o complete  The s e n i o r  and  If  then  lesson  grade  teacher.  o f f e r e d by the teacher  I n the case  attempts,  to enable  both  been  i t .  geometry  beneficial  assistance  i n a higher  the help  h e c a n go t o h i s g r a d e  assistance  member  pupils  s o l u t i o n , and a p u p i l  repeated  the next  class,  i n s t r u c t e d t o do so b y t h e  force  groups.  take  a  i n i t -  by the This  pro-  c a n be  66 c a r r i e d out i n a very i n f o r m a l way, and the teacher may take a very  small p a r t or no p a r t at a l l i n the d i s c u s s i o n .  The  problem f o r d i s c u s s i o n may be one that has been a s s i g n e d f o r homework; one which a p u p i l d i s c o v e r e d  on an o l d examination  paper; .one which was g i v e n to a b r i g h t p u p i l as a s p e c i a l p r i v i l e g e ; or one from some o u t s i d e  source a l t o g e t h e r .  The  diagram i s p l a c e d on the board and the students cooperate Informally  i n obtaining a solution.  The w r i t e r has seen such  d i s c u s s i o n s as these c a r r i e d on so long a f t e r school c a r e t a k e r has had to t e l l l o c k e d i n the school  that the  the students e i t h e r to go home o r be  a l l night.  The number t a k i n g p a r t I n  these d i s c u s s i o n s may be only a small f r a c t i o n o f the c l a s s , but n e v e r t h e l e s s indeed  the m o t i v a t i n g  upon t h a t few i s  valuable.  The m o t i v a t i n g  f o r c e o f games i s one which i s extremely  powerful i n a l l grades o f h i g h vary  Influence  considerably  school.  The types o f games may  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 j u s t as much a l i v e i n the grade XII student as i n the p u p i l o f grade IX.  The method o f u t i l i z i n g t h i s motiva-  t i n g f o r c e i n the t e a c h i n g  o f grade XII geometry i s c h i e f l y  by the j u d i c i o u s use o f s u i t a b l e mathematical r e c r e a t i o n s . I t I s , o f course, i n a d v i s a b l e t o make these r e c r e a t i o n s a main, f e a t u r e o f the course, but they are o f immense value i n arousing  the I n t e r e s t i n t h i n g s o f a mathematical nature i n  those p u p i l s who are l e s s r e s p o n s i v e motivation.  to the other  forms o f  A good s e l e c t i o n o f these mathematical r e c r e a t i o n s  67 in  geometry  and  i f they  most  i s a valuable  asset  are  at  presented  desirable effects  upon  The  a  suitable  f o l l o w i n g are f o r use  i n the  2.  3.  4.  the  suitable the  number  teacher times  of  they  mathematics, may  produce  pupils. of mathematical  teaching of  Mathematical 1.  to  grade  XII  recreations geometry:  Recreations  A house and b a r n are 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 r- ^ 5 rods from a brook running - - 25 in a straight line. What i s the 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 the house to get a p a i l of — i — w a t e r from the b r o o k and 03take i t to the barn? To c o n s t r u c t t h e f a m o u s N i n e - P o i n t C i r c l e . i.e. If 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 pass through the middle .points o f the l i n e s drawn from the o r t h o c e n t r e to the v e r t i c e s o f the t r i a n g l e , and through the middle p o i n t s o f the s i d e s o f the t r i a n g l e , ' i n a l l , through nine p o i n t s . T o p r o v e t h a t i t i s p o s s i b l e t o 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 . T a k e two i n t e r s e c t i n g circles w i t h c e n t r e s 0, 0 . Let one p o i n t o f i n t e r s e c t i o n b e P, and d r a w t h e d i a m e t e r s PM, FN. Draw MN, c u t t i n g the circumferences at A, B. J o i n PA, PB. Proof: S i n c e / PBM is inscribed in a semicircle i t Is a r i g h t angle. A l s o , s i n c e / PAN i s i n scribed i n a semicircle, i t i s a right angle.  To  prove  that  part  of  an  . . PA a n d PB a r e b o t h j_ to MN. Where I s t h e f a l l a c y ? angle equals the whole a n g l e . T a k e a s q u a r e ABCD a n d d r a w MM'P, the p e r p e n d i c u l a r b i s e c t o r o f CD. T h e n MM'P Is also the perpendicular bisector of AB. F r o m B d r a w a n y l i n e BX e q u a l t o AB. D r a w DX, and b i s e c t i t by the p e r p e n d i c u l a r NP. S i n c e DX--inter-  68 s e c t s CD, t h e p e r p e n d i c u l a r s cannot be p a r a l l e l a n d must m e e t a t P. D r a w PA, PD, PG, PX, P B . Proof: 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 a n d PD = P X . \ P X B PD m P C • B u t BX - BC b y c o n s t r u c t i o n a n d PB i s common t o t r i a n g l e s PBX, P B C .". / \ P B X 5 A P B C (3 s i d e s 3 sides) .'. 2 _ X B P = / C B P . The w h o l e / XBP e q u a l s i t s p a r t t h e / CBP. Find the f a l l a c y . Given a piece o f cardboard i n the form o f an e q u i l a t e r a l triangle. 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 b e put together t o form a perfect square. Given, f i v e squares o f cardboard a l i k e i n s i z e . Required, t o c u t t h e m 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 form one l a r g e square. . s  =  s  68A Theory without dead,"  practice  i s as f a i t h without works -  says James Strachan when w r i t 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 t h e statement A boy or g i r l  c e r t a i n l y c o n t a i n s much t r u t h .  who never sees any p r a c t i c a l a p p l i c a t i o n  o f the  f a c t s l e a r n e d i n h i s t h e o r e t i c a l study o f geometry w i l l never a p p r e c i a t e the f u l l v a l u e o f h i s mathematical e d u c a t i o n . Moreover, the v e r y 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 s i t u a t i o n s adds to the i n t e r e s t o f the s u b j e c t immensely and becomes a genuine m o t i v a t i n g  force.  There a r e three very a p p r o p r i a t e ways by which t h i s m o t i v a t i n g f o r c e may be a p p l i e d  to the t e a c h i n g of geometry; namely,  (1) by u t i l i z i n g the geometry o f the manual t r a i n i n g shops; (2) by u t i l i z i n g the geometry o f outdoor measurements; and (3) by u t i l i z i n g the geometry o f a r c h i t e c t u r a l forms. following  l i s t o f examples w i l l i n d i c a t e  The  certain specific  methods by which these three forms o f m o t i v a t i o n might be brought i n t o o p e r a t i o n : 1. B i s e c t i n g an angle by the c a r p e n t e r ' s square. To b i s e c t the angle A, take AC - AD. Place t h e square so t h a t BC BD. Prove AB i s the b i s e c t o r of angle A.  R  =  (1) Adams, J . , The New Teaching.  P. 108.  68B 2. E x p l a i n how the p a r a l l e l r u l e r may be used t o draw l i n e s p a r a l l e l to a g i v e n l i n e . The r u l e r moves f r e e l y about the p o i n t s A, B, C, and D. 3. To f i n d the c e n t r e o f a c i r c l e the c a r p e n t e r ' s square may be used as i n the f i g u r e and the l i n e AB draWn. Move the square and draw another diameter i n t e r s e c t i n g AB a t the c e n t r e . B  4. To f i n d the d i s t a n c e between two p o i n t s on o p p o s i t e s i d e s o f a river. To f i n d AB, l a y o f f any convenient d i s t a n c e AC perpend i c u l a r to AB, and CD p e r p e n d i c u l a r t o BC to meet BA"produced at D. Prove AB = A C - AD. 2  (1) E x e r c i s e s from Morgan, Foberg, Breckenridge, O p c i t . P.413. For more complete l i s t see Morgan, Foberg, B r e c k e n r i d g e , Opcit. Appendix.  68C I f A i s the c e n t r e of the arc BC, B the c e n t r e o f the a r c 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 f i g u r e thus formed by AB, arc AC and a r c BC i s an e q u i l a t e r a l Gothic a r c h . AB i s i t s span. I f a window has the form o f an e q u i l a t e r a l Gothic a r c h w i t h a span of 4 f t . , f i n d the area o f the g l a s s i n the window.  69 CHAPTER V MOTIVATION IN THE TEACHING GRADE IX  OF  ALGEBRA  The v a r i o u s forms o f m o t i v a t i o n s u i t a b l e f o r use i n the e a r l y grades o f h i g h school have a l r e a d y been d i s c u s s e d i n chapter I I .  The present chapter w i l l o u t l i n e s p e c i f i c methods  by which the d i f f e r e n t t o p i c s i n grade IX a l g e b r a may be p r e sented fullest  so as to u t i l i z e c e r t a i n m o t i v a t i n g f o r c e s to t h e i r extent.  When we know what m o t i v a t i n g f o r c e s are a t  our d i s p o s a l , the q u e s t i o n is", "By what means can we b r i n g these m o t i v a t i n g f o r c e s i n t o f u l l  operation?"  When commencing the study of a l g e b r a i n grade IX, the p u p i l i s n a t u r a l l y i n t e r e s t e d i n i t on account o f i t s n o v e l t y . I t i s something e n t i r e l y new to him to f i n d out t h a t i n t h i s p e c u l i a r s u b j e c t he i s going to use l e t t e r s as w e l l as numbers to r e p r e s e n t q u a n t i t i e s .  He i s amazed when he i s t o l d t h a t  i n a very short time he w i l l be able to add, s u b t r a c t , m u l t i p l y and d i v i d e by u s i n g l e t t e r s o f the alphabet  throughout.  The  teacher can c a p i t a l i z e upon t h i s n o v e l t y 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 a l l o w i n g the p u p i l s 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 t h e I n t r o d u c t i o n to the use o f symbols, the m o t i v a t i n g f o r c e of the n o v e l t y o f the s u b j e c t may g r a d u a l l y be  supple-  mented by the a r o u s a l of the p u p i l s ' i n t e r e s t i n the a c t u a l  70 situations studied. it  i s necessary  For the i l l u s t r a t i o n of the use  to use v a r i o u s types of l i t t l e  which l e t t e r s , a r e employed i n s t e a d of numbers. problems are made " r e a l " and  of symbols  problems i n I f these  " l i v i n g " to the p u p i l , he w i l l  f i n d h i m s e l f i n t e n s e l y i n t e r e s t e d i n the  .,  a c t u a l m a t e r i a l at  hand because of i t s a s s o c i a t i o n with the more p l e a s u r a b l e experiences  of h i s l i f e .  swimming, running,  Problems i n v o l v i n g b a s e b a l l ,  s k a t i n g , b u i l d i n g and numerous other  p l e a s u r a b l e a c t i v i t i e s , make an appeal  to the p u p i l which more  a b s t r a c t problems can never make.  f o l l o w i n g i s a compari-  The  son between some " l i v i n g " problems and or a b s t r a c t 1.  some of  the,"lifeless"  types:  (a) D i v i d e 84 i n t o three p a r t s so t h a t two of the p a r t s are equal and the. other p a r t f i v e times as great as e i t h e r of the equal p a r t s . ( 1 ) (b) Three boys went f o r a d r i v e i n an automobile a d i s t a n c e of 84 m i l e s . George and Jack each drove the same d i s t a n c e , but H a r o l d drove f i v e times as f a r as e i t h e r of the other two. How many m i l e s d i d each boy d r i v e ? (a) How many square f e e t are there i n a r e c t a n g l e which has adjacent s i d e s measuring 2p-t- q, and 3p-4q f e e t r e s p e c t i v e l y ? (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 f e e t long, and 3p-4q f e e t wide. What i s the a r e a of the field? How many f e e t of sawdust d i d he l a y making the outside l i n e s ? :  2.  It that he  i s a great i n c e n t i v e to i n d u s t r y , when a p u p i l f e e l s i s mastering  d i f f i c u l t to master.  something which i s r e a l l y somewhat The  e a r l y problems i n a l g e b r a  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. (1) H.S.Hall, Elementary Algebra, (2) I b i d , P. 43.  P.  20.  I f the teacher i s  71 i s observant of  as to which problems seem to the beginner  to be  a r a t h e r advanced nature, then he can be l i b e r a l w i t h h i s  p r a i s e when s o l u t i o n s to these problems are reached by the p u p i l ; and the combined e f f e c t of the p r a i s e o f the teacher and the r e a l i z a t i o n t h a t he has r e a l l y made a d e f i n i t e  accomplish-  ment, i s an e x c e e d i n g l y strong m o t i v a t i n g f o r c e i n d u c i n g f u r t h e r e f f o r t on the p a r t o f the p u p i l .  The f o l l o w i n g are  examples of problems which might appe ar to the beginner to be r a t h e r complicated, and f o r the s o l u t i o n o f v/hich l i b e r a l p r a i s e might be g i v e n : 1. A man who balances h i s accounts at the end o f every q u a r t e r f i n d s that he has three gains f o l l o w e d by a l o s s . The t h i r d g a i n i s 4 times the second, and the second i s three times the f i r s t . The l o s s i s twice the f i r s t g a i n . 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 p l a y marbles w i t h x marbles. He wins y more and then l o s e s x. He takes h i s marbles home and g i v e s h i s l i t t l e b r o t h e r o n e - t h i r d o f them. How many marbles has he now? 3. A c i r c u l a r r a c e t r a c k i s m yards around. One boy r i d e s around i t on h i s b i c y c l e n times and another boy r i d e s around i t n more times than the f i r s t . How many yards d i d both boys together r i d e ? 4. I f Jack can r u n k m i l e s per hour, and George twice as f a s t , w h i l e Henry can r u n only h a l f as f a s t as Jack, how many m i l e s w i l l the three boys together cover i n x hours? When the study of the t o p i c of s u b s t i t u t i o n i s reached, there i s a grave tendency wane.  f o r the i n t e r e s t o f the p u p i l to  The n o v e l t y o f b e g i n n i n g a new s u b j e c t has worn o f f to  a l a r g e extent, and the t o p i c o f s u b s t i t u t i o n does not l e n d I t s e l f very r e a d i l y to the use o f problems which may be made " r e a l " to the p u p i l .  Moreover, there are c e r t a i n types o f  s u b s t i t u t i o n questions which may q u i t e e a s i l y cause even advanced p u p i l s to become " t a n g l e d up".  72 For these reasons  I t i s a d v i s a b l e to g i v e the beginner the  g e n e r a l Idea behind the process of s u b s t i t u t i o n , but not cause him  to become confused by c o n f r o n t i n g him v/ith the more  technical points involved In substitution. work i n the fundamental processes  He can proceed to  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 w r e s t l e d w i t h the t e c h n i c a l i t i e s o f s u b s t i t u t i o n , and then when these fundament a l s have been mastered, he w i l l be able to come back t o s u b s t i t u t i o n and)understand  i t much more r e a d i l y .  For example,  on page 12 o f H a l l ' s School Algebra, t h e r e i s a whole page o f questions on s u b s t i t u t i o n , t h i r t y - t h r e e i n a l l ,  some o f which  are as f o l l o w s : 1. I f a  2, b  =  9**  =  1, c  27  *  =  3, x  =  4, y  =  6, z = 0, f i n d the v a l u e  2  x  2. With same values as above, f i n d the v a l u e o f a^b^ Questions  (b -t c - z )  2  such as these c o u l d v e r y c o n v e n i e n t l y be omitted  u n t i l the study o f the fundamentals ha"Se been completed, and the p o s s i b i l i t y o f a l i f e l e s s , u n i n t e r e s t i n g s e c t i o n o f the work k i l l i n g  the enthusiasm  o f the p u p i l s w i l l be e l i m i n a t e d .  Even i n the development o f the g e n e r a l i d e a o f s u b s t i t u t i o n , s p e c i f i c forms o f m o t i v a t i o n might be employed i n order t o m a i n t a i n the i n t e r e s t o f the p u p i l .  Races may be h e l d to f i n d  the v a l u e o f a c e r t a i n e x p r e s s i o n , such as: a + 2ab + 4b-2a +• b 2  2  when a - 2 and b = 3.  A l s o the use of u n i t y and zero may be made r a t h e r e f f e c t i v e , as  73 a p u p i l i s somewhat amazed to f i n d that some complicated l o o k i n g e x p r e s s i o n reduces  r i g h t down to u n i t y or z e r o .  In the t e a c h i n g of a d d i t i o n i n a l g e b r a , many o f the g e n e r a l types of m o t i v a t i o n s u i t a b l e f o r grade IX may be used, such as i n t e r e s t i n competitions, d e s i r e f o r good marks, d e s i r e f o r p r a i s e ; and these may be supplemented by other methods p a r t i c u l a r l y s u i t e d to the t e a c h i n g o f t h i s s e c t i o n of the s u b j e c t .  I n the t e a c h i n g o f a d d i t i o n there i s an  e x c e l l e n t o p p o r t u n i t y f o r the use o f problems which are o f r e a l i n t e r e s t to the p u p i l . questions which are connected of  the p u p i l ' s l i f e .  which are admirably  Here i t i s p o s s i b l e to i n t r o d u c e w i t h the most i n t e r e s t i n g phases  A few examples o f questions i n a d d i t i o n s u i t e d to p u p i l s of grade IX are as  follows: 1. A man on a motor t r i p covers x m i l e s the f i r s t day, three times as many the second day, y m i l e s the t h i r d day and z more m i l e s the f o u r t h day than on the t h i r d . Express the number of m i l e s covered i n the four days. 2. The hockey team on which Jack p l a y s scored k g o a l s i n t h e i r f i r s t game, two l e s s i n t h e i r second game, three more i n t h e i r t h i r d game than i n t h e i r second, and twice as many i n t h e i r f o u r t h game as i n t h e i r f i r s t . Express the t o t a l number o f goals scored. 3. A boy, f l y i n g h i s model a i r p l a n e , found that i t flew m f e e t on i t s f i r s t f l i g h t and 57 f e e t f u r t h e r on i t s second flight. On i t s t h i r d f l i g h t i t flew 10 f e e t l e s s than on i t s second, and on i t s f o u r t h 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 d i s t a n c e flown. The working o f the more mechanical questions can a l s o be motivated  types o f a d d i t i o n  somewhat by making use of the  f a c t t h a t a p u p i l d e r i v e s great s a t i s f a c t i o n from b e i n g able to work " b i g - l o o k i n g " questions c o r r e c t l y . a d d i t i o n occupy a l a r g e amount o f space,  Some questions i n  and appear v e r y  74 formidable indeed, but t h e i r a c t u a l working simple.  A p u p i l ' s confidence i n h i s own  i s comparatively  a b i l i t y i s greatly  strengthened when he d i s c o v e r s t h a t he has o b t a i n e d the c o r r e c t answer to such an enormous l o o k i n g q u e s t i o n as the following: Add together the f o l l o w i n g e x p r e s s i o n s : 7x y z-6x y z -H 5x y z ; 6  4  5  3  2  4  4  5x y z-8x y z +  3  6  4  4  4  3  2x y z ; 3  3  2  -12x y z + 5 x y z i - 3 x y z - 3 x y z . 6  4  5  3  2  4  4  3  3  3  2  The t o p i c o f s u b t r a c t i o n i s one of the most d i f f i c u l t s e c t i o n s of grade IX a l g e b r a to handle  satisfactorily.  The  r u l e f o r s u b t r a c t i o n i s indeed v e r y simple, but the e x p l a n a t i o n of  the r u l e i s a d i f f e r e n t matter.  The method of  working  s u b t r a c t i o n questions can be l e a r n e d very r a p i d l y by the p u p i l s , and yet i t Is the source of numerous e r r o r s 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 o f m o t i v a t i n g t h i s s e c t i o n of the work.  The d i f f i c u l t y which p u p i l s experience i n  understanding the r u l e f o r s u b t r a c t i o n may  be overcome to some  extent by the use o f f i t t i n g I l l u s t r a t i o n s such as the f o l l o w ing:  (1) Taking away h o l e s i n the ground - by adding  (2) Taking away debts - by adding money. c h i l l from water.  soil.  (3) Taking away the  (4) Taking away the need f o r food.  As a means of impressing the r u l e f o r s u b t r a c t i o n more f i r m l y upon the minds of the p u p i l s , they may  be t o l d to go  home and t e l l t h e i r parents that they have d i s c o v e r e d that there i s no such t h i n g as s u b t r a c t i o n ; and when the parents  75 q u e s t i o n the t r u t h of such a statement, i t can he e x p l a i n e d that i t i s simply a d d i t i o n w i t h the s i g n changed. t i o n w i t h t h i s the p u p i l s may ten  to make i t n i n e .  reason thus:  connec-  be asked what must be added to  A f t e r the answer "minus one" has been  o b t a i n e d the teacher may his  In  argue that i t should be "one", g i v i n g  Add I to X and you o b t a i n IX.  As a means of overcoming  the d i f f i c u l t y p r e s e n t e d by the  f a c t t h a t the mechanical method of working  subtraction  ques-  t i o n s seems so very easy to l e a r n and a t the same time v e r y l i k e l y to cause mistakes, i t i s necessary to motivate the mechanical process of s u b t r a c t i o n so as to induce the p u p i l s to  do s u f f i c i e n t p r a c t i c e i n i t without that p r a c t i c e becoming  tiresome. to  One means of accomplishing t h i s i s f o r the teacher  p l a c e a c e r t a i n number of s u b t r a c t i o n q u e s t i o n s on the  board and have the p u p i l s compete a g a i n s t one another to see who  can o b t a i n the l a r g e s t number of c o r r e c t answers i n a  c e r t a i n s p e c i f i e d time.  T h i s procedure may  be v a r i e d somewhat  by h a v i n g the p u p i l s p l a c e t h e i r names on the b l a c k b o a r d i n order as they f i n i s h , and then i t can be a s c e r t a i n e d who the f i r s t  one f i n i s h e d  w i t h a l l answers o o r r e c t .  as these develop the necessary mechanical  was  Such methods  s k i l l i n subtraction  and a t the same time prevent the t o p i c from becoming d u l l  and  lifeless. In  the t e a c h i n g o f ' m u l t i p l i c a t i o n i n a l g e b r a , the  d i f f i c u l t i e s are v e r y s i m i l a r to those encountered i n the t e a c h i n g of s u b t r a c t i o n , and they can be overcome v e r y l a r g e l y  '76 by the same methods of m o t i v a t i o n as were o u t l i n e d 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  o p p o r t u n i t y f o r developing  the p u p i l ' s i n t e r e s t i n the m a t e r i a l  at hand, because i n t h i s phase o f the work there i s an e x c e l l e n t o p p o r t u n i t y f o r i n t r o d u c i n g problems based on i n t e r e s t i n g experiences.  Problems such as the f o l l o w i n g g i v e the  life  necessary  p r a c t i c e i n m u l t i p l i c a t i o n and at the same time arouse the I n t e r e s t of the p u p i l  immediately:  1. I f an a i r p l a n e can f l y 3x-t- 2y m i l e s 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 d i s t a n c e around a s k a t i n g r i n k i s 4a-7b f e e t , how f a r 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 f e e t . How f a r w i l l the automobile t r a v e l w h i l e the wheel i s making 6p -t- 5q r e v o l u t i o n s ? 4. In one b a s e b a l l there are 2a-3b+4c yards of t h r e a d . How many yards of thread would there be i n 3a + b-2c b a s e b a l l s ? In the t e a c h i n g of d i v i s i o n the u s u a l forms of m o t i v a t i o n can be  employed, but here there Is an e x c e l l e n t o p p o r t u n i t y to  add i n t e r e s t to the t o p i c by u t i l i z i n g of  competition.  a c e r t a i n amount o f p r a c t i c e given i n i t s  a p p l i c a t i o n , the teacher may  select a certain f a i r l y  q u e s t i o n and r e q u i r e the f i r s t person board, w r i t e the q u e s t i o n on the board of o p e r a t i o n s  m u l t i p l y i n g and his  subtracting.)  long  to go to the  and perform  the  term i n the  When the f i r s t p u p i l  p a r t , the next p u p i l i n the row  correct.  i n each row  ( I . e . w r i t i n g the f i r s t  first quotient,  finishes  goes to the board  performs the second step, p r o v i d i n g he is  effect  V\Jhen the p r i n c i p l e s of l o n g d i v i s i o n have  been taught, and  set  the m o t i v a t i n g  and  t h i n k s the f i r s t  I f he has n o t i c e d ai y mistake i n the f i r s t  step step,  77 he must c o r r e c t i t b e f o r e proceeding.  T h i s procedure  continued u n t i l each of the questions i s f i n i s h e d The row  which o b t a i n s the c o r r e c t answer f i r s t  winning  row,  of  is  correctly.  i s d e c l a r e d the  and the others i n order as they f i n i s h .  T h i s form  c o m p e t i t i o n adds v e r y g r e a t l y to the i n t e r e s t which p u p i l s  take i n l o n g d i v i s i o n , and i t encourages each one develop h i s s k i l l  to t r y to  i n t h i s type of q u e s t i o n so as to make a  good showing when h i s t u r n comes to go to the board.  The  f a c t t h a t each p u p i l going to the board has to c o r r e c t any mistake  i n the q u e s t i o n , keeps p u p i l s f o l l o w i n g the working  v e r y c a r e f u l l y r i g h t from the b e g i n n i n g .  Since long d i v i s i o n  i n v o l v e s short d i v i s i o n , m u l t i p l i c a t i o n and s u b t r a c t i o n , t h i s type o f contest may  be used as a review of these  other  o p e r a t i o n s as w e l l as p r a c t i c e i n l o n g d i v i s i o n . q u e s t i o n f o r such a c o m p e t i t i o n would be one  A  suitable  l i k e any of the  following: 1. D i v i d e 6 x - x + 4x -5x -x-15  by  2. D i v i d e a  a + 2a b +b .  5  1 2  4  3  2  4-2a b H b 6  6  1 2  by  2x -x + 3. 2  4  2  3. D i v i d e x - 2 y - 7 x y - 7 x y + 1 4 x y 7  1 4  5  4  l 2  3  2  by  8  4  x-2y . 2  The m o t i v a t i n g 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 may  be  u t i l i z e d to a c e r t a i n extent i n the t e a c h i n g of l o n g d i v i s i o n . The  teacher may  as a s p e c i a l p r i v i l e g e f o r work w e l l done,  a l l o w c e r t a i n p u p i l s to t r y t h e i r s k i l l  at working a q u e s t i o n  which w i l l occupy p r a c t i c a l l y a whole page. as, D i v i d e x - 2 y - 7 x y - 7 x y W - 1 4 x y 7  14  5  4  l2  3  8  by  A question  such  X - 2 V might be used 8  f o r t h i s purpose, and the p u p i l s w i l l d e r i v e c o n s i d e r a b l e  78 enjoyment from s e c u r i n g the answer to a q u e s t i o n of  such  length. When d e a l i n g w i t h the removal of b r a c k e t s , a f t e r  the  process has been e x p l a i n e d as r e a l l y a form of a d d i t i o n and s u b t r a c t i o n , the r u l e f o r removal of b r a c k e t s may  be  impressed  upon the minds of the p u p i l s by comparing the removal of b r a c k e t s to the removal of the wrappings from a parcel.  In the case of b r a c k e t s , the innermost  Christmas ones come o f f  f i r s t , but i n the case of the p a r c e l the outer Wrappings are removed b e f o r e the  innermost.  The removal of b r a c k e t s lends i t s e l f very c o n v e n i e n t l y to  the employment of c o m p e t i t i o n by rows such as was  out i n l o n g d i v i s i o n .  In t h i s case, however, each  going to the board removes one  carried person  set of b r a c k e t s , and then  i s f o l l o w e d by the next person i n the row.  he  This i s a very  e f f e c t i v e method f o r i n c r e a s i n g i n t e r e s t In the l o n g e r  types  of  and  questions which might e a s i l y become v e r y mechanical  monotonous.  A q u e s t i o n such as the f o l l o w i n g would be  very  s u i t a b l e as the subject of a competition by rows: l _ - ( l - a + a ) - ^ L - ( a - a + a /} - [ l - ^ a - ( a - a 3 2  2  3  2  a  a ) 4  There i s a c e r t a i n f a s c i n a t i o n about removing b r a c k e t s from a l o n g and a p p a r e n t l y complicated e x p r e s s i o n , and that i s a l l reduces or z e r o .  finding  down t o a v e r y simple answer such as a,  I f questions such as these are s e l e c t e d by  1,  the  teacher, the p u p i l s w i l l d e r i v e t h a t e x t r a b i t of enjoyment which comes from the working o f such q u e s t i o n s .  The f o l l o w i n g  79 q u e s t i o n i s an example o f the type which might be used f o r t h i s purpose: Simplify: 2c -d(3c+ d)- £ c - d ( 4 c - d ) j + { 2d -e(c + d ) J 2  2  2  (Answer i s 0.) In the treatment  of b r a c k e t s , the teacher may develop a  f o r w a r d - l o o k i n g a t t i t u d e i n the p u p i l s by showing them how knowledge o f t h i s process i s a p r e r e q u i s i t e to the study o f equations.  P u p i l s may be allowed to glance ahead to grade X  work i n equations  t o see how f u l l o f b r a c k e t s some o f those  equations a r e . The v a r i o u s i n c i d e n t a l b i t s o f m o t i v a t i o n as o u t l i n e d immediately of  above a l l h e l p to s t i m u l a t e I n t e r e s t i n the study  a l g e b r a , and encourage the p u p i l to develop h i s a b i l i t y t o  i t s h i g h e s t degree along these l i n e s .  Nevertheless,  although  a l l these forms o f motivation' be a p p l i e d to the t e a c h i n g o f grade IX a l g e b r a , and although the p u p i l be encouraged by these means t o develop  a keen i n t e r e s t i n the s u b j e c t , t h i s  i n t e r e s t w i l l not endure u n l e s s i t i s coupled w i t h the s a t i s f a c t i o n that the m a t e r i a l covered has been mastered.  thoroughly  There i s no m o t i v a t i n g f o r c e more conducive  to the  development o f l a s t i n g i n t e r e s t i n a s u b j e c t than the sense o f s a t i s f a c t i o n d e r i v e d from a thorough knowledge o f the s u b j e c t matter s t u d i e d .  An o p p o r t u n i t y f o r e x p e r i e n c i n g t h i s sense o f  s a t i s f a c t i o n can by g i v e n to the p u p i l s by making use o f s p e c i a l review questions a t the b e g i n n i n g o f each p e r i o d .  (1) C f . Pp. 74, 76, H a l l ' s School  Algebra.  80 A short s e r i e s o f c a r e f u l l y s e l e c t e d q u e s t i o n s , b e a r i n g on t o p i c s p r e v i o u s l y s t u d i e d , i s p l a c e d on the board.  The  questions should be c a r e f u l l y graded, the f i r s t -ones b e i n g comparatively difficulty.  simple and the f o l l o w i n g ones o f i n c r e a s i n g The l a s t one o r two might be r e a l l y  ones I n t h e i r c l a s s .  difficult  The f o l l o w i n g i s an example o f a s e t o f  q u e s t i o n s which would be s u i t a b l e f o r t h i s purpose a f t e r the t o p i c s o f 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 s t u d i e d : 1. M u l t i p l y 2. D i v i d e  3a b -5a b+ ab 3  ™  4  j  4. D i v i d e  6  2  4  by  4  3  3  a b-3ac + 7 b c 2  ,  c  2  8 6  o  c  2  12 9 „  K  2  by  8  by  3  4a b .  3a-r4b  6. D i v i d e 6x -7x-3 2  7. M u l t i p l y  -ax  2  by  3  4x y . 3  2  -3a b c.  4 12  3  ,  15x y -25x y -35x y  5. M u l t i p l y  8. D i v i d e  2  12x y -8x y + 20x y  3. M u l t i p l y A  2  by  4  _ 43 -5x y .  5a-2b.  by 2x-3.  +• 3 a x y - 9 a y 2  by  4  15+-2m -31m + 9m + 4m + m 4  2  3  -ax-3ay . 2  5  by  3-2m-m . 2  By working a s e t o f questions such as these, each p u p i l has  the o p p o r t u n i t y o f determining the degree o f mastery which  he has developed  up t o t h i s stage.  I f marks are a s s i g n e d to  these q u e s t i o n s , and the marks are compared by the method o u t l i n e d i n chapter I I I , then the p u p i l s become m o t i v a t e d t o g r e a t e r e f f o r t s i n t h e i r study o f a l g e b r a a t the b e g i n n i n g o f each p e r i o d ; and a t the same time they are l a y i n g f o r thems e l v e s a strong f o u n d a t i o n upon which, t o b u i l d t h e i r mathemat i c a l superstructure. There a r e u s u a l l y a few p u p i l s i n each grade IX c l a s s who  81 master the work very q u i c k l y and are eager to go ahead to work.  new  Such p u p i l s d e r i v e a great d e a l of enjoyment from  knowing t h a t they are c o n s i d e r a b l y ahead of the remainder of the c l a s s , and by a l l o w i n g these p u p i l s to proceed  in this  manner, the teacher i s m o t i v a t i n g the work f o r them to a v e r y c o n s i d e r a b l e degree.  In a l a r g e c l a s s , however, i t would  o b v i o u s l y 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 i n t o  opera-  t i o n to any great extent, because of the i m p o s s i b i l i t y of  one  teacher g i v i n g i n d i v i d u a l i n s t r u c t i o n to a l a r g e number of p u p i l s working a t d i f f e r e n t p l a c e s i n the t e s t ; n e v e r t h e l e s s , the same m o t i v a t i n g e f f e c t can be o b t a i n e d i f the teacher  has  at h i s d i s p o s a l a l a r g e amount of e x t r a m a t e r i a l to supplement the work of the t e x t .  T h i s m a t e r i a l may  s e r i e s of more d i f f i c u l t  be i n the form o f a  types of questions on the v a r i o u s  t o p i c s s t u d i e d ; a number o f supplementary t e x t s i n which c e r t a i n questions are marked; mimeographed sheets of q u e s t i o n s ; or a supply of o l d examination  papers.  review  When a  b r i g h t 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 , may  be g i v e n the p r i v i l e g e of s e l e c t i n g one  mentary groups of q u e s t i o n s . of  The  of these  f o l l o w i n g are a few  suppleexamples  supplementary questions which are v e r y v a l u a b l e i n b r i n g i n g  i n t o o p e r a t i o n the type of m o t i v a t i o n r e f e r r e d to here: 1. C o l l e c t terms: 2. C o l l e c t terms: 3. 4. 5. 6. 7. 8.  he  5a-6 +9-8a+ a +8 2x-5y+ 3y-3x+x-y  C o l l e c t terms: 2 a b - 5 a b - a b + a^b Add: 2 x + 3 ; 4-x; -3x *• 1; -5 + 4x Add: 2c-3d; -2d*»c; 5d-2c S u b t r a c t : 3x-2 from 3-2x From ~2a 3b take 4b-3a A Take 3x -5x * 1 from 2x -f6x~5 2  2  2  2  2  82 9. S u b t r a c t 2 a b - 3 a b 2  10. F i n d the sum 11. M u l t i p l y  from  2  4ab + 3a b ' 2  2  of 3x + 2 x y - y j 2  (~6x y)  by  2  ~6x -»- 3 y ;  2  2  5xy-4y  2  2  (4x)  12. M u l t i p l y (~4cd ) by (^3c d ) 15. F i n d the product of -5xy and 1 2 2  2  14. Give the square of  3  -4a  15. Give the cube o f 3x 16. Give the cube of  -2a  17. S i m p l i f y 18. S i m p l i f y  (-5x ) (-4xy )  19. M u l t i p l y  (-3a+4)  20. M u l t i p l y 21. M u l t i p l y  3  4  2  2  (2xy-3)  3  by  by  (3a b-2)  4a -4x  by  2  (-4ab ) 2  Simplify: ( 6 x - f 3 ) (6x-3 )  22. -3a(-4a + 6b)  23.  24.  (7x-2y)(ox+y)  25.  (3a-2b) .  26.  (l-3x )  27.  (9a b ) ~  29.  C-3xy+6y) -  2 8  «  3  "5x y 2  2  3  -5y 31. -5a + 6 a -3a  J  32.  37. 38.  (3y)  3a -2a b 3  '  2  36.  (-9b)  2  2  33. 50x ~5x-6 5x + 2  35.  2  30. x -6x+ 8 x  3  34.  2  2  "  a  8a -27 2a-3 The number 78 i s d i v i d e d i n t o two p a r t s so that one i s f i v e times the o t h e r . What i s the s m a l l e r p a r t ? I f 90 cents be d i v i d e d among A, B, and C so t h a t A gets three times as much as G and B twice as much as C, what i s A's share? If x 3 and y = 4, f i n d value o f 2 y - x y I f x = -2 and y - 1, f i n d value of ( x - y ) 3  39. I f a  2  s  2  :  2, b a 3, o  s  0,  f i n d value of - 3 ( b - a c ) 2  40. I f x = -1 and y = 2, f i n d value of 2 x - 4 y 3  41. F i n d the sum of ( 2 x - 3 ) and (3x + l ) 42. Add (2a-3b)(a + b) to (2a+b) 2  2  43. Take 3x-2y from  zero  2  2  83 44. S u b t r a c t 2 a - 3 b 2  2  from 1  45. I f a boy i s 10 years o l d 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 o l d was he y years ago?. 47. How many inches a l t o g e t h e r i n x yards y f e e t ? 48. At the r a t e o f x m i l e s per hour, how many m i l e s are t r a v e l l e d i n b minutes? 49. At the r a t e o f c m i l e s per minute how many minutes are taken to go d m i l e s ? 50. What i s the t o t a l c o s t i n d o l l a r s o f x books a t y cents e a ch. Wha-t- -ia- -ay- -to-t-aJU -lo-as-?51. I buy k books a t b cents each and s e l l a l l o f them a t y cents each. -What i s my t o t a l l o s s ? 52. F i n d t h e sum o f three consecutive numbers, the s m a l l e s t o f which i s x. 53. F i n d the sum of two consecutive even numbers i f the l a r g e r one i s y. 54. What must be added to 3x-2y to g i v e 2x-*-5y? 55. What must be s u b t r a c t e d from 4x-3y to leave 5x+ 2y? 56. From what must -3a +b be taken to leave 4 a f b? 57. What.must be s u b t r a c t e d from zero, to l e a v e 3x? 58. A r e c t a n g l e i s x f e e t l o n g and y inches wide. Give the number o f Inches i n i t s p e r i m e t e r . 59. The perimeter o f a square i s y i n c h e s . Give i t s area i n square inches. 60. Express the area i n square f e e t o f a r e c t a n g l e b f e e t l o n g and 4 inches wide. 61. I n a school w i t h 212 students, 3x t 5 are boys and 4x-3 a r e girls. How many g i r l s are there? Simplify: 62.  (la-3b) + ( a - l b )  63. ( x - l y ) t (lx 1 y) t ( x - l y )  64.  (2a -^)-(^a -l)  65.  66.  (|a-2b)  67.  68.  (3a -2gb) - (§)  2  2  2  2  fy )(-^) 2  (Jx-3y)(gx-t y )  69. (2x -r2x-§) - (x-1) 2  70. g(5a-6) -t g(2a-3)  71. 3x(lx-1 )-4x(ix-1)  72. 9(a * b ) 3(a t b)  vs. 2x- ( 3 - ( - x + 2 )  2  (3-2x)}  74. 2 |5x-2(-x+ y)J  75. Add 5(a + b)-2(a-b) to 4(a + b) f 3(a-b) 76. S u b t r a c t 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 v a l u e of l x y 3 2 2 ' 2  3  84 78. % e n  a  1 g i v e value of l - 3 a +- 3 a - a 2  s  3  2 79. When b a -| give v a l u e of b - 5 b + 1 3  80. Expand ( x - 2 )  2  3  81. S i m p l i f y 82. S i m p l i f y 3£lZ +  7  1=5 ^ x  3  y  83. P i l l i n b r a c k e t s xa-3ya-a - a( ) 84. P i l l i n b r a c k e t s ax-2bx-x -x( ) 85. I f A 3x-2 and B = 2x +3 f i n d value of A -AB s  =  86. S i m p l i f y (a-2b) -3(2a-b)(a+b) 87. I bought an a r t i c l e f o r x d o l l a r s and s o l d i t at a g a i n of 5% of c o s t . F i n d the s e l l i n g p r i c e . 88. I s o l d 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 . F i n d the g a i n % reckoned on c o s t . 89. I f I t takes y seconds to t r a v e l c f e e t , express the r a t e i n yards per minute. 2  90. I f a a -3, b a 1, f i n d value 91. I f x  s  a -5b 2  -3, y a 3, f i n d v a l u e of  92. F i n d value of (2a + l )  a  2  x  (3x -2xy + 3) 2  when a - 2  93. F i n d the sum of the numerical s i o n of 2 xy-y2)2 (  3  2  c o e f f i c i e n t s i n the expan-  +  94. A d e a l e r 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 p r i c e , making a t o t a l g a i n of b d o l l a r s . For what does he s e l l each book? P u p i l s i n grade IX are ever ready to attempt the s o l u t i o n of a p u z z l e , or to j o i n i n some form of mathematical r e c r e a tion.  The use of such m a t e r i a l i n grade IX, i f c a r e f u l l y  s e l e c t e d and presented  at s u i t a b l e times, has  a very  valuable  m o t i v a t i n g e f f e c t upon the p u p i l s .  I t g i v e s them an oppor-  t u n i t y of e x p e r i e n c i n g t h a t " t h r i l l  of d i s c o v e r y " which comes  from the s o l u t i o n of mathematical problems, and  t h i s l e a d s to  the development of that i n e x p l i c a b l e f e e l i n g of  satisfaction  which comes from the working of questions of a mathematical nature.  The  f o l l o w i n g are some of the simpler forms of  85 mathematical r e c r e a t i o n s , which are s u i t a b l e f o r i n t r o d u c t i o n i n t o grade IX work i n a l g e b r a : Mathematical  Recreations  1. I f you d r i v e from Vancouver to San F r a n c i s c o at an average speed of 30 m i l e s per hour, and r e t u r n at an average speed of 00 m i l e s per hour, what i s your average speed f o r the whole journey? 2. Pat and Mike, two p a i n t e r s , were g i v e n the c o n t r a c t to p a i n t the lamp-posts on a c e r t a i n s t r e e t . Pat was to p a i n t the posts on the ftorth s i d e and Mike the posts on the south side. There were the same number o f posts on each s i d e . Pat a r r i v e d e a r l y and began to p a i n t . ?fnen Mike a r r i v e d he found t h a t Pat was p a i n t i n g and was j u s t completing the t h i r d post on the south s i d e . Mike p o i n t e d out the e r r o r to Pat and sent him over to h i s own s i d e where he again commenced to p a i n t at post number one. Mike completed h i s posts f i r s t , so i n order to h e l p Pat he went over and p a i n t e d s i x posts f o r Pat. Now, Pat p a i n t e d three posts f o r Mike i n the morning, and Mike p a i n t e d s i x f o r Pat i n the evening; who p a i n t e d the l a r g e r number of p o s t s , and ronhow many more than the o t h e r . 3. A farmer d i e d and l e f t h i s stock to h i s t h r e e sons. The o l d e s t son was to r e c e i v e o n e - h a l f of the stock, the second son was to r e c e i v e o n e - t h i r d o f the stock; and the youngest son one-ninth of the stock. When the stock was counted there were seventeen head o f c a t t l e . How were they d i v i d e d , none b e i n g k i l l e d ? 4. Express a l l the numbers from one to twenty-one, each time u s i n g f o u r 4's and a r i t h m e t i c a l s i g n s , e.g. 4 x 4 -4-4 4 5. I f a f i s h weighs 13 l b s . and h a l f I t s own weight, what i s the.weight of the f i s h ? 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 . S u b t r a c t the s m a l l e r of these two numbers from the l a r g e r . 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 s t r u c k out. 7. Which would you p r e f e r : a h a l f ton of sovereigns or a ton of h a l f - s o v e r e i g n s ? 8. A man went i n t o a shoe s t o r e and bought a p a i r of shoes. He gave the shoe merchant a ten d o l l a r b i l l , but the shoes c o s t o n l y $5.00. The merchant c o u l d not change the b i l l , so he went over to the drug s t o r e and changed i t . He gave the customer $5.00 change and the shoes. A f t e r the customer had gone, the drug c l e r k came over and s a i d t h a t the $10.00 b i l l t h a t the shoe merchant had g i v e n him was a c o u n t e r f e i t ; and i t was. The shoe merchant then had to g i v e the drug c l e r k a good ten d o l l a r s f o r the c o u n t e r f e i t b i l l . How much money as w e l l as the shoes d i d the shoe merchant l o s e ? 5  86 9. I f 6 c a t s eat 6 r a t s i n 6 minutes, how many c a t s w i l l i t take to eat 100 r a t s i n 100 minutes. 10. The head of a f i s h i s 9 i n . l o n g . The t a i l i s as l o n g as the head and one-half of the "body, and the body i s as l o n g as the head and t a i l . What i s the l e n g t h of the f i s h ? 11. I f a l o g s t a r t s from the source of a r i v e r on F r i d a y and f l o a t s 80 m i l e s down the stream d u r i n g the day, but comes back 40 m i l e s d u r i n g the n i g h t w i t h the r e t u r n 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 m i l e s long? 12. A k i n g has a horse shod and agrees to pay 1 cent f o r the f i r s t n a i l , 2 cents f o r the second, 4 cents f o r the t h i r d , d o u b l i n g each time. What w i l l the shoeing w i t h 32 n a i l s cost? 13. A hare i s 10 rods i n f r o n t of a hound, and the hound can r u n 10 rods w h i l e the hare runs 1 r o d . Prove t h a t the hound w i l l never c a t c h the h a r e . 14. A and B have an 8 - g a l l o n can of m i l k and wish to d i v i d e the m i l k i n t o two equal p a r t s . The o n l y measures they have are a 5 - g a l l o n can and a 3 - g a l l o n can. How can they d i v i d e the milk? 15. I bought a horse f o r $90 and s o l d i t f o r $100, and soon repurchased i t f o r $80. How much d i d I g a i n by t r a d i n g ? 16. • Three books were p l a c e d on a s h e l f i n proper order as shown In the diagram. Each book was three inches t h i c k i n c l u d i n g the covers, each of which was 1/8 of an i n c h Vol. Vol. Vol. thick. A bookworm bored a h o l e from the f i r s t page of volume I , Jl 1Ill s t r a i g h t through to the l a s t page of volume I I I . How f a r d i d he travel?  87 CHAPTER MOTIVATION IN THE  VI TEACHING OP  GRADE XII ALGEBRA In chapter I I , the v a r i o u s forms o f m o t i v a t i o n used i n the h i g h e r grades of h i g h s c h o o l were o u t l i n e d .  In'."the  p r e s e n t chapter those, types which are e s p e c i a l l y a p p l i c a b l e to  the t e a c h i n g of grade X I I a l g e b r a w i l l be d i s c u s s e d more  fully,  and s p e c i f i c methods w i l l be o u t l i n e d by which these  forms of m o t i v a t i o n may  be u t i l i z e d to t h e i r f u l l e s t  extent.  In the f i r s t p l a c e , we must r e a l i z e that i n t h i s grade there i s ever present t h a t g r e a t m o t i v a t i n g f o r c e of m a t r i c u l a t i o n examinations. at  the end of the s c h o o l year he w i l l be r e q u i r e d to w r i t e  departmental him  The very f a c t t h a t a p u p i l knows t h a t  examinations  i s sometimes s u f f i c i e n t to cause  to put f o r t h a maximum amount of e f f o r t , and r e a c h h i s  l i m i t of mastery. examinations p u p i l may  But  the presence  of m a t r i c u l a t i o n  i s by no means an i d e a l form o f m o t i v a t i o n .  A  work a t the h i g h e s t l e v e l o f h i s a b i l i t y I n order to  a v o i d f a i l u r e , but a t the same time he may I n t e r e s t i n the work which he i s doing.  have v e r y l i t t l e  The  f i r s t step towards  m o t i v a t i o n i n grade X I I , t h e r e f o r e , should be the c r e a t i o n o f a proper a t t i t u d e towards m a t r i c u l a t i o n examinations.. teacher should i n s t i l examinations  The  i n the minds o f the p u p i l s the i d e a t h a t  are c a r r i e d on not f o r the express purpose of  t o r t u r i n g p u p i l s , but r a t h e r because they are a d e s i r a b l e means  88 of a l l o w i n g a p u p i l the o p p o r t u n i t y o f s a t i s f y i n g h i m s e l f t h a t he has reached a h i g h l e v e l o f mastery i n the work which he has been doing.  Any p u p i l who has taken a keen i n t e r e s t i n  h i s school work, and has d i s c o v e r e d the p o s s i b i l i t y o f s e c u r i n g a great amount o f p l e a s u r e from a t t a i n i n g mastery over s u c c e s s i v e u n i t s o f work, w i l l look on the m a t r i c u l a t i o n examinations as another o p p o r t u n i t y f o r s e c u r i n g t h a t s t i m u l a t i n g s e n s a t i o n o f s a t i s f a c t i o n through mastery. From t h i s I t must not be i n f e r r e d that the present  system  of examinations f o r grade X I I p u p i l s i s a good one, because i t i s obvious that m a t r i c u l a t i o n examinations as at present c o n s t i t u t e d have a great many very u n d e s i r a b l e f e a t u r e s . p s y c h o l o g i c a l e f f e c t upon h i g h school students  i s often far  from b e n e f i c i a l and sometimes d e c i d e d l y h a r m f u l . l o n g as the present  Their  However, as  system i s i n vogue, i t i s imperative  that  h i g h s c h o o l teachers make the b e s t of the s i t u a t i o n as i t e x i s t s and encourage the p u p i l s to r e g a r d examinations i n such a way t h a t they may be a f f e c t e d by whatever b e n e f i c i a l i n f l u e n c e s are c o n t a i n e d i n them. Having c r e a t e d a proper remains f o r the teacher  a t t i t u d e towards examinations, i t  to cause the p u p i l s to develop an  i n t e r e s t i n t h e i r work, simply  f o r the work's sake.  be done very c o n v e n i e n t l y i n a l g e b r a by a s p e c i a l  T h i s can  adaptation  of the u n i t system.  I n grade XII a l g e b r a the t o p i c s s t u d i e d  are o f such a nature  t h a t there i s very l i t t l e  between them.  i n common  I n any one s e c t i o n there i s but s l i g h t  reference  89 to ly,  the work o f s e c t i o n s immediately  there i s p r a c t i c a l l y no o p p o r t u n i t y f o r r e v i e w i n g a previous  u n i t w h i l e s t u d y i n g a new one. r a t i o and graphs are almost for  preceding; and, consequent-  Quadratics, indices,  surds,  complete u n i t s i n themselves, and  t h i s reason i t i s convenient to operate three  u n i t systems d u r i n g the course o f the y e a r .  separate  During the f i r s t  p a r t o f the year, the v a r i o u s t o p i c s are s t u d i e d from the t e x t book.  I t would r e q u i r e f a r too much time, and be too l a b o r -  i o u s a task f o r a p u p i l to work every q u e s t i o n i n every s e t , but c a r e f u l l y s e l e c t e d and w e l l - g r a d e d examples are chosen to comprise of  each u n i t o f the t e x t .  When the t e s t s a t the end  each u n i t i n d i c a t e t h a t the m a j o r i t y o f the p u p i l s have  a t t a i n e d a s u f f i c i e n t degree o f mastery In t h a t t o p i c , the f o l l o w i n g u n i t i s t r e a t e d i n a s i m i l a r manner. a l l the u n i t s can be covered i n approximately  By t h i s means  two-thirds of  the school y e a r . S i n c e each u n i t i n grade XII a l g e b r a i s s t u d i e d as a separate e n t i t y , when a l l have been completed  i t i s necessary  to have some form of review and a l s o some means o f c o o r d i n a t i n g the v a r i o u s u n i t s .  These' two o b j e c t i v e s - review and  c o o r d i n a t i o n - can be reached by making use o f two separate s e t s o f u n i t sheets. c a r e f u l l y graded were s t u d i e d .  The f i r s t  s e t c o n s i s t s o f a s e r i e s of  questions arranged i n the same order as they  I n t h i s way a p u p i l can check up on h i s  knowledge o f the v a r i o u s u n i t s , and thereby a s c e r t a i n the exact spot where concentrated review i s necessary,  .in t h e  90 appendix a s e r i e s of questions i s g i v e n which were drawn up for  t h i s purpose.  The p u p i l s are allowed to work the  questions as q u i c k l y as they wish, w i t h a minimum number to be done each week.  I f the s e r i e s i s begun twelve weeks b e f o r e  the end of the s c h o o l year, the teacher may into  d i v i d e the  list  about t e n e q u a l p a r t s and make one p a r t a minimum f o r  each week.  At the end of each week the answers are g i v e n to  the q u e s t i o n s i n the s e c t i o n j u s t completed,  and  special  a t t e n t i o n i s p a i d to those questions with which the p u p i l s experienced  difficulty.  The above-mentioned s e r i e s of q u e s t i o n s , the s e r i e s f o r review,  i s supplemented by a e e r i e s f o r c o o r d i n a t i o n .  This  c o n s i s t s of t e n separate sheets designed f o r the purpose of r e l a t i n g the v a r i o u s t o p i c s and p r o v i d i n g a g e n e r a l survey o f the whole s u b j e c t .  Each sheet c o n t a i n s t e n questions d e a l i n g  w i t h d i f f e r e n t s e c t i o n s of the work,?..and the v a r i o u s types are arranged i n a d i f f e r e n t order on each sheet.  These q u e s t i o n s  vary from comparatively easy ones to those of c o n s i d e r a b l e difficulty.  A good many of these q u e s t i o n s are chosen from  p r e v i o u s examination of  this fact  papers,  and i f the p u p i l s are  informed  they show an added i n t e r e s t i n the q u e s t i o n s on  t h a t account.  When working on one of these sheets, i f a p u p i l  f i n d s t h a t he i s unable  to master one of the q u e s t i o n s , he  r e f e r s 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 The  question.  s e r i e s f o r c o o r d i n a t i o n i s a l s o given i n the  appendix,  and  these questions may  be g i v e n d u r i n g  the  SET'  l a s t ten weeks o f the school year at the r a t e of one^per week. The  a l g e b r a p e r i o d on one  may  be  p a r t i c u l a r day,  F r i d a y f o r example,  set a s i d e f o r the d i s c u s s i o n o f these papers.  f i r s t F r i d a y , sheet number one work these questions  is distributed.  The  the  pupils  d u r i n g the week, and hand i n t h e i r  s o l u t i o n s on the f o l l o w i n g F r i d a y . they r e c e i v e sheet number two.  At the end of t h i s  marked, c o l l e c t s the s o l u t i o n s to sheet sheet number t h r e e .  procedure i s f o l l o w e d ,  period  On F r i d a y o f the next week  the teacher hands back the s o l u t i o n s to sheet number  out  On  number two,  On each succeeding the teacher  one, and hands  F r i d a y the same  c o l l e c t i n g one  set of  answers; handing back another s e t , marked; and d i s t r i b u t i n g a new  set f o r the f o l l o w i n g week.  Most of the a l g e b r a  period  each F r i d a y i s devoted to d i s c u s s i n g the papers which have j u s t been r e t u r n e d ,  and  i n examining the questions  which gave  the p u p i l s most t r o u b l e . The  adoption  of t h i s t h r e e - u n i t method - the u n i t f o r  l e a r n i n g , the u n i t f o r review, and the u n i t f o r c o o r d i n a t i o n has  a great m o t i v a t i n g  e f f e c t upon the p u p i l s , and  coupled  with the development of a proper a t t i t u d e toward m a t r i c u l a t i o n examinations, i t i s almost s u f f i c i e n t m o t i v a t i o n (1)  See  p.  124  i n i t s e l f for  92 making the grade XII course i n a l g e b r a extremely and s a t i s f y i n g .  interesting  I t b r i n g s i n t o o p e r a t i o n most e f f e c t i v e l y  the  m o t i v a t i n g f o r c e s of s a t i s f a c t i o n through mastery, and o f the development of an I n t e r e s t i n a l g e b r a i c a l c a l c u l a t i o n s as a new  and d e l i g h t f u l The  experience.  t h r e e - u n i t system of t e a c h i n g a l g e b r a i n grade XII  can be made even more s u c c e s s f u l by overcoming one weakness which i s common to most u n i t a r y methods of i n s t r u c t i o n ; namely, that when one u n i t 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 g r a d u a l l y fades u n l e s s some means i s adopted  f o r keeping In constant contact w i t h i t .  This  weakness can be overcome by making use of a set of review questions s e l e c t e d to cover a l l the s a l i e n t p o i n t s i n the .various u n i t s without  I n v o l v i n g very l e n g t h y c a l c u l a t i o n s .  few o f these questions can be a s s i g n e d f o r homework each  A  day,  handed i n by the p u p i l s , 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. • f i r s t p a r t of each l e s s o n (about ten minutes) may  be  The  devoted  to going over the questions which the teacher has marked and Is r e t u r n i n g to the p u p i l s .  Such a system keeps a p u p i l  c o n s t a n t l y i n touch with the m a t e r i a l c o n t a i n e d i n p r e v i o u s u n i t s ; and when the time comes to commence the hundred review q u e s t i o n s , he f i n d s them much more f a m i l i a r than he would have done had he not had list  t h i s d a i l y review.  The f o l l o w i n g  of questions w i l l g i v e an i d e a of the type of q u e s t i o n  suitable for this  purpose:  93 1. S o l v e  / 2x-l - ( —  c 5  x  ,") „  +  3  X  +  x + 2 ., „ ~2~ +  7  2. A man bought a number o f a r t i c l e s f o r $200. He kept f i v e and s o l d the remainder f o r $180, g a i n i n g $2 on each a r t i c l e s o l d . How many d i d he buy? 3. Solve  5x -9x-4 a 0. 2  4. The hypotenuse o f a r i g h t - a n g l e d t r i a n g l e i s 25 i n . and the perimeter i s 56 i n . F i n d the remaining s i d e s . p 5. Solve  x -xy = y -3xy  =10  2  6. F a c t o r  6  (a) 4m -21m n + n . 4  2  2  4  (b) a f b-*- a + b 3  3  (c)  12x -27a x . 4  (d)  2  2  4(2a-3n) -(3a-7b) 2  2  7. By use o f f a c t o r s f i n d the product o f :  fx -2(Xrl)/«n-^+ 2(x-l)J 2  8. I f x + y 9. Solve  1, prove x ( y + 1 ) - y ( x + 1)-x ry = 0 3  s  3  x - 3 x = 0. 2  10. Solve 4-9x = 1 3 x . 2  11. Solve  6y-4x _ 5z - x _ y - 2z 3x-4y ~ 2y-3z ~ 3y-2z ~ 4 x r x-1.  12. Solve  2  13. Solve  42x -28c  14. Solve  x + 2x  2  2 3x .  4  15. S i m p l i f y  3  x  2  - 25cx.  2  s  a  -  4 x  2  a~2  :  94 17. S o l v e :  + -5 » 4 I y^ *  x^  1 - 1 18. S o l v e :  7y x  19. Express  2  2 r  l  15xy  + 2xy + 2 y  i  s  -68  =  17  2  i n i t s s i m p l e s t form:  4( /~~5-f 1)  [5-1  _ - 2 •* /~5~  2- fi  20. R a t i o n a l i z e the denominator o f :  21. F i n d t h e square 22. Solve:  root o f :  tp48~ - "i/ii"" a  -j/I+a t f x ^ b =  a V5T 23. The p r i c e o f photographs i s r a i s e d $3 per dozen; and cus• tomers consequently r e c e i v e seven l e s s p i c t u r e s than b e f o r e f o r $21. What was the o r i g i n a l p r i c e of the p i c t u r e s per dozen? 24. S o l v e :  x ^ l — — — — —  «  „,  fxV 1  Or  —  —  2  if^-i  25. Write down the r o o t s o f : (a) ( x - a + b ) ( x - a - b )  =  0  (b) x + 2x - 0 2  26. S i m p l i f y : ( b"  27. S i m p l i f y : 28. S o l v e :  1  0  2<fE+-  25x -y 2  - 84  2  5x - y  i 10  =  6  29. Show the meaning of a ~  n  —  95 30. S i m p l i f y :  / ^ ] /  j^g-  4x  31. Find, the square o f :  :  .2a°b-.3ab°  33. R a t i o n a l i z e the denominator of 'i/j—^ s i m p l e s t form. v  34.  0  ,  s.nd express i n  8  F i n d the square root o f 17-12 J~2 •  35. F i n d the square of e + e " x  36. S i m p l i f y :  a -g-f- a 5  z  5  a°*-a ^ 37. S o l v e :  x  2  y 38. S i m p l i f y :  2  8  14-f-xy  r y <T2(5 /3> x  1  0  fz)  / § ) - J3(2 f2-  39. S i m p l i f y by removing negative  i n d i c e s : , a ^ - f b"^" a *V b ^  40.  F i n d the value o f : x - f x + x - / - l when x  41.  Simplify:  2  3  s  1  42. A man has h hours a t h i s d i s p o s a l . How many m i l e s can he r i d e out a t r m i l e s per hour i f he must walk back a t w m i l e s per hour? 43.  F i n d the mean p r o p o r t i o n a l between /27^3 fz and  44.  T  If a  a+b-f c t d  a-b-r- c-d  — ~ — prove a^b-c-d a-b-c-fd.  a  ^ 7 - r 3«T¥  c  T - ~} d.  45. F i n d the square r o o t o f | -t fb~ 46. Using  the s c a l e 1 u n i t  \ i n c h , f i n d the p o i n t o f i n t e r 5 s e c t i o n o f the graphs o f 2x-3y 24 and — _ X _ 12 3 2 47. F i n d the equation f o r the s t r a i g h t l i n e p a s s i n g fchfcough the p o i n t s (3,4), (-2,5) 48. Solve f o r a and f o r n: S - ^(B^-f£) 49. Solve f o r r and f o r v: F gr =  s  m  v  2  96 50. Solve f o r x: x(a-x) c . Give the n u m e r i c a l value o f the r o o t s when a 16 and c = 6. =  =  There are s e v e r a l other minor, but n e v e r t h e l e s s  important,  means by which the t e a c h i n g o f grade XII a l g e b r a may be motivated to a f u l l e r extent, and one o f these i s a c a r e f u l treatment to develop  of a l g e b r a i c a l problems.  A number o f students seem  somewhat o f a d i s l i k e f o r problems, and become  possessed with, the i d e a t h a t grade XII problems are t r i c k y b i t s o f mathematical student's a b i l i t y .  r e a s o n i n g almost beyond any o r d i n a r y The development o f t h i s erroneous  impres-  s i o n can be prevented by p r e f a c i n g the work on problems w i t h a few c a r e f u l l y s e l e c t e d p u z z l e s which r e q u i r e c o n s i d e r a b l e thought  f o r t h e i r s o l u t i o n , but which are by no means out o f  the range o f any average  student.  I f these p u z z l e s are  p r e s e n t e d so as to g i v e each, p u p i l the i d e a that they a r e a c h a l l e n g e to h i s i n g e n u i t y and r e s o u r c e f u l n e s s , then he w i l l respond The  to the c h a l l e n g e and do h i s utmost to o b t a i n s o l u t i o n s .  f o l l o w i n g are examples o f r a t h e r simple p u z z l e s which  might very w e l l be used as a p r e f a c e to a l e s s o n on problems: 1. A b l a c k s m i t h had a stone weighing 40 pounds, and a s k i l l e d mason broke i t i n t o 4 p i e c e s whereby any number of pounds from l ' t o 40 c o u l d be weighed on s c a l e s . F i n d the weight of each of the f o u r p i e c e s . 2. Two automobiles 20 m i l e s apart a r e approaching each other, each t r a v e l l i n g 10 m i l e s per hour. A bee, which f l i e s a t the r a t e of 15 m i l e s per hour s t a r t s at the r a d i a t o r o f one automobile and f l i e s back and f o r t h between t h e i r r a d i a t o r s u n t i l the automobiles meet. How f a r does the bee fly? 3. When I was born my s i s t e r was o n e - f o u r t h mother's age, but she i s now o n e - t h i r d f a t h e r ' s age. I am now o n e - f o u r t h mother's age, but i n f o u r years I s h a l l be o n e - f o u r t h f a t h e r ' s age. How o l d I s each o f us?  97 4. How s h a l l we buy 12 eggs f o r e i g h t y c e n t s , i f hen eggs s e l l at 5^ each, duck eggs at 7^ each, and turkey eggs a t Qtf each, and i f we buy some of each? Puzzles l i k e ing  the ones g i v e n above are extremely i n t e r e s t -  to the h i g h s c h o o l student, and he soon b e g i n s to r e a l i z e  that r e a l enjoyment can be d e r i v e d from attempting to s o l v e problems o f v a r i o u s k i n d s .  U s i n g such p u z z l e s as these as an  i n t r o d u c t i o n , the teacher can l e a d the c l a s s on to the i n v e s t i g a t i o n of problems which i n v o l v e the use o f equations in  t h e i r s o l u t i o n , and once a p u p i l experiences the " t h r i l l o f  d i s c o v e r y " d e r i v e d from the s o l u t i o n o f a mathematical problem, he i s eager to t a c k l e more problems which w i l l enagle him to experience that t h r i l l more o f t e n . There are many ether forms of mathematical r e c r e a t i o n s , b e s i d e s the s t r i c t l y  "problem ' type, which can be used very 1  c o n v e n i e n t l y as a means of m o t i v a t i n g the work i n grade XII algebra.  The f o l l o w i n g are some examples o f such: Mathematical R e c r e a t i o n s  1. A s h i p i s twice as o l d as i t s engine was when the s h i p was as o l d as I t s engine i s now. T h e i r combined ages are 42. ~ How o l d are they? 2. A farmer bought one hundred head o f 3tock c o n s i s t i n g of c a l v e s , sheep and lambs. The c a l v e s cost $10 each; the sheep $3 each; and the lambs, 50^ each. A l t o g e t h e r the hundred head cost him one hundred d o l l a r s . How many of each d i d he buy? 3. My f a t h e r was born on 2  (l<f>*  JU  Efl  2  J L  (2.3.3 .6 )-3 J)' What was h i s age on August 10th, .1935? 4. I f Dr. Jones l o s e s 3 p a t i e n t s out of 7; Dr. Smith, 4 out of 13; and Dr. Brown, 5 out of 19; what chance has a s i c k man f o r h i s l i f e who i s dosed by the three d o c t o r s f o r the same disease? s  2  98 5. Mr. Dough, a b u s i n e s s manager, wished to h i r e one of three men. In order to d e c i d e which was the smartest, he adopted the f o l l o w i n g system. He c a l l e d the three men to him and showed them f i v e p i e c e s o f paper on h i s desk. Two o f the papers were b l a c k and three were white. He then t o l d the men to t u r n around, and he pinned a paper on each man's back, p u t t i n g the two remaining papers i n h i s pocket. He t o l d the men to look at the papers on each o t h e r ' s backs, and by so doing d e c i d e what c o l o r o f paper was on t h e i r own backs. The f i r s t man to d e c i d e the c o l o r of the paper on h i s back was to come up to the desk and e x p l a i n how he made the d e c i s i o n . I f h i s e x p l a n a t i o n were sound he would get the p o s i t i o n . One of the a p p l i c a n t s was s u c c e s s f u l i n determining the c o l o r o f paper on h i s own back. Which one c o u l d i t have been, and how d i d he decide? 6. An eagle and a sparrow are i n the a i r , the e a g l e 100 f t . above the sparrow. I f the sparrow f l i e s s t r a i g h t forward i n a h o r i z o n t a l 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 b e f o r e the eagle reaches the sparrow"? 7. To prove that 1 2 Let x - 1 Then x" - x . . x ~ l - x-1 s  2  Factoring (x-l)(x-t-l) Dividing x+1 1 but x 1 . [ . 1 + 1 = 1  =  x-1  =  =  . . . S a l  8. To prove t h a t -1 = 1 i I - a _ a* 3  ~? "~ "I?  . '. m  ?i 1  (-ai)  2  - (a*)  2  -a - a . . —1 s 1 Where i s the f a l l a c y ? 9. F i n d the keyword i n the f o l l o w i n g problem i n " L e t t e r D i v i sion": 0PN)A0UIERT(PCAAU CPN PIUI PUCM RRIE RHAH REER RHAH RIRT RCUN  99 A  teacher  algebra  has  teaching  teaches  at h i s  to  rivalry  who  more  than  d i s p o s a l another  some e x t e n t ;  namely,  the  classes,  and  p u p i l s know b e f o r e h a n d  obtained  i n the  each age  pupil of  the  The has  will  a record and  even  best can  minds  the  of be  that  more  Interest In  taken,  itself;  element  of  few  the  to  both  results  compared,  pulling  too,  This of  will  to  then  down t h e  aver-  nevertheless,  the  the  100$  to  a  teacher  to  surpass  certain  in  their matriculation  class a  the  than  strive  If  will  former  make  up  p e r f e c t mark. p u p i l s from  in  the  study  introduction of  i s a v a l u a b l e means  of  a  Care  developing of  the  slight  motivating  the  algebra. force to of  a  s c h o l a r s h i p s and  motivation  class,  by  of  certain  students  informed  class  prevent  when  previous matriculation  obtain  marks  form  by  i n the  examination  i s operative  be  friendly  i s given  that  results.  obtained  course,  XII  outstanding  should  they,  motivating  percentage  his  i s employed  present  students  competition  i n grade  grade.  of motivating  to be  avoid  obtained  previous  good  of  subject  which  the  the  be  The  results  n a m e d who  must  work  to  XII  same t e s t  going  of motivation  encourages h i s  examinations, their  h i s utmost  type of  classes,  students  do  grade  class.  same  the  classes are  of  stimulation of  classes.  two  the  class  means  between the  If  one  the  but to  extent affects  their  teacher  as  only  a  very  small  i n encouraging  very to  one  i n the m a t r i c u l a t i o n  i t Is'value do  prizes i s  best.  what  These  the pupils  scholarships  and  100 prizes  are  impressed  offered upon  to m a t r i c u l a t i o n pupils,  them  that as  awards  just  as  well  in  case  of  rivalry  the  tion  should  behind major this  a  not  be  student's  forms  of  chapter.  they  any  are  other  between  allowed  but  to  student  classes,  t o become  efforts,  motivation  able  i t should  o b t a i n one i n the  this  the  should  and  d i s c u s s e d i n the  these  province.  form  of  dominating  only  of  As  motivaforce  supplement earlier  be  part  the of  101 CHAPTER VII SOME EXPERIMENTAL EVIDENCE TO SHOW THE EFFECTS OF MOTIVATION Every teacher of mathematics has n o t i c e d t h a t c e r t a i n of his  l e s s o n s have seemed very d u l l and u n i n t e r e s t i n g b o t h to  h i m s e l f and to h i s p u p i l s , w h i l e other l e s s o n s have been'exceedingly enjoyable.  When.the f i r s t  type of l e s s o n i s  complet-  ed the teacher experiences a sense of r e l i e f , and the p u p i l s l o s e no time i n g e t t i n g away t h e i r mathematics books  and  p r e p a r i n g f o r the next l e s s o n ; but when the second type of l e s s o n i s f i n i s h e d the teacher experiences a sense of g r e a t s a t i s f a c t i o n , and the p u p i l s continue working even a f t e r  the  b e l l has rung, t r y i n g to complete as many q u e s t i o n s as p o s s i b l e b e f o r e the next l e s s o n b e g i n s . i s obvious.  The f i r s t  The reason f o r the d i f f e r e n c e  type of l e s s o n was  t i n g value, w h i l e the second had  l a c k i n g In motiva-  some f e a t u r e s about i t which  aroused and h e l d the i n t e r e s t of both p u p i l s and t e a c h e r . b e n e f i c i a l r e s u l t s d e r i v e d from t h i s second l e s s o n would doubtedly outnumber those d e r i v e d from the f i r s t ;  The un4  and at the  same time, the m a t e r i a l s t u d i e d i n the h i g h l y m o t i v a t e d l e s s o n would be b e t t e r understood  and more e a s i l y remembered than  that s t u d i e d i n the d u l l mechanical l e s s o n . The w r i t e r has r e c e n t l y completed  a three-year  experiment  with methods of m o t i v a t i n g the t e a c h i n g of h i g h s c h o o l mathematics.  Although  the r e s u l t s of t h i s experiment  cannot  be accepted as c o n c l u s i v e because of the smallness of the  o  102 and because of the f a c t that a number o f v a r i a b l e f a c t o r s were not c o n t r o l l e d , n e v e r t h e l e s s they may be c o n s i d e r e d as evidence to show t h a t e f f i c i e n c y and I n t e r e s t can be developed  i n the  s u b j e c t s o f a l g e b r a and geometry to a marked degree by the use o f a h i g h l y motivated form o f t e a c h i n g . The w r i t e r conducted 1931  and 1934.  the experiment between the years  He taught the experimental  c l a s s f o r three  years i n s u c c e s s i o n i n grades Ten, E l e v e n and Twelve.  The  c l a s s was an extremely heterogeneous one, b e i n g composed o f p u p i l s of many d i f f e r e n t types and o f s e v e r a l d i f f e r e n t nationalities.  Some o f the p u p i l s had done t h e i r grade IX  work i n j u n i o r h i g h s c h o o l and others had come d i r e c t l y t o the senior h i g h s c h o o l from grade V I I I i n the elementary  school.  There was a wide v a r i a t i o n i n t h e i r academic s t a n d i n g , some advancing  t o grade X w i t h a very good s c h o l a s t i c r e c o r d and  others b e i n g promoted only on t r i a l .  The c l a s s as a whole  was n o t c o n s i d e r e d a good c l a s s , the p r i n c i p a l ' s o p i n i o n b e i n g t h a t i t was a very weak one. On b e i n g a s s i g n e d as teacher to the above-mentioned c l a s s , the w r i t e r attempted to develop  the p u p i l s '  interest  i n mathematics and at the same time to b r i n g them to a h i g h 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 o f the methods mentioned i n t h i s t h e s i s .  The r e s u l t s o f  the experiment can best be judged from the t a b l e s which follow.  In the week p r e v i o u s to the m a t r i c u l a t i o n examina-  t i o n s , the p u p i l s o f the c l a s s were asked t o w r i t e down the  103 names o f the three s u b j e c t s which they had enjoyed  the most  d u r i n g t h e i r h i g h s c h o o l course - these names to be i n order of m e r i t .  The r e s u l t s o f t h i s vote, g i v i n g F i v e f o r f i r s t  c h o i c e , Three f o r second choice and One f o r t h i r d , are as follows* Algebra . . . . Chemistry . . . Physics . . . . Geometry . . . French . . . . Literature . . Social Studies. Composition . . Physiology . . Grammar . . . . The  above procedure  the w r i t e r conducted by the p u p i l s .  54 51 31 24 17 19 16 3 1 0.  had i t s l i m i t a t i o n s i n as much as  the v o t i n g and the b a l l o t s were signed  However, the p u p i l s were I n s t r u c t e d v e r y  e m p h a t i c a l l y t o overlook the p e r s o n a l element e n t i r e l y and g i v e exact statements  of their preferences.  At the same time the p u p i l s were asked to w r i t e down the names of the three s u b j e c t s i n which they thought had o b t a i n e d the h i g h e s t degree o f e f f i c i e n c y ,  they  and i n which  they were roost c o n f i d e n t o f making good marks i n the f o r t h coming m a t r i c u l a t i o n examinations. follows:  The r e s u l t s were as ;  Algebra . . . . Geometry . . . Chemistry . . . Literature . . Grammar . . . . Physics . . . . S o c i a l Studies French . . . . Composition . . Physiology . .  54 44 28 22 20 19 13 10 4 1  104 The marks a c t u a l l y o b t a i n e d by the p u p i l s of t h i s c l a s s i n the Junior Matriculation'examinations  are shown i n the t a b l e given  below. TABLE I I .  •  ~  C l a s s marks - J u n i o r M a t r i c u l a t i o n examinations Pupil A B C D E P G H I• J K L M N 0 P Q R S 'T U V W X Y Z a Notes:  The  Algebra  Geometry  92 75 19 74 95 83 84 83 85 44 80 84 87 63 83 94 68 58 94 92 81 55 100 94 62 56 82  92 71 17 50 81 79 61 70 61 45 69 78 80 60 83 92 67 74 83 72 69 57 82 89 67 55 73  1. P u p i l C, whose marks, brought the c l a s s average down c o n s i d e r a b l y , was o n l y a c o n d i t i o n e d student, not h a v i n g passed grade XI; and d u r i n g the s p r i n g term he was i n j u r e d i n a t r a f f i c a c c i d e n t which caused him to be absent from s c h o o l f o r s i x weeks. 2. There was one " r e p e a t e r " i n the c l a s s .  f o l l o w i n g t a b l e compares^the averages of the other  c l a s s e s i n the same school and w i t h the C i t y (Vancouver) and  '  105 Provincial  (B.C.) averages: TABLE I I I Comparison o f Averages i n S u b j e c t s A l l Other A l g e b r a Geometry S u b j e c t s  Averages of Experimental  Class  Averages o f other Classes i n Same School C i t y averages (Vancouver) P r o v i n c i a l Averages (B.C.)  76.6  69.5  59.5  57.2  66.8  60.2  63.  65.8  59.8  63.5  63.9  60.7  In the above t a b l e c e r t a i n p o i n t s should be n o t i c e d when c o n s i d e r i n g the r e s u l t s of u s i n g a s p e c i a l l y motivated of t e a c h i n g .  form  I n the f i r s t p l a c e , the t a b l e i n d i c a t e s very  c l e a r l y that the c l a s s was c e r t a i n l y not a " p i c k e d "  class.  The average of the c l a s s In a l l s u b j e c t s other than a l g e b r a and geometry i s 59.5$ which i s below the average of the three other grade X I I c l a s s e s i n .the same s c h o o l ;  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.  T h i s compari-  son i n d i c a t e s t h a t the c l a s s as a whole was r a t h e r on the weak side/.-  In contrast  with t h i s comparison, the a l g e b r a and  geometry averages f o r the experimental  c l a s s are considerably  above the averages f o r other c l a s s e s i n the same s c h o o l , a s . as w e l l as b e i n g much h i g h e r averages f o r these  subjects.  than both the c i t y and p r o v i n c i a l I t should be n o t i c e d  still  f u r t h e r that the average In a l g e b r a f o r the experimental c l a s s i s e x c e p t i o n a l l y h i g h , and when the v o t i n g was taken  106 b e f o r e the m a t r i c u l a t i o n examinations were h e l d , the p u p i l s expressed  themselves as f e e l i n g b e t t e r prepared  a l g e b r a examination than f o r any o t h e r .  for their  They a l s o i n d i c a t e d  by t h e i r b a l l o t s t h a t they had r e c e i v e d more enjoyment from the study o f a l g e b r a than from the study o f any other i n t h e i r h i g h school The  evidence  subject  course.  seems to i n d i c a t e , t h e r e f o r e , t h a t the  t e a c h i n g o f mathematics can be made much more e f f e c t i v e by a wider a p p l i c a t i o n o f the p r i n c i p l e s o f m o t i v a t i o n .  I t seems  safe to i n f e r t h a t , by a c a r e f u l study of the v a r i o u s means of m o t i v a t i n g  specific  the t e a c h i n g o f mathematics In h i g h  s c h o o l , average and even weak c l a s s e s may be brought up to a h i g h l e v e l o f e f f i c i e n c y i n both a l g e b r a and geometry; and a l s o t h a t the enjoyment of these be i n c r e a s e d very g r e a t l y .  s u b j e c t s by the p u p i l s may  107 CHAPTER  VIII  GENERAL CONCLUSIQNS I n c l o s i n g t h i s t r e a t i s e on M o t i v a t i o n i n the Teaching of High School Mathematics, i t i s apparent c o n c l u s i o n s might he reached;  t h a t the f o l l o w i n g  namely, (1) t h a t mathematics i s  a v e r y e s s e n t i a l p a r t o f an i n d i v i d u a l ^ education;  (2) t h a t  the t e a c h i n g o f mathematics i n h i g h school i s a t present not of  a sufficiently  h i g h standard to enable  students to d e r i v e  the maximum amount of b e n e f i t from t h e i r mathematical s t u d i e s ; and  (3) t h a t the s i t u a t i o n can be improved tremendously by a  more e x t e n s i v e a p p l i c a t i o n of the p r i n c i p l e s o f m o t i v a t i o n . The  statement  t h a t mathematics i s an e s s e n t i a l p a r t of an  i n d i v i d u a l ' s e d u c a t i o n i s c o r r o b o r a t e d by the statements o f numerous a u t h o r i t i e s on the s u b j e c t , who commend i t s study both from a u t i l i t a r i a n  and an a e s t h e t i c p o i n t of view.  a s s e r t i o n o f Comte t h a t " A l l s c i e n t i f i c  The  e d u c a t i o n which does  not commence w i t h mathematics i s , o f n e c e s s i t y , d e f e c t i v e at i t s foundation" ^  i s emphatic indeed; while W. A. M i l l i s  s t a t e s t h a t "For a l g e b r a there i s no s u b s t i t u t e .  The e l i m i n a -  t i o n o f a l g e b r a as a pure s c i e n c e from the c u r r i c u l u m would (2) nut the f o u n d a t i o n from under a l l s c i e n t i f i c  procedure."  D a v i d Eugene Smith speaks o f geometry i n m e t a p h o r i c a l as f o l l o w s : "Geometry i s a mountain. (1) Jones, Mathematical (2) M i l l i s , O p c i t .  Wrinkles.  P. 240.  language  V i g o r i s needed f o r i t s  P.^s".  108 ascent.  The views a l l a l o n g the road are m a g n i f i c e n t .  e f f o r t of climbing i s stimulating.  A guide who  The  p o i n t s out  the  grandeur and the s p e c i a l p l a c e s o f i n t e r e s t commands the a d m i r a t i o n of h i s group o f p i l g r i m s . " q u o t a t i o n s might be g i v e n emphasizing  Numerous o t h e r the same f a c t - t h a t  the study of mathematics i s a most e s s e n t i a l p a r t of a h i g h s c h o o l student's  education.  In support of the c o n c l u s i o n t h a t the t e a c h i n g of mathematics i s a t p r e s e n t not of a s u f f i c i e n t l y h i g h to enable  the students  standard  to d e r i v e the maximum amount of b e n e f i t  from t h e i r mathematical s t u d i e s , we have the statement Wi  A» l i i i i s - i - - t h a t  "The  reason f o r the present u n s a t i s f a c -  t o r y s t a t u s of mathematics i s poor t e a c h i n g . " a l s o the s t a t i s t i c a l  evidence  that the percentages  of f a i l u r e s i n mathematics, and  drawals due  of  There i s  g i v e n by P r o f e s s o r Judd showing with-  (3 to l a c k of i n t e r e s t i n mathematics, i s v e r y h i g h ;  To these might be added s t a t i s t i c a l  evidence g i v e n by  Millis  emphasizing  the same p o i n t and o u t l i n e d i n h i s "Teaching (4) High School S u b j e c t s " . E. R. B r i s l o c k s t a t e s t h a t  of  (5) "Algebra i s a s u b j e c t d i f f i c u l t to l e a r n and to t e a c h , " and P r o f e s s o r Judd w r i t e s t h a t "Mathematics must be  recognized  as one of the most d i f f i c u l t s u b j e c t s i n the h i g h s c h o o l (1) (2) (3) (4) (5)  Jones, I b i d . P. a&'va i l l l s V e , O p c i t . P. 233 Judd, C H . Psychology of H.S.Subjects. P. 18. M i l l i e , O p c i t , P. 232 B r e s l i c h , Problems i n Teaching Secondary School Mathematics. P. ToTI  109 course."  Both, these men f o l l o w up t h a i * statements hy  the c o n c l u s i o n s t h a t the t e a c h i n g o f mathematics I s a t present not up t o the standard necessary  f o r producing  best  results  i n the s u b j e c t . The  t h i r d o f our g e n e r a l c o n c l u s i o n s , t h a t the t e a c h i n g  of mathematics can be improved tremendously by a more e x t e n s i v e a p p l i c a t i o n o f the p r i n c i p l e s o f m o t i v a t i o n , i s supported  to a l a r g e extent by the evidence  preceding  chapters  o f t h i s t h e s i s , but these may be supplemen-  t e d by the o p i n i o n s o f a u t h o r i t i e s on these P r o f e s s o r Judd says t h a t "We i n such negative  presented i n  questions.  s h a l l c e r t a i n l y need to i n q u i r e ,  cases, what i s the p s y c h o l o g i c a l c h a r a c t e r  of p l e a s u r e , and what the p o s s i b i l i t y o f so r e a d j u s t i n g the s i t u a t i o n as to produce p l e a s u r e through the study o f (2)  geometry."  v  F. W. Westaway a l s o s t r e s s e s the importance  of i n c r e a s i n g the i n t e r e s t o f students  i n the s u b j e c t o f  mathematics, and he suggests s e v e r a l ways by which t h i s may be brought about.  He says that "A young boy's (13-14) n a t u r a l  fondness f o r puzzles o f a l l k i n d s may o f t e n u s e f u l l y be (3) employed f o r f u r t h e r i n g h i s i n t e r e s t i n geometry."  v  ;  Westaway a l s o s t a t e s t h a t "No boy can become a s u c c e s s f u l mathematician u n l e s s he g i g h t s h a r d b a t t l e s on h i s own b e h a l f , " and (1) (2) (3)  t h a t "There i s no b e t t e r means o f g i v i n g a boy a permanent Judd. O p c i t . P. 17. Judd. O p c i t . P. 89. Westaway, Craftsmanship i n the Teaching of Elementary Mathematics. P. 229  110 i n t e r e s t i n mathematics than to h e l p him  to a c h i e v e  of the commoner forms of mathematical p u z z l e s and The  e f f e c t s of paying  fallacies.  s p e c i a l a t t e n t i o n to m o t i v a t i o n i n t  the t e a c h i n g of h i g h s c h o o l mathematics may g r a p h i c a l l y as shown on p. The  a mastery  he  illustrated  111.  g r a p h i c a l I l l u s t r a t i o n s g i v e n 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 g i v e us  a v i v i d comparison between the amounts of i n t e r e s t  created.  I t w i l l be n o t i c e d t h a t graph I I i n d i c a t e s t h a t even though a teacher of mathematics i g n o r e s the p o s s i b i l i t i e s m o t i v a t i o n , some students develop the s u b j e c t .  of  a considerable i n t e r e s t i n  These students, however, do not reach, the same  h e i g h t of i n t e r e s t which they would r e a c h under more f a v o r a b l e conditions.  The m a j o r i t y of students under the c o n d i t i o n s of  graph I I w i l l undoubtedly develop  a d i s l i k e f o r mathematics,  and i n a few cases t h i s d i s l i k e may  be  exceedingly g r e a t .  In  graph I I I i n d i c a t i n g the amount o f i n t e r e s t c r e a t e d by a teacher u s i n g a h i g h l y motivated  form of t e a c h i n g , we  notice  that the m a j o r i t y of p u p i l s under such c o n d i t i o n s develop r e a l i n t e r e s t In the s u b j e c t , and exceptionally high l e v e l . p u p i l s develop  some of these r e a c h  Only a small percentage of  an a c t u a l d i s l i k e f o r mathematics.  shows us the s i t u a t i o n which might be reached a l l y w e l l developed  (1) Westaway, I b i d .  system of motivated  P.  11.  i f an  a  an the  Graph IV exception-  t e a c h i n g were employed.  112 In an average c l a s s i t should be p o s s i b l e to have every  member  i n t e r e s t e d to some extent i n the s u b j e c t of mathematics.  If  the c o n d i t i o n s i n d i c a t e d i n graph IV were reached,, then students would not only g a i n much p o 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 f o l l o w to some extent  the l e v e l of i n t e r e s t , and the whole s t a t u s of  mathematics i n our h i g h schools would be tremendously Improved. There i s , t h e r e f o r e , an urgent  need f o r a g r e a t e r a t t e n -  t i o n to m o t i v a t i o n i n the t e a c h i n g o f h i g h s c h o o l mathematics, and i t has been p o i n t e d out i n t h i s t h e s i s t h a t there a r e v a r i o u s s p e c i f i c means by which the p r i n c i p l e s o f m o t i v a t i o n may be a p p l i e d to the t e a c h i n g o f a l g e b r a and geometry.  By  employing c e r t a i n g e n e r a l methods o f m o t i v a t i o n throughout a l l the grades, and by supplementing these by more d e f i n i t e methods e s p e c i a l l y s u i t a b l e f o r c e r t a i n grades, the s u b j e c t of mathematics c o u l d be made I n t e n s e l y i n t e r e s t i n g , both to the p u p i l s and to the teacher. lowed by more teachers  ^  I f t h i s procedure were  fol-  o f h i g h school mathematics, then the  i n c r e a s e i n i n t e r e s t , f o l l o w e d by a n a t u r a l i n c r e a s e i n e f f i c i e n c y , would r e g a i n f o r mathematics some o f the p r e s t i g e i t has l o s t due to the use o f mechanical methods o f t e a c h i n g seriously d e f i c i e n t i n motivating  value.  (1) The importance o f c r e a t i n g i n t e r e s t i s d i s c u s s e d by Stormzand, and he a r r i v e s a t the c o n c l u s i o n t h a t "the problem o f I n t e r e s t p l a y s such an important p a r t i n education because success i n a l l t e a c h i n g i n v o l v e s the a r o u s i n g o f sufficient interest." Stormzand, P r o g r e s s i v e Methods o f Teaching. P. 129.  113 APPENDIX I GRADE X I I ALGEBRA QUESTIONS SERIES FOR REVIEW 1. S o l v e :  2 _ 3 41 x 2y - 35  2| . 3j _ _73 2x y ~ "70 2. S o l v e :  2y - x - 4xy 4 _ 3 x °  y 3. S o l v e :  |x  9  -  gy + gZ  8  =  1 1 1 gx - §y -+ gz - 5 c  l x 4- l y - z  -7  4. S o l v e :  x-f-z-1 a |(x=*-4z-8)  5. S o l v e :  I -t- | 2 3 6 1^-1 - 2 2 3 6 " =  5+2  3 6. S o l v e :  1  =  |(x-*-9z~27)  =  P  8  _io 2 -  x-8  x  4  1  x-10  x-5 . x-7 _  x- 6  ^  x-7  x-9  7. o l v e :  5x-64 _ 2 x - l l _ 4x-55 _ x-6 x-13 x-6 - x-14 x^V  8. S o l v e :  x x-2  9. S o l v e :  x x-v-b-a  s  9~x  r» 1 T^x - x - 1 b . _ x -t-b-c ~  10. S o l v e :  x-bc . x-ca  11. S o l v e :  5 x-t-2a  L  x-ab  _8 1 x-a ~~ a  8-x B^x  y  114 12. S o l v e :  _£,x-  ^ _  4  Sd  4 c  I "~ cx  die  13. S o l v e :  jx -6ax+9a -b  14. S o l v e :  m(x-t-y)4h n(x-y) 2mn m(x-+ y)-n(x-y) - mn l x - my § x y  15. Solve:  2  2  0  2 s  s  c  16. Solve:'  y  *±JL  s  £.  +  2 3 y-+ b , a 2 3 +  X  17. S o l v e :  a  +  2x , 5 x - l x-1  18. S o l v e :  5x-ll  x + 2 -  x-2  (a) 4x -10x = -5 2  (b)  x -f 6X-J-3 - c 2  19. Write the general equation f o r q u a d r a t i c s and s o l v e i t f o r x. 20. S o l v e :  (a) x -»- a x - a 2  - 0  2  (b) x(a-x)  c  =  2  21. F i n d two v a l u e s f o r x which w i l l make x ( 3 x - l ) equal to 0.362, g i v i n g each value to the n e a r e s t hundredth. 2 2 " 22. S o l v e : (a) x -»= a -hb 2  2  (b) ( x + 2 ) 2  (c) x(x-2x)  29(x -H2)-198  2  2  s  8 =  &  +-7a  2  -Sax  (a) x -3x-2 - 0 3  (b) x -t- 2x -  3x  (c) x + 6 a  7a x  4  3  24. S o l v e :  4  g  x 23. S o l v e :  2  3  3 P 4x -15X^H- 1  2  2  =  =  0  having g i v e n one r o o t as x s -  115 - g ( x - l H l O  25.  Solve:  (x-1)  26.  Solve:  ( a ) 3x-+ 3y  2  2xy-+y 27. S o l v e :  _ = 178  y'  x2 1 x  29.  Solve:  (a) ^  Solve:  1  =  1*  2  3xy+ y  4  18  2 =  y + 1 0 y s 1 4 5 ( ) 2x -3xy+ 2 y 2xy 2  7+ y  b  2  - 2  2  4  2  2  o  x^-i- xy +-y  x - 4 x y + y^-H f = 0 2  z 24  2  (a) x - + x y - t y  4  p  - 21 ( b ) 4 x - 2 x y + y 8x -t- y = 7 2  3  ( b ) x-+ 3xy  3  3  s  35 =22  Solve:  (a)  32.  Solve:  ( a ) 2J/X-1 - V/4X-11  ( b ) 2^/5x-35  33.  Solve:  (a) |/xT~a-)/x-a l/a  (b) \^x+ 7+|/x + 2  34.  Solve:  (a) \/x -fr 3  ( b ) )/x -t- a + / x T b  x ry = 243 xy(y-x) s 162 3  y-t- 2xy  s  3j/x-5  31 217  2  31.  3  4  y  2 x  x  30.  =  2  _1  (b) 4x -+ xy -. 7  ( a ) x -2xy - 24 .xy-2y  35 12 ZE  =  0>) -1  39xy 28. S o l v e :  xy 1' 1 x y =  - 7y-2  (a) 1 . 1 x .  To)  4  :  2  &  *  s  *• 5^/.2x-7 y/6x-r 13  =  - — V'X-»- a  \/x - 2 ~ 3[/x-13 35.  Solve:  ( ) 6 \/x-7 _  7 t/x-26  t/x-1  7 Vx-21  &  (b)  /'2  2*  x  5a l/a*< x  36. S o l v e :  (a)  2  5  \/9 t 2x - )/2x *  |/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. M u l t i p l y : 39. D i v i d e :  3m -3m  2m  3  x x  - y  4  5m -f  by  -2y  by x  4  -x A  40. F i n d the sq. root o f :  A T  (a) 9x-12x -t-10-~ •+• = s  VX  x  Cb ) 12a -t- 4 - 6 a + a x  r  41. S i m p l i f y :  (b)  , \JH \^  ( o j \  '  .  3x  Q~ '  /j  4  x  + 5a  2 x  ) ^  l / ^ W r ^ " )  /3  -2 42. S i m p l i f y :  ^  (  - j  -r (  43. S i m p l i f y : (a)  ^  ^  3  (a -b )2x(a^-b)^x(a-b)§ 2  2  l/(a b -4 a  (b)  3  3  f/Cb^a^ )3  6  1  44. S i m p l i f y : (a)  3.2"-4.2^g 2 .2 " n  n  a  4  ^ .3 "  ~  n  n  1  V l  6  (a) x«2 \/x~15  x  5 ~"  n  1  (b) x - 4 9 a  A  (c)  3a +5a -»-2  (e)  x" -+ x ~ ~ 2 c  s  4  2c  7 x  *  x«5 Vx-14  _ -  (d) x -+ 27 3m  c  47. Express i n i t s simplest ( )  2f_l£ „ 6^111.  15~ ~"^  1  45. S i m p l i f y :  46. F a c t o r :  (b)  form f r e e f r o m _ r a d i c a l  ^ J L 1-t V*  w  l  (b) J->-ab  signsi  \/iT  a  ab~b  l/a-b  48. F i n d the value o f : (a)  3J/147-7\/U  - i i 1/1  (b)  |/l8p q -pj/8pq -qj/50p q 3  3  3  3  117 49. F i n d the v a l u e o f the f o l l o w i n g to two decimal p l a c e s : (a) 54 2 1/3  (b) 7 1/3 - 5 ]/2  7-4 ]/3  ]/48 + V18  50. F i n d the value o f : 3cx  v  4cx  3c  8c  v  51. F i n d the v a l u e of : (a)  \/2__ 3~\/2  2+ x\/2 .7  (b)  52. Express w i t h r a t i o n a l (a)  22 3\/2*|/7  (yf W7  .)  8  denominators:  V& -t- x 4  \/a~x \/a -»• x- V^a-x  (b) 2+ |/3 -»|/7  <>  1  c  \/3 + Vs" - v^s" 53. Express i n i t s simplest form: (a) l/x -y -» x _ 2  x -» y >-y  2  i / 2 2 |/x -+ y 4-y  2  (b) 4(^34-1)  2  , / 2 2 x- |/x -y  2+-1/3  1/3»1  2~ V/3  v  /  54. F i n d the square r o o t o f : (a)  49^20/6  (b) \/175 - ^147~  (c)  3x-l-r 2 \/2x -3x-2 2  55. Express i n i t s s i m p l e s t  (d) 2m-t- 2 \/m -9n 2  2  form:  (a) \ / l 9 -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 t h a t j / a + y b cannot be expressed i n the form |/x + |/y 2 u n l e s s a -b i s a p e r f e c t square. 57. (a) F i n d the r a t i o compounded o f the d u p l i c a t e r a t i o 3:7 and the r a t i o o f 35:27. (b) F i n d the r a t i o of x:y from the e q u a t i o n - 221 2ax * b y ax 58. (a) I f m:n i s the d u p l i c a t e r a t i o o f m-f-x:n-Kx, prove that 2 x mn. d  =  a  x  b  y  119 58.  (b) I f a and b are unequal, and ab(c -j- d ) - b c + a d shew t h a t the r a t i o of a to b i s the d u p l i c a t e r a t i o of c to d. 2  59. If 2 q 60.  i  S s  a  B  prove that  _  u  a  I f |« c g = |  3^2  2  2  2  2  2  H±  u 2  q  s  u  prove that each r a t i o i s equal to 5/ 6 a c e - c e f 4-7ac 2 2 4 2 5 6b d f - d f + 7ad 2  2  4  5  61. i f  22 :• shew t h a t a p + b q - c r bz-cy "" cx-t-az ~ ay-Hbx xp-^yq-4-zr 0.  =  0 and  3  5 2  '  I  f  o,Til ^y**ox Prove  I oz« f r ly_L S = Iz—x ? = 1  t h a t each of these r a t i o s i s  equal to £; hence show t h a t e i t h e r x - y or z  a  xty.  .y. If g = § b d  63. (a) (b)  If  b  S prove t h a t . d  a  b  prove t h a t  s  &  =  c  "~ £  £ + £ d  d  64. '(a) F i n d the f o u r t h p r o p o r t i o n a l  to 12x , 3  (b) F i n d the t h i r d p r o p o r t i o n a l to 5  9ax , 8a x. 2  3  1/15.  (c) F i n d the mean p r o p o r t i o n a l to 2 1/18; 3 1/128. 65. I f a, b, c are three p r o p o r t i o n a l s ,  shew that  ( b + be + c ) ( a c - b c t c ) - b + a c + - c . 3 3 3 66. I f a:b - c:d, prove t h a t ^ ~° ^ -HI b^c - c d ad • 2  2  2  4  3  4  a  =  3  2  67. I f a, b, c, d are i n c o n t i n u e d p r o p o r t i o n , a:d  a +b + c 3  s  3  prove  that  b +c -r-d .  :  3  3  3  3  68. I f (a + b-3c-3d)(2a-2b-c-+d) (2a + 2b-c-d)(a-b-3c -f3d) prove that a, b, c, d are p r o p o r t i o n a l s . 69. I f b + c i s the mean p r o p o r t i o n a l between a + b and c + a show that b + c : c - + a c-a:a-b. 3  =  70. I f 2 ,  &  then  * -*y+y 2  2  _  x  2  J  a+b 2 . . , - 2 - ~T> ^ a ab+b a 71. I f 12:x = x:y - y:z z:18, c a l c u l a t e the v a l u e of x to two decimal p l a c e s , and show t h a t x +y +z - (x + y v-z )(x -y +z ) s  4  4  4  2  2  2  2  2  2  120 72. I f a+-x:a~x i s the d u p l i c a t e r a t i o o f a+b:a-b, then x-b:a-x 5 b ( a + b ) : a ( a - b ) . 73. Resolve i n t o f a c t o r s : (c)  c d e -l 0 p x^-y^-t- x H-y  (e)  2cd-2xy+x -»-y -c -d  (g)  c  (I)  4m -21m n + n  (a)  3  6  9  2  2  2  2  +-9(d ~a ) + 6 c d  2  2  4  2  2  (k) - x - y 8  2  4  8  (m)  (c -hd) -»- ( c - d )  (0)  6a -f  (q)  x -49  (s)  x- +x- -20  3  3  a ~19a -t-6  3  2  a  2 c  C  (b)  a - 3 a ~ a b -+ 3 a b  (d)  9(a-2b) -(4a-7b)  (f)  a -6a -a -2ab-b -+ 9.  (h.)  x  (j)  30a -*-37ab-84b  (1)  3(2b ~l)-7b  (n)  x -37x~84  (P)  a-7 V a -t-12  (r)  3a«-+ 5 ^ 4 - 2  (t)  x  4  3  3  2  6  3  2  2  +- 2 5 y - 1 9 x y  4  4  2  2  3  -t  3 m  27  74. S i m p l i f y : (a)  1-t 8 x 4 x - x _ (l-2x) -h 2x 2 * 2 • 2 (2-x) l-4x 2-5x-t-2x 3  3  2  v  x  ( ) b  x (x~4) 2  . (x ~4x)  2  2  (x + 4 r ~ 4 x  3  y  64-x  (x-f4)'  16-x  3  2  75. S i m p l i f y : (a)  a  3a 2  x -+• ax-2a  (a-x); (h)  1 a-x  2  2a+x  2x 2 , 2 a + x  1 a -v x  4X " 1  (x -a ) 2  76. F i n d the value o f : t \ 2 2 (a) -m n r n rm . (rrr»n)(m-r) (n-r)(n-m) L  (h)  q -t- r  (x-y)(x-z)  +  77. F i n d the v a l u e o f :  2  2  PxJPlIX (r-m)(r-n) 2  r-4- p  p-» q 4 (y-z)Cy-x) (z-x)(z~y)  (a)  1 -  2  x-» y x+y-  2  2  2  2  121 77.  (b)  2  12-  -1-x3  d  78.  (a) D i v i d e  ^  I6x-27 2 x^ -16  by x-1 +  n e  (b) M u l t i p l y 79.  (a) S i m p l i f y :  x + 2 a - ^ 3  (  8(l-x) (b ) S i m p l i f y :  80.  .  .  13 — ±  by  2 x - a . ^  1  jUx  8(l+x)  4(l-tx ) 2  x+-2y  5x +63xy-4-70y  •tjrx-y  2 x H-  2  2  3xy-35y  82.  83.  2  4  =  85. 86.  87.  a  2  2  b  p  2  84.  2  F i n d the value of f - t * + ~ — • + „ when x » - ^ r 2b.-x 2b-+x 7L -^b a-+-b Draw q u a d r i l a t e r a l ABCD,"the c o o r d i n a t e s of i t s angular p o i n t s b e i n g A(12,4)j B(-2,8); C ( - 8 , - l l ) ; D(9,-9). Calculate i t s area. I f the u n i t used i s 1/10", what i s the area of the q u a d r i l a t e r a l i n square i n c h e s ? • Draw graphs f o r the f o l l o w i n g l i n e s : (a) x + 7 = 0 (b) y-9 = 0 (c) 3x a 4y (d) 2x-3y - 6 (by i n t e r c e p t method) (e) -3y~4x-*5 - 0. 5x-+-17 Draw a graph of the f u n c t i o n — and from the graph read the value of the f u n c t i o n when x - 5 and when x-8. Draw a graph to show the v a r i a t i o n s of the f u n c t i o n s 1.2x-3 and 3.5-3.8x between the v a l u e s 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 e q u a t i o n 1.2X-3 3.5-3.8X. Solve g r a p h i c a l l y 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 ) 4 x + y = 10) (a) Draw the t r i a n g l e whose s i d e s are g i v e n by the equations 3y-x = 9; x + 7y a 11; 3x-fy - 13, and f i n d the c o o r d i n a t e s of i t s v e r t i c e s . (b) F i n d the e q u a t i o n f o r the l i n e j o i n i n g the p o i n t s (4,5), (11,11). Shew t h a t the p o i n t (-3,-1) a l s o l i e s on t h i s l i n e . (c) Prove t h a t the p o i n t s (2,4), (-3,8), (12,-4) l i e on a s t r a i g h t l i n e which cuts the a x i s of x at a d i s t a n c e of 7 u n i t s from the o r i g i n . (a) Solve g r a p h i c a l l y : x -»-y - 41 y - 2x-3 ~ (b) P l o t the graph of the f u n c t i o n y <* x 2x-4 and g i v e the c o o r d i n a t e s of i t s t u r n i n g p o i n t . d  81.  4(x -l)  2  2  d  122 87.  ( c ) F i n d g r a p h i c a l l y the r o o t s of the f o l l o w i n g equation to two decimal p l a c e s : 4X -16XH-9 a 0 2  88. Draw the graph^of: ('a)  y = 2x-  (To)  x -y 2  =  4~2x  In each case g i v e the maximum or minimum v a l u e f o r x or y. 89. (a) F i n d g r a p h i c a l l y the r o o t s o f the f o l l o w i n g equations: x -7x4-ll =0 ^ (b) What i s the minimum v a l u e o f the e x p r e s s i o n x -7x +11? 2  90.  (a) Show g r a p h i c a l l y that the e x p r e s s i o n x -2x-8 i s n e g a t i v e f o r a l l v a l u e s o f x between -2 and 4, and p o s i t i v e f o r a l l values f o r x o u t s i d e these l i m i t s . o p  (b) Solve g r a p h i c a l l y and t e s t a l g e b r a i c a l l y : x  91.  92.  93.  94. 95.  96. 97.  98.  y - 53) y-x = 5) An income o f $160 I s d e r i v e d p a r t l y from money i n v e s t e d at 3\% and p a r t l y from money i n v e s t e d a t 3%, I f the Investments were interchanged the income would be '$165. How much i s i n v e s t e d at each rate? The p r o f i t s o f a b u s i n e s s 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 . I n the f o u r t h year they were three times as much as i n the f i r s t two years t o g e t h e r . The t o t a l p r o f i t i n a l l f o u r years was h a l f as much again as i n the f i r s t and f o u r t h years t o g e t h e r . F i n d the t o t a l p r o f i t s . A man has a number of c o i n s which he t r i e s t o 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 i n c r e a s e s the s i d e o f the square by 3 c o i n s he wants 25 t o complete the square.. How many c o i n s has he? An o f f i c e r forms h i s men i n t o a hollow square f o u r deep. I f he has 1392 men, f i n d how many there w i l l be i n f r o n t . Two men,- A and B, t r a v e l i n o p p o s i t e d i r e c t i o n s a l o n g a i?oad 180 m i l e s l o n g , s t a r t i n g s i m u l t a n e o u s l y from the ends of the road. A t r a v e l s 6 m i l e s a day more ,than B, and the number of m i l e s t r a v e l l e d each day by B i s equal to double the number of days b e f o r e they meet. F i n d the number o f m i l e s which each t r a v e l s i n a day. F i n d two numbers such that t h e i r product m u l t i p l i e d by t h e 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 d i f f e r e n c e i s 30. The i n t e r e s t on a sum o f money f o r 1 year i s £31 17s 6d., i f the r a t e o f i n t e r e s t were l s 3 s by \ per cent i t would be necessary to i n v e s t £100 more t o produce the same' amount of i n t e r e s t . F i n d the sum Invested at f i r s t . There i s a number c o n s i s t i n g of two d i g i t s such t h a t the d i f f e r e n c e of the cubes o f the d i g i t s i s 109 times the d i f f e r e n c e o f the d i g i t s . A l s o the number exceeds twice the product of I t s d i g i t s by the d i g i t In the u n i t s p l a c e . F i n d the number.  123 (a) A boat goes up-stream 30 m i l e s and then downstream 44 m i l e s i n 10 hours, and i t a l s o goes upstream 40 m i l e s and downstream 55 m i l e s i n 15 hours. F i n d the r a t e of the stream and of the boat. (b) How l o n g w i l l i t take each of two p i p e s to f i l l a c i s t e r n I f one of them alone takes 27 minutes l o n g e r to f i l l i t than the other and 75 minutes l o n g e r than the two together? (a) A man buys 99 oranges at a c e r t a i n p r i c e ; they would have c o s t him 1 s h i l l i n g l e s s i f he had o b t a i n e d f o r 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 p r i c e d i d he pay? (b) A r e c t a n g u l a r p l o t of ground i s surrounded by a g r a v e l walk 4 f t . wide. The area of the p l o t i s 1200 s q . f t . , and the area of the walk Is 624 sq. f t . F i n d the dimensions of the p l o t . (c) A man i n v e s t s some money i n 3 per cent stook; i f the p r i c e were £10 more, he would r e c e i v e 1 per cent l e s s f o r h i s money; at what p r i c e d i d he buy the stock?  124 SERIES FOR  COORDINATION  Grade XII  Algebra I  1. S i m p l i f y :  ^ a * KTS 1 1 a b+ c  2. S o l v e :  x  +  g  1 0  x-2 3. S o l v e :  .32x .05  b o 2bc 2  I  1  _ 2 x  4-x  1  -0  2  0  x ~4  2  2  .045x .125  +  /  X  ~  13.52  4. Seven years ago a boy was h a l f as o l d as he w i l l be one year hence. How o l d Is he now? 5. A c o l l e c t i o n of f i v e - c e n t p i e c e s and q u a r t e r s c o n t a i n s 80 coins. T h e i r t o t a l v a l u e i s $16. How many are there of, each? 6. S o l v e :  5x-+ 2y~3z ~ 160 3x -+ 9y+8z 115 2x-3y-5z a 45 7. F a c t o r : (a) x + y - 7 x y . s  4  (b) x  4  2  2  64  3  <T  512  I  (c) x - r x - 1 6 x - 1 6 . 7  4  3  8. (a) Write the g e n e r a l equation f o r a q u a d r a t i c . (b) Write the s o l u t i o n of your equation. (c) By means o f t h i s formula s o l v e : 8x -4x~t-3 a 0 9. A l a b o u r e r worked a number o f days and r e c e i v e d f o r h i s l a b o u r $36. Had h i s wages been 20^ per day more, he would have r e c e i v e d the same amount f o r two days' l e s s l a b o u r . What were h i s d a i l y wages, and how many days d i d he work? 10. S o l v e : x -+ y + x + y - 18 xy = 6 2  p  p  II 1. D i v i d e :  16a~ -6a~ -»- 5a" -*- 6 3  2  1  2. F a c t o r : (a) 6x -fc- 9x-8xy-12y. 2  by  H-2a  _ 1  125 2.  (b) 4 a - V 1 5 a b - 4 b 4  (c) 9 r  2 x  2  -25x  2  6 x  4  .  3. S i m p l i f y by l e t t i n g x 4. S o l v e :  s  .4.  3x-2y-z 4x-3y + 4z 2x-vy-5z  t s  1 -3 -2  =  5. In the formula T » (a) Solve f o r H i n terms of the other  letters.  (b) I f 11=3, iHt^K-nJ* 7 W - « f i n d H. 6. A man s t a r t s from a c e r t a i n p l a c e and walks at the r a t e o f a m i l e s an hour, b hours l a t e r another man s t a r t s from the same p l a c e and r i d e s i n the same d i r e c t i o n at the r a t e of c m i l e s an hour. In how many hours w i l l the second man overtake the f i r s t ? 2 7. S o l v e : b ( a + x ) - ( a + x) (b-x) - x _ j - -22  8. Simpl:^-9. S o l v e :  ' °- "  2 \/x -  4  1  x  b  '  \/4x-3 |/4x-3  10. S o l v e :  4x + y 2  2 a  IV;  2xH-y = 5,  III 1. D i v i d e :  g|a  - §ffax  5  4  by |a * |  x  2. A man walks from A to B i n h hours. I f he had walked'V m i l e s an hour f a s t e r he would have been'tf hours l e s s on the r o a d . F i n d the d i s t a n c e from A to B and the r a t e o f walking. 3. A grocer spent $120 i n buying tea at 60^ a pound, and 100 l b s . of c o f f e e . He s o l d the t e a at an advance of 25$ on cost and the c o f f e e at an advance o f 20$ on c o s t . The t o t a l s e l l i n g p r i c e was $148. F i n d the number of l b s . of tea purchased, and the cost of the c o f f e e per pound. 4. A man spent $90 f o r wood, and f i n d s when the p r i c e i s i n creased $1.50 per l o a d he w i l l get 3 l o a d s l e s s f o r the same money. What was the p r i c e per load? 5. S o l v e : x * x y + y 21 x + xy+ y - 7 4  2  2  4  s  2  2  126 6. S o l v e :  7.  3(z-l) 4(y-+x) 7(5x-3z)  =  2(y-l) 9z-4 2y-9  s =  (a) F i n d t h e square r o o t of 83+  12\/55.  (b) The area of a r e c t a n g l e i s 16 \/10 - 25 and the l e n g t h of one s i d e i s 3 \/5 - \/2. F i n d the other s i d e i n i t s s i m p l e s t form. 8. S i m p l i f y : ( ^-2x%&+ y) S  x  O  a  3t  (x -y ) 9. I f a, b, c are i n continued p r o p o r t i o n , prove t h a t a -f- a c + c s ( a + b - t - c ) (a -b -t- c ) 10. In a c e r t a i n examination, the number of those who passed was three times the number o f those who f a i l e d . I f there had been 16 fewer candidates and i f s i x more had f a i l e d , the t o t a l number of candidates would have been to the number who f a i l e d as 2 to 1. F i n d the number of candidates. 4  2  2  4  2  2  2  2  2  2  IV 1. S i m p l i f y : | | j ( a-b )-8(b-c )] -  ^  - \ {c-a-§< a-b)}  2. (a) For what value of k does (3,2) l i e on the l i n e 4 x + k y 2? (b) Solve g r a p h i c a l l y : y = x|\2 x-2 and y - 2x-3. 3. F a c t o r :  (a)  x -6x -+- 12x-8. 3  (b) 24ax-16a -9x  2  2  2  '(c) x y - + 3 x y - 3 y ~ y 2  2  2  4. Supply the m i s s i n g term so as to make a p e r f e c t square: (a) 9m n -( 2  ) + 4.  2  5. S i m p l i f y :  4 f t  2,  4 f t+  1  2 , 4a hi  6. S i m p l i f y :  ^ ^  (b) &  *~ „ 3 ,  8a -1  25x -h30xH- ( 2  ).  2^ a  2~  -4  13+  7. S o l v e : .05x-1.82-.7x .008x-.504. 8. The p r e s s u r e of water on a pipe t h a t w i l l not break the pipe i s c a l l e d the safe working p r e s s u r e . In c a s t i r o n p i p e s , such as c a r r y water i n a c i t y water-system, the safe working p r e s s u r e i s g i v e n by the formula „ " 7200 . ~ 1-.01B, " ^ =  127 i n which P = the pressure per square i n c h , D - the i n s i d e diameter of pipe i n i n c h e s , T s t h i c k n e s s of i r o n s h e l l i n Inches. Given D r 3 f t ; T a 1§ i n . , f i n d P. 9. A man r e c e i v e s $140 per year as i n t e r e s t on $2,500. $500 i s i n v e s t e d a t 5$; p a r t of the remainder at 5§%; and the remainder at 6%, How much has he i n v e s t e d at 5§$? 10. S o l v e : .2^ ;  2j  =  x +2y 2  2 Q  s  2  18. V  1. Bracket the powers of x so t h a t the signe b e f o r e a l l the b r a c k e t s s h a l l be n e g a t i v e : 3b x - b x - a x - c x - c x - 7 x . 2. A boy can row 10 m i l e s down stream i n 2 hours and r e t u r n i n 3 1/3 hours. F i n d the r a t e which he rows In s t i l l water, and a l s o the r a t e of the stream. 3. F i n d two numbers whose d i f f e r e n c e i s 4 and the sum of whose r e c i p r o c a l s i s 3/8. 4. S i m p l i f y : ^n -t-.l g U •+1 2  2 (4 - ) n  n  L  "  n  ( 4  n  +  4  ±  )  4  n  -  4  2  4  1  5. I f a, b, c, d are i n continued p r o p o r t i o n , prove t h a t i s the mean p r o p o r t i o n a l between (a i-b) and ( c + d ) . 6. S o l v e g r a p h i c a l l y : o o x +j - 100 x-y .2 7. F a c t o r : (a) 12-15a +16x-20ax (b) 27y -512 J  d  d  3  8.  Simplify:  (c) 9 x  5 n  -x  9. S i m p l i f y :  2  t  4ab Solve:  x 3  3  4  S r  2x  2  2x -2  1. S o l v e :  2  (*) -(*) -12(*) ~ 2  10.  (d) x - 4 y 4 -  3 n  3x 4- 3x +• 5  x-(3x- £ ^  2  =  6x + 6  i(2x-57)-|  2  9-6x  (b-i-c)  128 2. S o l v e : .12(2x-t-.05) - .15(1.5x~2) 0.246. 3. One-sixth o f a man's age 8 years ago equals 1/8 o f h i s age 12 years hence. What Is h i s age now? 4. I f i t ' c o s t s as much to sod a square p i e c e o f ground a t 20^ per sq. y d . as to fence i t a t 80(Zf a yd., f i n d the s i d e of the square. 5. S o l v e : 1 1 _ 1 _ i m n p "* 1,1 i1 2 =  —  I —  m  n  T  1 - 1  m 6. S o l v e :  n  2^  2x  4-  -  O  p 1  s  -  3 0  p l l x _ 15 _  7. One s i d e o f a r i g h t a n g l e d t r i a n g l e i s 7 f e e t s h o r t e r than the other, and the area i s 30 sq. f t . F i n d the two s i d e s and the hypotenuse. 8. S i m p l i f y : y ^ J T ^ * v A £ o f xy V xy 9. F i n d a q u a n t i t y such t h a t when i t . i s added to each o f the numbers 11, 17, 19, 23, the r e s u l t s are i n p r o p o r t i o n . 10. (a) I s - t h e p o i n t (3,4) on the l i n e whose equation i s 3x~4y s 12? Give the reason f o r your answer, (b) S o l v e : 2x-t y - 1 y + 4x - 17 Check your r e s u l t . 2  VII 1.  Solve:  .25(x-3)-f .3(x-4) .125 °x-iy» 2. S i m p l i f y : xyiz VxyVz X--=r •2 -1 3 6 1 3 y ' 3. S o l v e : (8x)" "-(8x-15) V8x^15 a. What a r e the c o o r d i n a t e s o f the o r i g i n ? b. What i s the graph o f x 2? c. What I s the d i s t a n c e of any p o i n t P(x,y) from the o r i g i n ? d. Draw t h e graph o f 4y 8x-f-x =  3  yx Vx  *" V ** s  2  =  2  B  Simplify:  6  -^"7 . " **x  5QJC  3x  129 6. S o l v e :  x-2 _ x-4 _ ^ .05 .0625 " 7. The denominator of a f r a c t i o n exceeds the numerator by 4; and i f 5 be taken from each, the sum o f the r e c i p r o c a l of the new f r a c t i o n and 4 times the o r i g i n a l f r a c t i o n i s 5. F i n d the o r i g i n a l f r a c t i o n . =  8. S o l v e : (a)  22x  - 2ax-i-7a  2  (b) 5 x  2  - 17x-10  2  9. S o l v e : 7xy-8x 10; 8y -9xy 18 10. Around a r e c t a n g u l a r f l o w e r bed, which i s 3 yds. by 4 yds., there extends a border o f t u r f which i s everywhere equal i n b r e a d t h and whose area i s 10 times the a r e a o f the bed. How wide Is I t ? 2  2  a  =  VIII 1. F a c t o r :  (a) 2ax -2bx -6ax +-6bx-8a-t- 8b. 2  2  (b) a +  b ~c -9-2ab+-6c.  2  (c)  2  ab(x +1)+x(a -t-b ) 2  2. S i m p l i f y :  ^  ~  3. S i m p l i f y :  ^  ^^  4. S o l v e :  2  •  7  6 \/x-7 j .  ^  4  Vx  X  y g -  _ 7 ^x-26  5  |/x-i  2  2  '  ~~  yyi~2i  5. (a) What do you know about the graph o f the f o l l o w i n g equations: y .5x; y m 5x-4j y = 5x4-6. a  (b) S o l v e g r a p h i c a l l y : 6. S i m p l i f y :  1 x  2  =r  ,  m  2i- -  2x4-y 1  .  0;  =  y  s  i(x-f-5) 3  x  x +4 2x - -r—=r  +7  7. S o l v e : .01(2x -f .205)-.0125(1.5x-.5} 8. S o l v e : x , x a b=  s  .01955  1  3a 6b 3" 9. I n a c o n c e r t h a l l , 800 people a r e seated on benches o f equal l e n g t h . I f t h e r e were 20 benches fewer, two persons more would have t o s i t on each bench. F i n d the number of benches.  130 10.  Solve:  7  IX 1. The s u r f a c e of a sphere of r a d i u s ' V i s g i v e n by the formula S a 4~Jj~r F i n d (a) The surface of a sphere o f r a d i u s 1.4". (b) The r a d i u s o f a sphere whose s u r f a c e i s 38-g- s q . f t . 2. S i m p)llii ff y /, _ ~% - / xj \/ yfa y :: / ->/ 3,—^ \'/ 3  3. F a c t o r :  3 3 m n  (a)  (b) x°-8y -27z -18xyz  729"  (c) x - 1 5 x y + 9 y ; 4  2  2  4  4. a. S o l v e : 22x 3mn+7m . b. The sum o f the r e c i p r o c a l s o f two consecutive 5/6. F i n d the numbers. 5 . F i n d to t h r e e p l a c e s o f decimals the v a l u e o f : 5 4- y l d 2  2  =  4  6.  + 1/18  l^5  Solve g r a p h i c a l l y :  7. F a c t o r :  g  x  +  xf 4  Q C  (a)  a -a -9-2a b -i-b 4-6a  (b)  a 4- 3 a b -*-3ab + 2 b .  4  2  2  2  3  4  2  2  3  (c) a b~a b -a b 4-ab . 8. F i n d as simply as p o s s i b l e the value o f : 4  2  3  2  3  4  .5453  9. D i v i d e :  2 2 x -y  x-y  10. Reduce:  2  X  3/ —  /3  - ^  x  +  y  2  - .4567  .5433 - .4567  „2 x 4 y  numbers i s  x -y 2  2  2  131 X 1. In-the formula F F 200, f i n d v. 2. Show t h a t 1  2 ^I_> g i v e n m = 12.075, r = 3, g . 32.2, S 1 _  3  r  B  +  V16 .+ 2 \/63 3. (a) S i m p l i f y : ^ g (b) S o l v e :  1  +  7  V l 6 ^ 2 \/63 ^  ^ I ^ f  x  y/^11 ^ g y ^ l  6  3 1/x  2  4. S o l v e : (a)  5x -15x-Ml  (b) x -xy y ^ 3xy  0  2  s  2  -6 .10  5. Th.e l e n g t h 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 o f the same perimeter, i t s area would be 1/24 g r e a t e r . F i n d the s i d e s . 6. S o l v e :  2*-27 ^ Z=±2 + x-14 x-8 x-13 x-9 7. J. number has three d i g i t s , the u n i t s b e i n g J the tens and 1/3 of the hundreds. I f 396 be s u b t r a c t e d from the number, the d i g i t s are r e v e r s e d . F i n d the number. =  8. Prove t h a t the p o i n t s (3,2), (8,8), (-2,-4) l i e on a straight l i n e . F i n d i t s equation. Prove a l g e b r a i c a l l y and g r a p h i c a l l y t h a t i t cuts the x a x i s a t a d i s t a n c e o f 1 1/3 from the o r i g i n . 9. S o l v e : (b) x ^ - x y t 2y* = 4 x -3xy 2  2  —  1  4  + — +  5  = 12 .  10. Given /5 - 2.23607 f i n d to 4.places of: 7 yg" t l 5 \/5~2  \/5-l  c-2  3 4 1/5  o f decimals  the v a l u e  132 APPENDIX I I Answers to Review Questions 1. x -  -  2. x  2  V--2 - 2  9. x  s  12. x  P  Tc ^ 15. x  s  0  b  )  o  20. x  r  6. x  ( " ) a a  5  5  13. x  7. x - 10  3a-*-b o r 3a-b - „ 7a+8b 16. x 9 8a j-7b J V 9 19. x " "A b  2  1  2  |(a - ^a -4c)  22. x *3 (b) or 44  21. x  =  o  r  -J  x = 2;-l x - 0; 1; -2 (c) x = 2a; a; -3a , c 26. (a) x = 5; I  27. (a) x  ;  31. (a) x . 6; 3 y s -3; -6 2§  -  ' -3  29. (a) x =^5; - 12 3_  d 2 3 ; X12  y - t3; X J  t f 2  B  ^  =  28. (b) x = t l ;  y = * i b *3  32. (b) x  -8, *  s  y = 3J y ^ ; -  %6 .  t2  o*-.f(/5fl)  22. x - - a (a) o r - b  2 2  y  y  3  o  a  -  29.(b) x  a  <>  27. (b) x J; -2 y = 2; - § =  4  17. x . 4 18. x - 1.81 or 3/2 (a) o r .64 20. x »|(/5-l)  (lo)  ,  28. (a) x  v  23. a  24. x - 3.73; .27 25. x . 4; -1; 3; 26. < b ) x = 5 ; 7  11. x ss 13a or -a  a  .55 "*  22. x - 4a ( c ) or -2a or a  =  8. x - 4  14. x =. 3n+m » 4 „ 3n-m y —7— *  s  =  2  s  8  =  =  10. x - bc+ c a n ab  b  -2c-d or-ec-»-d ab-bm — - — cl  -  4. x ~ 6 y = 11 Z 6  -  ^ - 6  y - al-bm cm 18. x - -5.449 (  I  y a z = 12  s  3. x - 12 z7 r 18 6  y - -  J  5. x  J  a  30.(a) x - i l ; i 2 y s *2; H  (b) x = 3; % y = 1; 6  31. (b) x - 5; - | y - 2; - I i 33.(a)  x  m  2*  '  3  S  -  3  2  (  b  ,  (  )  X (  x  m  «J  g  e  x  3  t  .  }  133 34. a) x - 49 36. a)  x  ab , a 4-b  34.(b) x - 36.(b) x  - 8  =  35.(a) x - 64  8  35.(b) x - i £ " 3  37. a) 38. 15m-3m -2m- -t i A 40. a) 3x -2 4-x~ s  40. (b) a - 3 a - 2 2 x  41. (b)  43. b) a b ( b - a ) 6  b)  o9. X 4 x y +> 2y  a  41. a) b  46. a)  8m  6  x^  42. c ^ "  44.(a)  / 3  x  4.  43. (a) (a-b)  44.(b) 864.  :  45.  1  (Vx + 3 ) (Vx-5 )  (d)  ( x 4 - 3 ) ( x - 3 x + 9)  7)(x *'-7)  (e)  (x~  V  (X*  m  2 m  c t  m  5)(x-°-4)  c) (3a^4-2)(a^-hl) 47. a)  1.  8  s  (b)  49. a) 117.8897 51. a)  1.  (b) .4524  (b)  58. a) aj-^a^-x*  (b) 2 \/5 +3- 1/21  1  (b)  1.  (b)  64. a) 6 a . 4  73. a) b) c) d) e) f)  g) h) i) 4)  k) 1) m) n)  ( c ) 2 [/3 + 3 |/2t[/30  3  -8  1.  :—r£2—~  54.(a) 5-2^6  54. (c) \/2x +1 + Yx-2 55. a)  7 j/2.e  50.  2.  x  53. a)  4.  ( c ) 12 \/2"  ( c d e - l ) ( c d e 4- c d e + 1) a (a-3)(a-b) x4-y)(x-y +1) 3  2  4  6  2  3  2  b-a)(7a-l3b) a - x - a - b ) ( a - 3 + a + b) c 4-3d-3a)(c 4-3d 43a) 3  3  x - 5 y - 3 x y ) ( x - 5 y + 3xy) 2m 4-n -5mn)(2m 4- n -v 5mn) 6a-yb)(5a + I2b) x-y)(x4-y)(x -^y )(x +-y ) 3b -M)(2b-3) 2 c ( c l-3d ) (x 4 3 ) ( x - 4 ) ( x + 7) 2  2  2  2  2  2  2  2  2  2  2  2  4  WffityT  ~ Vl)  54.(d)l/m + 3n 4 l/m-3n  57. (a) 5:21  (b) \/3 2  (b) -4pq /2pq  48.(a) 19 VZ  4  /•u X D OD ID \ J ? — — v> oi* S?~ ^ - t — 3b y a 2a u x  134 (0)  (a 4-2)(3a-l)(2a-3)  (P)  (|/a-4)()'a-3)  (q)  ( x ^ 7 ) ( x S 7)  (r)  (3a^+2)(a^+1)  (s)  (i"  (t)  (x" ^ 3 ) ( 2 - 3 x - t - 9 )  -4)(x" f5)  f i  c  1  2m  m  74. (a) x(2-+x)  (b) x  75. (b)  T  x  (  x  * , 2^ 2 2 2,2 (a -t- x ) ( a -x ) 8  78. (a)  "  4  x u ^ P  )  a  76.la)  W  76. (b)  75. (a)  4  - ( y - z ) + q(z-x) + r ( x - y ) (y-z)(z-x)(x-y) x-3  (b)  °-  77. (a) i  2 t  2 y  y  (b) 6 ( x - l ) x-4  g  2  (2x +5a)(x-2)  x=¥ 79. (a)  2H-X  +-3x  (b)  g  2(1-x )  16x  4 8  83. 21:28.5  85. (a) x ± -2 y = 5  84.  (4,1);  (3,4)  (b) (-1,-5)  1.3.  88. (a) max.  (b) min. - -5  ( c ) x - 3.32 or .68  s  4  2.38 or 4.62  (b) min. - -1.25 91. |3,000 @ Z% $2,000 % 3 J #  90. (b) x - 2, -7 y - 7, -2 93. 600 c o i n s . 96. 6, 5  94. 91 men. 97. £750  99. (a) 3 - P * « m  t  (b) 7y-6x = 11  87. (a) x s 4, -1.6 y 5, -6.2  =  Q  (b) x c 3 y - -2  86. (a) (-3,2);  89. (a) x  <  8  81. 2.525 s q . l n .  =  0  x -256  4  h  100. (a) 5s. 6d.  98.  and 8 « P » » m  (b)  h  92. |l,800  95. A, 18 m i l e s ; B, 12 m i l e s 75 (b) 108 min.; 135 min.  30 f t . x 40 f t .  (c) £50.  135 ANSWERS TO ALGEBRA QUESTIONS FOR COORDINATION Paper 1. 8 a - - 7 a 2  2- (a) (  1  I  +6  +3)(3x-4y)  2 x  (b)  (2a-b)(2a^b)(a +b )  (c)  (3r -5x )(3r -t-5x )  2  x  3 x  2  x  3 x  (d) x ( l + 2 a y ) ( l - 2 a y 4-4a y ) 2  2  3. 2=2. x-1 4. x s"'-2; y a -3; z a -1 5. ( a )fl« R  6. — ^ h r s .  (b) H ss-s*  2 II R  c-a « be a 8. x. YJ  9. x - 1 10. x |, 2 7 = 4, 1 =  Paper I I 2 1. (a-t-b 4-c) 2bc  2. x - -6  3. x  5. 20 @ 5^ 6. x 60 % 2H y 9 9 9 9 7. (a) (x<M- y^-3xy)(x^>- y ^ + Sxy) /-x /X 4 x , 1 . 16\ Z  s s  s  20 15 =-  10  2  w  (  b  )  (  8 - ^  ) (  64 '2 -^ +  4  )  (c) ( x - 2 ) ( x + - 2 ) ( x i - 4 ) ( x -}-l)(x -x i-1) 2  2  8. ( c ) ll-\/..5 10. x - 2;  3;  9  1/3-t 3;  )/3-3  y - 3; 2', ^3-3; - 1/3-3  >  g  0  d  a  y  s  @ $1.80.  2  4. 15 y r s .  136 Paper I I I 1. | a - t | a x - t - j | a x + j l - i a x 4  3  2  2  2. ah(h-b)  3  3. 1 3 3 j l b s . @ 40^. 3_ 5. x M -2; -1 y i l ; *2  4. $6.  7. (a) 3 l / 7 + 2]/5  8. x*-y^  6. x = 2; y o 4-|; z a 3^  s  (b) 5 \/2 - J/5  10. 56. Paper IV  1. 3a-13b -tlOc 3  2. (a) k = 5  (b) x - 2; y  s  1  3. (a) (x-2)(x-2)(x-2) 3. (b) (3x-4a)(4a-3x) 4. (a) 12mn.  6. _J>Z 115  (b)  ( c ) (y-3x) (x-y) (x -ty)  9.  5.  7. x - -2  s  7  =  3  2  ( a * 2)(2a + l ) ( 4 a + 2a +1) 2  8. p = 1 5 7 i o 10. x  l p a 4 - a -2  9. $1,000.  i 4 ; - UZ - (-3.299) 3 _ £1; ; 1 / (-1.885) 2  4  g  Paper  V  1. -x (a-3b -+ c-v7)-x(b + c ) 4  2  3. 8, 4.  2. 4 m.p.h. and. 1 m.p.h.  2  4. 3g  6. x r 8; -5 y x 6; -8  8  7. (a) (3-t 4x)(4-5a) 7. (b) ( 3 y - 8 ) ( 9 y + 24y4 64)  (c) x  2  7. (d) (x-3-t-2y)(x-3-2y) o  (  . || i ?  8.  4.  3 n  ( 3x i-1) (3x -l) n  n  9.  4a  8 )  Paper VI  1.  x  =  5  2. x  =  -4  3. 68 y r s .  4. 16 y d s .  137 5. m - 2; n = 3; p = -6.  6. x - i§, -1  7. 12'; 5'; 13»-.  8. ( x + y ) \ / x y  10. (a) No.  9. -35.  (b) x 2, -2 y = -3, 5 a  Paper V I I 1. x - 5g  2.  X  £ y  3. x = 8  A  4. (b) |/x -»-y ~ 2  5. 2 1 x - 9 x  2  7. j |  2  8. (a) x - .61a; -.52a  9. x I' -5; i s y = -6; *3  4. (a) (0,0) 6. x = 8  3  (b) x = 2.643; .7566.  10. 4 y d s . Paper V I I I  1. (a) 2 ( a - b ) ( x - 4 ) ( x +1)  (b)|a-b+ c~3)(a-b-c +3)  1.' (c) (a-tbx)b-+ax) 4. x = 64. 7. x  =  2. x  n  "* +x ~ n  n  5. (b) (-2,4)  9.  8. x  _  3a; y  =  6.  -2b  =  1_ x +1  9. 100 benches.  4  10. x s i ; 3 Y - 1; * ~ 2'  3. 2-1/3  1  2  -1 3  Paper 1. (a) 24.64 s q . i n .  IX  (b) i f f t .  „ / » ,mn w m n .' -^ mnX, H , , I J U H M A-+--^-+l) a. (a) (-g--!H 2  2  .  n  8  1  a  2.  A a  13 X  T4"¥  (b) ( x - 2 y - 3 z ) ( x + 4y -+-9z 4-2xy +3xz-6yz) 2  2  (c) ( x - 3 y - 3 x y ) ( x - 3 y + 3xy) 2  2  2  2  4. (a) x - ^E;. - ** 11 2 6. x  =  -8; 0  (b) 2;3  7. (a) ( a - b - a + 5 ) ( a - b 4 - a-3) 2  2  7. (b) ( a 4 - 2 b ) ( a + a b + b ) 2  8.  1.  9. x.  5. 2.236. 2  ( c ) ab(a-b)(a-b)(a-+ b)  2  10.  2  x -2 %  x  138 Paper 1. v - 40 4. (a) x  3. (a)  7.  (b)  6. x - 11 x  a  5. 60 y d . x 90 y d . 7. 642  =  3  y =z 1  (b) x =  2  3  10.  x 9  1.7236; 1.2764  a  (b) x - i l ; ±3. y - t5; *1.  9. (a)  X  1.11803.  -I; 23 t|; +  y  =  8. 6x-5y = 8 ±2 +  tl  139 BIBLIOGRAPHY The New Teaching. Hodder and Stoughton, Toronto, (1918) Mathematical R e c r e a t i o n s and E s s a y s . B a l l , W.W.R.: Macmillan, Toronto, 1928. A Short Account of the H i s t o r y of B a l l , W.W.R.: Mathematics. Macmillan, Toronto, 1912. B r e s l i c h , E.R.: Problems i n Teaching Secondary School .Mathematics. U n i v e r s i t y o f Chicago P r e s s , Chicago, 1931. 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