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UBC Theses and Dissertations

Effects of different schedules and availability of answers on learning Kokoskin, William Joseph Alexander 1973

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THE EFFECTS OF DIFFERENT SCHEDULES AND AVAILABILITY OF ANSWERS ON LEARNING by WILLIAM JOSEPH ALEXANDER KOKOSKIN B . S c , M c G i l l U n i v e r s i t y , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department o f Mathematics E d u c a t i o n We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of " T ^ l ^ c V C — ^ X c c ^ ^^-^-c^gJCc^^. The University of British Columbia Vancouver 8, Canada ABSTRACT T h i s study was motivated by the author's b e l i e f b o t h as a t e a c h e r and r e s e a r c h e r t h a t the method o f p r e s e n t i n g answers to students who are i n the p r o c e s s o f l e a r n i n g i s important t o t h e i r l e a r n i n g . The two main v a r i a b l e s s t u d i e d were the a v a i l -a b i l i t y o f answers and the p r o p o r t i o n o f answers. A b a s i c 3x2 f a c t o r i a l d e s i g n was used w i t h two l e v e l s o f a v a i l -a b i l i t y 1) a v a i l a b l e 2) n o n - a v a i l a b l e and t h r e e l e v e l s o f p r o p o r t i o n of answers 1) 80% o f the answers g i v e n 2) 20% i n c o r r e c t answers g i v e n 3) a l l the answers g i v e n . Two t o p i c s , s o l v i n g e q u a t i o n s o f the f i r s t degree and s i m p l i f y i n g e x p o n e n t i a l e x p r e s s i o n s were used as the s u b j e c t m a t e r i a l . Four grade e l e v e n c l a s s e s were g i v e n i n s t r u c t i o n and then p r e t e s t e d on the p a r t i c u l a r t o p i c . Then the s i x treatments were randomly a s s i g n e d i n a s t r a t i -f i e d way and the students proceeded t o the e x e r c i s e s p e r t a i n i n g t o the p a r t i c u l a r t o p i c . A p o s t t e s t was admin-i s t e r e d a f t e r a p a r t i c u l a r t o p i c was completed. Then the students were r e - a s s i g n e d t o p i c s w h i l e m a i n t a i n i n g t h e i r same treatment. The students f o l l o w e d the same procedure as above o f l e c t u r e , p r e t e s t , e x e r c i s e s and p o s t t e s t . An a n a l y s i s of c o - v a r i a n c e was c a l c u l a t e d on each o f the groups a c c o r d i n g t o t o p i c , o r d e r o f u n i t , and p a r t i -c u l a r group. The p r e t e s t was c o v a r i e d on the p o s t t e s t . Students were a l s o asked to rank-order t h e i r p r e f e r e n c e f o r answer sheets o r treatments. The r e s u l t s i n d i c a t e d t h a t the a v a i l a b i l i t y f a c t o r had no s t a t i s t i c a l l y s i g n i f i c a n t e f f e c t on the s t u d e n t ' s performance on any t o p i c a l t h o u g h the students f a v o r e d the n o n - a v a i l a b l e answers. The r e s u l t s a l s o showed t h a t the students who were g i v e n 80% o f the answers and those who were g i v e n 2 0% i n c o r r e c t answers, performed b e t t e r than the group who was g i v e n a l l the answers. T h i s was s t a t i s t i c a l l y s i g n i f i c a n t w i t h the group who f o l l o w e d the sequence s o l v i n g e q u a t i o n s of the f i r s t degree then s i m p l i f y i n g e x p o n e n t i a l e x p r e s s i o n s and o n l y p e r t a i n e d to the s i m p l i f y i n g o f e x p o n e n t i a l i e x p r e s s i o n s . I The r e s u l t s were d i s c u s s e d i n terms o f the p a r t i -c u l a r t o p i c s i n v o l v e d and the v a r i a b l e s t h a t were under study. Suggestions f o r f u r t h e r s t u d i e s were made i n terms of the r e s u l t s o f t h i s study. i i i TABLE OF CONTENTS CHAPTER Page I THE PROBLEM 1 I Background ... 1 Statement o f Problem 4 I I Review o f L i t e r a t u r e 4 E x t e r n a l Reinforcement 4 Immediate Reinforcement 5 Q u a n t i t y o f Reinforcement 7 Type o f Reinforcement 8 Summary 9 Statement o f Hypotheses .......*.... 10 I I THE DESIGN OF THE STUDY 11 I I n t r o d u c t i o n • 11 I I The Treatments 12 I I I D i s t r i b u t i o n o f Treatments 14 The P o p u l a t i o n 14 Time o f Study 14 The Sample 14 Assignment o f Treatments 15 Experimental Procedure 15 IV Development of M a t e r i a l s 15 i v CHAPTER Page Su b j e c t M a t e r i a l 15 Treatment M a t e r i a l 17 T e s t Instruments 18 V Procedure 20 VI S t a t i s t i c a l A n a l y s i s 24 Data 24 N u l l Hypotheses 24 S t a t i s t i c a l Treatment 25 I I I ANALYSIS OF THE RESULTS 27 I T e s t i n g o f the Hypotheses 27 Hypothesis One • 32 Hypothesis Two 33 Hypot h e s i s Three ••• 33 I I F u r t h e r A n a l y s i s 34 Comparison o f V a r i a b l e A3 and V a r i a b l e A 2 35 Comparison of V a r i a b l e A2 and V a r i a b l e Aj_ 36 Comparison of V a r i a b l e A-^  and A 2 . . 36 I I I The Hawthorne E f f e c t s 36 IV C o n c l u s i o n s 37 V CHAPTER Page V A n a l y s i s o f A d d i t i o n a l Data 38 Students' O p i n i o n s 38 IV IMPLICATIONS OF THE STUDY 42 I I n t r o d u c t i o n 42 I I D i s c u s s i o n o f the C o n c l u s i o n s 43 F i r s t Hypothesis Questioned 43 Second Hypothesis 43 I n t e r a c t i o n Among Students 45 I I I L i m i t a t i o n s o f the Study 45 The Sample 45 The T o p i c s . 46 The Method o f I n s t r u c t i o n ........... 46 Long-Term E f f e c t s 47 Grade and A b i l i t y L e v e l 47 IV Suggestions f o r F u r t h e r Study 47 S i m i l a r Research 47 Other Research 49 V Summary • 50 FOOTNOTES 52 BIBLIOGRAPHY 56 APPENDIX A 64 v i Page APPENDIX B 68 APPENDIX C 73 APPENDIX D 92 APPENDIX E 105 APPENDIX F 110 APPENDIX G 112 v i i LIST OF TABLES TABLE Page 1. R e l i a b i l i t y of T e s t s 21 2. ANOVA f o r E q u a t i o n then Exponents group -Equ a t i o n s o n l y 27 3. ANOVA f o r Eq u a t i o n s then Exponents group -Exponents o n l y 28 4. ANOVA f o r Exponents then E q u a t i o n group -Exponents o n l y 28 5. ANOVA f o r Exponents then E q u a t i o n s group -Equ a t i o n s o n l y 29 6. ANOVA f o r Eq u a t i o n s then Exponents group 2 9 7. ANOVA f o r Exponents then E q u a t i o n s group 30 8. ANOVA f o r T o p i c o f Exponents 30 9. ANOVA f o r T o p i c of Eq u a t i o n s 31 10. ANOVA f o r F i r s t U n i t of Work 31 11. ANOVA f o r Second U n i t o f Work 32 12. Students' V o t i n g 40 13. Students' V o t i n g Grouped A c c o r d i n g t o V a r i a b l e B 40 14. Students' V o t i n g Grouped A c c o r d i n g t o V a r i a b l e A 41 v i i i LIST OF FIGURES FIGURE Page 1. Design Paradigm 13 2. The E x p e r i m e n t a l Procedure 16 3. Summary of Treatments 19 4. D i f f e r e n t Group A n a l y s i s 2 6 i x ACKNOWLEDGEMENT The author wishes to thank h i s w i f e , Sandra Lynne K o k o s k i n . The author a l s o wishes to thank the p r i n c i p a l , t e a c h e r s and students who p a r t i c i p a t e d i n the experiment and the members of h i s t h e s i s committee, Dr. G. S p i t l e r (Chairman), Dr. E . MacPherson, Dr. T. Howitz, and Dr. L. Rousseau. I CHAPTER I THE PROBLEM I . BACKGROUND I t i s the o p i n i o n o f many educators t h a t "know-ledge of r e s u l t s " i s an important f a c t o r i n f l u e n c i n g the achievement and m o t i v a t i o n o f s t u d e n t s . Most of the r e s e a r c h t h a t has been done on "knowledge of r e s u l t s " has ce n t e r e d on the knowledge of r e s u l t s o f t e s t s and i t s i n f l u e n c e on such f a c t o r s as achievement, m o t i v a t i o n and subsequent l e a r n i n g . L i t t l e r e s e a r c h has been done on how knowledge of r e s u l t s of answers, while the student i s a c t i v e l y engaged i n l e a r n i n g , i n f l u e n c e s h i s achievement. The author became i n t e r e s t e d i n the ar e a of"know-ledge o f r e s u l t s " when he f i r s t taught h i g h s c h o o l mathe-m a t i c s . He n o t i c e d t h a t j u n i o r h i g h mathematics t e x t s , from grades e i g h t t o ten, d i d not c o n t a i n answers, w h i l e the grade eleven, twelve s e n i o r t e x t s d i d c o n t a i n the answers t o the odd-numbered problems. F u r t h e r i n v e s t i g a t i o n has shown t h a t textbooks can be purchased with d i f f e r e n t schedules o f answers, a c c o r d i n g 1 2 to one of the f o l l o w i n g c a t e g o r i e s : (a) a l l answers are p r o v i d e d (b) even or odd answers are p r o v i d e d (c) s e l e c t e d answers are p r o v i d e d (d) no answers are p r o v i d e d . Very few textbooks mix the answers o r p r e s e n t the answers i n such a way so t h a t the subsequent answer to a q u e s t i o n i s not immediately a v a i l a b l e . There i s no a v a i l a b l e evidence t h a t mathematics t e a c h e r s who a s s i g n s e t s o f e x e r c i s e s pay any s p e c i a l a t t e n t i o n t o how the answers are p r e s e n t e d . The r e s e a r c h c o n c e r n i n g the use of reward, r e i n -forcement, knowledge o f r e s u l t s , and feedback can be summarized as a few g e n e r a l p r i n c i p l e s :^ (1) Ensure f r e q u e n t r e g u l a r e x p e r i e n c e of success (or r e i n f o r c e m e n t ) e s p e c i a l l y i n the e a r l i e r phases of l e a r n i n g . (2) G i v e knowledge o f r e s u l t s t o (a) p r o v i d e i n f o r m a t i o n so the student can gauge and a d j u s t h i s performance (b) motivate the student, e s p e c i a l l y i f the s u b j e c t matter does not appear to be i n t r i n s i c a l l y m o t i v a t i n g . (3) Use a p p r o p r i a t e schedules of r e i n f o r c e m e n t p a r t i c u l a r l y i n d r i l l t e chniques o r s i t u a t i o n s . When these p r i n c i p l e s are a p p l i e d to students working on a s e t o f e x e r c i s e s a number o f p o s s i b i l i t i e s 3 a r i s e as to how the te a c h e r or textbook s h o u l d p r e s e n t the answers to students so t h a t they w i l l r e c e i v e maximum b e n e f i t . Two p o s s i b i l i t i e s t h a t do a r i s e a r e : (1) the a v a i l a b i l i t y o f answers and (2) the p r o p o r t i o n o f answers to be g i v e n . The a v a i l a b i l i t y of answers can be d e f i n e d as whether or not a student can see the answer to each sub-sequent q u e s t i o n . Most of the answers i n a textbook are sequenced so t h a t once the student has looked up the answer to q u e s t i o n number one, he cannot h e l p but see the answer to q u e s t i o n number two. Thus a student might b e n e f i t more from answers i f the answer to a subsequent q u e s t i o n was hidden i n some way. A mapping scheme c o u l d be arranged whereby a student was g i v e n a key f o r a p a r t i c u l a r q u e s t i o n and hence c o u l d f i n d h i s answer amongst a l l the answers. T h i s roapped-out scheme ( n o n - a v a i l a b l e ) c o u l d be compared to the s e q u e n t i a l - l i s t ( a v a i l a b l e ) where one answer would appear a f t e r another, as i s the case i n most t e x t b o o k s . The p r o p o r t i o n of answers to be g i v e n can be d e f i n e d as how many c o r r e c t answers a student i s g i v e n . A student c o u l d l e a r n more as measured by a p o s t t e s t i f he r e c e i v e s 100% o f the answers to a g i v e n s e t o f e x e r c i s e s as compared to 80%.of the answers. The 20% t h a t are m i s s i n g 4 c o u l d be r e p l a c e d w i t h 2 0% i n c o r r e c t answers, and the student c o u l d be t o l d t h a t t h i s p r o p o r t i o n was i n c o r r e c t . Statement of Problem Thus the purpose of t h i s study was to determine which k i n d o f schedule o f answers and how they s h o u l d be pr e s e n t e d , b e s t h e l p s the student l e a r n . I I REVIEW OF THE LITERATURE There has been a c o n s i d e r a b l e amount o f r e s e a r c h i n the l i t e r a t u r e on the e f f e c t s o f r e i n f o r c e m e n t and v a r i a t i o n i n i t s s c h e d u l i n g on l e a r n i n g by animals l i k e monkeys and r a t s , but not as many s t u d i e s have manipulated r e i n f o r c e m e n t f a c t o r s i n the p r a c t i c a l i n s t r u c t i o n s i t u a t i o n of the c l a s s r o o m . E x t e r n a l Reinforcement Research has focused i t s a t t e n t i o n on c o n t r i v e d methods of r e i n f o r c e m e n t . M c M i l l a n has shown t h a t students r e c e i v i n g monetary r e i n f o r c e m e n t show a h i g h e r r a t e of response than students w i t h no monetary r e i n f o r c e m e n t . 3 Luty used hardboard s l a t e s as a feedback d e v i c e as opposed to p e n c i l and paper to work problems. T h i s enabled the teacher t o r e i n f o r c e the students more f r e q u e n t l y , as he was a b l e to see a l l o f the answers a t once by a s k i n g the students 5 to h o l d up t h e i r s l a t e s . Other r e s e a r c h e r s have t r i e d d i f f e r e n t i a l feedback. Wiggins^ had moderate success w i t h the g e n e r a l h y p o t h e s i s t h a t d i f f e r e n t i a l feedback e f f e c t s a s p e c t s o f b e l i e f s about the problem s o l v i n g p r o c e s s , and e f f e c t s the student's s e l f - c o n f i d e n c e i n understanding sub-sequent problem s o l v i n g a c t i v i t i e s . But r e i n f o r c e m e n t techniques do not study the students r e i n f o r c i n g h i m s e l f as he i s a c t i v e l y engaged i n the l e a r n i n g p r o c e s s . When a student completes a q u e s t i o n and looks up the answer and f i n d s t h a t he i s c o r r e c t , he has r e i n f o r c e d h i m s e l f and t h i s i s more e f f e c t i v e than somebody e l s e r e i n f o r c i n g the s t u d e n t . I n f a c t the te a c h e r cannot p o s s i b l y r e i n f o r c e t h i r t y o r more stu d e n t s when they are working on a s e t of t h i r t y o r more examples. T h i s suggests t h a t c u r r i c u l u m d e s i g n a l l o w the student to p a r t i c i p a t e more d i r e c t l y i n the immediate p r o c e s s o f s e l f - e v a l u a t i o n and s e l f - d i r e c t i o n through techniques which a l l o w f o r immediate knowledge o f r e s u l t s . Immediate Reinforcement Research has i n d i c a t e d t h a t "immediate knowledge of r e s u l t s may a i d l e a r n i n g by r e i n f o r c i n g a c o r r e c t response and immediately c o r r e c t i n g one t h a t i s i n c o r r e c t . In t h i s way, the student i s informed o f how w e l l he i s doing and 6 keeps him from compounding h i s e r r o r s . " ^ Immediate knowledge of r e s u l t s appears t o s i g n i f i c a n t l y e f f e c t immediate r e t e n t i o n (one day), but not lo n g term r e t e n t i o n ( t h r e e weeks) S i m i l a r r e s e a r c h has shown t h a t students do b e t t e r on f i n a l examinations when they are exposed t o immediate knowledge of q u i z r e s u l t s 7 r a t h e r than delayed knowledge o f r e s u l t s . However s t u d i e s as the one by H e a l d , ^ on the e f f e c t o f immediate knowledge of r e s u l t s and t e x t a n x i e t y upon t e s t performance, have been i n c o n c l u s i v e as t o whether h i g h a n x i e t y students do b e t t e r than low a n x i e t y students when both are g i v e n knowledge of r e s u l t s o f p r e v i o u s t a s k s . q Skinner a l s o p o i n t s out t h a t the most s e r i o u s c r i t i c i s m o f contemporary i n s t r u c t i o n i s the r e l a t i v e i n -frequency of r e i n f o r c e m e n t . I n a t y p i c a l classroom long p e r i o d s o f time c u s t o m a r i l y e l a p s e between response ( f i g u r i n g out an answer t o a p a r t i c u l a r question) and the re i n f o r c e m e n t (the answer t h a t i s g i v e n by the t e a c h e r ) . Sometimes up t o twenty-four hours go by b e f o r e s t u d e n t s are presented w i t h c o r r e c t answers. I f one ac c e p t s S k i n n e r ' s i n s i g h t then immediacy of r e i n f o r c e m e n t suggests t h a t answers should be pro v i d e d t o a q u e s t i o n immediately a f t e r the student has completed the q u e s t i o n , r a t h e r than w a i t i n g a day o r two and h a v i n g the te a c h e r g i v e a l l the answers to a p a r t i c u l a r e x e r c i s e . Q u a n t i t y o f Reinforcement I t seems c l e a r t h a t i n the a r e a o f programmed i n s t r u c t i o n t h e r e i s a d e f i n i t e s u p e r i o r i t y i n p o s t t e s t s c o r e s f o r programs w i t h answers a v a i l a b l e over programs without answers a v a i l a b l e . Moreover the a v a i l a b i l i t y o f answers i n a program p r i m a r i l y p r e v e n t s the student from making the same e r r o r on the p o s t t e s t t h a t he made i n the program. A study by R u b i n ^ where f o r t y u n i v e r s i t y student completed a s e r i e s of programmed m a t e r i a l s over s e v e r a l days, supports the h y p o t h e s i s t h a t students do not lo o k a t a l l the answers i n a program, but p r i m a r i l y a t items which were answered i n c o r r e c t l y . H i s study however was concerned w i t h programmed i n s t r u c t i o n as opposed t o a classroom s i t u -a t i o n w i t h textbooks and answers. F u r t h e r r e s e a r c h not i n the area of programmed i n s t r u c t i o n , has been done on how much o f an answer s h o u l d be g i v e n a f t e r each l e a r n i n g s i t u a t i o n . I t appears to be b e t t e r to r e i n f o r c e someone by g i v i n g the whole answer than j u s t i n d i c a t i n g whether the person i s " r i g h t " o r "wrong".1""1* Research has been done i n the a r e a of programmed l e a r n i n g and t e a c h i n g machines as to what ki n d s of schedule 8 13 of r e i n f o r c e m e n t t o use. In one experiment by Dickey, t h r e e d i f f e r e n t schedules of r e i n f o r c e m e n t (O'A, 33%, 100%) were s t u d i e d as to t h e i r e f f e c t upon achievement and r e t e n t i o n i n a l i n e a r program i n c o l l e g e b u s i n e s s mathematics, as w e l l as upon performance on immediate t e s t s . The r e s u l t s show t h a t the s u b j e c t s who r e c e i v e d o n e - t h i r d r e i n f o r c e m e n t a c h i e v e d s i g n i f i c a n t l y h i g h e r t r u e mean g a i n s from p r e t e s t to p o s t t e s t than the o t h e r two groups, and a l s o made s i g n i -f i c a n t l y h i g h e r t r u e mean ga i n s from p r e t e s t to r e t e n t i o n t e s t s . There was no s i g n i f i c a n t d i f f e r e n c e among treatments on e i t h e r p o s t t e s t o r r e t e n t i o n t e s t s . The r e s e a r c h suggests t h a t i t i s b e t t e r to g i v e answers and g i v e a complete answer to a p a r t i c u l a r q u e s t i o n , but the r e s e a r c h i s n ' t c l e a r what p r o p o r t i o n o f the t o t a l number of answers s h o u l d be made a v a i l a b l e . Type o f Reinforcement The students can be r e i n f o r c e d w i t h success or 14 . f a i l u r e . F e ather i n g e n e r a l , p o i n t s out t h a t success r a t h e r than f a i l u r e has the e f f e c t of i n c r e a s i n g performance s c o r e s on subsequent t a s k s . One study which i l l u s t r a t e s t h i s i s by Bridgeman 1" 5 i n which success feedback r e s u l t e d i n s i g n i f i c a n t l y h i g h e r s c o r e s than f a i l u r e feedback (p^.02). T h i s would tend t o suggest t h a t when students are 9 doing a set of exercises, i t would be i n t h e i r best i n t e r e s t to give them correct answers, rather than i n c o r r e c t ones. 1 fi Another study by Wentbing also points out that when students are given the correct response th e i r attitude toward i n s t r u -c t i o n and immediate achievement i s much higher than students who are not given the correct response. Summary In summary, the research says that: ( 1 ) Answers should be given. (2) Answers should be given immediately a f t e r the question i s completed. (3) Students are concerned with t h e i r i n c o r r e c t answers. (4) Students' attitudes towards subsequent learning are better when they are p o s i t i v e l y r einforced. ( 5 ) I t i s better to give a proportion of the correct answers than give 1 0 0 % of the answers, but i t i s unclear which proportion i s best. L i t t l e research has been done on whether the student's learning i s affected while doing a set of exercises when the answers i n the textbook are given sequentially so that each subsequent answer can be seen v i s u a l l y . I t could 10 make a d i f f e r e n c e to the student i f he doesn't immediately see the subsequent answer to a g i v e n q u e s t i o n . The r e s e a r c h i s l a c k i n g w i t h r e s p e c t to i n c o r r e c t answers. I f 2 0% o f the answers were g i v e n as i n c o r r e c t and the student was t o l d t h a t 2 0% were i n c o r r e c t , the student might take the time and e f f o r t to i n v e s t i g a t e h i s i n c o r r e c t answers. T h i s c o u l d be p a r t i c u l a r l y u s e f u l w i t h r e g a r d to e q u a t i o n s o l v i n g , when the student c o u l d p o s i t i v e l y f i n d out i f the answer gi v e n i s c o r r e c t o r i n c o r r e c t by s u b s t i t u t -i n g i t back i n t o the o r i g i n a l e q u a t i o n . Statement o f H y p o t h e s i s I t i s h y p o t h e s i z e d t h a t grade e l e v e n students who r e c e i v e hidden answers ( n o n - a v a i l a b l e ) to s e t s o f e x e r c i s e s on two t o p i c s w i l l l e a r n more than those who r e c e i v e the s t a n d a r d textbook format o f sequenced answers ( a v a i l a b l e ) . I t i s h y p o t h e s i z e d t h a t those students who r e c e i v e 80% o f the answers w i l l l e a r n more than those who r e c e i v e 100% o f the answers. CHAPTER I I THE DESIGN OF THE STUDY I . INTRODUCTION The study took p l a c e a t a h i g h s c h o o l i n the Vancouver area u s i n g students from grade e l e v e n . Two t o p i c s , r e l a t i v e l y independent c o n s i s t i n g o f the s o l u t i o n o f l i n e a r e q u a t i o n s , and the s i m p l i f i c a t i o n o f e x p o n e n t i a l e x p r e s s i o n s , were used i n the study to d e f i n e two b a s i c groups of s t u d e n t s . An i n s t r u c t i o n a l p e r i o d was h e l d f o r b o t h groups on t h e i r p a r t i c u l a r t o p i c . A t the end o f the i n s t r u c t i o n a l phase a p r e t e s t was a d m i n i s t e r e d . Subsequently a s e t o f e x e r c i s e s based on the p a r t i c u l a r t o p i c was g i v e n to the s t u d e n t s . S i x d i f f e r e n t treatments, s e t s of answers, were a s s i g n e d to the students doing these e x e r c i s e s . When the group o f students completed the e x e r c i s e s , a p o s t t e s t was a d m i n i s t e r e d t o the group. The two groups then i n t e r c h a n g e d the t o p i c s . A g a i n an i n s t r u c t i o n a l p e r i o d was h e l d f o r each group on t h e i r new t o p i c , a f t e r which a p r e t e s t was a d m i n i s t e r e d . The students were g i v e n e x e r c i s e s based on t h e i r new t o p i c . The s i x d i f f e r e n t treatments o r s e t s of answers were a g a i n g i v e n to the students but each student kept h i s p r e v i o u s 11 12 a s s i g n e d treatment from the f i r s t t o p i c . A f i n a l p o s t t e s t was a d m i n i s t e r e d f o r the new t o p i c to each group and a student o p i n i o n q u e s t i o n n a i r e was a l s o g i v e n t o a l l s t u d e n t s . The p r e t e s t was used as a c o v a r i a t e on the p o s t -t e s t and a n a l y s i s o f c o v a r i a n c e was used to compare the means of the groups, to see which of the treatments had the g r e a t e s t e f f e c t on l e a r n i n g . I I . THE TREATMENTS As mentioned i n the i n t r o d u c t i o n , s i x d i f f e r e n t treatments or s e t s o f answers ( l i s t e d below), were g i v e n t o stud e n t s doing the same s e t of e x e r c i s e s . The o n l y d i f f e r -ence among the groups was the s e t o f answers t h a t the students had w h i l e they were doing a p a r t i c u l a r s e t o f e x e r c i s e s . One of the h y p o t h e s i s a t the end of Chapter I was concerned w i t h the p r o p o r t i o n o f answers t h a t were g i v e n . The o t h e r h y p o t h e s i s was concerned w i t h the a v a i l a b i l i t y of the answers. Thus the independent v a r i a b l e s i n the study were: A - the p r o p o r t i o n o f answers g i v e n . B - the a v a i l a b i l i t y o f the answers. 13 The v a r i a b l e A was f u r t h e r s u b d i v i d e d i n t o t h r e e p a r t s : A3 - a l l answers were g i v e n . A 2 - a l l the answers were g i v e n but 2 0% were i n c o r r e c t . A-j_ - o n l y 80% o f the answers were g i v e n . The v a r i a b l e B was s u b d i v i d e d i n t o two p a r t s : B^ - each subsequent answer i s a v a i l a b l e as i n a textbook. B2 - each subsequent answer i s n o n - a v a i l a b l e . The b a s i c d e s i g n of the study i s a two-way f a c t o r i a l d e s i g n . The d e s i g n paradigm i l l u s t r a t i n g the s i x d i f f e r e n t treatments i s g i v e n i n F i g u r e 1. FIGURE 1 DESIGN PARADIGM The P r o p o r t i o n o f Answers Given A A l l Answers Given A 3 A l l Answers but 20% I n c o r r e c t A 2 Only 80% of Answers A l 1 A v a i l -a b i l i t y o f Answers B A v a i l a b l e B l N o n - a v a i l -a b l e B 2 14 I I I . DISTRIBUTION OF TREATMENTS The P o p u l a t i o n The p o p u l a t i o n c o n s i s t e d o f grade e l e v e n students from a secondary s c h o o l . The students were f o l l o w i n g the r e g u l a r B r i t i s h Columbia Program o f S t u d i e s , f o r grade e l e v e n . The students c o u l d be expected t o have had some background f o r the two t o p i c s o f equations and exponents. Time of Study The study was i n i t i a t e d on the f i r s t f u l l day of s c h o o l on September 8, 1971 and took twelve days to complete. The Sample Four grade e l e v e n c l a s s e s were used i n t h i s experiment. Two o f the c l a s s e s were r e g u l a r c l a s s e s o f the experimenter's w h i l e two were borrowed from another t e a c h e r ' s schedule without the s t u d e n t s ' knowledge of how lo n g the experimenter was to teach these two oth e r c l a s s e s . The s t u d e n t s o f the oth e r teacher were t o l d t h a t an a d m i n i s t r a t i v e change had been made and t h a t the e x p e r i -menter was now t e a c h i n g f o u r grade e l e v e n c l a s s e s . T h i s e l i m i n a t e d any e f f e c t t h a t the experimenter would have had as a s u b s t i t u t e . 15 The study s t a r t e d w i t h approximately 144 students but a student was o m i t t e d from the study i f he was absent one or more days. A few students were randomly o m i t t e d from the study t o s i m p l i f y the a n a l y s i s o f the d a t a . A t o t a l o f 120 students were used i n the a n a l y s i s of the d a t a . Assignment o f Treatments The treatments were a s s i g n e d randomly i n a 1 7 s t r a t i f i e d way, u s i n g a t a b l e o f random numbers, to ensure t h a t each c l a s s c o n t a i n e d approximately the same number of d i f f e r e n t t r e a t m e n t s . Two o f the c l a s s e s s t a r t e d w i t h the e q u a t i o n t o p i c and then proceeded to the exponent t o p i c . The o t h e r two c l a s s e s s t a r t e d w i t h the exponent t o p i c , then proceeded t o the e q u a t i o n t o p i c . E x p e r i m e n t a l Procedure The e x p e r i m e n t a l procedure which was f o l l o w e d appears i n F i g u r e 2. IV. DEVELOPMENT OF MATERIALS S u b j e c t M a t t e r The s u b j e c t matter f o r the t o p i c o f equations t h a t was used i n the e x e r c i s e s was taken from t h r e e sources'*" i Q or) • L Z >' . The e x e r c i s e s were o r g a n i z e d from easy to d i f f i c u l t . 16 FIGURE 2 THE EXPERIMENTAL PROCEDURE Day Two C l a s s e s F o l l o w e d Two C l a s s e s Followed T h i s Procedure This Procedure 1 I n s t r u c t i o n a l P e r i o d on I n s t r u c t i o n a l P e r i o d on Equations Exponents 2 P r e t e s t on Equations P r e t e s t on Exponents 3 E x e r c i s e s on Equations E x e r c i s e s on Exponents 4 E x e r c i s e s on Equations E x e r c i s e s on Exponents 5 E x e r c i s e s on Equations P o s t t e s t on Exponents 6 E x e r c i s e s on Equations I n s t r u c t i o n a l P e r i o d on Equations 7 P o s t t e s t on Equations P r e t e s t on Equations 8 I n s t r u c t i o n a l P e r i o d on E x e r c i s e s on Equations Exponents 9 P r e t e s t on Exponents E x e r c i s e s on Equations 10 E x e r c i s e s on Exponents E x e r c i s e s on Equations 11 E x e r c i s e s on Exponents E x e r c i s e s on Equations 12 P o s t t e s t on Exponents P o s t t e s t on Equations 17 The a c t u a l t h r e e e x e r c i s e s used appear i n Appendix A. The s u b j e c t matter f o r the t o p i c of exponents t h a t was used i n the e x e r c i s e s was taken from these same s o u r c e s . The a c t u a l two e x e r c i s e s used appear i n Appendix B. The i n s t r u c t i o n a l p e r i o d s on the two t o p i c s were re p e a t e d twice as shown i n F i g u r e 2 o f the E x p e r i m e n t a l Procedure. The experimenter t r i e d to keep the r e p e t i t i o n s as i d e n t i c a l as p o s s i b l e by f o l l o w i n g the same format and u s i n g the same examples i n h i s p r e s e n t a t i o n of the two t o p i c s . The answers to the e x e r c i s e sheets were c o l o u r coded. A d i f f e r e n t c o l o u r was used f o r each o f the s i x d i f f e r e n t treatments o r schedules of answers. Treatment M a t e r i a l The t h r e e treatments i n which the answers were a v a i l a b l e one a f t e r another, were p r e s e n t e d i n a column o r l i s t type f a s h i o n . The t h r e e treatments i n which the answers were n o n - a v a i l a b l e were p r e s e n t e d i n a mapping type o f f a s h i o n wherein the student was g i v e n a key f o r a p a r t i c u l a r q u e s t i o n and hence found h i s answer among a l l the o t h e r answers. Thus the students u s i n g these t h r e e treatments d i d not have the o p p o r t u n i t y to see the answer to the next q u e s t i o n 18 without working through with the key. See Appendix C and D f o r examples of the d i f f e r e n t t r e a t m e n t s . Out o f the s i x treatments two c o n t a i n e d a l l the answers, two c o n t a i n e d 80% of the answers, and two c o n t a i n e d a l l the answers but 20% were i n c o r r e c t . The treatment t h a t had 80% of the answers was c o n s t r u c t e d by e l i m i n a t i n g at random 20% o f the answers. The treatment t h a t had 20% i n c o r r e c t answers was c o n s t r u c t e d by changing 20% of the answers, at random, t o be i n c o r r e c t . The 20% m i s s i n g t r e a t -ment and the 20% i n c o r r e c t treatment d i d not have the same omitted or i n c o r r e c t q u e s t i o n numbers. The f u l l s e t of answers t o the e q u a t i o n worksheets appears i n Appendix C, whi l e the f u l l s e t of answers t o the exponent worksheets appears i n Appendix D. Note t h a t c o l o u r coding was used f o r each t r e a t m e n t . A summary of the treatments appears i n F i g u r e 3 . T e s t Instruments The p r e t e s t and p o s t t e s t were the same t e s t but i n d i f f e r e n t c o l o u r . The p r e t e s t was ad m i n i s t e r e d on white paper w h i l e the p o s t t e s t was ad m i n i s t e r e d on green paper. To assure the content v a l i d i t y of the t e x t , the items were e i t h e r chosen from the three main sources 18 19 20 mentioned p r e v i o u s l y ' or were m o d i f i c a t i o n s o f some 1 9 FIGURE 3 SUMMARY OF TREATMENTS Di f f e r e n t Treatments Colour Variable B Variable A A Green Mapped-Out (Non-Available) 20% missing B Yellow Mapped-Out (Non-Available) 2 0% in c o r r e c t Pink Mapped-Out (Non-Available) Complete, Correct Blue L i s t e d Sequentially (Available) 20% missing E White L i s t e d Sequentially (Available) 2 0% in c o r r e c t Orange L i s t e d Sequentially (Available) Complete, Correct 20 of the items i n the e x e r c i s e s . A t e s t l e n g t h o f twenty items was chosen to assure the experimenter t h a t i f the r e l i a b i l i t y was low, then items w i t h a low p o i n t - b i s e r i a l c o r r e l a t i o n c o u l d be d i s c a r d e d . The r e l i a b i l i t y o f the t e s t i s p r e s e n t e d i n terms of a KR2 0. An i n t e r n a l c o n s i s t e n c y measure was chosen because a c e r t a i n s t a t e and not a change over time was b e i n g measured. The KR2 0 i s a l s o i n t e n d e d to be used on homo-geneous t e s t s . S i n c e the r e l i a b i l i t y o f the t e s t s a f t e r the experiment came out to be adequately h i g h , over .80, an item a n a l y s i s on the da t a was not needed. The o r i g i n a l twenty items t h a t were s e l e c t e d were s u f f i c i e n t l y good enough t o be used as measures o f a student's knowledge of t h a t p a r t i c u l a r t o p i c . The r e l i a b i l i t y of the t e s t s i s r e p o r t e d i n Tab l e 1 w h i l e the o r i g i n a l t e s t s appear i n Appendix E. V. PROCEDURE The two exp e r i m e n t a l procedures were f o l l o w e d as mentioned p r e v i o u s l y , one f o r each o f the experimenter's c l a s s e s and one f o r each o f the other t e a c h e r ' s c l a s s e s . The students f o l l o w e d t h e i r r e g u l a r t i m e t a b l e s TABLE 1 RELIABILITY OF TESTS 21 Eq u a t i o n s Exponents P r e - t e s t KR20 = .814 N = 135 KR20 = .895 N = 136 P o s t - t e s t KR20 = .862 KR20 = .860 N = 136 N = 134 22 w i t h r o t a t i n g b l o c k s and met a t d i f f e r e n t tiroes throughout t h e twelve days o f the study. The i n s t r u c t i o n a l p e r i o d and a d m i n i s t r a t i o n o f the p r e t e s t took p l a c e on days one and two. On day th r e e the e x e r c i s e s were s t a r t e d . Everyone i n each c l a s s was handed the same e x e r c i s e sheet. Then the treatments were handed out a c c o r d i n g to a p r e v i o u s s e l e c t e d random, o r d e r . A t t h i s p o i n t the students had n o t i c e d t h a t some of the sheets t h a t t h e i r neighbour had was d i f f e r e n t than h i s i n c o l o u r . The experimenter then announced t h a t he was t r y i n g to f i n d out which s e t of answers would be b e s t f o r f u t u r e e x e r c i s e s . The experimenter had to h e l p a few students who had the n o n - a v a i l a b l e answers use the answer s h e e t s . The experimenter had mentioned t h a t t h e r e was a d i f f e r e n c e among answer sheets and t h a t he would a p p r e c i a t e i t very much i f everyone d i d h i s own work. The students cooperated extremely w e l l i n t h i s r e s p e c t and a f t e r the f i r s t two days the n o v e l t y o f having d i f f e r e n t answer sheets wore o f f . A t the end of a p a r t i c u l a r day t h a t the students were working on e x e r c i s e s , a l l the e x e r c i s e sheets and answers were c o l l e c t e d from the s t u d e n t s . In t h i s way the students c o u l d not compare the d i f f e r e n t answer s h e e t s . The next day they were handed out a g a i n . T h i s procedure 23 took f o u r to f i v e minutes i n a l l , each day. The students thus d i d not have any homework d u r i n g the whole experiment p e r i o d of twelve days. When everyone had f i n i s h e d a p a r t i c u l a r e x e r c i s e sheet, another was handed out t o g e t h e r w i t h the a p p r o p r i a t e answer sheet. The o l d e x o r c i s e sheet and answers were c o l l e c t e d and kept. Some students f i n i s h e d b e f o r e o t h e r s . They were t o l d to re a d some l i t e r a t u r e . Only a few students f i n i s h e d t h i r t y minutes ahead of everyone e l s e and the m a j o r i t y f i n i s h e d w i t h i n f i f t e e n to twenty minutes of each o t h e r . A f t e r the f i n a l e x e r c i s e sheet i n each o f the two t o p i c areas was completed by the stu d e n t s , the p o s t t e s t was a d m i n i s t e r e d . The time a l l o t e d to the students to w r i t e the p r e -t e s t and p o s t t e s t f o r equations was f o r t y - f i v e minutes w h i l e the time a l l o t e d f o r p r e t e s t and p o s t t e s t f o r exponents was f o r t y minutes. A f t e r the p o s t t e s t s were a d m i n i s t e r e d , the next p e r i o d was an i n s t r u c t i o n a l p e r i o d on the a l t e r n a t e t o p i c . On the day a f t e r the i n s t r u c t i o n a l p e r i o d the p r e t e s t was a d m i n i s t e r e d . T h i s took a f u l l p e r i o d . Then the same procedure f o r e x e r c i s e s and answer sheets was f o l l o w e d u n t i l the second t o p i c was completed. The f i n a l p o s t t e s t was a d m i n i s t e r e d on day 12 i n both c a s e s . The r e s u l t s o f the p r e t e s t and p o s t t e s t were not handed out u n t i l the complete experiment was f i n i s h e d . The experimenter then sought out the s t u d e n t s 1 o p i n i o n on what type of answer sheet he p r e f e r r e d . A f t e r a b r i e f e x p l a n a t i o n o f the d i f f e r e n t answer sheets t h a t had been used by a l l s t u d e n t s , the students were asked to answer the q u e s t i o n s t h a t appear i n Appendix F. V I . STATISTICAL ANALYSIS Data For each student, a p r e t e s t and p o s t t e s t score was r e c o r d e d f o r each of the two t o p i c s . Appendix G l i s t s the e x p e r i m e n t a l d a t a . N u l l Hypotheses HJL : There i s no s i g n i f i c a n t d i f f e r e n c e between the mean sc o r e s o f the students whose answers were l i s t e d s e q u e n t i a l l y and the mean sco r e s o f the students whose answers were found by means of a mapping. ( V a r i a b l e B) H2 : There i s no s i g n i f i c a n t d i f f e r e n c e among the mean scor e s o f the students who had a d i f f e r e n t p r o p o r t i o n o f answers. ( V a r i a b l e A) H-J : There i s no s i g n i f i c a n t d i f f e r e n c e among the mean scor e s of the students when the i n t e r a c t i o n e f f e c t 25 15 taken i n t o c o n s i d e r a t i o n . ( V a r i a b l e AxB) The t h r e e hypotheses were t e s t e d f o r each o f the f o l l o w i n g : (a) Group (1) - the group t h a t had equations f i r s t . (b) Group (2) - the group t h a t had exponents second. (c) Group (3) - the group t h a t had exponents f i r s t . (d) Group (4) - the group t h a t had equations second. (e) Group (1) and Group (2) - the group t h a t f o l l o w e d the procedure o f equations then exponents. (f) Group (3) and Group (4) - the group t h a t f o l l o w e d the procedure o f exponents then e q u a t i o n s . (g) Group (2) and Group (3) - the group t h a t s t u d i e d the t o p i c o f exponents. (h) Group (1) and Group (4) - the group t h a t s t u d i e d the t o p i c of e q u a t i o n s . ( i ) Group (1) and Group (3) - the group t h a t completed the f i r s t u n i t . (j) Group (2) and Group (4) - the group t h a t completed the second u n i t . S t a t i s t i c a l Treatment An a n a l y s i s o f c o v a r i a n c e was c a r r i e d out f o r each o f the ten groups l i s t e d (a) to ( j ) , which i s summarized i n F i g u r e 4. The p r e t e s t s c o r e was used as the c o v a r i a t e on 2 6 the p o s t t e s t s c o r e . The U n i v e r s i t y o f B r i t i s h Columbia's MFACO computer's program was used f o r the a n a l y s i s of c o v a r i a n c e . FIGURE 4 DIFFERENT GROUP ANALYSIS 1. Equations (a) <-(e) 2. Exponents (i ) 3 . Exponents -> (c) (f) (d) 4-. Equations 21 CHAPTER I I I ANALYSIS OF THE RESULTS I . TESTING OF HYPOTHESES Each of the t h r e e hypotheses were t e s t e d on the te n groups as f o l l o w s : Group (a) (60 o b s e r v a t i o n s - 10 per c e l l ) TABLE 2 ANALYSIS OF VARIANCE Equa t i o n s then Exponents group - Equations o n l y Source D.F. Sum of Squares Mean Square F-R a t i o P roba-b i l i t y B - V i s u a l 1 .0262 .0262 0.01 0.901 A - Answers 2 5.552 2 .278 0.55 0.588 A x B 2 .0789 .0395 0.01 0.981 E r r o r 53 269.4 27 28 Group (b) (60 o b s e r v a t i o n s - 10 per c e l l ) TABLE 3 ANALYSIS OF VARIANCE Eq u a t i o n s then Exponents - Exponents o n l y Sum of Source D.F. Squares Mean F-Square R a t i o P roba-b i l i t y B - V i s u a l 1 8.492 8.492 1.49 0.226 A - Answers 2 58.70 29.35 5.13 * 0.009 A x B 2 7.028 3.514 0.61 0.550 E r r o r 53 303.0 5.720 * The p r o b a b i l i t y of a c h i e v i n g an F-value as h i g h by chance, i s low enough t o conclude t h a t t h e r e s i g n i f i c a n t d i f f e r e n c e s . as t h i s are Group (c) (60 o b s e r v a t i o n s - 10 per c e l l ) TABLE 4 ANALYSIS OF VARIANCE TABLE Exponents then E q u a t i o n s • - Exponents o n l y Sum of Source D.F. Squares Mean F-Square R a t i o Proba-b i l i t y B - V i s u a l 1 1.221 1.221 0.25 0.628 A - Answers 2 9.126 4.563 0.92 0.409 A x B 2 7.680 3.840 0.77 0.472 E r r o r 53 264.1 4.982 29 Group (d) (60 o b s e r v a t i o n s - 10 per c e l l ) TABLE 5 ANALYSIS OF VARIANCE Exponents then Equations - Eq u a t i o n s o n l y Source D.F Sum of Squares Mean Square F-R a t i o Proba-b i l i t y B - V i s u a l 1 2 .401 2 .401 0.38 0.545 A - Answers 2 25.60 12.80 2 .05 0.137 A x B 2 9.982 4.991 0.80 0.459 E r r o r 53 331.3 6.252 Group (e) (120 o b s e r v a t i o n s - 20 per c e l l ) TABLE 6 ANALYSIS OF VARIANCE Equa t i o n s then Exponents Source D.F Sum of Squares Mean Square F-R a t i o Proba-b i l i t y B - V i s u a l 1 5 .978 5.978 1.09 0.300 A - Answers 2 30.14 15.07 2 .74 0.067 ** A x B 2 3.497 1.749 0.32 0.732 E r r o r 113 30 Group (f) (120 o b s e r v a t i o n s - 20 per c e l l ) TABLE 7 ANALYSIS OF VARIANCE Exponents then Equations Source D.F. Sum of Squares Mean Square F-R a t i o Proba-b i l i t y B - V i s u a l 1 3.355 3.355 0.60 0.447 A - Answers 2 6.866 3.433 0.61 0.550 A x B 2 12.07 6.036 1.07 0.346 E r r o r 113 635.1 5.620 Group (g) (120 o b s e r v a t i o n s - 20 per c e l l ) TABLE 8 ANALYSIS OF VARIANCE T o p i c of Exponents Sum of Mean F- Proba-Source D.F. Squares Square R a t i o b i l i t y B - V i s u a l 1 11.07 11.07 1.94 0.162 A - Answers 2 20.75 10.37 1.82 0.164 A x B 2 2.394 1.197 0.21 0.812 E r r o r 113 643.8 5.697 31 Group (h) (120 o b s e r v a t i o n s - 20 per c e l l ) TABLE 9 ANALYSIS OF VARIANCE T o p i c of Eq u a t i o n s Sum of Mean F- Proba-Source D.F. Squares Square R a t i o b i l i t y B - V i s u a l 1 1.475 1.475 0.27 0.612 A - Answers 2 24.30 12.15 2 .21 0.112 A x B 2 3.120 15.60 0.28 0.757 E r r o r 113 621.2 5.500 Group ( i ) (120 > o b s e r v a t i o n s - 20 per c e l l ) TABLE 10 ANALYSIS OF VARIANCE F i r s t U n i t of Work Sum of Me an F - Proba-Source D.F. Squares Square R a t i o b i l i t y B - V i s u a l 1 0.9981 0.9981 0.20 0.657 A - Answers 2 2 .449 1.224 0.25 0.782 A x B 2 3.960 1.980 0.40 0.675 E r r o r 113 554.7 4.909 32 Group (j) (120 o b s e r v a t i o n s - 2 0 per c e l l ) TABLE 11 ANALYSIS OF VARIANCE Second U n i t o f Work Source D.F. Sum of Squares Mean Square F-R a t i o Proba-b i l i t y B - V i s u a l 1 14.06 14.06 2.22 0.135 A - Answers 2 32.01 16.01 2.52 0.083 ** A x B 2 14.01 7.006 1.10 0.336 E r r o r 113 716.8 6.343 Thus from the above t a b l e s , the o n l y s t a t i s t i c a l l y s i g n i f i c a n t r e s u l t a t the 5X> l e v e l o f s i g n i f i c a n c e , comes from T a b l e 3. A l t h o u g h i t should be noted t h a t the valu e s i n T a b l e 6 and Table 11 are c l o s e to the 5% l e v e l of s i g n i f i c a n c e ; note double s t a r r e d (**) e n t r i e s . Hypothesis One Hypothesis one says t h a t "there i s no s i g n i f i c a n t d i f f e r e n c e between the mean s c o r e s o f the students whose answers were l i s t e d s e q u e n t i a l l y and the mean scores of the students whose answers were found by means o f a mapping ( V a r i a b l e B ) . " 33 I t can be seen from the above t a b l e s t h a t the o b t a i n e d value was always l e s s than the c r i t i c a l value as noted by the p r o b a b i l i t y v a l u e . T h e r e f o r e i t was concluded t h a t t h e r e was no s i g n i f i c a n t d i f f e r e n c e o f the mean sco r e s due to the v i s i b i l i t y o f the answers. Hypothesis Two Hypoth e s i s two was t h a t "there i s no s i g n i f i c a n t d i f f e r e n c e among the mean s c o r e s o f the students who had a d i f f e r e n t p r o p o r t i o n of answers ( V a r i a b l e A ) . " I t can be seen from the t a b l e s above t h a t the o b t a i n e d v a l u e was h i g h e r than the c r i t i c a l v a lue as noted by the p r o b a b i l i t y value i n o n l y one t a b l e , t h a t o f Tab l e 3. T h e r e f o r e i t was concluded t h a t i n the group t h a t d i d equations f i r s t and then exponents, and when o n l y c o n s i d e r -i n g the exponent s e c t i o n o f m a t e r i a l , t h a t t h e r e i s a s i g n i -f i c a n t d i f f e r e n c e among the mean sco r e s o f the students who had a d i f f e r e n t p r o p o r t i o n of answers. But i n a l l o t h e r cases i t was concluded t h a t there was no s i g n i f i c a n t d i f f e r -ence of the mean sco r e s due to the p r o p o r t i o n o f answers. Hypothesis Three Hypothesis t h r e e was t h a t "there i s no s i g n i f i c a n t d i f f e r e n c e among the mean sco r e s of the students when the i n t e r a c t i o n of v a r i a b l e A and v a r i a b l e B i s taken i n t o 3 4 c o n s i d e r a t i o n . I t can be seen from the above t a b l e s t h a t the o b t a i n e d v a l u e was always l e s s than the c r i t i c a l value as noted by the p r o b a b i l i t y v a l u e . T h e r e f o r e i t was con-c l u d e d t h a t there was no s i g n i f i c a n t d i f f e r e n c e i n the mean sco r e s due to the i n t e r a c t i o n e f f e c t o f v a r i a b l e A and v a r i a b l e B. I I . FURTHER ANALYSIS S i n c e t h e r e was a s i g n i f i c a n t d i f f e r e n c e among the mean s c o r e s f o r the p r o p o r t i o n o f answers as t e s t e d by h y p o t h e s i s two, f u r t h e r a n a l y s i s was done on t h i s v a r i a b l e A, to f i n d out which o f the th r e e v a r i a b l e s v a r i a b l e A 3 - where a l l the answers were g i v e n v a r i a b l e A 2 - where a l l the answers were g i v e n but 2 0 % were i n c o r r e c t v a r i a b l e A^ - where o n l y 80X> o f t h e answers were g i v e n was most s i g n i f i c a n t . A comparison t e s t on the a d j u s t e d treatment means 21 was done as Edwards d e s c r i b e s : The t e s t f o r s i g n i f i c a n c e i s a t - t e s t where I —» w i t h k ( n - l ) - 1 degrees of freedom. An a d j u s t e d treatment mean on v a r i a b l e Y i s g i v e n by The e f f e c t i v e r e s i d u a l e r r o r mean square f o r a l l comparisons on the a d j u s t e d treatment mean i s g i v e n by where S 2 i s the e r r o r mean square o f the a n a l y s i s o f c o -v a r i a n c e . From the data the e f f e c t i v e r e s i d u a l mean square ( S e ) was e q u a l t o 5.754 a l s o ^/K, = 13.65 yj = 14.74 % = 12-32 Comparison o f V a r i a b l e A^ and V a r i a b l e A2 The above i n f o r m a t i o n was used to o b t a i n a t - v a l u e o f 3.21. T h i s o b t a i n e d v a l u e was compared w i t h the c r i t i c a l v a l u e o f t w i t h 176 d . f . S i n c e the o b t a i n e d v a l u e was h i g h e r than the c r i t i c a l v a l u e , i t was concluded t h a t t h e r e was a s i g n i f i c a n t d i f f e r e n c e between the treatment means of V a r i -a ble A3 and A 2 a t the 5% l e v e l o f s i g n i f i c a n c e . 3 6 Comparison of V a r i a b l e A3 and V a r i a b l e A^ The above i n f o r m a t i o n was used to o b t a i n a t -val u e o f 1.73. T h i s o b t a i n e d value was compared w i t h the c r i t i c a l v a l u e o f t w i t h 176 d . f . S i n c e the o b t a i n e d value was g r e a t e r than the c r i t i c a l v a l u e , i t was concluded t h a t t h e r e was a s i g n i f i c a n t d i f f e r e n c e between the treatment means o f V a r i a b l e A3 and A^ a t the 5% l e v e l o f s i g n i f i c a n c e . Comparison o f V a r i a b l e A^ and A 2 The above i n f o r m a t i o n was used to o b t a i n a t -v a l u e o f 1.43. T h i s o b t a i n e d value was compared w i t h the c r i t i c a l v a l u e o f t w i t h 176 d . f . S i n c e the o b t a i n e d value was l e s s than the c r i t i c a l v a l u e , i t was concluded t h a t t h e r e was no s i g n i f i c a n t d i f f e r e n c e between the treatment means o f v a r i a b l e A^ or A 2 a t the 5% l e v e l of s i g n i f i c a n c e . I I I . THE HAWTHORNE EFFECTS The a t t e n t i o n Hawthorne e f f e c t was the same f o r a l l s tudents t a k i n g p a r t i n the experiment. None of the students knew, to the b e s t of the i n v e s t i g a t o r ' s knowledge, th a t they were a c t u a l l y t a k i n g p a r t i n an experiment. The experimenter had o n l y mentioned i n Day 3 t h a t he was " t r y i n g to f i n d out which answer sheet was the b e s t . " 37 The n o v e l t y Hawthorne e f f e c t was c o n t r o l l e d p a r t i a l l y by a l l the c l a s s e s h a v i n g the same t e a c h e r . While some of the answer sheets were new, the n o n - a v a i l a b l e ones, the experiment was of s u f f i c i e n t l e n g t h and d u r a t i o n , t h a t t h i s e f f e c t wore o f f , and the students accepted the n o n - a v a i l a b l e answer sheet, as j u s t another answer sheet. Perhaps t h i s c o u l d be one reason t h a t n o n - s i g n i f i c a n t r e s u l t s were o b t a i n e d between the n o n - a v a i l a b l e answer s h e e t and the s t a n d a r d textbook format answer s h e e t . The i d e n t i f i c a t i o n Hawthorne e f f e c t would have been v e r y hard t o manipulate by the experimenter i f he had t r i e d to manipulate i t . S i n c e one classroom c o n t a i n e d a l l o f the treatments, the experimenter c o u l d not s i n g l e out any s p e c i f i c group and i n f l u e n c e them by h i s p r e c o n c e i v e d thoughts or o p i n i o n s on which treatment would be b e s t . IV. CONCLUSIONS There was no s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e between the group of students whose answers were a v a i l a b l e , one a f t e r another, and the group of students whose answers were n o n - a v a i l a b l e , mapped-out. There was a s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e f o r the group of students who d i d the equations f i r s t and 38 then the exponents, and o n l y on the exponent s e c t i o n t h a t they worked on, among the mean scores of the t h r e e t r e a t -ments. F u r t h e r a n a l y s i s has shown t h a t the group which had a l l the answers was s t a t i s t i c a l l y s i g n i f i c a n t l y lower than the other two groups, the one which had a l l the answers but 20% were i n c o r r e c t , and the one which had o n l y 80% o f the answers. But t h e r e was no s t a t i s t i c a l d i f f e r e n c e between the l a t t e r two groups. There was no s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e f o r any o f the o t h e r groups i n r e s p e c t to the p r o p o r t i o n of answers t h a t were g i v e n . V. ANALYSIS OF ADDITIONAL DATA Students' O p i n i o n The students were asked to f i l l out the q u e s t i o n -n a i r e as i t appears i n Appendix F. Amongest a l l the 144 s t u d e n t s o n l y 3 of them had d i f f e r e n t c h o i c e s f o r q u e s t i o n I I I and q u e s t i o n IV. The student's c h o i c e s were ranked w i t h a f i v e g i v e n to f i r s t c h o i c e , a f o u r f o r second c h o i c e , e t c . and a zero f o r l a s t c h o i c e . The t o t a l v a l u e s o f the r a n k i n g s c a n be seen i n T a b l e 12. 39 I t can be seen from Table 12 t h a t the students gave the l a r g e s t vote to the group to which they had belonged. From T a b l e 12, the votes were combined i n t o two groups, a c c o r d i n g to v a r i a b l e B. T a b l e 13 g i v e s the g r o u p i n g . I t can be seen t h a t 60% of the vote went to the n o n - a v a i l a b l e answer s h e e t s . Perhaps the p o p u l a r i t y of the n o n - a v a i l a b l e answer sheets can be a t t r i b u t e d to t h e i r n o v e l t y , or perhaps the students thought t h a t w i t h o u t s e e i n g the next answer t o the q u e s t i o n , they would do much b e t t e r than i f they had seen the answer. From T a b l e 12, the votes were combined a c c o r d i n g t o v a r i a b l e A, or the p r o p o r t i o n of answers. Ta b l e 14 g i v e s the r e s u l t s . I t s h o u l d be noted t h a t the percentages a r e approximately a l l the same, and the students d i d not r e a l l y have a p r e f e r e n c e on how many or how c o r r e c t the answers were. 40 TABLE 12 STUDENTS VOTING Student's Choice A B C D E F A 72 50 59 47 31 26 Student B 59 79 56 32 40 19 Belonged C 63 39 85 30 20 48 to D 59 28 76 54 20 48 Group E 40 43 65 31 60 46 F 47 29 71 46 26 66 TABLE 13 STUDENTS VOTING GROUPED ACCORDING TO VISIBILITY (VARIABLE B) A v a i l a b l e B-^  N o n - A v a i l a b l e B 2 A, B, C D, E, F T o t a l Vote % of t o t a l 690 40% 1020 60% 41 TABLE 14 STUDENTS VOTING GROUPED ACCORDING TO PROPORTION OF ANSWERS (VARIABLE B) A3 a l l the answers A 2 a l l the answers but 2 0 % i n c o r r e c t AJL 80 % of the answers Tot a l Vote % of t o t a l 665 3 9 % 465 2 7 % 580 3 4 % 4 ^ CHAPTER IV IMPLICATIONS OF THE STUDY I . INTRODUCTION Two main hypotheses were i n v o l v e d i n t h i s study. F i r s t i t was h y p o t h e s i z e d t h a t grade e l e v e n students who r e c e i v e d n o n - a v a i l a b l e answers to s e t s of e x e r c i s e s on two t o p i c s would l e a r n more than those who r e c e i v e d the s t a n d a r d textbook format. A n o n - s i g n i f i c a n t d i f f e r e n c e between the two groups was found on a c r i t e r i o n t e s t . But s t u d e n t s seemed to l i k e the n o v e l approach to the method o f n o n - a v a i l a b l e answers as demonstrated by t h e i r o p i n i o n s . Second i t was h y p o t h e s i z e d t h a t those students who r e c e i v e d 80% r e i n f o r c e m e n t would l e a r n more, s c o r e h i g h e r on the c r i t e r i o n t e s t , than those who r e c e i v e d a l l the answers. A s i g n i f i c a n t d i f f e r e n c e was found f o r the group of students who worked on equations f i r s t and then on exponents, and o n l y on the exponent p o r t i o n of the e x e r c i s e s . A l s o the students d i d not demonstrate any r e a l p r e f e r e n c e to the q u a n t i t y of answers t h a t they were g e t t i n g . 42 43 I I . DISCUSSION OF THE CONCLUSIONS F i r s t Hypothesis Questioned From t h i s study i t appears t h a t making answers n o n - a v a i l a b l e does not s i g n i f i c a n t l y improve a student's performance when working on a p a r t i c u l a r e x e r c i s e o r t o p i c . However the study does show t h a t the students do f a v o r t h e s e n o n - a v a i l a b l e answers. Perhaps the teacher c o u l d use t h i s approach of c o n c e a l i n g answers f o r m a i n t a i n i n g i n t e r e s t and s t i m u l a t i n g the s t u d e n t s . I n s t e a d of simply l i s t i n g , "here are the answers," the t e a c h e r might p r e s e n t them i n t h i s c o n c e a l e d f a s h i o n . Second Hypothesis From t h i s study i t appears t h a t p a r t i a l r e i n -forcement may be b e t t e r than complete r e i n f o r c e m e n t as f a r as g i v i n g students answers to e x e r c i s e s . A s i g n i f i c a n t r e s u l t was o b t a i n e d o n l y f o r the group who f o l l o w e d the sequence s o l v i n g equations o f the f i r s t degree then s i m p l i -f y i n g e x p o n e n t i a l e x p r e s s i o n s , and o n l y r e t a i n e d to the s i m p l i f y i n g o f e x p o n e n t i a l e x p r e s s i o n s f o r a number o f r e a s o n s . The equation s e c t i o n was a l i t t l e l o n g e r than the exponent s e c t i o n and students were more f a m i l i a r w i t h the e q u a t i o n s . I t i s p o s s i b l e t h a t the s t u d e n t s got used to 44 u s i n g the answer sheets w h i l e working on the equations and thus used t h i s approach of q u e s t i o n i n g and checking when answers were m i s s i n g . The exponent s e c t i o n was a r e l a t i v e l y new t o p i c . The equation t o p i c a l s o lends i t s e l f t o s e l f - c h e c k i n g . For example, o f one answer i s m i s s i n g or i n c o r r e c t i t i s easy to recheck i n the o r i g i n a l e q u a t i o n . The group t h a t d i d exponents f i r s t and then e q u a t i o n s , d i d not show any d i f f e r e n c e i n t h e i r performance on the e q u a t i o n s e c t i o n p r o b a b l y because they d i d not q u e s t i o n o r check the answer i n the s h o r t two-worksheet exponent t o p i c . The r e s u l t s a l s o showed t h a t a c l o s e r r e s u l t to s u b s t a n t i a t i n g h y p o t h e s i s two a t the 5% l e v e l o f s i g n i f i c a n c e was a c h i e v e d from the e q u a t i o n t o p i c r a t h e r than the exponent t o p i c . The main reason f o r t h i s i s p r o b a b l y due t o the f a c t t h a t the e q u a t i o n t o p i c l e nds i t s e l f to l e a v i n g out answers and p u t t i n g i n i n c o r r e c t answers because the answers can always be checked i n the o r i g i n a l e q u a t i o n . The r e s u l t s showed t h a t when c o n s i d e r i n g f i r s t u n i t s o f work, and second u n i t s o f work, a c l o s e r r e s u l t to s u b s t a n t i a t i n g h y p o t h e s i s two at the 5 % l e v e l o f s i g n i f i -cance was a c h i e v e d from the second u n i t o f work. The reason f o r t h i s may be due to the f i r s t u n i t of work b e i n g a t r a i n i n g p e r i o d whereby the students l e a r n e d to cope w i t h m i s s i n g and i n c o r r e c t answers. . The study tends to show s t a t i s t i c a l l y t h a t the e q u a t i o n t o p i c lends i t s e l f as a f i r s t u n i t o f work to achieve h i g h e r r e s u l t s on a new t o p i c l i k e exponents. I n t e r a c t i o n Among Students As mentioned i n the c h a p t e r on the d e s i g n of the study, the procedure s e c t i o n , the students were urged not t o i n t e r a c t w i t h each o t h e r . A t f i r s t the students attempted to seek out the answers i f t h e i r s was m i s s i n g . But i f the whole c l a s s or an i s o l a t e d group were g i v e n one s e t o f answers they might i n t e r a c t b e n e f i c i a l l y by showing each o t h e r not o n l y the m i s s i n g answers but the method by which they o b t a i n e d i t . A f o l l o w - u p study c o u l d be done whereby a whole c l a s s would be i n v o l v e d w i t h a s p e c i f i c t reatment. I I I . LIMITATIONS OF THE STUDY The Sample The students were chosen as a convenience r a t h e r than a random sample from a p o p u l a t i o n o f a l l students i n B r i t i s h Columbia. Four c l a s s e s were used from the same s c h o o l and were taught by the same t e a c h e r . Perhaps o t h e r 46 s t u d i e s c o u l d be done u s i n g other grade e l e v e n students and u s i n g another t e a c h e r . The T o p i c s Only two t o p i c s , t h a t of equations and exponents were used i n t h i s experiment. The equations were of the f i r s t degree and i n one v a r i a b l e . Perhaps t h i s a b i l i t y to c h e c k back i s o n l y i n h e r e n t and u s e f u l i n s o l v i n g e q u a t i o n s . Other t o p i c s c o u l d be t e s t e d . The Method o f I n s t r u c t i o n D u r i n g the experiment a l i m i t e d number o f t e a c h i n g techniques were used by the i n s t r u c t o r . The " i n s t r u c t i o n a l p e r i o d " which proceeded the e x e r c i s e sheets was ir re or l e s s a l e c t u r e p e r i o d . A l t h o u g h the students were g i v e n time to do a few examples from the board a f t e r which the e x p e r i m e n t e r went through these examples, t h e r e was l i t t l e i n t e r a c t i o n between the t e a c h e r and s t u d e n t s . During the e x e r c i s e sheets t h e r e was a l s o l i t t l e i n t e r a c t i o n between the teacher and s t u d e n t s . The e x e r c i s e sheets were more or l e s s a d r i l l t echnique whereby the s t u d e n t s c o u l d l e a r n by themselves. A few examples were done by the t e a c h e r f o r a l l c l a s s e s but the experimenter was more o r l e s s i n h i b i t e d and f o r c e d not to i n t e r a c t w i t h t h e c l a s s . 47 Long-term E f f e c t s R e t e n t i o n was not i n v e s t i g a t e d i n t h i s study. I t c o u l d have been t h a t a l t h o u g h the students performed b e t t e r o n the c r i t e r i o n t e s t immediately a f t e r the e x e r c i s e sheet because of a s p e c i f i c treatment, they might not have i n a few weeks time. Thus the c o n c l u s i o n s t h a t were reached were o n l y v a l i d i n the s h o r t - r u n . Grade and A b i l i t y L e v e l The study was run on grade e l e v e n s t u d e n t s . Perhaps d i f f e r e n t r e s u l t s would have been a c h i e v e d at d i f f e r e n t grade l e v e l s . Perhaps the s t u d e n t s a t the grade e l e v e n l e v e l r e a c t d i f f e r e n t l y to c o n c e a l e d answers than do o t h e r students a t d i f f e r e n t age l e v e l s . IV. SUGGESTIONS FOR FURTHER RESEARCH S i m i l a r Research A study s i m i l a r to t h i s one c o u l d be done u s i n g a d i f f e r e n t sample, perhaps from a d i f f e r e n t socio-economic background and l o c a t i o n . I t would be i n t e r e s t i n g to i n v e s t i g a t e what the r e s u l t s would be w i t h a d i f f e r e n t t e a c h e r . Other t o p i c s , r a t h e r than equations of the f i r s t degree i n one v a r i a b l e , and exponents, c o u l d be s t u d i e d . 48 Perhaps equations w i t h two v a r i a b l e s c o u l d be used as a s u b s t i t u t e i f the i n v e s t i g a t o r was s t i l l i n t e r e s t e d i n m a i n t a i n i n g the c hecking back e f f e c t . Other s t u d i e s c o u l d be designed so t h a t a whole c l a s s would use one treatment. T h i s would i n v o l v e a broader sample and q u i t e a l a r g e r one. A l t h o u g h t h i s study produced n o n - s i g n i f i c a n t r e s u l t s w i t h the v a r i a b l e o f a v a i l a b i l i t y , perhaps the same v a r i a b l e c o u l d be s t u d i e d i n another l i g h t . The i n v e s t i g a t o r thought t h a t when the answers were p r e s e n t e d i n a c o n c e a l e d f a s h i o n t h a t the students would do b e t t e r on a c r i t e r i o n t e s t based on the e x e r c i s e s . I t c o u l d have been t h a t , a l t h o u g h the students d i d n ' t l e a r n more on the e x e r c i s e sheet t o p i c , they c o u l d have l e a r n e d something more v a l u a b l e ; t h a t of being able to use c o o r d i n a t e s and a c o - o r d i n a t e system i n mapping. Another study c o u l d be designed to study the e f f e c t of the v a r i a b l e of a v a i l a b i l i t y on b e i n g a b l e to use maps and c o - o r d i n a t e systems. Other s t u d i e s c o u l d be done at d i f f e r e n t grade l e v e l s . The experimenter b e l i e v e s t h a t the a v a i l a b i l i t y v a r i a b l e does have an e f f e c t , not o n l y on student o p i n i o n , but on o t h e r a r e a s . The experimenter d i d n ' t show any s t a t i s t i c a l l y s i g n i f i c a n t r e s u l t i n r e s p e c t to m a t e r i a l 49 l e a r n e d , but perhaps other i n v e s t i g a t o r s c o u l d show t h a t t h i s concealment v a r i a b i l i t y does have an e f f e c t on some a s p e c t of l e a r n i n g . Other Research T h i s study found t h a t students who were g i v e n 80% reinforcement, performed b e t t e r than the group w i t h 100% r e i n f o r c e m e n t (on a l i m i t e d t o p i c ) . In the "Review of the 13 L i t e r a t u r e , " Dickey had found t h a t 33 1/3% r e i n f o r c e m e n t was s u p e r i o r than 100%. Perhaps a study can be done w i t h v a r y i n g amounts o f answers on a broader s c a l e , from 0% t o 100% a t 5% i n t e r v a l s ? Research c o u l d be s t a r t e d i n classrooms on the g e n e r a l t o p i c o f r e i n f o r c e m e n t . Perhaps one type o f teacher r e i n f o r c e s i n a p a r t i c u l a r way as compared t o a d i f f e r e n t t e a c h e r , and one type o f r e i n f o r c e m e n t i s f a r more e f f e c t i v e f o r a p a r t i c u l a r s t u d e n t . For example, a t e a c h e r ' s a p p r o v a l o r d i s a p p r o v a l o f a c o r r e c t or i n c o r r e c t answer might have more e f f e c t on a student than the s a t i s -f a c t i o n a student might get from o b t a i n i n g a c o r r e c t answer. L i k e w i s e r e i n f o r c e m e n t should be s t u d i e d i n regards t o d i f f e r e n t s t u d e n t s . Perhaps some students are a f f e c t e d q u i t e d i f f e r e n t l y from a p a r t i c u l a r type of r e i n -forcement than other s t u d e n t s ? 50 V. SUMMARY The study was designed to i n v e s t i g a t e the e f f e c t s o f two v a r i a b l e s ; t h a t of the a v a i l a b i l i t y of answers, and t h e p r o p o r t i o n o f answers. P r e v i o u s f i n d i n g s by r e s e a r c h e r s had i n d i c a t e d t h a t p a r t i a l r e i n f o r c e m e n t would i n c r e a s e the amount l e a r n e d o f a p a r t i c u l a r t o p i c . There was no r e s e a r c h t h a t was r e l e v a n t to the a v a i l a b i l i t y v a r i a b l e . A b a s i c 3x2 f a c t o r i a l d e s i g n was used w i t h two l e v e l s of a v a i l a b i l i t y 1) a v a i l a b l e 2) n o n - a v a i l a b l e and t h r e e l e v e l s o f p r o p o r t i o n o f answers 1) 80% o f the answers g i v e n 2) 20% i n c o r r e c t answers 3) a l l the answers g i v e n . Two t o p i c s , t h a t o f equations and exponents were used as the s u b j e c t matter. Four c l a s s e s were g i v e n i n -s t r u c t i o n and then p r e t e s t e d on the p a r t i c u l a r t o p i c . Then t h e s i x treatments were randomly a s s i g n e d i n a s t r a t i f i e d way and the students proceeded on e x e r c i s e s p e r t a i n i n g the the p a r t i c u l a r t o p i c . A p o s t t e s t was a d m i n i s t e r e d a f t e r a p a r t i c u l a r t o p i c was f i n i s h e d . The students were r e -a s s i g n e d t o p i c s w h i l e m a i n t a i n i n g t h e i r same treatment. The students f o l l o w e d the same procedure of l e c t u r e , p r e -t e s t , e x e r c i s e s and p o s t t e s t . 51 An a n a l y s i s of c o - v a r i a n c e was c a l c u l a t e d on each o f the groups a c c o r d i n g to t o p i c , o r d e r o f u n i t , and p a r t i -c u l a r group. The c o - v a r i a t e was the p r e t e s t on the p o s t -t e s t . A l s o students were asked to rank o r d e r t h e i r p r e f e r -ence f o r answer s h e e t s . The r e s u l t s showed t h a t the a v a i l a b i l i t y f a c t o r h a d no s t a t i s t i c a l l y s i g n i f i c a n t e f f e c t on the student's performance on any t o p i c . A l t h o u g h the students f a v o r e d t h e n o n - a v a i l a b l e answers. The r e s u l t s a l s o showed t h a t the students who were g i v e n 30% of the answers and those who were g i v e n 2 0% i n c o r r e c t answers, both performed b e t t e r than the group who was g i v e n a l l the answers. T h i s was s t a t i s t i c a l l y s i g n i f i c a n t o n l y w i t h the group who f o l l o w e d the sequence s o l v i n g equations o f the f i r s t degree then s i m p l i f y i n g e x p o n e n t i a l e x p r e s s i o n s and o n l y p e r t a i n e d t o the s i m p l i f y -i n g o f e x p o n e n t i a l e x p r e s s i o n s . £2* FOOTNOTES 53 "^C. W. 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"The E f f e c t s of Delayed Knowledge o f R e s u l t s on L e a r n i n g and R e t e n t i o n f o r D i f f e r i n g Age L e v e l s of Sc h o o l Age C h i l d r e n , " D i s s e r t a t i o n A b s t r a c t s , 26:5858, 1965. Brown, F r a n c i s J . "Knowledge of R e s u l t s as an I n c e n t i v e i n School Room P r a c t i c e , " J o u r n a l o f E d u c a t i o n a l  Psychology, XXIII, (October, 1932) pp. 532-552. 58 Brown, R. S. "An Ex p e r i m e n t a l Study of Knowledge of R e s u l t s i n Complex Human L e a r n i n g , " D i s s e r t a t i o n  A b s t r a c t s , 24:2356, 1963. Chapdonis, A. "Knowledge of Performance as an I n c e n t i v e i n R e p e t i t i v e , Monotonous Tasks," J o u r n a l of A p p l i e d  Psychology, LXVIII (August 1964) pp. 263-267. C h i l d , I r v i n L . " S o c i a l i z a t i o n , " i n L i n d z e y and Gardner ( e d . ) . Handbook o f S o c i a l Psychology. V o l . 2, Addison-Wesley P u b l i s h i n g Company, Inc., 1954, pp. 655-92. Davi s , B e t t y . "The E f f e c t of Reinforcement i n Teaching A r i t h m e t i c on the Performance of F i f t h Grade Students," D i s s e r t a t i o n A b s t r a c t s , 22:178, 1961. D e l l a , P i o n a and M a r i o G a b r i e l . "Two E x p e r i m e n t a l Feedback P r o c e d u r e s — a Comparison of T h e i r E f f e c t s on the L e a r n i n g of Concepts," D i s s e r t a t i o n A b s t r a c t s , 16:910, 1956. Dickey, Ouida W. "A Study of the E f f e c t s o f Three Schedules of Reinforcement Upon Achievement and R e t e n t i o n i n a L i n e a r Program i n C o l l e g e B u s i n e s s Mathematics," D i s s e r t a t i o n A b s t r a c t s , 27:3363, 1966. D o l c i a n i , Mary P., Simon L . Berman and W i l l i a m Wooton. Modern A l g e b r a and Trigonometry, S t r u c t u r e and  Method, Book I I , Boston: Houghton M i f f l i n Company, 1963, pp. 39-117. 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"The E f f e c t of D i f f e r e n t i a l F a i l u r e on E x p e c t a t i o n of Success, Reported A n x i e t y and Response U n c e r t a i n t y , " J o u r n a l of P e r s o n a l i t y , XXXI, 1963, pp. 289-312. Feather, N. T. " E f f e c t s of Amount of P r i o r Success and F a i l u r e s on E x p e c t a t i o n s of Success and Subsequent Task Performance," J o u r n a l of P e r s o n a l i t y and S o c i a l  Psychology, V (February, 1967), pp. 226-232. Feather, N. T. "The R e l a t i o n s h i p of E x p e c t a t i o n of Success t o Need Achievement and T e s t A n x i e t y , " J o u r n a l of  P e r s o n a l i t y and S o c i a l Psychology, I (1965), pp. 118-126. Feather, N. T. " E f f e c t s of P r i o r Success and F a i l u r e or E x p e c t a t i o n s of Success and Subsequent Performance," J o u r n a l of P e r s o n a l i t y and S o c i a l Psychology, I I I (March 1966), pp. 287-298. Feather, N. T. "Performance a t a D i f f i c u l t Task i n R e l a t i o n t o I n i t i a l E x p e c t a t i o n of Success, T e s t A n x i e t y and Need Achievement," J o u r n a l o f P e r s o n a l i t y , XXXIII (1965), pp. 200-217. Feather, N. T. "Mowner's R e v i s e d Two-Factor Theory and the Motive Expectancy-Value Model," P s y c h o l o g i c a l Review, LXX (1963), pp. 500-515. Feather, N. T. "The R e l a t i o n s h i p of P e r s i s t e n c e at a Task t o E x p e c t a t i o n of Success and Achievement R e l a t e d Motives," J o u r n a l of Abnormal and S o c i a l Psychology, L X I I I (1961) pp. 552-561. Feather, N. T. " P e r s i s t e n c e a t a D i f f i c u l t Task w i t h A l t e r n a t i v e Task of I n t e r m e d i a t e D i f f i c u l t y , " J o u r n a l  of Abnormal and S o c i a l Psychology, LXVI (1963), pp. 604-609. 60 Feather, N. T. "The r e l a t i o n s h i p of e x p e c t a t i o n of success t o r e p o r t e d p r o b a b i l i t y , t a s k s t r u c t u r e and achievement r e l a t e d m o t i v a t i o n , " J o u r n a l of Abnormal and S o c i a l  Psychology, LXVI (1963), pp. 231-238. F e r s t e r , C h a r l e s , B. and Skinner, B. F. Schedules of Reinforcement. New York: A p p e l t o n - C e n t u r y - C r o f t s , (1957) p. 741. F r e d e r i c k , Franz, J . " E f f e c t s of Delay of Knowledge of R e s u l t s and Post Knowledge of R e s u l t s Delay on O b s e r v a t i o n a l P a i r e d - A s s o c i a t e L e a r n i n g , " D i s s e r t a t i o n A b s t r a c t s , 26:5911, 1965. H a r r i s , Chester W. (ed.) . E n c y c l o p e d i a of E d u c a t i o n a l R e s e a r c h . T h i r d E d i t i o n . New York: M a c M i l l a n , 1960. H i l g a r d , E r n e s t R. and D. H. R u s s e l l , " M o t i v a t i o n i n School L e a r n i n g , " i n L e a r n i n g and I n s t r u c t i o n , 49th Yearbook, P a r t I, N.S.S.E., U n i v e r s i t y of Chicago, 1950, pp. 36-68. H i r s c h , R. S. "The E f f e c t s of Knowledge of T e s t R e s u l t s on L e a r n i n g of M e a n i n g f u l M a t e r i a l , " ( P e n n s y l v a n i a S t a t e U n i v e r s i t y , I n s t r u c t i o n a l F i l m Research Program), P o r t V7ashington, N. Y., U.S. Naval T r a i n i n g Device Center, O f f i c e o f Naval Research, Tech. Rept. No. SDC-7-30, Sep t . 1952. Humphreys, L . G. "The E f f e c t of Random A l t e r n a t i o n of Reinforcement on the A c q u i s i t i o n and E x t i n c t i o n of C o n d i t i o n e d E y e l i d R e a c t i o n s , " J o u r n a l of E x p e r i m e n t a l  Psychology. XXV (1939), pp. 141-158. I r i o n , A. L . and L . J . B r i g g s . " L e a r n i n g Task and Mode of O p e r a t i o n V a r i a b l e s i n Use of the S u b j e c t Matter T r a i n e r , " L a c k l a n d A i r Force Base, A i r Force P e r s o n n e l and T r a i n i n g Research Center, Tech. Rept. AFPTRCTR-57-8 (ASTIA Doc. No. AD 134252), O c t . 1957. J e n k i n s , W. 0. and J . C. S t a n l e y . " P a r t i a l Reinforcement: A Review and C r i t i q u e , " P s y c h o l o g i c a l B u l l e t i n , XLVII (1950), pp. 193-234. Judd, C. H. " P r a c t i c e Without Knowledge of R e s u l t s , " P s y c h o l o g i c a l Monographs S u p p l . V I I , No. 1, (1905) pp. 185-198. 61 Karabinus, Robert A. " D i f f e r e n t i a l Examination-Feedback Treatments, L e a r n i n g and A t t i t u d e , " D i s s e r t a t i o n  A b s t r a c t s , 27:3724, 1966. Keim, F l o y d N. "The E f f e c t s of D i f f e r e n t Kinds and Amounts of W r i t t e n Feedback on the V e r b a l Teaching Behavior, A t t i t u d e s and O p i n i o n s o f Tenth Grade Teachers," D i s s e r t a t i o n A b s t r a c t s , 28:1657, 1967. K i r k , Roger E . E x p e r i m e n t a l Design: Procedures f o r the B e h a v i o r a l S c i e n c e s . Belmont: Brooks C o l e P u b l i s h i n g Company, 1968, pp. 520-521. Lennes, N. J . and J . W. Maucker. A F i r s t Course i n A l g e b r a , Toronto: Macmillan Company of Canada L t d . , 1954, pp. 105-133, 219-253. Mandler, G. and S. B. S a r a s o n . "A Study of A n x i e t y and L e a r n i n g , " J o u r n a l of Abnormal and S o c i a l  Psychology. XXXXVII (1952), pp. 166-173. M c C l e l l a n d , D. C , J . W. A t k i n s o n , R. A. C l a r k and E . L . L o w e l l . The Achievement Motive, New York: A p p l e t o n -C e n t u r y - C r o f t s , 1953. M c L a u l i n , J . C. "In I n v e s t i g a t i o n of the I n f l u e n c e o f Knowledge of R e s u l t s on L e v e l of M o t i v a t i o n During Problem S o l v i n g , " D i s s e r t a t i o n A b s t r a c t s , 22:2308, 1961. M i c h a e l , D. N. and N. Maccaby. " F a c t o r s I n f l u e n c i n g V e r b a l L e a r n i n g from F i l m s Under V a r y i n g C o n d i t i o n s of Audience P a r t i c i p a t i o n , " J o u r n a l of E x p e r i m e n t a l  Psychology. XLVT (1953) pp. 411-418. Monaghon, F l o y d V. "The E f f e c t of Delay of Knowledge of T e s t R e s u l t s on L e a r n i n g i n a N a t u r a l S c i e n c e Course," D i s s e r t a t i o n A b s t r a c t s , 22:2723, 1961. Oppenhiem, Don B. "The R e l a t i o n Between I n t e l l i g e n c e and D i f f e r e n t P a t t e r n s of Feedback i n a L i n e a r Teaching Program," D i s s e r t a t i o n A b s t r a c t s , 25:2857, 1964. 62 O t t i n a , J . R. "The E f f e c t s of Delay i n Knowledge of R e s u l t s on the Amount Learned i n T e a c h i n g Machine Programs o f D i f f e r i n g Cue Content," D i s s e r t a t i o n A b s t r a c t s , 25:1753, 1964. P i l e c k i , F r a n c i s J . "An I n v e s t i g a t i o n of the P r e d i c t i v e Value of I n t e r m i t t e n t Feedback and R e l a y Feedback i n Task Accomplishment," D i s s e r t a t i o n A b s t r a c t s , 28: 0086, 1967. P i t k i n , Walter B. The Psychology of Achievement. New York: Simon and S c h u s t e r , 1930. P o r t e r , D. A. "A C r i t i c a l Review of a P o r t i o n o f the L i t e r a t u r e on T e a c h i n g Devices," Harvard E d u c a t i o n a l  Review, XXVII (1957) pp. 126-147. Press y , S. L . "Development and A p p r a i s a l of D e v i c e s P r o v i d -i n g Immediate Automatic S c o r i n g o f O b j e c t i v e T e s t s and Concomitant S e l f I n s e r a c t i o n , " J o u r n a l of Psych-ology, XXIX (1950) pp. 417-447. Skinner, S. F. The Technology of T e a c h i n g . New York: A p p e l t o n - C e n t u r y - C r o f t s , 1968. Skinner, B. F. S c i e n c e and Human Behavior . New York: M a c M i l l a n , 1953, p. 461. S k i n n e r , B. F. "The S c i e n c e of L e a r n i n g and the A r t o f Teaching," Harvard E d u c a t i o n a l Review, XXIV (1954), pp. 86-97. Smith, P h i l i p . D. "Knowledge of R e s u l t s and C o n t i n u i t y of V a r i o u s Techniques i n P r e s e n t i n g a F i l m s t r i p as F a c t o r s i n Immediate L e a r n i n g and R e t e n t i o n , " D i s s e r t a t i o n A b s t r a c t s , 25:2385, 1964. Twelker, P a u l A. "Rules, Answers, and Feedback i n L e a r n i n g , R e t e n t i o n and T r a n s f e r o f Concepts," D i s s e r t a t i o n  A b s t r a c t s , 25:5131, 1964. Wallen, N. E . and R. M. T r a v e r s . " A n a l y s i s and I n v e s t i -g a t i o n of Teaching Methods," i n N. L . Gage (ed.) . Handbook of Research on T e a c h i n g . Chicago: Rand McNally, 1963. 63 APPENDICES 6 4 APPENDIX A E x e r c i s e Sheets on the E q u a t i o n T o p i c (a) Worksheet #1 on E q u a t i o n s (b) Worksheet #2 on E q u a t i o n s (c) Worksheet #3 on E q u a t i o n s 65 (a) Worksheet #1 on Equ a t i o n s Mathematics 11 Name Sol v e f o r the v a r i a b l e i n r e d , or f o r the i n d i c a t e d v a r i a b l e 1. x + 5 = -7 8. t = -9 15. 3x = a 2. x + 3 = -8 -5 16. x = b 3. y - 8 = -12 9. 5k = 4k-1 2 4. y - 1 = -29 10. -2 = -2 + 5k 17. x = be 5. -2z = 8 11. -12m + 8 = -13m a 6. -3z = 15 12. -6m + 2 = -7m 18. hy = a 7. t = -11 13. 5r + 5 = 0 19. I = p r t 3 14. p = a +b +c 20. I = p r t - hy F i n d the r o o t o f each of the f o l l o w i n g e q u a t i o n s . 21. u + 4 = -8 31. hu = -3 7 9 2 22. w - 3_ = -5_ 32. 9c = -3. 2 7 2 11 23. -11 = x + 7 33. 5x = -1_ 24. x - (-11) = 14 7 8 25. (-3.5) - y = 7.5 34. .07y = 8.4 26. w + 4 = 5. 35. -4.7 = 9.4 9 7 y 27. -1 = z = 4 36. -4.2 =2.1 5 3 x 28. -5.9 = x + (-3.4) 37. -4 = 3x 29. 5. - x = 6 9 11 11 7 38. .36 = -7.2 30. -3.9 - y = -5.9 x 39. 8 y = -5 13 7 40. - 1.3 = -3.9 t F i n d the s o l u t i o n s e t f o r the f o l l o w i n g e q u a t i o n s . 41. 7x + 8 = 4x + 20 46. 9x + 20 - 2x = 2x + 3 42. 13 + 8x - 4 = 87 - 5x 47. 6 - 4x + 3x - 3 = 3 43. x - 5 = 15 - x 48. x/3 + 12 = 13 44. 11 + 9x - 3 = 7x + 16 49. 4x - 9 + 2x + 1 = 4 45. 9 + 4x = 3 = 24 + 2x 50. 18y = 203 - l l y 66 (b) Worksheet #2 on Equations Mathematics 11 Name S o l v e f o r the v a r i a b l e i n r e d , or f o r the i n d i c a t e d v a r i a b l e 1. x + 6a = 7a 11. 1 8 + 2 (x-4) = 26 2. w + 1 = 3w - 21 12. 3 (2+x) + 7x = 2 (x+6) 3. y - a = 2 b 13. 9 (2x+2) - 30 = x + 5 4. 3a = p_ + 2c 14. 3 (3-2x) +3 (3-x) = 24 5. 3x - 7b = x 15. 3_ (4-2x) + 5x = -11 6. p = w 2 y 16. 2 (3-3x) +3 (4-2x) = 18 7. v = Bh 2 3 17. 5r + (r+15) -3 (r+2) = 0 8. 4 (x+2) = 3 (x+4) 18. 3 (x-9) = 7 4 - 2 (61+2x) 9. 16 (2x - 36) - 44 = x 19. 2w - 4 (w+2) =5 (w+4) 10. 3x + 14 (x-1) = 105 20. 5(x+l) =3 (x+2) So l v e each o f the f o l l o w i n g e q u a t i o n s . 21. n = 10 26. 7x-3 = 22 22. h& = .7 27. 14 = 4-z 23. 5+y = 3 28. 6^ y_ = 7 24. 2b - 1 = 9 -5 25. 3n-^ = 2 29. 2-v = 2 4 3 30. 7x-13 = 4 -3 Determine the r o o t s o f the f o l l o w i n g e q u a t i o n s . 31. 7_ (7-y) = -21 36. x+13 = -7x 3 -5 32. y-6 = -3 37. 7 = - x+3 y X 33. 7 = 3+S 38. - 5 = -7 S x+9 34. _ y _ = 7 39. 5 x +13 -7 x = 15 y-b 3 3 3 35. 6-x = 4x 40. - 4 x + 5 = 7 x + 7 7 3 9 (c) Worksheet #3 on Eq u a t i o n s Mathematics 11 67 Name Solve the f o l l o w i n g e q u a t i o n s . Assume t h a t the replacement s e t i s the s e t of r e a l numbers. 1. 3 (x-6) = 5 2. -2 (2t-3) = 7 3. 3 (2a+5) -7 (a -6) = 4a-6 4. 12x - (4x-6) = 3x - (9x-27) 5. 6z - (19-3z) = 6 - 3 (z+2) 6. -1 + 3r = 3 (r -1) 7. 7 + 6x _ 5 = -3 (2-2x) + 18 8. 3 (x-3) = 4 (3 -x) 9. y - 1 = -y - (y+1) + 2y 10. v - 3 = k V 11. 4 (x-1) = 2x - 3 2 12 . 4 + 3 + 7.= 6 x x 8 x 13. 7 = 11 y+2 y 14. 3 = 3 2-v v-2 15. x-3 = 3x + 4 7 16. 3 = 1 4-S 2S-5 17. 5 = 9 r-3 3 r - l 18. 7 + 4 = 1 - 2 y y Y 19. 1 - 3x = 1 -7x 5 20. 4 (S-3) = S+2 - (6-S) 3 3 21. 4 (3v+2) = 5 (v+8) + 7v 5 22. %k (2k-8) = -k (4-k) 23. 9-y (9-y) = 3y (1/3 y - 3) 24. q (3q+2) + 4 = 3 ( q 2 + q) 25. 7 = -5 2x+4 x-3 APPENDIX B E x e r c i s e Sheets on the Exponents T o p i c (a) Worksheet #1 on Exponents (b) Worksheet #2 on Exponents 69 (a) Worksheet #1 on Exponents Mathematics 11 F i n d the i n d i c a t e d p roducts and q u o t i e n t s . W r i t e a l l answers with p o s i t i v e exponents. Assume t h a t v a r i a b l e s i n denominators do not equal z e r o . 1. a5 . a 6 11. R 4 21. ( c 3 ) 3 R^ 2. y 3 • y 4 22 . ( y 2 ) 5 12. k l - x # k l + x 3. 3 X . 3y 13. 5 S 23 . 4. b 4 . b* 24. ( - 5 x 4 ) 2 5. 2 3 y • y • y J 14. x 2 - a x a 25. ( _ 4 y 5 ) 2 6. t x - l ^ t x + l 15. x a . x b 26. (1 ) 4 7. m? i 2 m3 16. (2xy)- 3 27. ( 4 B 3 ) 2 8. n6 17. (-3ab) 3 n^ 28. ( 2 S 3 ) 3 18. (-2RST) 2 9. 72R 29. ( - c d 2 ) 4 7* 19. (3 k l m ) 2 30. (-5R 2S 3) 10. y 5 b 20. ( a ) 2 y 7 b 3 31. (-3RS) ( 5 R S 2 ) R 4 37. htx {htx) 32. ( - 6 x y ) ( 8 x 2 y ) y 3 38. RS (RS) ( -R) 33. (-4a) (3ab) (-5b) 39. -Rt (R2t)(-Rt) 34. (6n)(-6n)(-6n) 40. -3x ( 4 x 3 y 2 ) ( - y 4 x 2 ) 2 35. (-7a) (-b) (-2c) 41. ( .2 2R 3S 7) ( .4 2RS 8) 36. (8RS 2T) (-6R2S) . 42. ( .3 2 z v 5 ) ( 9 2 z 3 v 4 ) 70 43. ( c 2 d 4 f 3 ) (-de7 f 3 ) 44. (-w2 x 5 y 4 ) (-x7 y 2 w3) 45. (-a)2 (a 3b) (-b) 3 46. (-2m 2) 3 47. ( - 3 S 2 ) 3 48. ( - 3 x ) ( 4 x y ) 2 49. -6n (5mn) 2 50 . ( -b) ( -3b) 3 51. (-R)(-6R) 3 52. 8m (2mn) 3 53. 3y ( S x y ) 3 54. 5a ( - 2 a 2 b ) 2 55. ( - 7 b ) ( . 2 y 2 x ) 3 (b) Worksheet #2 on Exponents Mathematics 11 71 Name F i n d the i n d i c a t e d products and q u o t i e n t s . W r i t e a l l answers with p o s i t i v e exponents. Assume t h a t v a r i a b l e s i n denomin-a t o r s do not eq u a l z e r o . 1. 3x 3 . 8x 3 13. 6 4 a 3 b 2 4ab 2 2. 4 x 3 . 5x 14. _ 1 5 R 4 c 3 3. (-5a 2b)(-3ab 2) 5R4 C 4. (-9abc)(3a 2b 3c) 15. 28mp2 -7mJp 5. (12RST 2)(-3R 2S 2T) 16. 5 k 5 v 4 6. (-2a) (-3a 2) (-4a 3) - 4 5 k 6 v 8 7. (-3xy)(-5x)(5y 3) 17. ( - 2 x 2 y ) 3 (6xy^) 'z 8. (-2m 2) 3 (3mn 2) 2 18. ( 3 a b 2 ) 4 9. ( - 3 S 3 ) 2 ( 2 S y ) 4 (-12a 2b) 2 10. 5ab ( - 2 a 2 b ) 3 19. ( 1 2 c 2 b 3 ) 3 ( 2 0 c 4 b 2 ) 3 11. ( 2 R S 2 t 3 ) 2 ( - 3 R 2 S 3 t ) 3 20. ( l S d ^ n 8 ) 2 ) 12. ( 6 a 2 b c 2 ) 3 ( - 2 a b 3 c 2 ) 2 ( 2 7d 3h& ) S i m p l i f y each of the f o l l o w i n g e x p r e s s i o n s 21. (-7a 2)(5b)(-2b) + (4ab)(-2ab) 22. (5p 2) (-3q) (-qp 2) - ( 7 q p 3 ) ( 7 p q ) 23. (-3RST) (4R 2ST) + (-6S 2) (2RT 2) (-R2) 24. (9v 2wz 3)(-2vw 2z) + ( - 3 z 2 ) ( - 6 v 3 w 2 ) ( w z 2 ) . 72 25. - ( 2 2 c 5 d 3 ) ( 5 c d 7 ) ( - 5 e ) + ( 3 2 d 6 c 3 e ) ( - d 2 c 2 ) c d 2 ) 26. (6 2m 3n 2p)(-2m 4p 4n5) - ( - p 3m 2n 3)(8 2m 5n 4p 2) 27. ( 3 x y ) 2 + (-5x) (-4xy) (-y) 28. (-2c) (-3cd) 3 + (3c) 2 ( c 2 d 2 ) (-6d) 29. ( - a ) 2 a + (2a 2) ( - 2 a 2 ) 3 - ( 2 a 2 ) 4 30. ( 2 b ) 2 (-25b 4) + ( - 4 b ) ( - b ) 2 + (-10b 3) 2 31. ( - z t ) 3 ( . 2 z 2 t ) + ( t 2 z ) 2 (.3z 3) + ( t 2 z ) ( z 2 t ) 2 32. (-3b) 3 (6b 2) - (2b 4) (5b) APPENDIX C F u l l s et of Answers f o r the E q u a t i o n Worksheets Answers t o Worksheet #1 i (a) Green i i (b) Y e l l o w i i i (c) P ink i v (d) Blue V (e) White v i (f) Orange Answers t o Worksheet #2 v i i (a) Green v i i i (b) Yellow i x (c) P ink x (d) B l u e x i (e) White x i i (f) Orange Answers t o Worksheet #3 x i i i (a) Green x i v (b) Yellow XV (c) P i n k x v i (d) Blue x v i i (e) White x v i i i (f) Orange 74 Answers t o Worksheet #1 i (a) Green Name: Equat i o n s 1. B6 11. D5 21. B4 31. CI 41. D9 2. C7 12. E l 22. A8 32. D7 42. E7 3 . DI 13. CIO 23. D4 33 . E10 43 . C4 4. A9 14. B2 24. D8 34. E5 44. D3 5. E6 15. E9 25. A10 35. B8 45. C6 6. D10 16. A3 26. C5 36. E2 46. D6 7. A l 17. E3 27. A2 37. BI 47. E8 8. B9 18. B7 28. E4 38. A4 48. BIO 9. A5 19. C8 29. B3 39. C9 49. A6 10. C3 20. D2 30, A7 40. C2 50. B5 There are 10 answers on t h i s worksheet t h a t are m i s s i n g . A B C D E 1 -6 -4 2 1/3 I RT -2 3 -31 77 0 4 X ab 4 -20 -92 63 -18 5 -1 7 17 63 120 6 2 -12 9 -3 2 5 -4 7 y -11 -2 33 6 8 11 44 -.5 I PR 3 0 9 -28 45 -65 56" 4 a 3 10 -11 3 -1 -5 -49/40 Answers to Worksheet #1 75 i i (b) Yellow Equations Name: 1. B6 11. D5 21. B4 31. C l 41. D9 2. C7 12. E l 22 . A8 32 . D7 42. E7 3. DI 13. CIO 23. D4 33. E10 43. C4 4. A9 14. B2 24. D8 34. E5 44. D3 5. E6 15. E9 25. A10 35. B8 45. C6 6. D10 16. A3 26. C5 36. E2 46. D6 7. A l 17. E3 27. A2 37. B l 47. E8 8. B9 18. B7 28. E4 38. A4 48. B10 9. A5 19. C8 29. B3 39. C9 49. A6 10. C3 20. D2 30. A7 40. C2 50. B5 There are 10 answers on t h i s worksheet t h a t are i n c o r r e c t A B C E P 1 -33 -44 -6 -4 -2 27 2 3_ p-a-c 1_ _I_ -% 2 3 R T 3 2b -31_ 0 \ x 77 ab 4 -20 -92. 10 -18 -2.5 63 5 - 1 7 17. 8 120 63 6 2 -12 9 -3 2. -4 5 7 2 y -5 -2_ 6 33 8 11. -.5 _ £ 3 14 P R 9 -28 5. -65 4 a 9 56 3 10 4 13 -1 -5 35 56 Answers t o Worksheet #1 76 i i i (c) Pink E q u a t i o n s Name: 1. B6 11. D5 21. B4 31. C l 41. 2. C7 12. E l 22 . A8 32 . D7 42. 3. DI 13. CIO 23. D4 33 . E10 43. 4. A9 14. B2 24. D8 34. E5 44. 5. E6 15. E9 25. A10 35. B8 45. 6. D10 16. A3 26. C5 36. E2 46. 7. A l 17. E3 27. A2 37. B l 47. 3. B9 18. B7 28. E4 38. A4 48. 9. A5 19. C8 29. B3 39. C9 49. 10. C3 20. D2 30. A7 40. C2 50. D9 E7 C4 D3 C6 D6 E8 BIO A6 B5 B 1 -33 -44 -6 -4 -2 27 2 41_ p-a-c 1 _I_ -2 15 3 RT 3 2b -31 0 4 x 77 ab 4 -20 -92 10 -18 -2.5 63 5 - 1 7 17 -8 120 63 6 2 -12 9 -3 2. -4 5 7 2 y -11 -_2 6 33 8 11 -.5. _L 3 0 14 PR 9 -28 45 -65 4 a 56 3 10 -11 3 -1 -5 -49 40 Answers t o Worksheet #1 i v (d) Blue Equations Name: There are 10 answers on t h i s worksheet t h a t are m i s s i n g 1. -12 19. _ I 37. PR 2. -11 20. _I_ 38. -20 RT 3. -4 21. -92 39. -65_ 63 56 4. -28 22. 11 40. 1 14 3 5. -4 23. -18 41. 4 6. -5 24. 3 42. 6 7. 25. -11 43. 8. 45 26. 17. 44. 4 63 9. -1 27. 45. 9 10. 0 28. 46. -3 2_ 5 11. 2 9 . - 3 1 47. 0 77 12. 30. 48. 3 13. -1 3 1 . - 6 49. 2 14. 3 2 . - 2 . 50. 7 33 15. 9 33. -49 3 40 16. 34. 120 17. x_ 35. -.5 ab 18. y 36. -2 Answers t o Worksheet #1 v (e) White E q u a t i o n s Name: There are 10 answers on t h i s worksheet t h a t are i n c o r r e c t 1. -12 19. I 37. -44 PR 27 2. -5 20. _X 38. -20 RT 3. -4 21. -92 39. -65 63 56 4. -28 22. 11 40. 1 14 3 5. -4 23. -18 41. 4 6. -5 24. 3 42. 6 7. -33 25. 4 43. 10 8. 5_ 26. 11 44. h 9 63 9. -1 27. 3_ 45. 9 2 10. 0 28. -2.5 46. -3 2 5 11. 8 29. -31. 47. 77 12. -2 30. 2 48. 13 13. -1 3 1 . - 6 49. 2 14. p-a-c 3 2 . - 2 . 50. 7 33 15. a 33. 35 3 56 16. 2b 34. 120 17. x 35. -.5 ab 18. y 36. -h Answers to Worksheet #1 v i (f) Orange Equations Name: 1. -12 19. I 37. -44 PR 27 2. -11 20. I _ 38. -20 RT 3. -4 21. -92_ 39. -65 63 56 4. -28 22. 11 40. 1 14 3 5. -4 23. -18 41. 4 6. -5 24. 3 42. 6 7. -33 25. -11 43. 10 8. 45 26. 17_ 44. 4 63 9. -1 27. 41 45. 9 15 10. 0 28. -2.5 46. -3 2_ 5 11. -8 29. -31_ 47. 0 77 12. -2 30. 2 48. 3 13. -1 3 1 . - 6 49. 2 14. p-a-c 32. -2 50. 7 33 15. a 33. -49 3 40 16. 2b 34. 120 17. x 35. -.5 ab 18. y 36. -2 80 Answers t o Worksheet #2 v i i (a) Green E q u a t i o n s Name: 1. B2 11. C8 21. C6 31. C9 2. C7 12. B3 22 . A2 32 . A10 3. D3 13. A l 23. B9 33. B l 4. A6 14. D2 24. A8 34. B6 5. C l 15. CIO 25. D10 35. A7 6. BIO 16. B4 26. C2 36. D9 7. A3 17. A9 27. B5 37. B8 8. D5 18. D4 28. D7 38. A5 9. D8 19. C5 29. C4 39. C3 10. A4 20. B7 30. DI 40. D6 There are 8 answers on t h i s worksheet t h a t are m i s s i n g . B D 1 2 3 4 5 6 7 8 9. 10 .4,-1.4J 3v B 3a - 2c 6 29 I 5, -4 j -3 3_ 2 7b 2 I 25, - 19 S 7 7 3 4 0 !~10, 18] -3 8 f ~ 2 > ~ Q \ -2 3 V-a 2 1-4. 81 -4 4 ( 10, -lo\ -18 19 31 2 8 16 20 13. 34 I-5- ' ~k I 112 12 j 81 Answers t o Worksheet #2 v i i i (b) Y e l l o w E q u a t i o n s Name: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 B2 C7 D3 A6 C I BIO A3 D5 D8 A4 11 . 12. 13. 14. 15. 16. 17. 18. C8 B3 A l D2 CIO B4 A9 D4 19. C5 20. B7 2 1 . 22. 23 . 24. 25. 26. 27. 28', 29, 30 C6 A2 B9 A 8 D10 C2 B5 D7 C4 DI 31. 32. 33. 34. 35. 36. 37 . 38, 39, 40 C9 A10 B I B6 A7 D9 B8 A5 C3 D6 There a r e 8 answers on t h i s w o r k s h e e t t h a t a r e i n c o r r e c t . 82 Answers t o Worksheet #2 i x (c) Pink E q u a t i o n s Name: 1. B2 11. C8 21. C6 31. C9 2. C7 12. B3 22. A2 32 . A10 3. D3 13. A l 23. B9 33. B l 4. A6 14. D2 24. A8 34. B6 5. C l 15. CIO 25. D10 35. A7 6. BIO 16. B4 26. C2 36. D9 7. A3 17. A9 27. B5 37. B8 8. D5 18. D4 28. D7 38. A5 9. D8 19. C5 29. C4 39. C3 10. A4 20. B7 30. C l 40. D6 B C D 1 1 2 I 1.4, -1 3 3v B 4 7 5 -7 2 5 6 3a - 2c 7 6 29 8 f 5 ' " 9 -3 10 2 2 \12 12 <; 83 Answers t o Worksheet #2 x (d) B l u e Equations Name: There are 8 answers on t h i s worksheet t h a t are m i s s i n g 16. 0 30. 17. -3 31. 16 1. a 2. 3 1 2 3. y - a 2 4. 3a -5. 7b 2 6. 7. 3v B 8. 4 9. 20 10. 11. 8 12. 3. 4 13. 14. -2_ 3 15. 18. 32. 3_ 2 19. -4 33. h 20. h 34. 10*s 21. ] 10, -10^ 35. 6_ 22. { 1.4, -1.4 | 36. 13. 34 23. ? -2, -8 ^ 37. - 3_ , » 8 24. | 5, -4 4 38. 25. J__5, - !_ ( 3 9 . - 3 ' l 2 12 I 26. ]25, -19 I 40. -18 19 • ) _, -19 * 7 7 27. } -10, 18\ 28. 29. )-4, 8 ] Answers t o Worksheet #2 x i (e) White E q u a t i o n s Name: There are 8 answers on t h i s worksheet t h a t are i n c o r r e c t 1. a 2. 3 1 2 3. y-a 2 4. 3a - 2c 5. 6. 7. 8. 9. 10. 7 7b 2 7b 2 3v B 6 7 20 11. 8 12. 4 3 13. 1 14. -2_ 3 15. -83s 16. 0 17. -3 18. -3 19. -4 20. h 21. 1 10, -10 ] 22. \ 1.4, .35^ 23. { -2, -8 \ 24. ! 5, -4 I 25. ) 5_, - _ l ] t 12 12 I 26. | 25, -19 ^ 27. | -10, 18"$ 28. ^41, -41} 29. } -4, 8 ^  30 . / ' 1 , 31. 7 16 32 . 2 3 33. * 34. 23 2 35. 29 6 36. i l 36 37. - 3_ 8 38. - 7 39. -3 40. -18 19 Answers t o Worksheet #2 x i i (f) Orange Equ a t i o n s Name: 2. 3_1 2 2 4. 3a - 2c 5. 6. 7. 8. 9. 7b 2 P t 3v B 4 20 10. 7 11. 8 12. 3_ 4 13. 1 14. - 2 . 3 15. -8% 16. 0 17. -3 18. -3 19. -4 20, 21. ( 10, -10 | 22. \ 1.4, -1.41 23. |-2, -8] 24. 7 5, -4 ] 25. / 5_, -1_] \ 12 12 { 26. "j 25_ , - 19 | 27. } -10, 18 1 28. ] 41, -29 ) 29. \ -4, 8 ^  1 7' 30. ' 31. 16 35. 36. 40 3. 2 32 . 33. 34. 10*3 6_ 29 13. 34 37. - 3 8 38. -7 39. -3 -18 19 86 Answers t o Worksheet #3 v i i i (a) Green E q u a t i o n s Name: 1. B4 6. C l 11. A3 16. DI 21. D5 2 . E2 7. A5 12. E l 17. B5 22. B l 3. A l 8. B2 13. D4 18. E4 23. E3 4. C5 9. E5 14. D2 19. C2 24. A4 5. D3 10. C3 15. B3 20. A2 25. C4 There are 5 answers on t h i s worksheet t h a t are m i s s i n g . 4> ) r e a l nos.] B c D D r e a l nos •\ 4> Hi il l \ 3 l / " — l ( 20 ! HI | 3 ^ | r e a l nos ^  \ 0 ] 87 Answers t o Worksheet #3 x i v (b) Yellow E q u a t i o n s Name: 1. B4 2. E2 3. A l 4. C5 5. D3 6. 7. 8. 9. 10. CI A5 B2 E5 C3 11. A3 12. E l 13. D4 14. D2 15. B3 16. DI 17. B5 18. E4 19. C2 20. A2 21. D5 22. BI 23. E3 24. A4 25. C4 There are 5 answers on t h i s worksheet t h a t are i n c o r r e c t . A B c D E 1 \ 12 | | j r e a l nos •1 4 1"! [-8 1 * 7 t 2 { r e a l nos ,\ 3 <{> 1 20 \ 4 } 4 ] } 11 | ' 3 i 5 | r e a l nos ,| HI | r e a l nos •1 4 88 Answers t o Worksheet #3 xv (c) Pink E q u a t i o n Name: 1. B4 6. CI 11. A3 16. DI 21. D5 2. E2 7. A5 12. E l 17. B5 22. BI 3. A l 8. B2 13. D4 . 18. E4 23. E3 4. C5 9. E5 14. D2 19. C2 24. A4 5. D3 10. C3 15. B3 20. A2 25. C4 B | 12 2 ^ "j r e a l nos <ty | r e a l nos . ] ^ 3 | \ ~ ^ H 20 < * 3 } m ^ r e a l n o s . | HI 8 4 4> 2 2 r e a l nos -6 1 89 Answers t o Worksheet #3 x v i (d) Blue E q u a t i o n s Name: There are 5 answers on t h i s worksheet t h a t are mi s s i n g , 12 13 \ 23 j 14. <f> \ 15. 1 -31 ] ' 20 * 4. 1 3 } 17 6. 19. 7. ] r e a l n o s . ] 20. 8. } 3 ) 21. ^ r e a l n o s . | 9. ) 0 I 22. ^ r e a l n os. ^  10. ] 4 } 23. 11 <£> 24. I 4 } I -8 I 25. } 1_ * 7 * * 17 1 - 5 M 90 Answers t o Worksheet #3 « x v i i (e) White E q u a t i o n s Name: There are 5 answers on t h i s worksheet t h a t are i n c o r r e c t , 1. 3 \ 14. 4> 2 - \ 1 15. "J 31 ] » 20 *> 3. J 12 |} 16. | 1 | | 6. (J) 19. | 5 | 7. ^ r e a l nos.^ 20. ~] r e a l n o s . ] 8. 3 J 21. jj r e a l n os. ^  9. ^ 22. ^ r e a l n o s . ^ 10. "|4) 23. <j> 11. <p 24. | 4 f 13. f - 5*5 J 91 3. Answers t o Worksheet #3 x v i i i (f) Orange Equ a t i o n s Name: 1. \ 23.^ 14. 4* 2 . 1 -%) 15. {-31 | 20 16- H? 6. <£> 19. | 5 | 7. ^ r e a l nos 20. ^ r e a l n o s . | 8. 1 3 1 21. 1 r e a l nos.~| 9. ] 0 \ 22. I r e a l nos. | 10. ) 4 } 23. (J> 11. <fa 24. J 4 ^ 12. 25. | M 13. ^ - 5?3 ^ APPENDIX D F u l l s e t of Answers f o r t h e Exponent Worksheets Answers t o Worksheet #1 i (a) Green i i (b) Y e l l o w i i i (c) Pink i v (d) Blue V (e) White v i (f) Orange Answers t o Worksheet # 2 v i i U) Green v i i i (b) Y e l l o w i x (c) Pink X (d) Blue x i (e) White x i i (f) Orange 93 Answers t o Worksheet #1 i (a) Green Exponents Name: 1. B5 12. C2 23. A7 34. D2 45. D l l 2. A6 13. C9 24. B3 35. C8 46. C3 3. BIO 14. E l l 25. E4 36. B7 47. A5 4. E8 15. A4 26. A2 37. E3 48. B9 5. D6 16. BI 27. C5 38. A3 49. A l 6. C l l 17. E2 28. A10 39. A l l 50. D10 7. C6 18. B8 29. D8 40. E9 51. E6 8. D3 19. E5 30. A l l 41. B2 52 . A9 9. C4 20. E10 31. D4 42. CIO 53. DI 10. E7 21. D7 32. A8 43. E l 54. C7 11. D9 22 . CI 33. B4 44. D5 55. B6 There are 11 answers on t h i s worksheet t h a t are m i s s i n g : A B C D E 1 -150m 2n 3 y 1 0 - C 9 d 5 f 6 2 1 k 2 216n 3 - 2 7 a 3 b 3 3 -R 3S 2 2 5 x 8 1 n 3 4 x a + b 6 0 a 2 b 2 -15R 6S 3 1 6 y 1 0 5 a 1 1 16b 6 w 5 x 1 2 y 6 9 k 2 l 2 m 2 6 y 7 -.056bx 3y 6 216R 4 7 64 z^" - 4 8 R 3 S 3 t 2 0 a 5 b 2 C 1 - 2b 8 - 4 8 x 3 y 5 4 R 2 S 2 T 2 -14 abc b 4 + x 9 64m 4n 3 - 4 8 x 3 y 2 1 1 1 2 x 6 y 6 52s 10 8S 9 3 2 + y 7.29 v 9 z 4 a 2 9 11 25R4 S6 R4 f c3 t 2 * - a S b 4 „ 2 - 2 a 94 Answers t o Worksheet #1 i i (b) Yellow Exponents 'Name: 1. B5 12. C2 23 . A7 34. D2 45. D l l 2. A6 13. C9 24. B3 35. C8 46. C3 3. BIO 14. E l l 25. E4 36. B7 47. A5 4. E8 15. A4 26. A2 37. E3 48. B9 5. D6 16. B l 27. C5 38. A3 49. A l 6. C l l 17. E2 28. A10 39. B l l 50. D10 7. C6 18. B8 29. D8 40. E9 51. E6 8. D3 19. E5 30. A l l 41. B2 52. A9 9. C4 20. E10 31. D4 42. CIO 53. DI 10. E7 21. D7 32. A8 43. E l 54. C7 11. D9 22. C l 33. B4 44. D5 55. B6 There are 12 answers on t h i s worksheet t h a t are i n c o r r e c t . A B C D E 1 -150m 2n 3 8 x 3 y 3 y l O 1 5 x 3 y 4 -C 9d5 f6 2 3 1 R 6 -R3 S2 .0064R 4S 1 5 -25x 8 k 2 -8m6 216n 3 n 3 - 2 7 a 3 b 3 1_ t 2 x 2 4 na+b 1 2 a 2 b 2 7 R -15R 6S 3 16 16yl0 5 -27S 6 a 1 1 16b 6 w 5 x 1 2 y 6 3 k 2 l 2 m 2 6 Y 7 -.056bx 3y 6 m 4 y 6 216R 4 7 64 - 1 4 R 3 S 3 t 2 0 a 5 b 2 C 9 y 2 b 8 -48x y 5 4 R 2 S 2 T 2 -14 abc c 4 d 8 b4+x 9 64m 4n 3 - 4 8 x 3 y 2 1 52s 1 1 2 x 6 y 6 10 8S 9 3 X ^ 7 . 2 9 v 9 z 4 27b 4 a 2 11 2 5 R 4 S 6 R 4 t 3 t22 -a8b 2 9 x 2 a 95 Answers t o Worksheet #1 i i i (c) Pink Exponents Name: 1. B5 12. C2 23. A7 34. D2 45. D l l 2. A6 13. C9 24. B3 35. C8 46. C3 3. BIO 14. E l l 25. E4 36. B7 47. A5 4. E8 15. A4 26. A2 37. E3 48. B9 5. D6 16. BI 27. C5 38. A3 49. A l 6. C l l 17. E2 28. A10 39. B l l 50. D10 7. C6 18. B8 29. D8 40. E9 51. E6 8. D3 19. E5 30. A l l 41. B2 52. A9 9. C4 20. E10 31. D4 42 . CIO 53. DI 10. E7 21. D7 32. A8 43. E l 54. C7 11. D9 22 . CI 33. B4 44. D5 55. B6 A B C D E 1 -150m 2n 3 8 x 3 y 3 y 1 0 3 7 5 x 3 y 4 - c 9 d 5 f 6 2 1 .0064R 4S 1 5 k 2 216n 3 - 2 7 a 3 b 3 3 - * 3 S 2 2 5 x 8 -8m6 1 1 t 2 x 2 —- — ,,, - n J 16 4 x a + b 6 0 a 2 b 2 7* -15R 6S 3 16yl0 5 -27S 6 a 1 1 16b 6 w 5 x 1 2 y 6 9 k 2 l 2 m 2 6 y 7 -.056bx 3y 6 m 4 y 6 216 R 4 7 64 - 4 8 R 3 S 3 t 2 0 a 5 b 2 c 9 1 z 5" y 2 b 8 - 4 8 x 3 y 5 4 R 2 S 2 T 2 -14 abc c 4 d 8 b 4 + x 9 64m 4 n 3 - 4 8 x 3 y 2 _1_ i 1 2 x 6 y 6 52S 10 8 S 9 3 x + y 7 . 2 9 v 9 z 4 2 7 b 4 a?_ 9 11 2 5 R 4 S 6 R 4 t 3 t 2 x - a 5 b 4 x 2 " 2 a Answers t o Worksheet #1 i v (d) Blue Exponents Name: There are 11 answers on t h i s worksheet t h a t are m i s s i n g . 1. a 1 1 21. C 9 41. 2. Y 7 22. y 1 0 42. 7 . 2 9 v 9 z 4 3. 3 x + y 23. 64 43. - c 9 d 5 f 6 z 6 4. b X 24. 25x 8 44. w 5 x 1 2 y 6 5. 25. 1 6 y 1 0 45. - a 5 b 4 6. t 2 x 26. 1 46. 7. 27. 16b 6 47. 8. 1_ 28. 8 S 9 48. - 4 8 x 3 y 2 n 3 9. 29. 49. -150m 2n 3 10. 1 30. 2 5 R 4 S 6 50. y 2 b 11. 1 31. -15R 6S 3 51. 216R 4 12. k 2 32. - 4 8 x 3 y 5 52. 64m 4n 3 13. 1 33. 6 0 a 2 b 2 53. 2^~S 14. x 2 ~ 2 a 34. 216n 3 54. 2 0 a 5 b 2 15. x a + b 35. -14abc 55. -.056bx 3y 6 16. 36. - 4 8 R 3 S 3 t 17. - 2 7 a 3 b 3 37. 18. 4 R 2 S 2 T 2 38. -R 3S 2 19. 9 k 2 l 2 m 2 39. R 4 t 3 20. a i 40. 1 2 x 6 y 6 Answers to Worksheet #1 v (e) White Exponents Name: There are 12 answers on t h i s worksheet t h a t are i n c o r r e c t . 1. a 1 1 21. c 9 41. 0 0 6 4 R 4 S 1 5 2. y 7 22. y l O 42. 7.29 v 9 z 4 3. 3*y 23. 64 43. - c 9 d S f 6 4. b 4 + x 24. -25x 8 44. w 5 x 1 2 y 6 5. y 6 25. 1 6 y 1 0 45. - a 8 b 2 6. t 2 x 26. 1_ 46. -8m6 R 6 7. m 4 27. 16b 6 47. -27S 6 8. n3 28. 8 S 9 48. - 4 8 x 3 y 2 9. 7 R 29. c 4 d 8 49. -150m2n 3 10. y 2 b 30. 2 5 R 4 S 6 50. 27b 4 11. 1 31. -15R 6S 3 51. 216R 4 12. k 2 32. - 4 8 x 3 y 5 52. 64m 4n 3 13. 5 2 5 33. 1 2 a 2 b 2 53. 1 5 x 3 y 4 2a -j „ _ 14. x 34. 216n J 54. 2 0 a s b 2 a+b r> 15. x 35. -14 abc 55. -.056 b x J y 16. 8 x 3 y 3 36. - l 4 R 3 S 3 t 17. - 2 7 a 3 b 3 37. 1_ t 2 x 2 16 18. 4 R 2 S 2 T 2 38. -R 3S 2 19. 3 k 2 l 2 m 2 39. R 4 t 3 20. a 2, 40. 1 2 x 6 y 6 9 98 1. 2. 3. 4. 5. 6. 7. a 1 1 Y 7 3x+y b4+x y 6 t 2 x m 4 8. 1 9. 7^ 10. 1 ,2b 11. 1 12. k 2 13. 1 5 ^ 2-2a 14. x 15. x a + b 16. 8 x 3 y 3 17. - 2 7 a 3 b 3 18. 4 R 2 S 2 T 2 19. 9 k 2 l 2 m 2 20. a 2. 9 Answers t o Worksheet v i (f) Orange Exponents ,9 #1 21. 22 . 35. 36. 37. 38. ,10 23. 64 24. 25x 8 10 25. 16y 26. 1_ R8 27. 16b ( 28. 8s^ 29. c 4 d 8 30. 2 5 R 4 S 6 31. -15R 6S 3 32. - 4 8 x 3 y 5 33. 6 0 a 2 b 2 34. 216n 3 -14 abc - 4 8 R 3 S 3 t ] _ t 2 x 2 16 -R 3S 2 Name: 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. R 4 t 3 1 2 x 6 y 6 .0064R 4S 1 5 7.29 v 9 z 4 - c 9 d 5 f 6 w 5 x 1 2 y 6 -a^b 4 -8m6 -27S 6 - 4 8 x 3 y 2 -150m2n3 27b 4 216R 4 64m 4n 3 3 7 5 x 3 y 4 2 0 a 5 b 2 -.056bx 3y 6 99 Answers t o Worksheet #2 v i i (a) Green Exponents Name: 1. B4 9. CI 17. C6 25. C7 2. A8 10. D7 18. D6 26. D2 3. D4 11. B8 19. A6 27. B6 4. D8 12. BI 20. B5 28. C8 5. C5 13. A4 21. D5 29. DI 6. A2 14. C2 22 . B2 30. A l 7. B3 15. C4 23. A3 31. A5 8. A7 16. D3 24. B7 32 . C3 There are 6 answers on t h i s worksheet t h a t are m i s s i n g . A B C D 1 8 6 4 a 8 b 9 c 1 0 1 4 4 s 1 0 y 4 a 3 - 3 2 a 8 2 -24a 6 - 3 c 2 -8m 7n 7p 5 3 0 -172b 5 - 1 9 k v 4 4 16a 2 24x 6 ^4P 1 5 a 3 b 3 m 2 5 - 3 6 R 3 S 3 T 3 6 2 a 2 b 2 6 27b 3 - l l x 2 y 2 -2x 4 9 b 6 1 2 5 c 6 9 y 1 6 7 -72m 8n 4 0 9 1 c 6 d 1 0 e 8 20x 4 - 1 0 8 R 8 S 1 3 t 9 0 - 2 7 a 3 b 4 c 2 100 Answers t o Worksheet #2 v i i i (b) Yel l o w Exponents Name: 1. B4 9. CI 17. C6 25. C7 2 . A8 10. D7 18. D6 26. D2 3. D4 11. B8 19. A6 27. B6 4. D8 12. BI 20 . B5 28. C8 5. C5 13. A4 21. D5 29. DI 6. A2 14. C2 22 . B2 30. A l 7. B3 15. C4 23. A3 31. A5 8. A7 16. D3 24. B7 32. C3 There are 6 answers on t h i s worksheet t h a t are i n c o r r e c t . A B C D 1 -4b 3 2 -24a 6 3 0 4 16a 2 5 l . l z 5 t 4 6 27b 3 1 2 5 c 6 7 -72m8n 864a 8b9 c10 1 4 4 S 1 0 y 4 -31a 5 - 3 4 p 4 q 2 3 -8m 7n 7p5 C 2 - 6 0 x 2 y 4 -172b 5 - 1 9k v 4 24x 6 -4P 1 5 a 3 b 3 m 2 4 d 2 - 3 6 R 3 S 3 T 3 7 8 a 2 b 2 i h 2 " - l l x 2 y 2 -2x 4 9_ b 6 9y 16 0 9 1 c 6 d 1 0 e - 1 0 a 7 b 4 8 9 x 4 - 1 0 8 R 8 S 1 3 t 9 0 - 2 7 a 3 b 4 c 2 101 Answers t o Worksheet #2 x i v (c) Pink Exponents Name: 1. B4 9. C l 17. C6 25. C7 2. A8 10. D7 18. D6 26. D2 3. D4 11. B8 19. A6 27. B6 4. D8 12. B l 20. B5 28. C8 5. C5 13. A4 21. D5 29. DI 6. A2 14. C2 22 . B2 30. A l 7. B3 15. C4 23. A3 31. A5 8. A7 16. D3 24. B7 32 . C3 A B C D -4b 3 8 6 4 a 8 b 9 c 1 0 1 4 4 S 1 0 y 4 a 3 - 3 2 a 8 -24a 6 - 3 4 p 4 q 2 - 3 c 2 -8m 7n 7p 5 0 6 0 x 2 y 4 -172b 5 m 2 9k v 4 16a 2 24x 6 -4P 1 5 a 3 b 3 l . l z 5 t 4 4 d 2 - 3 6 R 3 S 3 T 3 6 2 a 2 b 2 9 h 2 27b 3 - l l x 2 y 2 - 2xf_ 9_ b 6 125c 6 9y 16 -72m 8n 4 0 9 1 c 6 d 1 0 e - 4 0 a 7 b 4 8 20x 4 - 1 0 8 R 8 S 1 3 t 9 0 - 2 7 a 3 b 4 c 2 102 Answers t o Worksheet #2 x (d) Blue Exponents Name: There are 6 answers on t h i s worksheet t h a t are m i s s i n g . 1. 24x 6 17. - 2 x 4 2. 20x 4 18. 9 b 6 3. 1 5 a 3 b 3 19. 16 27b 3 4. - 2 7 a 3 b 4 c 2 20. 125c 6 5. - 3 6 R 3 S 3 T 3 21. 6 2 a 2 b 2 6. -24a 6 22. 7. 23 . 0 8. -72m 8n 4 24. 0 9. 1 4 4 S 1 0 y 4 25. g i c ^ 1 0 ! 10. 26. -8m 7n 7p 11. - 1 0 8 R 8 S 1 3 t 9 27. - l l x 2 y 2 12. 8 6 4 a 8 b 9 c 1 0 28. 0 13. 16a 2 29. a 3 - 3 2 a 8 14. - 3 c 2 30. 15. -4P m 2 31. 16. - 1 9 k v 4 32 . -172b 5 Answers xj There are 6 answers on t h i s 1. 24x 6 2. 9 x 4 3. 15 a 3b3 4. - 2 7 a 3 b 4 c 2 5. - 3 6 R 3 S 3 T 3 6. -24a 6 7. - 6 0 x 2 y 4 8. -72m 8n 4 9. 1 4 4 S 1 0 y 4 10. - 1 0 a 7 b 4 11. - 1 0 8 R 8 S 1 3 t 9 12. 8 6 4 a 8 b 9 c 1 0 13. 16a 2 14. 3_ c 2 15. -4P 16. - 1 9 k v 4 : t o Worksheet #2 (e) White Exponents Name: worksheet t h a t are i n c o r r e c t . 17. - 2 x 4 9Y 18. 9_b 6 16 19. 27b 3 125c 6 20. 4 d 2 9 h 2 21. 7 8 a 2 b 2 22 . - 3 4 p 4 q 2 23. 0 24. 0 25. 9 1 c 6 d 1 0 , 26. -8m 7n 7p 27. - l l x 2 y 2 28. 0 29. -31a 5 30. -4b 3 31. l . l z 5 t 4 32 . -172b 5 104 Answers t o Worksheet #2 v i i (f) orange Exponents Name: 1. 24x 6 17. - 2 x 4 2. 2 Ox 4 18. 9y 9_b 6 3. 1 5 a 3 b 3 19. 16 27b 3 4. - 2 7 a 3 b 4 c 2 20. 1 2 5 c 6 4d2 5. - 3 6 R 3 S 3 T 3 21. 9h2 6 2 a 2 b 2 6. -24a 6 22 . -34p4 q2 7. 6 0 x 2 y 4 23. 0 8. -72m 8 n 4 24. 0 9. 144sl0y 4 25. 9 l c 6 d 1 0 e 10. -40a 7b4 26. -8m 7n 7p5 11. -108R 8sl3t9 27. - l l x 2 y 2 12. 8 6 4 a 8 b 9 c 1 0 28. 0 13. 1 6 a 2 29. a 3 - 3 2 a 8 14. - 3 c 2 30. -4b 3 15. -4P m 2 31. l . l z5 t 4 16. - 1 32. -172b 5 9kv4 APPENDIX E T e s t Instruments P r e t e s t and P o s t t e s t f o r E q u a t i o n s P r e t e s t and P o s t t e s t f o r Exponents 106 (a) P r e t e s t and P o s t t e s t f o r Equations Mathematics 11 Name: Solve the f o l l o w i n g e q u a t i o n s f o r the i n d i c a t e d v a r i a b l e or f o r the v a r i a b l e i n r e d . The replacement s e t f o r these equations i s the s e t of r e a l numbers. 1. -21.8 = x + 7.3 2. -7 y = -5 12 3 3. 2z + 3_ = ^ - z 4 4. 8x - 14 - 3x = 2x + 10 5. S = t + u + 6 6. 2 - 3 (2x - 5) = 2x - 7 7. w - 3_ = - 5. 4 9 8. -3_ = 1 13 t 9. C = 2zR 10. 3x - 11 = 8 (1 - 2x) 11. 7R - 3 = 1 7 12. 5(m+2) = 10 13. 7 (y-2) +3 (y+7) -3 (3-y) = 0 14. -4s+3 (2S-5) = 3(S-6) +h (4-2s) 15. 1 R + = 5 3 16. l z + 6 = 15 3 17. 3 (a+1) -2a (5a-4) = 10 (a-a 2) +a 3 18. z6 = _JL x-2 2-x 19. _6_ = -7 3-x x-5 20. 3x - 7 = -(7-3x) (b) P r e t e s t and P o s t t e s t f o r Exponents Mathematics 11 Name: F i n d the i n d i c a t e d p roducts and q u o t i e n t s . W r i t e a l l answers w i t h p o s i t i v e exponents. Assume t h a t v a r i a b l e s i n demoninators do not equal z e r o . 1. 5 X . 5 a  2. an  3. (-4m 2n 3) 3  4. ( 3 t k ) 2 (-2t 3k 4) 5. 3mn 3t 2 (-m 2t 3) 6. -2n 2  6n65 7. ( - 3 p 2 q 5 ) 4  8. (-R) 3 (2RS 2) (3R 2S 4) 9. S 4 1 1 " 1 . 8 3 n + 1 . 8 10. - 6m 3n 4  lSm^n 11. (-3x) 2 ( 2 x y 2 z 5 ) 3  12. ( 2 a b ) 3  13. ( 3 p 3 R 4 ) 2 ( - p 2 R 3 S ) s  14. ( 3 a b 3 ) 4  3 ( 6 a 2 b 4 c ) 2 15. ( - 5 a 2 x ) 3  109 16. 3x (4y 2) - (2y2) (5x) 17. ( 4 c 2 d 3 ) 3 2 ( 8 a c 4 d ) 2 18. z 4 z 5 z 6 19. 7x (-3x) + ( 2 x 2 ) 2 - (3x) ( 2 x ) 3 + (5x 4) 2 x 3 20. ml m^  APPENDIX P Student O p i n i o n Sheet I l l Student O p i n i o n Sheet I Name I I Colour I I I (a) I n the f i r s t s e t of e x e r c i s e s on (exponents) which type of answers would you have p r e f e r r e d t o have. L i s t them from f i r s t c h o i c e , second c h o i c e , e t c . (b) D i d you l i k e the s e t of answers you had or would you have wanted another one? IV (a) I n the second s e t of e x e r c i s e s on (equations) which type of answers would you have p r e f e r r e d t o have. L i s t them from f i r s t c h o i c e , second c h o i c e , e t c . (b) D i d you l i k e the s e t of answers you had or would you have wanted another one? APPENDIX G E x p e r i m e n t a l Data Scores f o r P r e t e s t and P o s t t e s t f o r the group f o l l o w i n g the sequence E q u a t i o n s then Exponents Scores f o r P r e t e s t and P o s t t e s t f o r the group f o l l o w i n g the sequence Exponents then E q u a t i o n s 113 (a) Scores f o r P r e t e s t and P o s t t e s t f o r the Group f o l l o w i n g the sequence Eq u a t i o n s then Exponents EQUATIONS A B c D E F 4 10 6 12 8 15. 4 10 11 16 6 10 9 14 9 13 3 7 5 13 7 11 8 15 6 12 5 10 7 13 14 19 16 16 11 15 14 17 13 18 11 16 12 114 10 14 8 14 15 16 11 17 7 12 4 14 13 17 7 16 11 14 4 8 8 12 13 14 12 16 5 17 16 18 12 18 8 12 5 13 13 11 6 7 9 16 8 11 16 20 7 13 13 18 14 17 9 14 6 10 6 12 9 18 5 12 17 14 14 20 6 12 10 14 9 10 11 16 18 17 EXPONENTS A B C D E F 8 10 11 15 9 15 1 10 8 17 6 17 10 15 8 14 7 9 11 14 3 12 10 13 6 9 11 14 4 4 16 17 13 16 6 12 16 16 5 16 9 17 15 15 11 16 6 11 15 18 3 14 9 11 4 12 17 17 7 12 10 14 6 12 11 10 5 16 11 16 11 12 17 19 14 16 5 11 3 6 9 18 5 9 11 15 14 13 17 18 4 10 7 14 11 14 8 14 7 14 11 11 11 15 1 6 8 12 10 16 11 15 14 11 7 14 8 17 19 18 The f i r s t s core i r i the number p a i r s r e p r e s e n t s the p r e t e s t w h ile the second s c o r e r e p r e s e n t s the p o s t t e s t . 114 (b) Scores f o r P r e t e s t and P o s t t e s t f o r the Group f o l l o w i n g the sequence Exponents then E q u a t i o n s EXPONENTS A B C D E F 6 15 4 11 5 14 13 17 12 18 8 16 5 11 2 6 12 13 7 17 15 18 9 15 2 7 12 17 5 17 13 15 10 13 9 17 1 10 15 17 6 15 3 13 15 18 10 14 8 11 4 9 8 15 4 13 5 13 5 11 10 13 14 15 8 14 8 10 8 11 3 15 13 17 8 13 6 12 5 9 4 10 10 12 2 7 9 15 3 10 13 16 5 14 15 19 14 19 6 14 5 11 6 9 5 13 14 16 2 10 5 12 5 13 13 17 4 15 2 6 EQUATIONS A B C D E F 10 18 5 8 9 13 7 18 12 19 17 20 7 14 2 7 12 15 8 11 14 16 11 13 5 12 18 20 11 17 12 18 12 12 15 19 8 15 17 18 5 9 6 12 18 19 17 19 6 12 6 6 3 5 8 16 11 15 8 16 18 18 14 16 7 14 7 8 4 10 5 10 16 20 11 16 7 14 8 15 8 16 8 14 1 8 16 19 7 11 9 14 •9 16 14 16 7 18 12 16 8 15 8 8 10 18 16 18 11 16 14 17 15 17 7 18 14 13 5 13 The f i r s t s core i n the number p a i r s r e p r e s e n t s the p r e t e s t score w h i l e the second s c o r e r e p r e s e n t s the p o s t t e s t . 

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