the standardization of a final test in elementary algebra

The standardization of a final test in elementary algebra

Item type text; Thesis-Reproduction (electronic)

Authors Futrell, Ralph Averille, 1906-

Publisher The University of Arizona.

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THE STANDARDIZATIOH OF A FINAL TEST IN ELEMENTARY ALGEBRA

by

Ralph A, F u t r e l l

A T h esis

su b m itted t o th e f a c u l ty o f th e

D epartm ent o f E d uca tion

in p a r t i a l f u l f i l lm e n t o f

th e req u irem en ts f o r th e d eg ree o f

M aster o f A rts

in th e G raduate C o llege

U n iv e rs ity o f A rizona

1946

a c to r o f T hesis

TABLE OF CONTENTS

Page

i l l

£f 97? / 7 9 4 5 -

/ d )

Acknowled ge merits

INTRODUCTION....................... .... , • • ................................................. 1The Problem S ta te d • • • • • • • • • • • • • • • • 5

CHAPTER I .................. * ............................................The O b jec tiv es o f E lem entary A lgebra A n a ly sis o f T ex t Books • • • • • • «S e le c t io n o f Q uestions • • • • • • •S t a t i s t i c a l T rea tm en t• # # * » # * * The R evised E xam ination . . . . . . .C o r re la t io n w ith an A ccepted T e s t. •The R evised T e s t A d m in is te red . « « « . . » . « • • 16R e l i a b i l i t y o f th e R evised T e s t. • • • • • • • • • 16S t a t i s t i c a l A n a ly sis 21

CONCLUSIONS....................... . ........................... ' . 4 ........................ 24

BIBLIOGRAPHY...................................................... .... • . . . . • • • 25

APPENDIX . ................................... 28

Jv

i

172491

^ V

> to U> <0 to

Number

TABLES

Page

I ANALYSIS OF TEXT BOOKS.......................................................... 6

I I FREQUENCY DISTRIBUTION OF EXPERIMENTAL TEST . . . 9

I I I EXPERIMENTAL ACHIEVEMENT TEST PERCENTILE TABLE. . 12

IV PER CENT OF PUPILS SOLVING EACH PROBLEM CORRECTLY 13

V TABLE SHOWING THE SCHOOLS WHICH COOPERATED INGIVING THE REVISED TEST. . . . . . . . . . . 15

VI FREQUENCY DISTRIBUTION OF ACHIEVEMENT ON THE. REVISED TEST ALL SCHOOLS . . . . . . . . . . XT

V II , REVISED ACHIEVEMENT TEST PERCENTILE TABLE . • . • 20

V II I , PER CENT OF CORRECT RESPONSES ON. REVISED TEST . • 22

. . FIGURES

1 . Frequency P o lygon . S cores Hide by 407 S tu d e n ts onth e E xperim en ta l Achievement T e s t . « • • • • • 10

2 . P e r c e n t i le Graph o f 407 S tu d en ts in E xperim en ta lA lgebra T es t » « 11

5 . Polygon, R ep re se n tin g 2043 S tu d en ts on R ev isedT e s t ....................... .................................................................. 18

4 , P e r c e n t i le Graph o f 2045 P u p ils on R evised T est . • 19

i i

ACKHOWLEDGBMEITTS

The w r i t e r acknowledges t i e c o u rte sy o f 1.5*. R obert D. Morrow,

S u p e rin ten d e n t o f P u b lic S ch o o ls , Tucson, A rizona; ISc* C* A. C arson ,

A s s is ta n t S u p e rin ten d e n t o f Schools in Tucson, A rizo n a , and P r in c ip a l

o f th e Tucson S en io r High S ch o o l; ZSr# Ray Webb, Head o f M athem atics

D ep& rtront, Tucson S e n io r Higfc S choo l; and W e P au l Be Carmony, D irec

t o r o f th e D epartm ent o f T e s t P ro d u c tio n o f Tucson S en io r Hlgpi School*

Acknowledgement i s a ls o made to th o many te a c h e rs who gave v e ry

h e lp fu l c r i t i c i s m s o f th e t e s t and to th o se who a d m in is te re d and graded

th e t e s t s * W ithout t h i s v a lu a b le a s s is ta n c e th e work would have boon

a lm o st im p o ss ib le .

The w r i t e r i s v e ry g r a te f u l to h is w if e , E th e l D. F u t r e l l , whose

a s s is ta n c e in cheek ing th o s t a t i s t i c a l d a ta on a l l ta b le s and c h a r ts

was v e ry h e lp fu l*

i l l

INTRODUCTION

The coming o f W orld War I I b ro u g h t w ith i t in c re a se d demands on

te a c h e rs o f m athem atics • To a g re a te r e x te n t th a n e v e r b e fo re boys and

g i r l s were going in to a v ia t io n , e n g in e e rin g , in v e n tio n , p ro d u c tio n , and

many o th e r l in e s o f endeavor in which a need fo r a knowledge o f mathe

m a tic s ranged a l l th e way from a v e ry meager knowledge to th e h ig h ly

s k i l l e d use o f lo g a rith m s and th e s l id e ru le * T h e re fo re , in keep ing w ith

th e se demands th e a u th o r embarked upon a program planned to improve th e

q u a l i ty o f te a c h in g in h ie c la s s e s * A p a r t o f t h i s program was th e de

velopm ent o f a t e s t i n g in s tru m e n t i n e lem en tary a lg e b ra , s in c e t h a t i s

w hat he was engaged in te a c h in g a t th e tim e* From t h i s d e s ir e to improve

th e q u a l i ty o f in s t r u c t io n a f i n a l t e s t i n e lem en ta ry a lg e b ra was con

s t r u c te d and t r i e d out*

The t e s t was n o t made to fo llo w any p a r t i c u l a r t e x t book, b u t i t was

com prehensive enough so t h a t good r e s u l t s would be o b ta in e d w ith th e t e s t

re g a rd le s s o f w hat t e x t was u s e d .

The o b je c t iv e s o f e lem en ta ry a lg e b ra were s e t up and c o n su lte d con

s ta n t ly in th e making o f th e t e s t to see t h a t th e y were b e in g a t ta in e d *

The c o n te n ts o f seven commonly used t e x t books were examined to

determ ine w hat to p ic s were most commonly t r e a te d and th e amount o f em

p h a s is g iven to each to p ic *

From 317 q u e s tio n s p re p a red and used a t th e com pletion o f th e d i f

f e r e n t to p ic s d u rin g th e y e a r 50 q u e s tio n s were chosen and made in to a

t e s t * No a tte m p t was made to a rran g e th e se q u e s tio n s a cco rd in g to d i f -

1

f i o u l t y s in c e th e r e l a t i v e d i f f i c u l t y o f each q u e s tio n vms n o t known#

C on sid erab le th o u g h t was g iven to th e s e le c t io n o f th e se q u e s tio n s and

an a tte m p t was made to r e p re s e n t ev ery p ro cess in th e e n t i r e c o u rse j how

e v e r , one problem cou ld r e p r e s e n t two to p ic s#

This t e s t was th e n used by seven a lg e b ra te a c h e r s , in c lu d in g th e

a u th o r , and was a d m in is te re d to 407 s tu d e n ts in th e Tucson S en io r High

School and in f iv e o f th e s ix ju n io r h igh sch o o ls in th e c i t y o f Tucson#

One o f th e ju n io r h ig h sch o o ls d id n o t o f f e r a lg e b ra t h a t year#

These te a c h e rs were asked to g ive t h e i r c r i t i c i s m s o f th e t e s t#

They recommended t h a t number 25 , ( |£ . + i h + - ) , be changed so t h a t

th e re would be on ly one l e t t e r i n th e n u m era to r, and th e y a ls o ag reed

t h a t number 50, (a graph showing d a i ly te m p e ra tu re s ) , had l i t t l e v a lu e

as a th o u g h t-p ro d u c in g q u estio n # A lthough i t was so lved by on ly 70$ o f

th e p u p ils i t was f e l t t h a t i t would have been so lv ed by a much l a r g e r

p e r c e n t had i t occupied a p lace n e a r th e b eg in n in g o f th e t e s t#

The t e s t was th e n su b m itted to s ix o th e r te a c h e rs o f m athem atics ,

none o f whom had used i t , f o r c r i t ic is m s # This group recommended th a t

number 45 , ( In m u l t ip l ic a t io n o f l e t t e r s , exponents a re added, e tc # ) , be

e lim in a te d s in c e th o a p p l ic a t io n o f th e p r in c ip le in v o lv ed in i t was used

r e p e a te d ly in th e t e s t . They a ls o recommended t h a t two to p ic s be s t r e s s e d

a l i t t l e more, v is# r a t i o n a l i s i n g a denom inator and m u l t ip l ic a t io n o f f r a c

t io n s j th e re fo re q u e s tio n s 49 and 50 were added to th e re v is e d t e s t t o

m eet t h i s recommendation#

Both g roups, however, ag reed t h a t th e t e s t was q u ite thorough and

t h a t th e q u e s tio n s were w e ll d i s t r ib u te d over th e e n t i r e co u rse o f

e l e m e n t a r y a l g e b r a .

The Problem S ta te d

Having c a r r ie d th e ex p erim en t th u s f a r i t became d e s i r a b le to con

t in u e th e s tu d y and to r e v is e and s ta n d a rd is e th e t e s t* The method used

in do ing so i s d e sc r ib e d in th e fo llo w in g c h a p te r and co p ie s o f th e o r ig

i n a l , th e r e v is e d , and th e unused s c a le d t e s t may be found in th e appen-

CHAPTER I

The f i r s t s te p ta k e n i n th e c o n s tru c tio n o f th e t e s t was th e s e t t in g

up o f th e o b je c t iv e s to bo aimed a t in th e study o f f i r s t y e a r a lg eb ra '.1

The so', a s m od ified from th e w r i t in g s o f S o h o rlin g , a re a s fo llo w s t

The O b je c tiv e s o f E lem entary A lgebra

I P r a c t i c a l O b je c tiv e s

A. The m astery o f th e fundam ental p ro c e sse s in e lem en ta rya lg e b ra i s sought#

B# An u n d e rs tan d in g o f th e language o f a lg e b ra i s an import a n t c o n s id e ra t io n in te a c h in g a lg eb ra#

C, An u n d e rs tan d in g o f th e fundam ental laws o f a r i th m e tic and a lg e b r a , and th e im portance th e y have in th e l iv e s o f th e in d iv id u a l i s to be k e p t in mind#

D# The a b i l i t y to u n d e rs tan d g ra p h ic a l r e p re s e n ta t io n s and f a c i l i t y in in te r p r e t in g them shou ld be gained by th e s tu d e n ts#

E# S k i l l i n th e u se o f th e to o ls o f m athem atics i s t o be exp e c te d o f each s tu d e n t , v i z , a d d i t io n , s u b tr a c t io n , m u l t ip l ic a t io n , and d iv is io n n o t on ly o f whole numb e r s , b u t a ls o o f common and decim al f r a c t io n s as t r a i l .

I I D is c ip l in a ry O b je c tiv e s

A# The a c q u is i t io n o f th o se id e a s o r concep ts in term s o f w hich th e q u a n t i ta t iv e th in k in g o f th e w orld i s dim e, such a s r a t i o and p ro p o r t io n , p o s i t iv e and n e g a tiv e numb e r s , and th e dependence o f one number upon a n o th e r , i s to be u rged upon each s tu d e n t .

B, The developm ent o f th e a b i l i t y to th in k c le a r ly in te rm s o fth e se id e a s and co n cep ts should be gained by each s tu d e n t .

1* S o h o rlin g , R aleigh# The Teaching o f M a th e m a t ic s .The Ann A rbor P re s s , 1936, p . 1#

6

C» The r e q u i s i t i o n o f m en ta l h a b i ts and a t t i t u d e s t h a t w i l l make th e above t r a i n i n g e f f e c t iv e in th e l i v e s o f th e in d i v id u a l i s to be e v e r k e p t in mind*

| D* T ra in in g in " fu n c t io n a l th in k in g * i s one o f th e m ost fundament a l d i s c ip l in a r y aims o f th e te a c h in g o f e lem en ta ry a lg eb ra*

I I I C u ltu ra l O b jec tiv es

A* Id e a ls o f p e r f e c t io n as to lo g ic a l s t r u c tu r e , p r e c is io n o f s ta te m e n t o f th o u g h t, lo g ic a l re a so n in g , and d is c r im in a t io n betw een th e t r u e and th e f a l s e should be a t ta in e d by eaeh s tu d en t*

B* A p p re c ia tio n o f th e power o f m athem atics a s a p p lie d to s c i e n ce , in d u s t ry , p h ilo so p h y , and e d u c a tio n a l r e s e a rc h should be g iv en to a l l who study a lg eb ra*

A n a ly sis o f T ext Books

, W ith th e se o b je c t iv e s in m ind, th e second s te p was th e a n a ly s is o f

t e x t books which a re a v a i la b le f o r use o f s tu d e n ts o f f i r s t y e a r a lg eb ra*

S ince a r a te d t e s t f o r f i r s t y e a r a lg e b ra m ust measure w e ll and c o n s is

t e n t l y th e work which i s b e in g ta u g h t , i t i s s a fe to assum e, t h a t i n

American sch o o ls th e co u rses o f fe re d a rc l a r g e ly determ ined by th e t e x t

book co n ten ts* The r e s u l t s o f th e a n a ly s is a re shown in Table I*

S e le c t io n o f Q uestion#

As was s ta t e d in th e in tr o d u c t io n , 50 q u e s tio n s were s e le c te d from. 517

u sed d u rin g th e y e a r in s h o r t to p ic t e s t s * These re p re s e n te d th e m a te r ia l

w hich was m ost commonly and m ost f u l l y t r e a te d by a l l o f th e t e x t books,

and were fa sh io n e d in to f iv e -m u l t ip lo re sp o n se item s f o r th e t r i a l t e s t

w hich was a d m in is te re d to 407 p u p ils in th e Tuoson p u b lic schoo ls*

S t a t i s t i c a l Treatm ent

The t h i r d s te p c o n s is te d o f th e s t a t i s t i c a l tre a tm e n t o f th e d a ta

TABLE I

MALTS IS OF TEXT BOOKS

Topiog t r e a t e d and p a g e s R iv e n to each

T i t l e o f bodk A uthor Bato o f

p u b l ic a t io n

E s s e n tia lo o f A lgebra , l e t C ourse , H a r t ,1941 16 40 45 44 34 52 20 25 29 SO 14 14 T 23 2 6 12 43*

F i r s t Y ear A lg e b ra ,S o h o r lin g , C la rk ,S m ith , 1943 22 48 24 42 20 52 35 26 34 16 26 6 22 22 11 9 49*

A lgebra & I t* Uses',S i l b e r e t e i n , H ew ell,H a rp e r , 1958 24 26 24 32 42 SO 28 28 28 36 32 12 26 36 30 6 20 432

Hew Course i n A lgebra,Hawke*, Louhy, *T eu ton , 1926 10 8 11 20 24 10 9 24 33 37 11 14 28 18 24 6 16 418

A F i r s t C ourse i n A lg e b ra , M hllo ry ,1943 62 40 12 35 26 20 30 26 30 20 6 8 20 10 26 12 13 506

( c o n t i n u e d )

TABLE 1 (c o n tin u e d )

ANALYSIS OF TEXT BOOKS

T i t l e o f bookA uthor Date o f

p u b l ic a t io n 5

I

T opics t r e a te d and page* g iven to each

I

a

Modern F i r s t T ea r A lg eb ra , L o l ls &H u rt, 1923

P ro g re ss iv e F i r s t A lg eb ra , H a r t , *

17 2 16 10 S3 IS 22 21 20 16 4 I f 8 9 23 5 12 843

1943 ' 15 39 35 7 46 IS 16 16 25 32 8 8 8 23 25 4 24 404

TOTAL 156 203 164 190 224 174 179 166 199 186 85 90 97 171 151 52 106 2998

A verage 22 29 21 27 32 25 25 23 28 27 12 13 14 23 22 8 15 428

♦Thoro i s a s l i g h t d iso re p an c y betw een tb e number o f pages i n th e book and th e t o t a l number o f pages..........rdevo ted to th e d i f f e r e n t to p ic s * T h is i s due t o th e o v e rla p p in g o f some to p i c s j e*g», Haekes Louby, and Teuton do n o t l i s t r a t i o and p ro p o r t io n a s a s e p a ra te to p ic b u t i t i s t r e a te d a s a p a r t o f E q u a tio n s Q o n ta in in g F ra c tio n s*

8

produced by th e e x p e rim en ta l t e s t* The d i s t r i b u t io n o f th e sc o re s i s

shown in Table I I , and th e polygon o f d i s t r i b u t io n w hich i s s l i g h t l y

skewed to th e l e f t o r n e g a tiv e ly i s shown in F ig u re 1* T his skewness to

th e l e f t and p i l i n g up to th e r i g h t in d ic a te s t h a t th e t e s t was r a th e r to e

easy fo r th e Tucson group# The mean was found t o be a t 33*9, w h ile id eality

i t shou ld be a t o r n e a r 25j t h a t i s , n e a r th e m id p o in t o f th e t e s t * The

p e r c e n t i l e graph i s shown in F ig u re 2 , w ith a p e r c e n t i l e t a b le shown In

Table I I I* Using th e s p l i t h a l f method and a p p ly in g th e Spearman-Brown

form ula th e r e l i a b i l i t y o f th e ex p e rim en ta l t e s t was found to be *856 ^ *012*

The p e r o en t o f p u p ils so lv in g each problem was n e x t found and is shown

in Table 17, columns 2 and 4 , w h ile th e new number g iven to each problem

in th e r e v is e d t e s t i s shown in column one o f th e same ta b le *

The two q u e s tio n s numbered 45 and 50 on th e o r ig in a l t e s t were om it

te d on th e re v is e d t e s t , and two new problem s wore added a s numbers 49

and 50*

The R evised Exam ination

The t e s t was now p re p a red fo r f u r th e r use by re a r ra n g in g th e ques

t i o n s , w ith th e e x c e p tio n o f 49 and 50 which were new and u n t r i e d , in th e

o rd e r o f in c re a s in g d i f f i c u l t y , and in t h i s form i t was a d m in is te red to

2045 s tu d e n ts in sch o o ls in A rizo n a , C a l i f o r n ia , M isso u ri, Oregon, T exas,

and W ashington.

The d i s t r i b u t i o n o f th e se s tu d e n ts I s shown in Table V,

C o r re la t io n w ith an Aooepted T est

2h o rd e r to le a r n more co n cern in g th e v a l i d i t y o f th e r e v is e d t e s t .

9

TABLE I I

FREQUENCY DISTRIBUTION OF EXPERIMENTAL TEST

S cores f F

48-50 15 407

45-47 26 594

42-44 44 568

59-41 50 324

56-58 47 274

55-55 52 227

50-52 54 176

27-29 41 121

24-26 28 80

21-25 SO 52

18-20 15 22

15-17 6 9

12-14 5 5

H = 407

Mean 55 .9 Median - 54.14

PEa. .57 PBmd - .38

Sigma 8 .17 Q - 6 .14

Sigma o f Sigma - .29 F H - .24

Frequency Po lygon . S co res Made t y 407 S tu d e n ts on th e E xperim en ta l Achievement T e s t .

S c o r e s F F rv r - s v ¥£>1 to oy s - v ? J y y <H~S

y z ~ w 3$ 1 - 4 ! 3 2* / 7 f . ; -

3 L - 3 2 Z 7 y 4 7.y •

3 3 ' 3 S x * 1 sS.-)

3 0 - 3 % n $ +0.$

l x / 14.7%*/—%(» 2 0 ;4 .7A / - Z J S % / f l .J

/ 2 - 2 o %% s - \ •

/s-n 7/ <5 •7

F ig u re 2

P e r c e n t i le Graph o f 407 S tu d e n ts in E xperim en ta l A lgebra T e s t

12

TABLE I I I

EXPERimFTAL ACHIEVEMENT TEST PERCENTILE TABLE

P erc e n t

U p p e r 'l im it = = o f (p e r c e n t)

sco re

100 4 9 .695 4 6 .690 4 4 .485 42 .980 4 1 .675 4 0 .470 59 .165 57 .960 56 .556 56 .560 5 4 ,145 52 .940 5 1 .855 50 .7SO 29 .625 28.120 27 .815 24 .410 22 ,4

5 20 .10 12 .5

I S

TABLE IV

PER CENT OF PUPILS SOLVING EACH PROBLEM CORRECTLY

Number o f p ro b lem ,

r e v is e d t e s t


o r ig i n a l t e s t

Rumbor o f s tu d e n ts -who

so lv ed e o r ro o tly

Per c e n t o f

t o t a l

1 2 384 94 ,32 1 364 91 .93 20 551 86 ,24 28 851 86.25 7 348 85.56 IS 348 85.57 11 340 83 ,48 23 358 83,049 12 836 82,5

10 17 336 82 .611 5 333 81 .812 6 329 80 .813 16 328 80 .614 18 319 78 ,315 19 306 75 .116 33 305 74 .917 45 304 74 .618 35 304 74 .619 34 297 72 .920 16 297 72 .921 31 289 70 .822 50 287 70 .323 4 286 70 ,0924 22 285 69.525 24 288 69 .226 8 269 66.0927 27 268 65 .028 42 266 66 .229 21 260 6 3 .830 38 258 63 .331 44 255 62 .632 36 261 61 .633 43 251 61 ,634 26 260 61 ,436 41 249 61 .136 26 246 60 .437 3 259 5 8 .7

(o o n tin u ed )

14

TABLE IV (c o n tin u e d )

PER CENT OP PUPILS SOLVING EACH PROBLEM CORRECTLY

Number o f p rob lem ,

r e v is e d t e s t


o r ig in a l t o s t

lum ber o f .............s tu d e n ts who

so lv ed c o r r e c t ly

P er c e n t o f

t o t a l

- 58 29 258 5 8 .459 14 255 67 .940 52 252 66 .741 10 250 56 .542 40 227 5 5 .743 57 224 55.0544 9 222 54 .545 49 210 51 .546 48 170 4 1 .747 39 165 40 .0 448 50 159 54 .149 47 155 52 .6so 46 124 5 0 .4

1 5

TABLE V

TABLE SHOWING THE SCHOOLS WHICH COOPERATED IK GIVING TOE REVISED TEST

S ta teName o fsch o o l

Andreas o f School

Number o f s tu d en t#

A rizona A m phitheater H i£ i School Tucson 26Brandss School Tucson 4C a ta lin a J u n io r High School Tucson 65Hayden Union High School Hayden 14M anesfeld J u n io r High School Tucson 194Hoga Union High School 15a sa 109Moreno1 High School Morenol 41N ogales High School Nogales 44P a tag o n ia Union High School P a tag o n ia 19Boskruge J u n io r High School Tuoson 79S a ffo rd J u n io r H ig t School Tuoson ' 41S c o tts d a le Union High School S c o tts d a le 28Tempo Union High School Tempo 61Tucson S en io r High School Tuoson 98W akefield Ju n io r High School Tucson 34W illiam s Union High School W illiam s 28Turn Union High School Yum 157

C a l i fo rn ia Lynwood High School Lynwood 124

M issouri Greenwood T ra in in g School S p r in g f ie ld 29S o u th e a s t High School Kansas C ity 187

Oregon Klamath F a l l s Union High School Klamath F a l l s 283

Texas A u stin High School 11 Paso 221

W ashington G a rf ie ld High School S e a t t l e 190

T o ta l 2045

16

th e C oopera tive A lgebra T e s t, w hich I s p rep a red under th e d i r e c t io n o f th e

American C ouncil on E d u ca tio n , tjus a d m in is te red to 249 s tu d e n ts who a ls o

took th e r e v is e d t e s t * The c o r r e la t io n betw een th e two was found to be

•56 #03*

The R evised T e s t A dm in istered

' The t e s t was a d m in is te re d by v a r io u s te a c h e rs in th e sch o o ls l i s t e d ,

was sco red by th e s e sane te a c h e r s , and th e n re tu rn e d to th e a u th o r who

checked f o r accu racy o f sco rin g # The freq u en cy d i s t r i b u t i o n , m easures

o f c e n t r a l ten d en cy , and o f sp read a re shown in Table VI#

I t i s w orthy o f n o te t h a t w ith t h i s w id er s e le c t io n o f th o se meas

u red th e mean f a l l s a t 27*36 and th e skewness o f th e d i s t r i b u t io n i s

sm a lle r th a n t h a t found in th e e x p e rim en ta l t e s t i n g o f 407 p u p ils a l l

from one school*

The freq u en cy polygon i s shown in F igure 3 and th e p e r c e n t i l e graph

in F igu re 4 w h ile a p e r c e n t i l e ta b le i s found in Table VII#

R e l i a b i l i t y o f th e R evised T es t

S ince on ly one form o f th e t e s t was a v a i la b le i t was n e c e ssa ry to

use th e s p l i t h a l f method o f f in d in g i t s r e l i a b i l i t y *

S cores on odd numbered q u e s tio n s were p a ire d w ith sco res on even

numbered q u e s tio n s made by each p u p il and th e c o r r e la t io n o f odds w ith

evens was found to be *79. The Spearman-Brown form ula fo r f in d in g th e

r e l i a b i l i t y o f th e f u l l le n g th t e s t was th e n a p p lie d and i t was found to

be ,882 £ ,0 0 6 .

17

TABLE VI

FREQUENCY DISTRIBUTION OF ACHIEVEMENT ON THE REVISED TEST

ALL SCHOOLS

S cores f F

49-61 4 204546-48 21 205945-45 62 20184 0 -4 2 90 196657-59 115 187654-56 181 176551-53 233 168228-50 284 154928-27 506 106622-24 284 76919-21 226 50616-18 154 27915-15 95 14610-12 56 62

7 - 9 15 174 - 6 4 4

11 s 2045

Ifcan - 27.36 Median - 27.07

Sigma - 8 .14 Q - 5 .66

Sigma o f tfcan - .18 PE ctdn - ,16

Sigma o f Sigma - .15 Sigma o f Q - .16

F ig a ro 3

Polygon, R ep re se n tin g 2043 S tu d en ts on R evised T e s t

ScoresW 4 - 5 I

| Af-3— V^>;

^ 7 - 3 ^

i j v - 3 413/-33\ a . l - 3 G

\ 2 S - 2 1

I ^ - a . y| / f - ^ /I / b — / ^

! / 3 —/ 0 ~ / 2 j

7 - 9\ 4 . -

Q 5 to !£ ?V n r 30 3? *0 '+r 5° S'S' Li 4-r 74

F ig u re 4

P e r c e n t i le Graph o f ^>43 P u p ils on R evised T e s t

2 0

TABLE V II

REVISED ACHIEVEMENT TEST PERCENTILE TABLE

Upper l im i tP er o f (p e r c e n t)cen t sooro

100 61.6095 41 .1690 58.5185 56.0680 54.5775 52.8670 51 .6465 50 .2760 29.1965 28.1250 27.0746 26 .7040 26.0755 25 .9850 22.7726 21 .5720 20.2215 18 .8610 16 .85

5 14.120 5 .50

n

S t a t i s t i c a l A nalyeia

An a n a ly s is o f th e su o o e ss fu l s o lu t io n o f th e v a r io u s item s in th e

t e s t i s shown in T able T i l l and i t w i l l be seen t h a t th e r e v is e d t e s t was

ap p ro x im a te ly , b u t on ly a p p ro x im a te ly , a rran g ed a s a s c a le d t e s t ex ten d

in g g ra d u a lly from v e ry easy to v e ry d i f f i c u l t p roblem s. Such an a p p ro x i

mate arrangem ent i s s a t i s f a c to r y i n a speed t e s t , b u t in an untim ed power

t e s t more a c c u ra te s e a l in g would be d e s i r a b le .

For t h i s re a so n . Column 1 in Table T i l l shows a rea rran g em en t o f th e

q u e s tio n s in to a s c a le d t e s t a s a c c u ra te ly a s can be made from th o sco re s

o b ta in e d from 2043 w id e ly s c a t te r e d h ig h schoo l p u p i l s . l a th e appendix

may bo seen co p ies o f th e E xperim en ta l T e s t , th e R evised T e s t, and th e

u n t r i e d , u n te s te d S ea led T e s t a s w e ll a s th e C ooperative A lgebra T es t

w ith which th o re v is e d t e s t was compared as one method o f d e te rm in in g i t s

v a l i d i t y .

Any e x a c t d u p l ic a t io n in th e t e s t o f any problem in any book i s a

c o in c id e n c e .

f t

TABLE V III

PER c m r OP CdlRECf RRSP0R8B8 OS REVISED TEST

Bunbor on s e a le d t e s t

ihintior on r e v is e d t o s t

Iluebor o f For c e n t o fc o r r e c t re sp o n se s* c o r r e c t re sp o n ses*

1 1 1858 90*9442 6 1826 89*578a 4 1781 87*1754 5 1740 85*1685 2 1720 84.1896 15 1608 78 .7077 5 1571 76 ,9908 26 1468 71,8559 10 1466 71 .767

10 14 1464 71,65911 7 1467 71,51612 15 I486 71,26715 19 1452 71.07114 28 1*62 71.071IS 25 1442 70,68216 2® 1412 69.11417 9 1595 68.10418 12 1546 66,80519 22 1556 65,59420 35 1292 64.51221 21 1272 62,26122 31 1265 61 ,91025 SO 1229 @0.15624 24 1208 59,12826 11 1169 67,21926 20 1154 66.48627 25 1140 65 ,80028 55 1128 55 .21229 8 1125 54.968SO 56 1120 64.821S I 45 1061 51 .95352 16 912 44 ,64055 27 887 43 .41654 18 879 45 .02456 42 876 42 ,87856 54 885 42 .241 -37 58 861 42 .14558 57 722 56.540

(c o n tin u e d )

85

TABLE V II I (o o n tim w d )

PER CERT OF CORRECT RKPOHSBS OE REVISED TEST

Number on s c a le d t e s t

“ Number on re v is e d t e s t

Number o fc o r r e c t re sp o n ses*

P er c e n t o f c o r r e c t re sp o n ses*

39 17 704 54.459' 40 32 703 54.410

41 39 677 53.13742 40 682 28.487

• 43 44 624 25.648• 44 41 505 24.620' 46 50 401 19.627' 46 47 370 18.110" 47 45 565 17.376" 48 48 552 17.229

49 46 344 16.837’ 60 49 296 14.439

* The number o f s tu d e n ts t h a t to o k th e t e s t was 2043• T h ere fo re 100#would be 8045e

COM3HJSIOB8

I t i s b e lie v e d t h a t th e r e v is e d 'to s t covers to e e n t i r e co u rse l a

e lem en tary a lg e b ra q u ite th o ro u g h ly , t h a t i t i c q u ite v a l id and h as a

r a th e r h ig h r e l i a b i l i t y # The t e s t w ith i t s f iv e ch o ice s o f an answ er

e lim in a te s g u ess in g on th e p a r t o f th e p u p il# I t len d s i t s e l f to e a sy

and ra p id s c o r in g by th e te a c h e r and can be a d m in is te re d in one c la s s

p e r io d in any h ig h sch o o l w hich has a c la s s p e r io d o f 46 m in u te s , o r o v e r,

as i s t r u e o f any sch o o l t h a t i s a member o f to e N orth C e n tra l A ssocia

t i o n o f High Schools and C o lleges#

The sc a le d t o s t h as been s e t up on th e b a s is o f to e achievem ent o f

2043 p u p ils in w id e ly s c a t te r e d h ig h s c h o o ls , b u t i t has n o t been g iv en

an a c tu a l t r i a l w hich would be n e c e s sa ry to f u l l y co n firm th e num bering

o f th e q u e s tio n s g iv en i n i t #

2 5

BIBLIOGRAPHY

T ext Books i n E lem entary A lnobr*

1 . H a r t, W alte r W.E s s e n tia l* o f A lgebra—F i r s t C ourse« .D,C, Heath and CoBgmny, B oston , 1941.

t* H a r t , W alte r W*P ro g re ss iv e F i r s t A lgebra*D.C, Heath and Gos^>any, B oston , 1945•

5* Hawk®s, H e rb e rt Be, Louby, W illiam A», and T euton , Frank C#Hew F i r s t Course in A lgefara,Ginn and Company, KeW Y ork, 1920*

4# IT illo ry , V i r g i l 8#A F i r s t Course i n A lg eb ra .Benjamin He Sanborn Company, C hicago, 1945#

5e S o h o rlin g , R a le i^ x f C la rk , John He, S m ith , H olland Fe F i r s t Y ear A lg eb ra .World Book Company, Yonkers on Hudson, 1945e

6# S i l b e r s t e in , H athani N ew ell, M arquis J # , and H arper, George A# A lgebra and I t s Uses#Row P e te rso n and Company, Hew Y ork, 1938

7# W ells , W ebster and H a r t , W alter W#Modern F i r s t Y ear A lg eb ra#De Ce Heath and b'ompany, !To\t York, 1925#

S t a t i s t i c a l T ex ts

Be G a r r e t t , Henry E#S t a t i s t i c s in Psychology and Education# L onp ians, Green and Company, Hew York, 1926#

9# Jerom e, H arryS t a t i s t i c a l MethodH arper B ro th e rs , P u b l i s h e r s , Hew York, 1924#

2 6

1 0 . L in d q u is t , E .F .A F i r s t Course in S t a t i s t i c s .Houghton, M if f l in Company, I.'otr York, 1938#

11 . Sm ith , James G.E lem entary S t a t i s t i c s .ifenry H o lt and Company, llerr Y ork, 1934*

12. Sorenson , H e rb e r t .S t a t i s t i c s f o r S tu d en ts o f Psychology and E duoatlon . HfoGraW-Hill Book Corapany, Hew York, 1956.

T e s ts and Measurements

13. B rooks, Samuel S . , and Buckingham, B.R.Im proving Schools by S tan d a rd ized T e s ts .Houghton M if f l in Ccmpany, Hew York, 1922.

14 . G reen, H arry A*, Jo rg e n se n , A lb e r t I . , and G erb erich , J . Raymond.Measurements and E v a lu a tio n s in th e Secondary School*

' Longmans, Green and Company, How York, 1943,

15. Green, H arry A ., and Jo rg en sen , A lb e r t H.The Use and I n t e r p r e ta t io n o f E d u ca tio n a l T e s ts .Longmans, Green and Company, Hew York, 1929*

16 . G reen, H arry A ., and Jo rg e n se n , A lb e r t H.flae Use and I n te r p r e ta t io n o f High School T e s ts . to n g n an s. Green and Company, Ren Y o r k ,1936.

17* Ikm kes, H e rb e rt E*, L in q u is t , E .F . , and Mum, C .R.The C o n s tru c tio n and Use o f Achievement E xam ina tions. Houghton M if f l in Company, llevr York", 1936.

18 . K e lley , T .L .I n t e r p r e ta t io n o f E d u c a tio n a l M aasurem ents.World Book Company, Hew York, 1927.

19 . Monroe, W alter S .An In tro d u c tio n to th e Theory o f E d u c a tio n a l M easurem ents. Houghton M if f l in Company, How York, 1923.

20. P re s se y , S idney L«, and C o le , I n e l l a .In tro d u o tio n to th e Uso o f S tandard T e s ts .W o rld k o o k Company, C hicago, 1922.

27

21* Ross* C*C«Measurements in Today^s S ch o o ls .& ren tloe H a l t , Htao«, New York, 1941.

22* Rueh, G.M*The O b jec tiv e o r TTew Typo Ezam lnatlm s.S o o tt F oresaan and Company, C h ic a g o ,1929*

25. Rueh, G.M., and S to d d a rd , George D.T es ts and Measurements in High School I n s t r u c t io n . World Soolic Company, Chloago, 1927.

24* R u s s e l l , C h arlesC lassroom T e s ts »Ginn and Company, B oston , 1926.

25. R u s s e l l , C h a r le s .S tan d ard T e s ts .Ginn and Company, lew Y ork, 1930*

26. S o h o rlin g , R a le ig h .The Teaching o f th o rn t i e s .!lhe Ann Arbor ^Press',' Ann A rbor, 1956#

27. T ie g s , E rn e s t W.T e s ts and Measurements f o r Teaohers*kqughton M if f l in Company, B oston, 1931.

28. T ie g s , E rn e s t W.T e s ts and M easurements in th e Improvement o f L ea rn in g . Houghton M if f l in Company, B oston , 193§.

APPENDIX

ACHIEVEMENT TEST IN ELEMENTARY ALGEBRA

BY

RALPH FUTRELL Tucson S e n io r High School

FINAL TEST

NAMBt DATE,

SCHOOLl SCOREt

CITY, STATE,

AGE, GRADEj SEX,

♦♦♦DIRECTIONS***

There Are 50 questions in th is t e s t .

You w i l l have 52 m inu tes in w hich to f i n i s h th e t e s t*

Do n o t tu r n th e page u n t i l you a re t o ld to do so .

W rite th e number in f r o n t o f th e o o r r e o t answ er in th e space a t th e r i g h t o f t h i s page t h a t i s p rov ided f o r you r answ er.

Utice no e x p la n a tio n s o r comments*

There i s one o o r re o t answ er and only one o o r r e o t answ er i n th e group o f answ ers ,

♦•♦EXAMPLES***

1* The sum o f 12 and 5 i s t(1 ) 15* (2 ) 9 , (3 ) 36 , (4 ) 4 , (5 ) 108 1 . __1____

2* The p ro d u c t o f x and y 1 s t(1 ) X T y* (2 ) x » y ; (3 ) x * y , (4 ) x y , (6 ) x*y* . t . 4

3 , The square o f 20 1 s t(1 ) 20* (Z) 400 , (5 ) 4 0 , (4 ) 80 , (5 ) 26 3 . 2

1 , The sub o f x and y le t(1 ) x • y* ( I ) x » y i (S ) x y j (4 ) x 4 y ; (5 ) xy2 1 .

2 . The p ro d u e t o f 5x*y and »6xy2 i s i(1 ) 18*yi (B) -18xy , (3 ) - 1 8 x V j (4 ) (5 ) 18y5 2 .

5e The sum o f two numbers i s 18* I f one number i s x , th e o th e r i s t(1 ) 18 - x , (2 ) 18 4 x i (3 ) 18xi (4 ) 18 $ x j (5 ) x -1 8 . 3 .

4 . The p ro d u c t o f 5x , «4xy and Sx2 i n(1 ) -SOx^y; ( 2 ) - l l x ^ y i (3 ) 4x*y; (4 ) 60x*yi (5 ) GOxy4 . 4 .

6 . I f -tSx^y® i s d iv id e d by 6xy th e r e s u l t w i l l b e t(1 ) 3x5y2? (g ) . S x V l (S ) - I 2 x 5y 2 | (4 ) 24xy; (6 ) - 6 . S .

6* I f 15x - 3 (x - 5 ) -5 (2 x 4 T) i s s i s p l i f i e d tite r e s u l t i s ;(1 ) 4x - 40i (2 ) 2x 4 20* (3 ) 2x - 20j (4 ) 6x 4 7 ; ( 5 ) 2 0 6 ,

7* I f -4 x * 36 , th e n x equals*(1 ) 9 i (2 ) 52i (3 ) -3 2 i (4 ) -1 4 4 i (5 ) -9 7*

8 . I f 6x - 3(4x - 8) e x - 5K), th e n x equals*(1 ) - 9 ; (2 ) 121 (3 ) 30; (4 ) 9 ; (5 ) -4 5 , 8 .

9 . I f % * 4 , and y - 3 , th e n 6xy - 7xy2 equals*(1 ) 192i (2 ) -1 9 2 | (3 ) -6 0 j (4 ) 252| (5 ) 186 9*

10, In th e e q u a tio n , (x - 5 ) 2 * (x 4 -3 ) • (x - 7) i» 4 , x eq u a ls *(1 ) w7; (2 ) 4 | (3 ) -4 s (4 ) 7 , (5 ) 25 . 10 .

11, The le n g th o f a re c ta n g le i s x in ch es and th e w id th i s y in c h e s . The a re a in square in ch es is*(1 ) x - y i (2 ) x 4 y i (3 ) x 4 y i (4 ) x y j (6 ) 2xy 11 .

12 , The le n g th o f a r e c ta n g le i s x in c h es and th e w id th i s y in c h e s .The: p e r im e te r in in ch es is*(1 ) x - y i (2 ) x f y i (3 ) 2x f 2ys (4 ) xy# (5 ) 2xy 12 .

13, S i th e e q u a tio n , 9x * 54 , x e q u a ls *(1 ) 5 4 | (2 ) 6 , (3 ) - 6 i (4 ) f» ; (5 ) 436 13 .

14 , th e v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n , x 2 - l2 x ts:28, a re*(1 ) 14, 2f (2 ) 14, -2 s (3 ) -1 4 , -2 s (4 ) -1 4 , 2s (5 ) 28 . 14 .

15, The square r o o t o f 59,049 is*(1 ) 423 , (2 ) 343 , (3 ) 243, (4 ) 233, (5 ) 543 . 15 .

16, In th e e q u a tio n . Sab - 5x % x - 16ab, x equ als*(1 ) Sab, (2 ) -G ab, (3 ) -1 8 ab , (4 ) -6 x ; (5 ) 15ab 16 .

17 . The p ro d u c t o f (3a 4*2b) (4a - 5b) is*(1 ) 12a2 - 2Sab - 10b2 , (2 ) 12a2 4 Tab - 10b2 , (3 ) 12a2 6 10b2 ,

(4 ) 12a2 4 2Sab 4 10b2, (5 ) 12a2 - Tab -10b2 17.

18. I f 20*2 . 4 Sxy 4 Sly* j,B d iv id e d by 8x • 7y. th e r e s u l t w i l l be*(1 ) 4x 4 36* ( 2 ) 4x - 3y* (3 ) 5x - 7y* (4 ) Sx 4 7 y ; (5 ) 4x . 18 .

19 . The f e e te r s f o r 6e 2 • 12a5b e ro s(1 ) 6e (a 4 2e 2b ) , (%) ( 6e * ) ( l 4 2&b); (3 ) 6a 2( l - 2a b ) ,(4 ) 6 a (l - 2 b )* (5 ) 6*2(1 - 2a) 19.

20. The f a c to r s o f Sx2 • iSx 4 6 a re*(1 ) (6x - 3) (x - 2)* (2 ) (5 x f 3 ) (x 4 2 ) ; (3 ) (5x - 3 ) (x 4 2 ) ;(4 ) ( 8x 4 2 ) (x - 2 )» 2 0 .

21 . The base o f a t r i a n g le i s x and th e a l t i te id e i s x • 3 . The a re a o f th e t r i a n g le is*(1 ) x 2 - 3 ; (2 ) X2 . Sx * (3 ) x 2 4-Sx * (4 ) x 2 - Sx* (5 ) x 2 . 3 . 2 1 .

2 " S . 2 ~

22* The sum o f x -7 y and 3x • 4y . s u b tra c te d from th e sum o fx • IQy and x 4 4v 1 s t ,(1 ) 4x • l ly * (2 ) -2 x 4“Syj (3 ) 2x - Gyt (4 ) 6x - 17yj (5 ) x z - 3 . 22.

2 3 , The sum o f 2a 3b y 4e l e i

( l ) 6a * 6b 4 16e * ( 2 ) 6a 4" 6b ^ 16c j (3 ) 6a 4 6b - 16e *12 1 ? ........ 12

(4 ) “ 6a 4 6b 4 12o * (5 ) Sab 4 Sab 4 16ae' l rg 12 23 .

24* The f a c to r s o f 63xf 4 34xy - 40y2 are*(1 ) (7x • 4y) (9x 4-lO y)* (2 ) (7X + 4y) (9x - lO y ), (3 ) (7x - 4y)(9x • 10y)* (4 ) (7x 4 1 0 y ); (5 ) ( 7 x - f 8y) (9x - By) 24 .

25* A ll th e v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n x® • 2x 2 - 63%, a re x equals*(1 ) 0 , 9 , *7 , (2 ) 0 . 9 . 7 , (3 ) 0 , - 9 , -7* ( 4 ) 0 , 7 . 9*(5 ) Or *2* “63 25 .

26 . A ll th e v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n , x 2 4 12x2 - 4 5 x 3 0 a re x e q u a ls t(1 ) 0 , 15, 3* (2 ) 0 , -1 6 , -3* (3 ) 0 , -1 5 , 3*(4 ) 0 , 16,-12* (5 ) 0 , - x , -4 5 . 26 .

27. In th e e q u a tio n , x x _ SO, th e v a lu e o f x is*I

(1 ) 30; (2 ) 6 ; (3 ) 38* (4 ) 36 , (5 ) 180 27 .

28 . A man i s x y e a rs o ld now, 2h y y e a rs from now he w i l l be*( 1 ) X r y* ( 2 ) X 4-y* (3 ) x - y* (4 ) xy* (5 ) x*y 28 ,

29 . I f x d iv id e d by 3 i s f iv e ( 5 ) , th e n lOx w i l l b e t(1 ) 16 , (2 ) 150* (3 ) 6 0 , (4 ) 35 , (5 ) 30 29 .

Pago 4

30. The p ro d u c t o f two numbers i s x and one o f them i s y ; th e o th e r 1 s t(1 ) x « y* ( t ) x 4 y j (3 ) x 5 y j (4 ) x y j (5 ) x2y 3 0 .

31. The square ro o t o f 98 i s t

(1 ) t l p h (2 ) 7 T T | (3 ) U Y T | (4 ) i J T ; (S ) 7 J 'u T 31.

32. The v a lu e s o f x and y t h a t w i l l s a t i s f y th e sim u ltaneous e q u a tio n s ,

M y l i * ' a r 8 '( 1 ) 12, 10} ( 2 ) 6 , 5} (3 ) 11 . - 1 ; (4 ) -1 1 , 1 , (5 ) 11 , 1 3 2 .

53, The p ro d u c t o f • 3 KzO i s i(1 ) 120 , ( 2 ) 100, (3 ) 12 , (4 ) 60 , (5 ) 1200

34. In th e e q u a tio n , 90 ‘ 3 ^ 40 13, x e q u a ls tx " T^x T

( 1 ) 50 , (2 ) 5 , (3 ) -SO, (4 ) - 5 , (5 ) 40

33.

3 4 .

35. The square r o o t o f x2 - 24x ^ 144 is*(1 ) x - 144, (2 ) x - 12, (3 ) x 4 12; (4 ) x - 24, (5 ) x 4 6 . 3 5 .

36 . The v a lu e s o f x and y t h a t w i l l s a t i s f y th e s im u ltaneous e q u a tio n s ,

(1 ) 0 , 4 , (2 ) 20, 16 , (3 ) 4 , 0 , (4 ) 16, 20 , ( 6 ) 4 . 4 . 36 .

37 . The v a lu e o f x t h a t w i l l s a t i s f y th e e q u a tio n , ,3x - 4*2 t 4x - 4 1 .2 , is*(1 ) -3 7 , (2 ) 1 0 , (3 ) 41 , (4 ) - 4 .2 ; (5 ) 4 . 3 7 .

58, I f y th e n x in 5x 4 - 4y s 23, i s t (1 ) 23, (2 ) 7; (3 ) - 7 , (4 ) -1 2 ; (5 ) 5 38 .

39 . I f x books c o s t y d o l la r s th e n t books w i l l c o s ts(1 ) x 4 y ; (2 ) x y t , (3 ) x y -e , (4 ) (x 1 y ) s , (5 ) x (y • x ) 39 .

40. The v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n , 2 ^ j&ZLy » a-r 6 i

(1 ) - 3 , - 2 , (2 ) - 5 , 2 , (3 ) 3 , 2 , (4 ) 3 , - 2 , (6 ) 2 , 2 . 4 0 .

41. Then L s- 12, r - 4 , and a - . 3 , th e n S ^ in S - Lr -^ a e q u a ls t

(1 ) 48 ; ( 2 ) -4 8 , (3 ) -1 5 , (4 ) 15, (5 ) 1 9 . 4 1 .

42. The b inom ia l x - 6 i s one f a c to r o f x* $ 30x - 175* The o th e r f a c to r is*(1 ) X * 35; (2 ) x - 175, (3 ) x - 30 , (4 ) x - 25 , (5 ) X - 170. 42*

43. The sum o f two numbers i s 25 . One fo u r th o f th e la r g e r p lu s one t h i r d o f th e sm a lle r i s 7 . The numbers a re #(1 ) 7 & 25; (2 ) 18 * 7 ; (3 ) 16 & 9 ; (4 ) 18 & 26, (5 ) 32 & 18 . 4 3 ,

44. The sum o f th re e co n se c u tiv e even numbers i s 96 . They a r e :(1 ) 30 , 33, 34 , (2 ) 30 , 32, 34 , (3 ) 31 , 32 , 33 ,(4 ) 30 , 32 , 36 , (5 ) 31 , 33 , 35 . 44

e» w

t* ►

o

4 5 . In z n u ltlp lio A tlo n o f l e t t e r s exponents s re*(1 ) added; ( f ) a u t t r e e te d ; (3 ) m u l t ip l ie d ; (4 ) d iv id e d ; (5 ) c a n o e le d . 4 5 .

46 . la th e e q u a tio n , a % n (a L ) , a equ als*

( l ) - n L J L l i t (2 ) nL " 2% ; (3 ) nL * 2s ; ( 4 ) nL - 2 s ; (5 ) nL - 2s 4 6 . __a 2 a a 2n ”

4 7 . I f a * 1 , b g 15. and o s 56, th e n x i n th e e q u a tio n , x • -b 1 V 4ap , w i l l b e t

2a(1 ) 8 , 7 , (2 ) - 7 , - 8 ; (3 ) - 7 , 8; (4 ) 7 , - 8 ; (5 ) * 8 4 7 . _

4 8 . There a re th re e co n secu tiv e even numbers such t h a t th e equare o f th e f i r s t i s 140 le a s th a n th e p ro d u c t o f th e second and t h i r d . The numbers are*(1 ) 20, 22, 24; (2 ) 22, 24, 26; (3 ) 20, 24, 26;(4 ) 22, 24, 28; (5 ) 22, 23 , 23 4 8 . _

4 9 . The e n ro llm en t in a c e r t a in sch o o l i s shown on th e graph below .

S tu d e n ts e n ro l le d l a 1939

12 1 ! ! ' . ! ! ! ! 1 ! ! ! ! ! " . ! !

i i ..........................................................................................

i o M i ; ;

9 ; ; ; ; ; ; ; ;

e A # x \ ( i . t M x ^ x ! ' i !

7 Ml'!»|)U)HVIjIliw)>invilmtuiiimVxiiliuMniiyivnl * ; ;30 40 V 200 ISO 160 ISO 220 250 270

The number e n ro l le d in th e sev en th grade was how many tim es th e number e n ro l le d in th e e le v e n th g ra d e t(1 ) 6 ; (2 ) S» (3 ) 7 , (4 ) 10; (5 ) 4 49 .

5 0 , The tem p era tu re f o r one day f o r a c e r t a in American c i t y i s shown m the graph *

1 1 5 ...................................................... .................................................

6 5 ........................................................... ....6 0 ......................................... ..............................................................5 5 ........................................................................................................5 0 ........................................................................................................A.M32 2 4 6 8 * 1 1 2 2 4 6 6 1 0 P.M.

th e tem p era tu re a t 8 A.M. was*(1 ) 85; (2 ) 60; (3 ) 110; (4 ) 90; (5 ) 50 50 .


B?

RALPH FCTRBLL Tucson S en io r High School

REVISED TEST

NAME* DATE*

ODD NOS. ( )SCHOOL* EVEN NOS. ( ) TOTAL*

CITY* STATE*

AGE* GRADE* SEX*

**♦ DIRECTIONS ***

There a re SO q u e s tio n s in t h i s t e s t *

You w i l l have 46 m inu tes in w hich to f i n i s h th e t e s t*

Do n e t tu rn th e page u n t i l you a re t o ld to do so*

Make no e x p la n a tio n s o r comments* Ask no q u estio n s*

There i s one c o r r e c t answ er and on ly one o o r r e o t answer in th e group o f answ ers.

W rite th e l e t t e r in f r o n t o f th e o o r r e o t answer i n th e space a t th e r i g h t o f th e page j u s t o p p o s ite th e q u e s tio n you a re an sw erin g .

see EXAMPLES ***

1 . The sum o f 12 and 3 is*(A) 9? (B) 15* (C) 36 , (D) 4 , (B) 10B. I . B

2* The p ro d u c t o f x and y is*(A) x r y , (B) x - y , (C) x f y , (D) x y , (B) 2xy. 2 . D

3 . The square o f 20 is*(A) 20, (B) 4 0 , (C) 400, (D) 80 , (E ) 1 0 . 3 . C

1 . The p ro d u c t o f Sx^y and »6xy^ is*(A) 18xy; (B) -18xy ; (C) -I8 x 3 y 3 , ( d) -18x5y$ (g ) -18y3 ,

2« The sum o f x and y is*(A) x - y ; (B) x f y j (C) x $ y ; (D) xy* (E) x2y .

3 . The f a c to r s o f 5x2 • 13x ■f 6 a re*

4* A man i s x y e a rs o ld now. In (n ) y e a rs from new h is age w i l l be* (A) (x • n ) 5 (B) x • 2nj (C) n • x j (D) x 4 2n; (E) x 4- n .

5 . I f - 4x ■=* 36 , th e n x e q u a ls t(A) -9 j (B) 32} (C) -32} (D) -1 4 4 ; (B) 9 . 5 .

6 . In th e eeuatitML 9x «. 54 , x equals*(A) 54; (B) 6 * (C) - 6 } (D) 9 ; (E) 4 5 . 6 .

7 . Tho le n g th o f a re c ta n g le i s x in ch es and i t s w id th i s y in c h e s .I t s a re a in square in ch es is*(A) xy ; (B) x > y} (C) x - y} (D) x $ y} (E) 2x | 2y . 7 .

8 . The sum o f 2a , 3a , 4a is*V T + J ~

(A) 7a; (B) 7a; (C) 7a; (D) 28a; (E) 14a. 8 .

9 . The le n g th o f a re c ta n g le i s x in ch es and i t s w id th i s y in c h e s .I t s p e r im e te r in in ch es is*(A) xy ; (B) x f y ; (C) 2x - 2y; (D) 2x 4 -2 y ; (E) 2xy 9 .

10 . The p ro d u c t o f (3a 4 2b ) (4a - 5b) i s ;(A) I2a2 - 23ab - 10b%; (B) I 2a 2 | 7ab - 10b2;(C) 12a2 4 10b2, (D) 12a2 | gBab 4 10b2 (E) 12a 2 . 7ab - 10b2. 10 .

11 . I f - 1 8 x ^ 3 £a d iv id o d by 6xy th e q u o t ie n t w i l l be*(A) 3xSy* (B) -3x3y2 ; (C) -12x3y2 ; (D) 24xy; (E) -24x5y4e 11 .

12. I f IGx - 5 (x - 5 ) -5 (2 x 4 7) i s s im p lif ie d th e r e s u l t w i l l b e t(A) 4 (x - 1 0 ); (B) 2 (x f 1 0 ); (C) 2(x - 1 0 ); (D) 6x 4 7}(E) 5x - 3 5 . 12 .

13. In th e e q u a tio n . Sab - 6x * x - 15ab, x e q u a ls *(A) Sab; (B) -a b ; (C) -18ab ; (D) -6 x ; (B) 15ab. 13 .

14 . I f 20x2 - 43xy 4 21y2 i s d iv id e d by 5x - 7y th e q u o t ie n t w i l l b e t(A) 4x 4 3y; (B) 4x - 3y; (C) 5x - 7y* (D) 5x 4 7y;(E) 15x - 14y. 14 .

16, The f a c to r s o f 6&2 - 12a® b a re s(A) 6a(a - S ab ) | (B) Sa2( 1 4> 2 ab )j (C) 6&2(l • Sab);(D) 3a2(2a - 4 a b ); (E) 6(a2 - 2a S ) .

16 , The p ro d u c t o f 4 ^ 6 , 3 VzO* i s t(A) ISO; (B) 1100; (C) 12; (D) 100; (E) 1200

17, The e x p re ss io n , (x ^ ^ _ 24) ^ ( i I . 24 ) >iri i t a s im p le s tform I s s x ' * x(A) - x ; (B) x 2 - 7x - 2x; (C) -x 2 j (D) x 2 ; (B) x .

15 ,

16.

17.

18. In th e e q u a tio n , 90 - 3 - 40 - 13 , x equals*

(A) 60; (B) 5; (C) -5 0 ; *(D ) -5 ; (E ) 4 0 , 18 ,

19 . The square r o o t o f 59,049 is*(A) 427; (B) 243; (C) 253; (D) 233; (B) 543. 19.

20. The square r o o t o f 98 is*(A) 2 V tJ (B) 7 Y f; (C) u Y f t (D) 7 ^ ; (E) 7 f u . 20.

21. The p ro d u c t o f 5x , -4xy and 3x2 i Bl(A) -6 0 x % ; (B) - l l x 2 y ; (c) 4 x ^ j (D) 60x*yj (E) 60xy4 , 21.

22. I f th e sum o f x - 7y and 3x - 4y i s s u b tra c te d from th e sum o f x - lOy and x ^ 4y th e r e s u l t w i l l be*(A) 4x - l l y ; (B) -2x 4 5y; (C) 2x - 6y; (D) 6x - 17y;(E) l l x - 4 y . 22.

23. One f a c to r of 49x2 - M xy - 15y2 i s 7x 4 %r; th e o th e r f a c to r is*(A) 7x - 3y; (B) 7x 4 5y; (C) 7x 4 (D) 7x - 6y;(E) 7x - 7y . 23.

24. I f 8x - 3(4x - 5) x - 30 th e n x equals*(A) - 9 ; (B) 12; (C) 30; (D) 9 ; (E ) -4 5 . 24.

25. In th e e q u a tio n x x - 30 , x equals*. • T ~ J m T

(A) 30; (B) - 6 ; (C) 6 ; (D) 36; (E ) ISO. 25 ,

26. The b in o m ia l x - 5 i s one f a c to r o f x2 4*30x - 175. The o th e rf a c to r is*(A) x f 35; (B) x - 175; (C) x - 30 ; (D) x - 25; (E) x - 35 . 26 .

27, The b ase o f a t r i a n g l e i s x and i t s a l t i t u d e i s x o f th e t r i a n g le is*(A) x2 - 3 ; (B) x 2 - 3x ; (C) X2 f 3x ; (D)

' i ' "" ' 2 2(E) x2 - 3 .

3 . The a re a

3x,

27.

2 8 ,28, I f y eq u a ls - 3 , th e v a lu e o f x in 5x f 4y * 23 is*

(A) 23; (B) 35; (C) 7 ; (D) -1 2 ; (B) 70 .

P a g e 3

;9 . Tho sum o f th re e co n secu tiv e even numbers i s 96 . They are*(A) 30, 32, 34; (B) 40 , 42, 44% (C) 31 , 32, 38;(D) 31, 33, 35; (B) 30 , 31 , 35 .

JO. The v a lu e s o f x and y t h a t w i l l s a t i s f y th e s im ultaneous e q u a tio n s , 5x - 3y =:20

4x 4 3y - 16 » r i(A) 0 , 4 ; (B) 20, 16; (C) 4 , 0 ; (D) 16 , 20; (B) 4 , 4 .

51. Tho sum o f two numbers i s 25 . One fo u r th o f th e l a r g e r p lu s one th i r d o f th e sm a lle r i s 7 . The numbers are*(A) 7 & 25; (B) 18 & 17; (C) 16 k 9% (D) 18 & 25; (B) 32 k 18 .

52. A ll tho v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n ,*3 _ 2x 2 - 63x , a re x equals*(A) 0 , 9 , -7 ; (B) 0 , 9 , 7 ; (C) 0 , - 9 , -7% (D) 0 , 7. 9;(E) 0 , -2 -6 3 .

53 . When L - 12, r % 4 and a - 3 th e n S , in S - Lr - a" r - 1

(A) 48; . (B) -4 8 ; (C) -1 5 ; (D) 15; (B) 1 9 .

e q u a ls *

3 4 . A ll th e v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n , x^ 4 " 12x2 - 45X - o , a re x equals*(A) 0 , 15, 3 ; (B) 0 , -1 5 , -3 ; (C) 0 , -1 5 , 5 ; (D) 0 , 15 , -1 2 ;(E) 0 - 5 , -1 5 .

35 . The sum o f two numbers i s 18 . I f one o f th e numbers i s x th e o th e r w i l l bo :(A) 18 - x ; (B) 18 4 x ; (C) 18X; (D) 18 : X; (B) x - 18 .

36. I f x d iv id e d by 3 i s 5 , th e n 10 x w i l l b e t(A) 15; (B) 1150; (C) 150; (D) 35 , (B) 30 .

37 . The v a lu es o f x t h a t w i l l s a t i s f y th e e q u a tio n , x 2 -l2 x % 28, are*(A) 14, 2 ; (B) 14, —2; (C) -1 4 , - 2 , (D) -1 4 , 2 ; (B) 28, 12 .

38 . The v a lu es o f x and y t h a t w i l l s a t i s f y th e e q u a tio n s ,

(x ♦ y 5 1 2 ) • a r a i(A) 12, 10; (B) 6 , 5 ; (C) 11 , - 1 ; (D) -1 1 , 1 ; (B) 11, 1 .

39 . The value o f x in th e e q u a tio n , (x - 5 )* — (x 4 3) (x - 7) 4 (A) - 7 , (B) 4 , (C) - 4 ; (D) 7 ; (B) 25 .

40 . The v a lu es o f x t h a t w i l l s a t i s f y th e e q u a tio n x ^ £ j.^5

(A) - 3 , - 2 ; (B) - 3 , 2 , (C) 3 , 2 , (D) 3 , - 2 , (B) 2, 2 .

41* The v a lu e o f x t h a t w i l l s a t i s f y th o e q u a tio n , •3x - 4*2 a 4x - 4 1 .2 , is*(A) -3 7 ; (B) 1 0 , (C) 4 1 , (D) - 4 .2 , ( B ) 4 .

29. __

30 . w

31 . _

32 . _

3 3 . __

34 . __

35. __

3 6 . _

37 . __

38 . _

4 , is*

39 . _

4 0 . _

41 .

CJ « <«i o

w

4 2 . I f x - 4 and y - 3 then 5xy - 7xy2 equals*(A) -1 9 2 ; (B) 192; (C) -6 0 ; (D) 262; (E) 186. 4 2 .

4 5 . The en ro llm en t in a c e r ta in schoo l i s shown on th e graph below*

Page 4

S tu d en ts e n ro l le d in 1939

12

11

10

9

8

7

::::::::

o 3D © © 120 bo 180 ao a«)The number e n ro l le d in th e sev en th grade was how many tim es th e number e n ro l le d in th e e le v e n th g rad e1 (A) 5 ; (B) 3 ; (C) 7; (D) 10; (E) 4 . 43 ,

44 . There a re th re e co n secu tiv e even numbers such t h a t th e square o f th e f i r s t i s 140 le s s th a n th e p ro d u c t o f th e second and t h i r d . Thenumbers are*(A) 20, 22, 24; (B) 22, 24, 26; (C) 20 , 24, 26; (D) 24, 26, 28;(E) 22, 23, 24. 44 .

45 . I f x books c o s t y d o l la r s th e n t books w i l l co st*(A) x - f y ; (B) xyz; (c) xy - z ; (D) (x + y ) s ; (B) z (y ? x ) 4 5 .

46 . The p ro d u c t o f two numbers i s x and one o f them i s y ; th e o th e r is*(A) x f y ; (B) x - y ; (C) x ? y ; (D) x y ; (E) x% . 4 6 ,

47 . I f a - 1 , b * 15, and c c 56 th e n x in th e e q u a tio n ,

* - z k ^ E Z 5 = . h„ TOlue„ o f l

(A) 8 , 7; (B) - 7 , - 8 ; (C) - 7 , 8 ; (D) 7 , - 8 ; (E) * 8 . 4 7 .

4 8 . In th e e q u a tio n , S * (a f L ), a equals*

(A) nL - 28; (B) nL - 28 ; (C) nL ¥ 2 8 ; (D) 2S - nL ;n 2 2n " ' n '

(E) 28 4 nLn 48 .

2549 . t o . o x p ro aa io c , ( 3 . — ) ; (1 ^ ) , „ duM d t e l t , . t o p i c ,*

form is*(A) 3x - 5 ; (B) 5x - 3 ; (C) 5 x ^ 3 ; (D) 3x ^ -5 ; (E) 9x2 - 26 . 49 .

50 . The p ro d u c t o f th e f r a c t i o n , V s

y r v r #(A) 100. (B) 2&; (C) 3*$ (D) 10; (E) 4

is*

50


BY

RALPH FUTRELL Tucson S en io r High School

SCALED TEST

NAME: DATE:

ODD NOS. (______ )SCHOOL:____________________________________ EVEN NOS, ( ) TOTAL:_____________

CITY:____________________________________________________ STATE:_________________________

AGE ^ ^ GRADE: SEX:

+** DIRECTIONS ***

There a re 50 q u e s tio n s in t h i s t e s t*

You w i l l have 45 m inu tes in w hich to f i n i s h th e te s t*

Do n o t tu rn th e page u n t i l you a re to ld to do so .

Make no e x p la n a tio n s o r comments. Ask no q u e s t io n s .

There i s one c o r r e c t answer and on ly one c o r r e c t answer in th e group o f an sw ers .

W rite th e l e t t e r in f r o n t o f th e c o r r e c t answ er in th e space a t th e r i g h t o f th e page j u s t o p p o s ite th e q u e s tio n you a re answ ering*

. *Do n o t s to p when you come to th e end o f a page, b u t go on to th e n e x t page*

I f you do n o t know th e answ er to a q u e s tio n , do n o t s to p b u t go on to ano ther*

*** EXAMPLES ***

1 . The sum o f 12 and 3 i s :(A) 9; (B) 15; (C) 36; (D) 4 ; (E) 108. 1 . ___ B

2* The p ro d u c t o f x and y i s ;(A) x r y ; (B) x - y ; (C) x + y ; (D) x y ; (B) 2xy.

3 . The square o f 20 is*(A) 20; (B) 40; (C) 400; (D) 80; (B) 10 .

2 # D

C3

].1. The p ro d u c t o f 5x^y and -6xy2 i 8s

(A) 18xyi (B) ••18xy; (C) -18x3y3$ (D) -18x3yj (E) -18y3 ,

2 . In th e e q u a tio n 9x s 54, x e q u a ls i(A) 54; (B) 6 ; (C) - 6 , (D) 9 ; (E) 45 SJ.

3 . A man i s x y e a rs o ld now. In (n ) y e a rs from now h is age w i l l b e t(A) (x - n ) j (B) x - 2n; (C) n - x ; (D) x -f-2n; (E) x + n . 5 .

4 . I f -4x m. 36, th e n x equals*(A) -9 ; (B) 3p; (C) -3 2 ; (D) -1 4 4 ; (B) 9 . 4 .

5 . The sum o f x and y is*(A) x - y ; (B) x r y ; (C) x + y ; (D) xy ; (B) x^y . 5 .

6 . In th e e q u a tio n . Sab - 5x *. x - 15ab, x eq u a ls *(A) Sab; (B) -a b ; (0 ) -18ab ; (D) -6 x ; (E) 15ab. 6 .

7 . The f a c to r s o f 5x2 - 13x 4 6 are*(A) (5x - 3) (x - 2 ) ; (B) (5x ^ 2) (5x f 2 ) ;(C) (5x f 3) (x - 2 ) ; (D) (5x - 3) (x f 2) ; (E) (5 x - 3 )2 7 .

8 . The b inom ial x - 5 i s one f a c to r o f x2 30x - 175. The o th e r f a c to r is*(A) x f 35; (B) x - 175; (c) x - 30; (D) x - 25; (B) x - 35. 8 .

9 . The p ro d u c t o f (3a 4 2b) (4a - 5b) is*(A) 12a2 - 23ab - 10b2. (B) ig a 2 + 7ab - 10b2,(C) 12a2 4 10b2 ; (D) 12a2 j 23ab f 10b2 , (E) ig a 2 - 7ab - 10b2 . 9 .

10 . I f 2Ox2 - 43xy 4 2 ly 2 i s d iv id e d 5x - 7y th e q u o t ie n t w i l l be*(A) 4x 4 3y; (B) 4x - 3y; (C) 5x - 7y, (D) 5x 4 7y;(E) 15x - 14y. 10.

11 , The le n g th o f a re c ta n g le i s x in ch es and i t s w id th i s y in c h e s .I t s a re a in square in ch es is*(A) xy ; (B) x f y ; (C) x 4 y ; (D) x - y ; (B) 2x 11.

12, The f a c to r s o f 6a2 - I2a3b are*(A) 6 a(a - 2 ab ); (B) 6a2( l 4 2 a b ); (C) 6a2( l - 2 ab );(D) 3a2(2a - 4 a b ) ;(E ) 6 (a 2 - 2 a 3 ). 12.

13, The square ro o t o f 59,049 i s ;(A) 427; (B) 243; (C) 253; (D) 233; (B) 543 13.

14 , I f y eq u a ls -3 , th e v a lu e o f x in 5x 4 4y e 23 is*(A) 23; (B) 35; (C) 7 ; (D) -1 2 ; (B) 70 14.

15, One f a c to r o f 49x2 - 14xy • 15y2 i s 7x 4r 3 y ;ih e o th e r f a c to r is*(A) 7x - 3y; (B) 7x f 3y; (C) 7x f 5y , (D) 7x - 5y;

7x - 7y . 15.

P a g e 2

16. The sum o f th re e oonseou tlvo even numbers i s 9 6 . They a re s(A) 30 , 32 , 34; (B) 40 , 42 , 44; (C) 31, 32, 33;(D) 31, 33 , 35; (B) 30, 31, 35 . 16.

17. The le n g th o f a re c ta n g le i s x in ch es and i t s w id th i s y in c h e s .I t s p e r im e te r in in ch es i s t(A) xy ; (B) x 4- y ; (C) 2x - 2y; (D) 2 x - f 2y; (E) 2xy 17.

18 . I f 15 x - 5 (x - 5) -5(2x4* 7) i s s im p lif ie d th e r e s u l t w i l l be*(A) 4 (x - 1 0 ); (B) 2(x t 1 0 ); (C) 2 (x - 1 0 ); (D) 6x 4* 7;(B) 5x - 35 . 18 .

19. I f th e sum o f x - 7y and 3x - 4y i s s u b tra c te d from th e sum o f x - lOy and x - f 4y th e r e s u l t w i l l be*(A) 4x - l l y ; (B) - 2 x - f 5y; ( c ) 2x - 6y; (D) 6x - 17y;(E) l l x - 4y . 19.

20. Vthen L * 12, r r 4 and a « 3 th e n S , in S - Lr - a , eq u a ls *' r - 1

(A) 48 ; (B) -4 8 ; (C) -1 5 ; (D) 15; (B) 19 . 20.

21. The p ro d u c t o f 5x , -4xy and 3x2 i s *(A) -60x4y . (B) - l l x 2 y . (c ) 4x4y ; (D) GOx^y. (e ) 60xy4 , 21.

22. The sum o f two numbers i s 25 . One fo u r th o f th e l a r g e r p lu s one th i r d o f th e sm a lle r i s 7 . The numbers a rc s(A) 7 & 25; (B) 18 & 7; (C) 16 & 9; (D) 18 & 25; (E) 32 & 18. 22 .

23. The v a lu e s o f x and y t h a t w i l l s a t i s f y th e sim u ltaneous e q u a tio n s ,5x - 3y n 20 4x 4 3y = 16 '(A) 0 , 4 ; (B) 20, 16; (C) 4 , 0 ; (D) 16, 20; (E) 4 , 4 . 23.

are*

24. I f 8x - 3(4x - 5 ) r x - 30 th e n x eq u a ls *(A) - 9 ; (B) 12; (C) 30; (D) 9 ; (E ) -4 5 . 24.

25, I f -18x4y3 ia d iv id ed by 6xy th e q u o t ie n t w i l l be*(A) 3x3y2; (B) -Sx^y2 ; (C) -12x3y 2 ; (D) 24xy; (B) -24xey4 . 25.

26. The aoimro r o o t o f 98 i s t; (B) 7 XT ; (c) m ZT (D) 7 X T (B) 7 X lT . 26.

27 . In th e e q u a tio n x x ^ 50, x equals*2 ? #

(A) 30; (B) «6 | (C) 6 ; (D) 36; (E) 180

28 . The- sum o f two numbers i s 18 . I f one o f th e numbers i s x th e o th e r w i l l b e t(A) 18 f x ; (B) 18 - x ; (C) 18x; (D) 18 \ x ; (B) x - 18,

29. The sum o f 2a , 3a - 4a i s tr + r + T

(A) 7a; (B) 7a; (C) 7a ; (D) 28a ; (B) 14a12 aT T T T T T

27.

28.

2 9 .

30• I f x d iv id e d by 3 i s 5 , th e n 10% w i l l b e t(A) 16) (B) 1160) (0 ) 150) (D) 35) (B) 30 . 30.

31 . The e n ro llm e n t in a c e r t a in schoo l i s shown on th e graph below .

6RADE

S tu d en ts e n ro l le d in 1939.

M :::::::::::::::::::::i i Iw W w i* ! ‘

io m r i m r u M m r m : : : : : : : : : : : : : :

8 ! ! ! ! ! ! ! !

a ; ; ; ;

0 33 63 90 120 150 180 &0 %0

The number e n ro l le d in th e sev en th grade was how many tim es th e number e n ro l le d in th e e le v e n th grade?(A) 5 ; (B) 3 ; (C) 7 ; (D) 10; (E) 4 . 31.

32 . The p ro d u c t o f 4X 5* • 3 1 s t(A) 120) (B) 1100; (C) 12; (D) 100) (E) 1200 32.

33 . The base o f a t r i a n g le i s x and i t s a l t i t u d e i s x - 3 . The a re a o f th e t r i a n g l e is*(A) * 2 - 3 ) (B) x 24 3x ) (C) x 2 - 3x ; (D) x2 - 3x;

2 2 2(B) x2 - 3 . 33 .

34 . In th e e q u a tio n , 90 3 40 13, x e q u a ls tx— ? = — ~ T

(A) 50) (B) 5 ) (C) -5 0 ) (D) -5 ) (B) 4 0 . 3 4 .

35 . I f x - 4 and y - 3 th e n 5xy - 7xy2 e q u a ls t(A) -192) (B) 192) (C) -60) (D) 282) (S ) 186. 36 .

36 . A ll th e v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n ,x* 4 12x2 - 45x 0 , a re x equals*(A) 0 , 15, 5) (B) 0 , -1 5 , -3 ) (C) 0 , -1 6 , Sj (D) 0 , 16, -12*(B) 0 , - 5 , - 1 5 , 8 6 . _

37 . The v a lu e s o f x and y t h a t w i l l s a t i s f y th e eq u a tio n s* (x -y » IO)are* (x 4 y * 1 2 )(A) 12, 10) (B) 6 , 6 * (C) 11 , -1 ) (D) -1 1 , 1) (E) 11, 1 . 3 7 . _

3 8 . The v a lu e s o f x t h a t w i l l s a t i s f y th e eq u a tio n X^ - I 2x %. 28, are*(A) 14, 2* (B) 14 , -2* (C) -1 4 , -2 ) (D) -1 4 , 2) (B) 28, 12 . 38 . _

6 8 * The ex p re ss io n # X_ - 7 „ 24) . _ 7 , 24 . in i t s s im p le s t form is *

(A) -x* (B) %$ - 7x - 2%) (C) - x 2 ) (D) x 2 * (B) x , 3 9 . ^

40. A ll th e v a lu e s o f x t h a t w i l l s a t i s f y th e e q u a tio n , x3 - 2x& # 63x, a re x e q u a ls t(A) 0 , 9 , -7* (B) 0 , 9 , 7j (C) 0 , -9 - 7 , (D) 0 , 7 , 9 $(E) 0 , -2 -63• 40 ,

41 . The v a lu e o f x in th e e q u a tio n , (x -S )2 5 (x f 3) (x - 7 ) f 4 , i s :(A) -7 ; (B) 4> (C) -4 ; (D) 7 ; (E) 25 . 41 .

x 5 ^ 542. The v a lu es o f x t h a t w i l l s a t i s f y th e e q u a tio n Y ^ x "z , a r e :

(A) - 3 , -2$ (B) - 3 , 2 ; (C) 3 , 2 ; (D) 3 , - 2 ; (B) 2, 2 . 42 .

43 . There a re th re e co n secu tiv e even numbers such t h a t th e square o f th e f i r s t i s 140 lo s s th an th e p ro d u c t o f th e second and t h i r d . The numbers a r e :(A) 20, 22, 24; (B) 22, 24, 26; (C) 20, 24, 26; (D) 24, 26, 28;(E) 22, 23, 24. 43 ,

4 4 . The va lue o f x t h a t w i l l s a t i s f y th e e q u a tio n , .Sx - 4 ,2 a. 4x - 41*2, i s :(A) -3 7 ; (B) 10; (C) 41 ; (D) -4 * 2 ; (E) 4 .

4 5 . The p ro d u c t o f tho f r a c t io n s V 5 Yen"7 f 7 ^ 181

(A) 100; (B) 2&; (C) 3&; (D )l0 ; (E) 4 .

44 .

45.

46 . I f a » 1 , b s i s l a n d c - 56 th en x in tho e q u a tio n , __ -b * y b<i - 4ac

x % ga s has values of:

(A) 8 , 7 ; (B) - 7 , - 8 ; (C) - 7 , 8 ; (D) 7 , - 8 $ (E) * ^ 8 .

4 7 . I f x books c o s t y d o l la r s th e n z books w i l l c o s t : .(A) x 4 y ; (B) x y s; (C) xy - z ; (D) (x 4 y ) s ; (B) z ( y - x )

4 8 . In th e e q u a tio n , S -

(A) nL - 28; (B) nL -n 2

4 L ), & e q u a ls :

(C) nL 4 2 8 ; (D) 28 - nL;2n n

(B) 28 4 nL n

46 ,

4 7 .

4 8 .

4 9 . Tho p ro d u c t o f two numbers i s x and one o f them i s y , th e o th e r i s :(A) X 4 y ; (B) x - y ; (C) x & y ; (D) x y ; <B) x%y. 4 9 .

50 . The e x p re s s io n , (« * „ 25) x (■» e 5 ) , reduced to i t s s im p le s t3x e 1 &

form i s 1(A) 3x - 5 ; (B) 6x - 3 ; (C) 5x * 3 ; (D) 3x { 5 ; (B) 9xz - 26 . 60 .

AMERICAN COUNCIL ON EDUCATION

COOPERATIVE ALGEBRA TESTELEMENTARY ALGEBRA THROUGH QUADRATICS

REVISED SERIES FORM S

byLEONE E. CHESIRE and MARGARET P. MARTIN, Cooperative Test Service;

and L. P. SICELOFF, Columbia Universitywith the editorial assistance of

HUBERT V. DAVIS, Cranbrook School; JO H N P. EVERETT, Western Michigan College of Education;RUTH VAN PELT, Western Washington College of Education; ALFRED G . WHITNEY, Melrose High School (Mass.);

and JACK WOLFE, Brooklyn College

Please print:

Name.__ Date___Middle

Grade or Class. Date of BirthYrs. Mos.

School.......................... ................................................C ity...........................................................Sex.M. or F.

Title of the algebra course you are now taking........................................... Instructor...........;................... .

In what grade did you begin the study of algebra?..........................................................................................

Number of years you have studied algebra (one semester = year; one quarter = % year):.......

General Directions: Do not turn this page until the examiner tells you to do sd. This examination consists of three parts, and requires 40 minutes of working time. The directions for each part are printed at the beginning of the part. Read them carefully, and proceed at once to answer the questions. DO NOT SPEND TOO MUCH TIME ON ANY ONE ITEM. ANSWER THE EASIER QUESTIONS FIRST; then return to the harder ones if you have time. There is a time limit for each part. You are not expected to answer all the questions in any part in the time limit; but if you should, go on to the next part. If you have not finished Part I when the time is up, stop work on that part and proceed at once to Part II. If you finish the last part before the time is up, you may go back and work on any part. No questions may be asked after the examination has begun.

You may answer questions even when you are not perfectly sure that your answers are correct, but you should avoid wild guessing, since wrong answers will result in a subtraction from the number of your correct answers.

Part i ii III Total

Minutes is 10 15 40

Scaled Score Percentile

Copyright, 1942, by the Cooperative Test Service. All Rights Reserved. Printed in U. S. A.IS Amsterdam Avenue, New York, N. Y.

- 2 -

PART I

(15 minutes)Directions: Each problem below is followed by five choices, one of which is the correct answer. By working each problem, find the correct answer and put its number in the parentheses at the right.

Sample0. If a is equal to 2 and b is equal to 5,

a + b equals 0-1 9 0-2 20-3 30-4 70-5 5 .................................................... 0( 4 )

Since the answer is 7, which is the fourth choice, the number 4 has been written in the parentheses at the right.

1. 8n — 2n equals1-1 — 4n1-2 6n1-3 61-4 44-5 - 16n . . . . . . . . . . . . 1( )

2. A square root of 16g1 2 is2—1 8q

. 2-2 4q2-3 4qK2-4 4q22-5 256q4 . . . ............................ .... 2( )

6. H p = — 1, q = — 2, and r = 3, the numerical value ol 2p — 3q — r is 6-1 16-2 - 11 •

6—3 — 76—4 . — 56-5 5 ..................................................6( )

7. Simplify 12g + 4 — 3(5p — 2) by removing parentheses and combining like terms.7-1 - 3flr + 107—2 — 3 q — 27-3 - 3 p + 27-4 27ff - 27-5 27p + 10 . . . . .....................7( )

8. The relationship between m and n is

indicated by the formula m = -n + 8.

When m equals 20, n equals S-l 8 8-2 18

8-3 2 l l

8-4 268-5 42 . . ....................................... . .8 ( )

3. What is the product of 4n2 and 3n4 *?3-13-23—33-4

3-5

12n812n67n67n83n?4 3(

9. What is the quotient when 21c,d* is divided by — 3c2dA?P-1 - 63cndlt9-2 - 7c3dP-3 - 7c8cf8P-4 - 7c4P-5 — 7c3 .............................................. 9( )

4. If 5w — 6

4-1

4-2

4-3

4-4

4-5

8, w equals

•a

4i6

10. If 3p + 5 equals — 9, what is the value of g?

W -l

10-2 210-3 - 210-4 7

10-5 l . . . 10( )

5. One of the factors of Z2 — 2tto + w2 is 5-1 2Z + w 5-2 2t — w 5-3 2w + t 5—4 Z -f-5-5 t — w ........................................ 5(

11. Subtract — 16a from — 7a. 11-1 911-2 - 9a11-3 9a 11-4 - 2344-5 - 2 3 a .................... IK )

Go on to the next page.

12. If .09m + 1.4 = 2.66, what does m equal?12-t i l l 12-2 2.8 12-3 1412-4 31.112-5 some other v a l u e .................... 12 ( )

13. What is the product of M 3 and M~10?13-1 M -1 13-2 Af-A 13-3 M -li 13-4 Af-*°13-5 2M-7 . .........................................13( )

14. | i equals

14-1

14-2

14-3

14-4

14-5

3s + 4w ws

12ws3s + 4w

iy + s 7

to + s3s + 4to 14( )

If m 15-1

15-2

15-3

• st + c, then t equals m — sc

sm — c m — c

s

15-4

15-5

2 - c5c — m

s 15( )

16. Simplify V2 — 4^2 + 5V2.16-1 0*16-2 216-3 V2 /tf-4 2V2itf-5 - 4 + 5 V 2 ............................. .... 16( )

18.

19.

If Sc — 2b ■= 7 and 2c + 3b what does b equal?18—1 118-2 - 118-3 1

3

18-4 911

18—5 919.........................

i f iU IS .-j-, v equals

19-1 ISrk 15 119-2 T ~ 115k19-3 r

19-4 1 15r k

19-5 15rk ........................

20. The formula for the circumference of a22

circle is C = 2«r. If is = —, and

C = 88, r equals 20-1 56 20-2 2 20-3 28 20-4 420-5 1 4 . . ........................................

21. D , 3c* - cdReduce w Z T d* to lowest terms.

2/-1

21-2

21-3

21-4

21—5

cc + d

c3 c + d- cd

3 - d 2c

3 c - d— c

3 - d

17. If the graph of the equation4x — my = 4 passes through the point (3, 4), the value of m is

17-1 6?17-2 2 17-3 3 17-4 417-5 5 .....................................

22.7n - 5 5n + 4

22-1

22-2

22-3

22-4

22-5

9, n equals

18( )

19( )

20( )

21( )

17( ) ............................ 22( )

Go on to the next part.

PART II

(10 minutes)Directions: Continue as in the preceding part.

1. If Phil has c marbles and gives John d of them, how many marbles does Phil have left?J -l cd

1—3 c — d 7-4 d — c7-5 c + d ............................................ 1( )

u 10

M T W Th F S Day of the Week

2. The graph above shows the number of magazines Bob sold each day from Monday through Saturday. How

. many magazines had he sold through Wednesday?2 -\ 152-2 20 2-3 25 2-4 452-5 60 ........................ .... .2 ( )

3. How long will it take a train to travel . 1000 mi at m mi per hr?

J -l 1000m J—2 1000 — Tti

3-3

3-4

J—5

1000mm

10001000 + m 3( )

If one side of a square is s, what is the perimeter?4-1 s2 4-2 2s 4-3 2s2 4-4 4s

4-5 | . . . . . . . . . . . . . .4 4 ( )

5. If Ellen starts a savings account by putting 50(1 in the bank the day she opens the account and deposits 106 each week after that, how much (in cents) will she have saved at the end of w weeks?5-15-25-35-4

5-5

w + 60 50

10zylOw — 50IQiy5010u> + 50 5( )

6. If Dan weighs 140 lb and is x lb heavier than Ed, how many pounds does Ed weigh?

6-1

^ A6-3 140 + % 6-4 x — 140 6-5 140 — x 6( )

7. If James buys b books for $10, what is the average price per book in dollars?

7-3 106 7-4 1 0 - 67-5 6 - 1 0 ........................ ... . . . 7( )

8. The sum of three angles of a triangle is 180°. If the triangle has two equal angles each denoted by x, how many degrees, in terms of x, are in the third angle?J -l 180 - 2x '8—2 180 — 3x ■

8-3 1 8 0 - 1

8-4

8-5

1802x2x 8( )


- 5 -

High School But Hot C o lleg e

25* Grades O nly

9. The graph above shows the amount of education of all persons in the United States who are over 25 years of age. What percent of them have had at least some elementary school education?0-1 1000-2 930-3 860-4 750-5 68

10. When Helen has gained one-fourth of her present weight, she will weigh 120 lb. What does she weigh now?10-1 90 lb10-2 96 lb10-3 100 lb10-4 150 lb10- 5 160 lb . . . . ......... .............10( )

11. George wants to cut a 10-ft board so that one piece will be 2 ft longer than the other. What will be the length of the longer piece?11- 1 6 ft11-2 7 ft11-3 8 ft11-1 4 ft

12. Six boys went on a camping trip.They agreed that the two boys who owned the tent should each pay only half as much as each of the others.The expenses of the trip were $40.How much did each of the boys who owned the tent have to pay?12-1 $3.3312-2 $4.0012-3 $6.6712-4 $8.0012- 5 $ 1 0 .0 0 .....................................12 ( )

13. The length of one rectangle is 4 times its width. A second rectangle is 1 unit wider and 6 units longer. If x represents the width of the first rectangle, the area of the second rectangle, in terms of x, is equal to13- 1 4x2 + 613-2 8x2 + 613-3 4x2 + 1213-4 (4x + l)(x + 6)13- 5 (4x + 6)(x 4 - 1 ) ............................13( )

14. A newsboy sold the Post for 5<i and the News for 3<L He sold 70 papers in all and received $2.50. How many Posts did he sell?14- 1 5714-2 5014-3 2314-4 2014- 5 5 .......................................... .14( )

15. A triangle whose base is 8 ft has the same area as a square whose side is 10 ft. What is the altitude of the tri-

1angle? (Area of a triangle equals ab.)

15- 1 10 ft

15-2 12 ft

15-3 20 ft15-4 25 ft15-5 50 f t ................................ .... . . 15( )

Go on to the next part.

- 6 -PART HI

(15 minutes)Directions: Continue as in the preceding part.

1. r2 means 1-1 2r 1-2 r + 2

1-4 r(r)1-5 Vr . . . . . . . . . . . . . -1( )

7. The number of feet in n yards is

7-2 3n 7—3 n + 3

7-4

7-5

3n12n 7( )

2. What is the sum of — 2k3 and — 12ft5?2-12-22-32-42—5

10ft5— 14ft3- 14ft8

24ft3 24ft9 2( )

3. If 3r — 2s = 10, what is the value of s when r equals 6?J - l3-2

3—3

3-43-5

14 - 8

i46 . 3( )

4. The product of 4V5 and 5V3 is 4-1 9 V54-2 9 V64-3 2 0 ^4-4 20\^64-5 20(V2 + V3) ....................................4( )

5. 2r divided by ( — r) equals 5-1 - 25-2 25-3 - 2 rJ5-4 r5-5 - r ......................... 5( )

6. 3ft + 5 — (— 4ft) — 6ft equals 6-1 ft + 5 6-2 - 7ft + 56-3 13ft+ 56-4 5 f t+ 56-5 27ft+ 5 . . . 6( )

8. If ^ = 24, what does t equal?

8-2 6 8-3 8<?-4 20 <?*~5 96 «( )

9. Find the value of xn + 5 when x = 2 and n = 3.

«-> 49-2 11 9-3 13 9-4 149-5 5 + V2 . . . ......................... .... 9( )

10. If n2 = 64c, one value of n is 70-1 32Vc J0-2 16VcJ0-3 8Vc JO-4 8cJ0-5 32c . . . . . . . . . . . . 10( )

11. “The sum of two numbers decreased by the square of the difference between the numbers,” may be expressed algebraically as JJ-1 a + b - (a2 - b'-)11-2 a + b - (a - b)2

11-3

11-411-5

a + b (a - b)2 a + b — a2b2 a + b — a2 — b2 H( )


- 7 -“The second power of t divided by the product of t by itself" equals 12- \ 1

0 2 ■ t t 2 12t

12-2

12-3

12-5 i i ................................................ 12( )

V35 ,equals

5V7V5o

7 V5V7

7 .

id -iid-2id-3id-4id-5

14. If c< + 4 = of, then t equals 414-1

14-2

14-3

14-4

a — c c — a

4 4

a — c

14-5 ^ - 4 14( )

15. The product of m1/4 and m1/4 is id-1 m1/is id-2 m1'8 id-3 2m1/2 id-4 2m1'4id-5 m,/2 . . . . . . . . . 15( )

16. Which one of the following is true?

16—3 — *r* h — — a a, , . a . . 1 16-4 t- — — = a o o

16-5 a 4-^ = b . .............................16( )

17. If z = ---- , then one value of d is

iZ-1 4nz — c

£

j7 - 3 V^-li7-4 Vcnz

^7-5 ( y ) 2 ........................................ 17( )

18. The roots of the equation x2 + 5x — 24 = 0 are id—1 — 4 and 618-2 - 2 and 1218-3 4 and 618—4 3 and — 818—5 — 3 and 8 ................................ 18( )

19. — Z~2d— eflua siP -1 10d4 + 2 19-2 - 5d4 + 219-3 - 5d% - 4d19-4 - 5d* + 219-5 Sd* — 2 . . ........................... 19( )

2 d + 3 Sd - 1 4 2

20-1 - 8d + 1 - 8d + 520-2

20-3

20-4

20-5

equals

4- 3 d + 2

6- 3 d + 4

2- 8d + 1 20( )

21. Vo* equals 2 i- l a5/s 2i-2 a* 2i-3 au 2i-4 a40 2i-5 a8'8 21( )

22- i f M = ^ - mequa,s 22-1 1 22-2 0 22-3 822-4 - 422-5 - 8 ........................ 22( )


— &

4 a r C D

2 -1

{Items 23 through 26 refer to the graph above.)

23. Which one of the following points lies on line BS'?23-1 (2, 1)ZJ-2 (2 ,4 )23—3 (— 3, — 2)2J-4 (4 ;2 )

23-S ( z l , - l ) .................... ... • 23( )

25. The solution of the pair of simultaneous equations represented by lines 4 4 ' and BE'Is 25-1 x = 0, y = 0 25—2 x = — 2, y = — 3 25—3 x = 3, y = 2 25-4 x = 2, y = — 325-5 x = — 3, y = 2 ........................ 25( )

For line 2)1)', the value of x when 26. Which line is the graph of the equationy = 2 is ' 2x — y = 4?24-1 1 26-1 AA' • '24-2 2 26-2 SB'24-3 3 26-3 CC'24-4 - 4 . . . . . 26-4 2)2)'24-5 26-5 E E ' ............................................ 26(

If you finish before the time is up, you may go back and work on any part.

Number wrong012

316

71

10

111

14

151

18

191

22

231

26

271

30

311

34

351

38

391

42

431

46

471

+Amount to be subtracted 0 1 2 3 4 5 6 7 8 9 10 11 12

Number right.

Subtract (See table above)

Raw Score = Difference.

Scaled Score . (See table on key)

[ 1 7 1 1 . n 45 -1 0 C5

a 3 9 0 0 1 0 0 1 2 8 5 5 38b

/ a2_

9 7 9 1 9 4 5

S E L L R A K S T A N O A R D I Z A T I O N O F A F I N A L T

I N S E R T B O O K IV",\STER C A R D F A C E UP IN FRONT SLO T O F S .R . PU N C H

MASTER CARDUNIVERSITY OF ARIZONA

LIBRARY

• \

f 6%^

1 / I

the standardization of a final test in elementary algebra

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