EVALUATTON PROBLEMS RELATING TO PLACEMENT OF FRESHMAN ENGiNEERING STUDENTS IN MATHEMATKCS Thosil far the Degree cf M. A. MICHIGAN STATE COLLEGE jane Albee 194i umwwi Michigan .State University MSU LIBRARIES ” ' RETURNING MATERIALS: Piece in book drop to remove this checkout from your record. FINES wil] be charged if book is returned after the date stamped below. EVALUATION PROBLEMS RELATING TO PLACEMENT OF FRESHMAN ENGINEERING STUDENTS IN MATHEMATICS by Jane Albee A THESIS Submitted to the Graduate School of Michigan State College of Agriculture and Applied Science in partial fulfilment of the requirements for the degree of MASTER OF ARTS Department of Mathematics 1941 INDEX Section 1. The History and Background of the Problem page at Michigan State College and at Other Colleges 1 Section 2. Questions Arising out of the Results of the Placement Test Given in September, 1940 at Michigan State College 5 Section 3. Types of Errors Made at Michigan State College and at Other Colleges 15 Section 4. Statistical Analysis of the Test Given September, 1940 ' 25 Section 5. A New Test 28 Bibliography Appendix (1) Evaluation Problems Relating to Placement of Freshman Engineering Students in Mathematics Section 1. The History and Background of the Problem at Michigan Statg_College and at Other Colleges. In September, 1940 during the Freshman Week all the engineering students of the freshman class at Michigwn State College were given a placement test in algebra,1. but no formal use was made of the results during this school year. Previously it had seemed evident that all the freshmen entering Michigan State College did not have equal backgrounds in mathematical preparation, even though offering similar entrance credits from the various high schools. As an experiment this algebra test was employed with the purpose of seeing if it could be used as a basis for the determination of students weak in mathematics. This giving of placement tests to incoming freshmen is not original with Michigan State College. In the tall of 1938 such tests were given at the University of North Carolina and at Washington and Jefferson College. sThese" were used to aid Mn dividing the students into three groups. ‘At Washington and Jefferson College, a small college for men with.an entering class of about 200, these who had high grades in four years of mathematics in preparatory school and also had a high score on the m Cooperative Intermediate Algebra Tat oFThe American Council on Education, form 1957, which includes quadratics and beyond. (2) placement test were encouraged to take the course in analytic geometry and calculus. There was no compulsion about this, the students made their own decisions, and the result was that about a fourth of the class enrolled in the advanced course with a high degree of success. Those Whose high school records were low or who did not score very high on the placement test were assigned to a mathematics course at a certain period. At the first meeting these students were given the choice of taking the regular freshman algebra course for college credit or of electing a semester course in high school algebra including logarithms and numerical trigonometry without college credit. This course was"designed to remedy faulty preparation and enable the student to carry for credit the regular freshman course in trigonometry during the second semester". This would mean that the regular first semester college algebra course would be made up in summer school or during later years. These freshman courses in mathematics were all scheduled at the same hour so that transfers could be made during the first month without a change of the student's whole program. "At end of the first month students failing in the ad- vances course are automatically shifted back to the reg- ular course, and likewise those failing in the regular course go back to the preparatorybourse. No further changes are then permitted." The results Of this pro- cedure were a marked decrease in the number of failures (5) in mathematics. On the other hand, the University of North Carolina with an entering class of 850 used a similar plan of a high school algebra course without credit which also resulted in a decrease in the number of failures. How- ever, at this University the poorer students in the reg- ular courses were given five hours of work with only three hours of credit. This works very successfully, but not if pursued in too many subjects. A student may be poor in several subjects and be assigned so many extra no- credit hours that he has not sufficient time for adequate preparation.2. Two other colleges, namely Purdue University and Wittenberg College give tests to their freshman mathe— matics students, although the use made of these tests was not stated in the reports of the results. At Purdue a diagnostic test is given to students entering first semester mathematics courses. This test was quite diffe erent from the one given at Michigan State College.3. At Wittenberg College an achievement test in algebra was given to students who had studied high school algebra for an average of 2% semesters and had completed a third Q. Dorwart, H.‘ET of Washington and jefferson College, w"Comments on the North Carolina Program in Freshman Math- ematics." The American Mathematical Monthly, January, 1941. 3. Keller, M.W., Shreve, D.R., and Remmers, H. H. of Pur- due University, "Diagnostic Testing PrOgram in Purdue University." The American Mathematical Monthly, January, 1941. (4) 4. of the first year's course in college algebra. Thus other colleges have recognized this problem of student differences in mathematical knowledge. Both washington and Jefferson College and the University of North Carolina have divided their students on the basis of a placement test and the high school record, and the results are satisfactory for these institutions. At Michigan State College only the engineering students are involved in this problem, whereas these two other colleges used their plan for all their students including those in the liberal arts field. At Purdue and Wittenberg the tests were not of the type to designate the poorer students in order to isolate them for remedial work, and the final use 'of their tests is not stated with any specific benefits. These facts and also the lack of details accompanying the results from whese other institutions seem to necessitate a complete and independent study of the problem at Mich- igan State College 47* Arnold, H.J. of Wittenberg college, TAbilities and Disabilities of College Students in Elementary Algebra". JOurnal of Educational Research, April, 1931. (5) Section 2. Questions Arising out of the Results of the Placement Test Given in September, 1940 at Michigan State College. After the results of the test given at Michigan State College in September, 1940 were compiled, several questions arose in regard to their future usefulness. First to be considered is the question whether this test can be depended upon to predict success in first term mathematics for engineers at Michigan State College, and if it is equally valuable for the students who take college algebra, trigonometry, and solid geometry. Second, if this test does not seem to predict with and degree of accuracy, what is the difficulty? Third, are the three different parts of the test of equal importance? Fourth, does the test differentiate to any great extent between the three groups of engineers, those signing up for the three different mathematics courses? Fifth, can we pick out the students who really have no need for college algebra and allow them to go into other mathematics? Sixth, can we pick out the students supposedly ready for college algebra who should take a review course of high school mathematics, without credit, instead? Seventh, would an ea81er test have given approximately the same distribution of grades? This last would mean that an easier test could be used that possibly would require less time to administer and would show the same results. (6) In answer to the first question as to whether the test can be used to predict success, it is found that the scores on the tests when compared with the final grades Of the first term of college mathematics do not shww that one can determine who will fail and who will succeed without making some errors. Thus some who failed re— ceived the same mark on this test as some who received D, some who received C, some who received B, and even a few who received A. Perhaps some of the A students didn't work up to their capacity on the test. On the other hand, some tt the F students who stood near the top in rank of all the F students may have failed their college mathe- matics course because of other reasons than that they did not have a thorough foundation in the fundamental prin— ciples of algebra. Such factors as illness, too heavy a schedule when combined with working for board or room or both, several years between high school and college, or some other contributing cause might have resulted in failure to do satisfactorily the work of the course. Be that as it may, the A group when considered as a whole was way ahead of the other four grade groups, the B group was as a whole ahead of the C, D, and F groups, and the C, D, and F groups when considered as a whole, each was lower than the one ahead of it. This was true for all three groups, that is those taking the three different courses in mathematics. This of course is not surprising and indeed.m1ght be said to be exactly (7) what would be expected. The second point to be considered is that of deter- mining the reasons for the test not being an accurate basis for prediction. Undoubtedly one reason is due to the time element entering into the picture. In the one and a half hours required to give this test, it is almost impossible to accurately rate these students so as to de- termine whether or not their ability to work algebraic problems is sufficient for college mathematics. As with.most tests of this type, the time specified by the authors does not permit any of them to finish all the examples in any one part. There are quite a few problems on the test which in themselves are easy to work but which do require some time for the student to figure out. This means that time is consumed on them, and some of the other types which are more difficult or involve diff- erent principles are never reached. On the other hand, there are too many hard questions that take up time and do not test for those fundamental processes to be dis- cussed later that are necessary for success in college mathematics. Such an example is the following problem which is number 26 in Part I: “V?-5x-61-1=x By the time the student has worked this problem he has spent a disproportionate amount of time on it. In this part of the test, those who tried this, whether they succeeded or not, did not do any of the following prob- (8) lems in this section of the test. Those who omitted it did get on to try several of the problems that followed. Still another cause for inaccurate prediction based on this test is that the answers alone do not make up a valid basis for such an evaluation of‘ability. In the hurry and excitement of taking this test, mistakes in arithmetic might have been made even though the fundaé mental algebraic facts were known.‘ This teat does not allow any space on it for calculations, and so the ex- aminer does not know in what manner the student missed the example. One way to test for arithmetic errors would be to have an additional section preceding this test devoted to arithmetic alone, and if the student made no errors here, he would presumably be less likely to make arithmetic errors in his algebra problems, so that his mistakes in this case could be considered due to a lack of knowledge of algebra. Another method of ‘finding errors caused by poor arithmetic would be to give another test that would require all the work to be done on the test paper and that would provide the necessary space for computation so that the examiner could see wherein the mistakes were made. This testing for mistakes in ariflmmetic would give another basis for determining causes of failures in college mathematics. The third question as to the values of the three parts of the test is in part a matter of Opinion. As the authors of the test specified different time limits (9) for the different parts, it must have been believed by them that the parts vary in importance. Part I is given 'the most time, 40 minutes, and with.reason. It deals with the fundamental processes of algebra. Part II, with 25 minutes, is a bit of application of this algebra. This section of the test has the so-called "story problems" Ihihh.test the ability of the student to set up the prob- lem and solve for one or more unknowns with various bits of information given and which require the use of algebra. Part III, also with 26 minutes, is a group of 42 multiple choice examples. The student is supposed to pick out the right one, but here errors are made when the student sees before him the right answer with many combinations that include the common errors. It is my belief that if a blank weredeft, many students would put in the correct answer where they pick out a wrong one in their haste, overlooking a negative sign or some other equally easy- to-make error. Not one of the students taking this test finished.more than two-thirds of Part III unless he omitted many of the ones in this section. Thus it is definitely too long. Of course most tests of this type are made up with the idea of keeping the student busy the whole time so that no one should finish the test, but with a third of it undone, it seems to me to be too long. In Part III some of the same algebraic fundamentals as are tested in Part I appear, though in a little differ- ent form. The correlation coefficients between the scores (10) of the three parts are not high enough to draw any con- clusions. For the158 students enrolling in college alge- bra the first term, the correlation between Part I and Part II is .59; between Parts II and_III it is also .59; but between Parts I and III it is .79. This means that. if a student does well on Part I he is more apt to do equally well on Part III than on Part II. For this same group of students, the correlation coefficients between the three parts of the test and the first term's grades are for Part I .71, for Part II .49, and for Part III .66. This indicates that Part I is the best single indication for success in college algebra and that Part III is almost as good, but that Part II which deals wholly with the so- called story problems is the least accurate. The ques- tion arises in my mind as to why the first and last sec- tions of the test couldn't be combined, that is the good portions of the first part and of the last part combined, and a better result might be achieved. The fourth point is to determine wherein the test differentiates between the three groups of engineering freshmen. It must be remembered that in September, 1940 when a student of the engineering division signed up for his first mathematics course at Michigan State College, his high school credits in mathematics were the basis upon which the selection was made. Thus if he came to college having had solid geometry and trigonometry he took the course in college algebra required of all en- (11) gineering students. If he had.had either solid geometry or trigonometry in high school, but not both, he took the one he hadn't had the first term and then followed it with college algebra the second.term. If he had had neither of these he took solid geometry the first term, trigonometry the second term and college algebra the third term. Regardless of the high school mathematics credits offered , all these students took the same test. Naturally those having a better foundation in mathematics did better on this test than the others when considered as a group. The mean raw score on this test as given in September, 1940 for all those enrolled in college algebra was 41.78, for those who enrolled in trigonometry it was 26.59, and for those who enrolled in solid geometry it was 27.27. It is interesting to note that those who enrolled in solid geometry or in trigonometry did about the same on this test, but were much lower than those who signed up for college algebra. However, it must not be overlooked that individuals in each.of the three groups had the same scores. In the algebra group the raw scores ran from 10 to 87, in the trigonometry group they were from 10 to 61, and in the solid geometry group they were from. 8 to 82. Thus it is evident that this test shows no sharp differentiation between the three groups of students with such differing preparation backgrounds. In the fifth question, as to whether this test could be used to determine what students do not need college (12) algebra and should be allowed to start analytic geometry, the conclusion drawn after careful inspection of the test and its results must be in the negative. If the students ‘Iere notified before coming down to East Lansing that they were to be given such a test, they could review and come prepared to take the test with a serious purpose in view. As it is now, the students do not really have a chance to show how much they know as the time limit is insufficient to allow them to get to the really difficult examples. Another point not to be forgotten is that the college algebra students who received an A the first term did not all know the material before taking the course. 80mm perhaps might have known the work beforehand, but there is no definite clue to this from.these tests, be- cause some of those standing quite high.on the test did receive low marks. To the sixth question as to whether or not we can pick out the students who, when conlidered on the basis “of high school credits, are supposedly ready for college algebre but who should actually take a review course of high school mathematics without credit before tackling the college algebra, the answer is yes, with certain 'reservations. Of this group of students who took college algebra the first term, those who received an A or a B for the term's work ranged on this test all the way from 35 to 87. Those who received an F in the same course ranged from 10 to 47 on this test. Thus some who re- (13) ceived equal grades on this test got an A or a B or an F. This would lead one to believe that one could not use this as a basis for deciding who should be advised to take another course before attempting college algebra. However, it is a question whether these students took this test seriously. If thdy were told that this test would be used as a basis for such a decision, they would have worked to the best of their abilities, and the re- sults would have been different. many studtnts would be willing to take a review course if they understood that it would result in greater success a term later in college algebra and in all the courses of mathematics that follow. Certainly below a certain point on this test, it could be safely said that a student would find it difficult to do the work. Take for an example the arbitrarily chosen raw score of thirty on this test. Of the 49 who received a G for the first term's work in college algebra only 11 of than got below 50 on this test. Of the 17 who received a D, 8 were below 50. Of the 22 who received F all but four got below 30. If all of these receiving a score be- 1ow 30 had been given a review course in mathematics, with- out a doubt they would all have fared better when they did try college algebra a term later. Another criterien that might be used in conjunction with such a chosen low score would be the high school record. With a poor high school rating in mathematics and a low score on this test con- sidered together as a basis for making up the enrollment (14) for the review course,:much remedial work could be done for these students that would certainly help them.in the college mathematics courses. This would help to avert later failures also, an a better foundation in the fun- damentals will be a help in the more advanced courses of mathematics. The seventh question as to whether the same dis- tribution of grades would have resulted if an easier test had been given can be answered tentatively by taking an arbitrarily chosen group of easy questions from the tesg and figuring the scores on them.alone. To do this, twenty ' questions from Part I were chosen. They were the first fifteen, the 17th, 18th, 20th, 24th, and 25th. The re- sults of this tabulation were almost the same as the original one, with the A's having the highest scores and the B's next, and so on down through.the F's. A study of the rank correlation coefficients between the original test and this easier one to determine whether each.student would be likely to come out with the same rank in his group was made, and the results are tabulated on page vii of the appendix. All of these are .6 or higher with the exception of 2, one with .546 and the other .392. This dOes not show that the easier test would rank the students in the same order, though there is some tendency towards this. A reason for this is that the students spent some of their time on other problems. An easier test if handled in the right fashion might give the same results. (15) Section 5. Types of Errors Made at nichigan State College and at Other Colleges. In analyzing these tests it is interesting to note the types of errors made by these freshmen. The students who failed college algebra the first term at Michigan State College showed a certain amount of lack of know- ledge in the fundamentals of algebra on this test. They were definitely unsuccessful in the handling of exponents. This even bothers some of the D, C, and even B students also. Certainly one cannot succeed in college algebra if this is not mastered. They also all failed to do the following problem in reducing a fraction by factoring and canceling: __16xy - 4y 16x2 - ye A great many of them canceldd, and didn't bother to factor first. They canceled out the 16x from the numerator and denominator, crossed out the y2 on the end in the numer- ator and denominator, and for an answer got y - 4 over 1, and presumably they thought they had done it correctly. As so many of them did it this same way, this is evidence enough that this was not well learned in the high school courses that they took. All of these F students also failed on the four problems dealing with the interpreta- tion of a graph,and they were all unable to use logar- ithms correctly or to use the binomial expansion. These (16) three types of errors in themselves are not as important as the ones mentioned before. They would not be so apt to cause one to fail a course in college algebra, but a knowledge of them might contribute to ones ability to do the work better and.more easily and to get a higher grade. In considering the rest of the errors made by the F students, we find that out of the 22 who took this test and received a failing grade the first term in college algebra, 20 could not simplify 27 raised to the 4/5 power. Many obtained 36 for an answer merely by multi- plying 27 by four-thirds instead of raising it to that power. Given the equation kx - 6y 2 3 whose graph passes through the point (5,5), 19 of them couldn't find the coefficient of x which is here denoted by k. 18 of the 22 had trouble with factoring, which certainly is a necessary skill if one is to do college algebra. 16 had difficulty solving "story problems", and the same number were unable to form an equation when given its roots. 15 of these F students could not reduce the square root of 80 to the simplest radical form, and a like number could not solve simple equations for one unknown. If they cannot solve for one unknown, what will they do when they have two, three, or more unknowns? 15 also could not find the product of two numbers involv- ing radicals. 14 out of the 22 could not solve the problem on the arithmetic series and only three of them (l7: solved it correctly, while the rest of them were so slow that they didn't get this far. 13 of the 22 were unsuccess- ful in finding the square root of a six digit number. If these engineers learn to use the slide rule, they may get by without mastering this, but what will they use to check their work when a check is necessary? Half of the number were unable to use the formula for solving quadratic equations and only one obtained thecorrect answer while the others didn't get this far on the test. Half this number who failed were unable to do the following problem: If a = 3 and b = -2, what is the numerical value of a5 - b5 - (a - b)5? Four of them said it equals 0 indicating that they did not know the expansion of (a - b)5, while the others obtained various and inexplicable wrong answers. Nine missed both ex- amples based on the fundamentals of trigonometry, but three of them got one right and ten got both right. This might well be due to the fact that trigonometry was the last mathematics studied in high school. Several made errors in taking the square toot of 36xl6. The majority of these errors involved extracting the square root of the exponent, thus showing a complete lack of understanding of the laws of exponents. A few got the exponent correct, but forgot to get the square root of the constant coefficient. This might be due entirely I to carelessness. In another example the student is asked to give the negative root of an equation which (18 is set equal to 0. Some gave the factor that had a negative sign in it, which shows that this mistake is due to a deficiency in their mathematical wocabulary. Other errors are not so easy to detect, but if a system for the test could be worked out whereby the work were ‘all on the paper, they could be isolated. From the above accounting of their errors it would appear that these students failed their first term of mathematics at Michigan State College because their foundation in high schohl algebra was so shaky that they could not do the advanced work required in college algebra, which.after all does use the fundamental principles learned in high school algebra. Bor purposes of comparison, let us consider the A students. Those who received an A for the first term of college algebra at Michigan State College made some of the same miStakes as those who got an F, but as a group a smaller percent of them.made these errors. Although none of the A students got all six of the problems on exponents, 40% of them got 5 out of the 6, and 85% of them.didn't miss more than 5, whereas the F students came out with 50% getting 1 right, and only 22.7% getting two of them right. 0n reducing the fraction 16xy - 4y2 over 16x2 - ye only 50% of the A students missed it whereas the F students missed it 100%. On interpreting the graph correctly 20% of the A's got all four of them right, and only 50% of them missed all four of them. Not .- *D a. - o .u. x ' m -.o as. 0' c“! m u I I U Egg». uni-rel. . if} _[I It \E’EL. \ Fl». \ ' (19) one of the F students got these four all right and.more than half of them, 54.5% failed to do even one of them. For the logarithm examples, 40% of the A's and 0%of the F's did them all satisfactorily. Those doing correctly at least three out of the five logarithm ones were 60% for the A's and 4.5% for the F's. Neither group had much success with the binomial expansion. 30% of the A's tried it but got the wrong constant coefficients, and 100% of the F's missed it entirely. The problem quoted above involving a radical sign and which required the student to solve for x by putting the items not under the radical sign on one side of the equal sign and then squaring and solving, was done successfully by 55% of the A's where the F's scored a miss 100%. It is interesting for the purpose of comparison to consider the errors made by students at other institutions. At Purdue fewer than 10% of all those in the first semester of college mathematics could solve a quadratic equation by the use of the quadratic formula, while at Michigan State College 19.57% of the engineering students taking college algebra could do the same type of problem. On these tests given by Purdue University many of the errors were due to a lack of "accuracy in manipulation". Some of the students understood the principles but were in- accurate in computations. The conclusion of those who administered the test was that this could be corrected (20) 5. by more drill work in the secondary schools. In comparing the errors made by the Wittenberg students with those made by the freshmen at Michigan State, the group that compares the most closely with the group at Wittenberg is the one made up of those students ' who entered college algebra their first term at Michigan State College. Even then it must be remembered that the Wittenberg group had completed a third of the first year's course. At Wittenberg 81% made errors in four addition and subtraction problems that involved fractions. 0n the test at M.S.C. only two problems were of this type, and one was missed by 70% of the group, and the other was missed by 59% of them. 86% at Wittenberg failed to solve a rather simple problem in addition of radicals while at Michigan State College only 10% missed a prob- lem of similar type. 44% failed.to multiply radicals correctly at Wittenberg, whereas 47% failed this at Michigan State. 58% of the Wittenberg students omitted half of the eight graph problems, and at Michigan State College 57% omitted half of the four graph problems. Thus it can be seen that students at these two institu- tions missed similar problems, though the percentages vary a bit. The following summary was made of the errors on the test given their students by those who administered the test at Wittenberg College: 1. Errors in clearing flactions apparently due to 5. ‘Keller, MSW?) Shreve, D.R., and Remmers, Ibid (21) inability to find the L.C.D. 2. Errors due to failure to heed changes of signs in combining and removing parentheses. 5. Incorrect reduction in the case of similar frac- tions and fractional answers. 4. Failure to invert the divisor. 5. Errors in removal of parentheses 6. Inability to formulate correct equations in connection with problem solving. 7. Inability to read comparatively simple graphs. 8. Marked inability to indicate relationship in terms of algebraic symbols was also found.6. 0n the test given at Michigan State, in Part I examples 2, 10, and 24 and in Part III example 10 involve the clearing of fractions. These were missed by 54%, 44%, 70%, and 59% respectively of the group that took college algebra the first term. Thus this point concerning errors made in clearing of fractions applies to the Michigan State group as well as to the Wittenberg group. Points 2 and 5 which deal with parentheses errors can not be applied to the group at Michigan State College as ex- amples of this type were not given to them. Point three and four may hold for the students at the Michigan insti- tution, but as none of the work is shown on the paper no definite checkup can be made. Point six compares to Part II as given to the Michigan State freshmen, and here many 6. [1710111, H.J., Iblde (22) of the students do show that they are not capable of setting up equations to solve this story-type of problem. All of Part II deals with this type of problem, and by making a close inspection of the detailed report of the number of the college algebra students getting these prob- lems right as is shown in the Appendix on pages v and vi, it can be readily seen that this is a common error made by many of the students at Michigan State College. Point seven about the graphs has already been discussed. Point eight compares to examples l4 and 22 in Part III of the test given at Michigan State College. 40% of the group missed the first one and 54% missed the second, which shows that they are not all well acquainted with the uSe of algebraic symbols. It is interesting to note that at other colleges, where tests of a similar nature have been given, students have made the same type of errors as those found on the tests given at Michigan State College. (23) Section 4. Statistical Analysis of the Test Given September, 1940 In considering the future use of a placement test at Michigan State College, it seemed eXpedient to invest- igate the correlations between its scores and the scores ,Of the psychological test which are given to every fresh- man as well as between each of these two and the first termfls grades. It was desirable to discover if the psy- cholOgical test adds to the information of the mathe- matical ability of the students. To ascertain this a group of 50 were chosen from the total ef 158 of the college algebra group. Some of these 50 received A, B, C, D, and F in college algebra at the end of the first term. The number of tests selected from.each grade group was proportional to the total number in each grade group. Thus from the 20 A students 7 were taken, from the 50 B students 11 were taken, from the 49 0 students 18 were taken, from the 17 D students 6 were chosen, and from the 22 F students 8 were taken. Within the grade group some papers with high schres on the placement test, some with low scores, and some with scores in between these were selected, so that as nearly as possible this would be a good representative group. The letter grades were assigned numerical values to facilitate the computation. A was 11, A- 10, B+ 9, B 8, B- 7 and so on down to D-1 and F -. Upon computing the correlation coefficients for these fifty papers, it was discovered that there was a much (24) higher correlation coefficient between the first term's grade and the score on the algebra placement test than there was between the psychological test score and the first term grade. The results are as follows: between first term's grade and psychological test .174 between first term's grade and algebra test .702 between algebra test and psychological test .264 The multiple correlation coefficient for this group of fifty was .7015 and the eXperimental error in predicting the first term's grade from the psychological test and the placement test was found to be 2.55. This means that if a grade of C were predicted, the probable range where the mark would fall would be from about D to about B. These statistics show that one's attainment in college mathe- matics is more dependent upon the mathematical prepara- tion in the secondary school than upon one's relative in- telligence as shown in the psychological test. As the correlation coefficient between the psychological test scores and the first term's grades was so low and because the psychological test results are not available at the beginning of the term, any further investigation involving the use of the psychological test as an aid in foretelling the possible accomplishments of students in mathematics was abandoned. An analysis of covariance was made of the results of the placement test in mathematics. The regression among the three groups, those signing up for college algebra, (25) for trigonometry, and for solid geometry, was signifi- cantly different from.the regression within groups. This arises from the fact that the three original groups differ in their knowledge of algebra, while the final grades for each group follows nearly the same distribution. The mean score on this test for the college algebra group was 41.78, for the trigonometry group it was 26.59, and for the solid geometry group it was 27.27. The mean grades for the three groups are respectively 1.065,.946, and 1.080 where a grade of 1 is equivalent to C, and 2 to B. Thus this appears to be due to the fact that originally the three groups differed greatly in average test score, and while the higher averages tend to be accompanied by slightly higher grades, a great difference in test average is needed to produce much change in average grade. A por- tion of the usual covariance technique yields a test of significance of the correlation and regression coefficients. The size of the correlations here and the size of the groups renders this test unimportant since we are certain of the significance in advance. The regression coefficients and the correlation coefficients of the various groups are quite different. For the algebra, trigonometry, and solid geometry groups the regression coefficients are .054, .048, and .055 respectively and the correlation coefficients are .71, .47, and .48 respectively. These differences are not found to be significant. Common sense leads one to dis- count this test to some extent for the fact that only three (26) groups are concerned and hence only two degrees of freedom are available which.makes it necessary to have extreme differences if it is to be significant. The size of the three correlations and regressions indicates that, as is to be expected, the test is far more effective in the college algebra group than in the others. There is sufficient difference in the results despite the lack of significance, to suggest that the groups should be treated separately. Moreover, while the correlations of the test scores with trigonometry and solid geometry grades are sizeable, these correlations are probably due to two factors, the use of algebra in these courses and the mathematics ability common to all subjects rather than the fact that algebra is taught in . these courses. This situation differentiates these two groups from the college algebra group. In connection with this analysis of covariance a test was made for the linear regression of the algebra group, and it turned out that the relationship of test scores to term grades was linear. The standard error of estimate for the predicting of the first term's grade in college algebra based on this test was .78. That is to say, if a grade were predicted, the standard error would only be .78 of one letter grade, as in this analysis grades were given a rating of 2 for A, 1 for B, 0 for C, -l for D, and -2 for F. To determine if the dispersion about this pre- dicting line was similar at all parts, the lowest 15 (2'7) scores of the algebra group with the first term's grades were plotted about this predicting line. The sum of the squares of the distances from.the predicting line divided by 15, the number in the group, was .95. For a similar number at the very highest end of the scores it Was .75, and for the fifteen at the very center of the distribution of scores on this test it was .60. Thus a score at the middle of the distribution scale would predict more accurately the first term's grade than at the bottom or at the top, although these two are not too far off. The majority of the test scores which are around the middle of the dispersion thus can bu used to predict success with a fair degree of accuracy. (28) Section 5. A New Test Supposedly all the students enrolled in solid geometry and trigonometry have had all the algebra :necessary for college algebra, but their scores are so Inuch lower than the algebra group that it is doubtful if the difference can be explained in terms of the few items involving trigonometry or solid geometry. It would, however, be interesting to see how much the students im- proved in algebra by taking solid geometry or trigonometry. However, the more immediate concern is with the use of the test for predicting success or difficulty in alge- bra. Hence we turn to a more detailed consideration of this phase of the problem. It is evident that the test given in September, 1940 does not produce good results. The best example of a test that could be used to determine the weak students in mathematics that has been used successfully is the Mathe- matics Attainment Test given by the College Entrance Exam- ination Board. It is an achievement test in the field of mathematics so designed that three different levels are tested by the same examination. The point at which the student starts is determined on the basis of his secondary shhool preparation. It is arranged according to a ladder scale so that each problem involves a bit more mathematical knowledge. The test is divided into two parts, one section having the essay type of questions and the other the objective type. "In each of these two parts the time (29) has been apportioned so that most students will stop working because of lack of ability rather than lack of time...The test is a test of ability or power rather than of speed." Those marking these examinations were unanimously of the opinion that on this test "a thor- ough foundation was more valuable to a candidate than was a little knowledge on a wider range of topics." The results of this test are sent to the colleges with an explanation so that their value can be interpreted corr- ectly. Thus a student with four years of high school mathematics is expected to stand far ahead of the student who has had only two years of it. The use of these tests is for admission to the various colleges and also for an aid in helping the students to plan their college work. A student who intended to major in literature would not need such a high score for success in college as would a student who intended to major in mathematics, physics, or some other science. It is felt by the colleges using these examinations that this is undoubtedly worthwhile for the sake of the future work of the student as well as for the sake of averting failures.7. Later, with the cooperation of five of the colleges using these tests for entrance qualifying, a study was made of the correlation coefficients between these tests and the records of the students in college courses of mathematics. The highest 7. College Entrance Examination Board, Research Bulletin number 7, Report on the Mathematics Attainment Test of June, 1956. Deaember 1956, New York, New York. (50) of these correlation coefficients are in the group having the greatest amount of secondary school mathematics. They areror one university as follows: for the one-semester course of coordinate geometry for freshmen .48, for the one-semester course of differential calculus for freshmen .67, for the one-semester course of integral calculus for sophomores .78, and for the one—semester aourse of differ- ential equations for sophomores .69. This type of test was given first in 1956, and the College Entrance Emamin- ation Board feels that it can be of great use to the colleges and_universities that require these tests from their students.8. Although this test was not used for the purpose er placement in college mathematics, the results show that it could be used for this purpose. Hewever, it is not a practicable plan for Michigan Statt College if considered on no other basis than the expense involved. The College Entdance Examination Board charges a fee of five dollars to every student taking these exam» inations. This includes the fee for correcting the tests, but they make no provision for individual institutions to use the tests and correct them at a lower price. Thus the expense that would result from the use of these tests is practically prohibitive. A new test could be evolved for use at Michigan State College using the good points of the test used in 1940 8. Cellege Entrance Examination Board, General Report on the Mathematics Attainment Test,_l958. ‘NewfiYork, New (51) with the necessary parts that do not appear in it added. If the items tested in Part I and Part III were combined so that the student would have the Opportunity to show the details of his calculations, a better result would be obtained. The possibility of including a section on arithmetic should not be neglected. If these students 'were notified beforehand of this test and were told that it would be used as a basis for selection of their first term's course in mathematics, they would all do their 'best on the test, and more accurate conclusions would be obtained. Such a test for future freshman engineering students at Michigan State College should include those fundamental concepts and processes of mathematics that are believed to be essential for success in college mathematics. With this in mind the following list of necessary principles and skills was compiled: 1. Removing parentheses preceded by a minus sign re- quiring the change of sign of all terms that were within the parentheses. 2. Multiplying or dividing the numerator and the denominator of a fraction by the same term without changing the value of the fraction. 5. The addition and subtraction of fractions, in- volving the L.C.D. 4. The multiplication and division of fractions, including those necessitating factoring to reduce (52) to lowest terms. 5. The solving of simple equations, involving one unknown. 6. The solving of a system of two linear equations in two unknowns by substitution, by addition, by subtaaction, and by the graph. r (‘2 7. Reading graphs and being able to pick out the line on a graph associated with.an equation. 8. The use of formulas for areas, volumes, etc. 9. Simplification, addition, subtraction, and Imursn; multiplication of radicals. 10. Factoring the difference of two squares, the product of two binomials, the square of a binomial, and the product of a monomial and a polynomial. 11. Multiplication, division, extracting roots, and raising to a power of a monomial and a binomial. 12. Imaginary numbers. 15. Solving the quadratic equation by factoring, by completing the square, and by the quadratic formula. 14. The following notation: (1,0) for a point, i for 1F5: 0, ( ), [ 3 , i i , ( )% (any fractional exponents), ()n.-.+.+.’V'.%"‘. 15. Terminology, that is mathematical vocabulary, including sum, difference, product, quotient, remainder, coefficient, root, to factor, axis, raise to a power, invert, inversion, ratio, square, cube, quadratic, monomial, binomial, trinomial, and polynomial. (33) Most of these fundamentals are evidently necessary for good work in college mathematics, with the possible exception of the item concerning graphs. This knowledge is not so important for the college algebra classes, but it certainly is quite desirable for the later course in analytic geometry which has much material concerning graphs. Each of these points has been carefully considered in constructing a new test. The examples are in the order of difficulty, so as not to discourage the students at the start. Examples in arithmetic are included as one more check on reasons for not being able to do college mathematics successfully. For rapid scoring this test provides a prOper place on the right hand side of the sheet for the required answers. There is also space provided for all necessary computations so that if a closer study is made of these tests the examiner can readily find the causes of mistakes. The problems are grouped somewhat according to the type, so that the students' minds will more readily follow from one prob- 1em to the next. This also has the advantage of requiring fewer directions and so less time is spent reading what to do next instead of doing it. Each problem tests for some specific skill or process, and so the Optimum con- ditions would be for each.student to work, or at least try to work, every problem. Otherwise one will not have a true picture of each boy's ability. At first this seems (54) almost impossible, but after careful thought the follow- ing plan seems feasible. The students are told that they will have two hours to do the test, but that at the end of one hour those who have finished may leave, and that at the end of an hour and a half there will be another Opportunity to leave. Thus the confusion of departing students will be concentrated into a few min- utes. Two hours should be sufficient to try all 65 ex- amples, and the poorer students will thus demonstrate. their inability to do the problems. Equally sensible is the idea of not being rushed as in a speed test, because haste Often results in silly, careless wrrors. This again will give a better picture of the students' capacities. The test does not purport to pick out the superior students, but it should determine those who have in- sufficient mathematical ability to do successfully the work of college algebra. After considering all these points very carefully, the following test was drawn up. cm”.- ——‘-fi 1.. Mathematics Test for Freshman in the Engineering Division Name I graduated from. High School in p_f_(city) in (year) In the following list indicate with a check:(yfl the amount of mathematics studied in high school. Algebra: 1 year ( ), 1% years( ), 2 years( ). Plane Geometry: % year ( ), 1 year( ). Solid Geometry: % year( ). Trigonometry: % year( ). Directions: Do all the work on the test paper in the space provided after each problem. Write the answers between the parentheses on the right side of the sheet. YOu will be allowed two hours to work the problems, but at the end of an hour those who have finished will be excused, and at the end of the next half hour those who have then finished will be given another Opportunity to leave. If you cannot do a problem, go on to the next one, and when you have finished the test go back to the ones that you had trouble with. (35) —\2 Find the answers to the following problems, expressing all results in simplest form. 1) 2) 3) 4) 5) 6) 7) 8) 2+5-6+5-7= 14_ 12'... 01K) e ~10) e elf. II 0101 d «It II Mathematics Test (sei— (57) 9) 41/5-wsz-2V3’= ( ) 10)J75= ( ) 11) [1%- ( ) 12) V32: ( ) 15) V1? ( ) 14) WE? . 4V5: ( ) 15) 5(x + 1) - 2(5x - 1) + (2x - 1) - ( ) 15) 5x - [s— (x +6)] = ( _ ) 17) 5a-6 __ ( ) TIT—8'.- 18) W _) II A 19) (2a - 5)(a + 5) -"-' ( ) (58) 20) If the area of a triangle is given by the formula A 3 %bh, what is A in square feet if b is 5 feet and h is 8 feet? ( ) 21) If the volume of a cone is given by the formula v - ijrszh and if v is 207) cubic inches and h is 5 5 ‘5 inches, how many inches long is r? Factor the following completely: 22) 4x 4 4 ( ) 23) x2 - 1 ( J_ ) 24) x2 + 4x + 4 ( ) 25) 22 + 4x - 5_ ( ) 26) 5x2 — 4x + 1 ( ) 27) 16x4 - 81y4 ( __) 2;: Solve the following for x: 28) 29) 50) 51) 52) 35) 54) 55) 56) (41:31 5x-2(x-l)=5 x-I-l 2x+5_ .___,7._. 5x2 + 7x +220 x2-5x-290 8=K + xt (39) Solve the following for x and y: 5x - y = 5 57) 4x - y B 2 _ x + y = 7 58) 4x +5y 26 Simplify the following: 59) x2 o x3 (x (Y (x (y \2v1 (41) 45) 1503-1 . 1"-4 ( ) 46) (x3)5 = ( ) 47) (xi) = ( ) a. 48) (27R5 . ( ) Graphs: 49) In making a graph of the equation 5x - y- 6, where does it once: the x axis? ( ) Where does it cross the y axis? ( ) a U f ,\ \. n 1' 1 . “ . -\ ,' yr \ 1‘ ‘ ' ' " / , 3 ‘ . ‘- f - , \ \ 6 ., "‘\ ' 3/ 3‘ R\\ a 3/ ~ In this figure the line for equation x + y = 2 is ( ) and for line 6x - 2y 3 5 is ( ) The common solution for these two equations is ( 51) 52) 55) The The The The The The (42) A fourth as many boys at a camp sleep in tents as sleep in cabins. If there are a hundred boys at the camp, how many boys sleep in tents? (_ ) Jimmy has twice as many pennies as Betty has, and he has three more nickels than she does. Betty has an equal number of nickels and pennies. If together _._...l they have $1.06, how many pennies does Jimmy have? ( How many nickels does he have? ( ) The length of a given rectangle is 8 feet greater than its width. If its width is increased by 2 feet and its length is increased.by 4 feet, the area is doubled. Let x = width, in feet, of the given rectangle length, in feet, of the given rectangle is t_____) area, in square feet, of given rectangle is L_____) width, in feet, of the enlarged rectangle is (_____) length, in feet of the enlarded rectangle is S ) area, in square feet, of enlarged rectangle is ( ) following equation can be used to flind the value of x (do not give the solution) (__ ) (43) 54) A passenger train and a freight train leave Detroit at the same time. The passenger train averages 15 fibre miles per hour than does the freight train. If they are both going to the same city 175 miles,away, and the passenger train arrives an hour and a half before the freight train, what is the speed of the freight train? Let x = average speed in miles per hour of the freight train. Average speed in miles per hour of the passenger train is (_ ) Number of hours for the freight train to arrive at the city 175 miles away is ( ) Number of hours for the passenger taain to arrive at the city 175 miles away is ( ) The following equation can be used to find x (do not give the solution) ( ) 55) If the ratio of two numbers is as 5 : 4 and the larger number is 16, what is the number? ( ) Perform the indicated operations: 56)x+1 2x-5 5x-1_ ( ) T‘T*“‘2“'- (44) 58) 4 3 2 '_ ( ) ;2-—-§'1'x'_ y +-y _ xv. ‘ y 59) 5x + 1 ._ 6x - 4 _, ( ) x2 4 6x + 8 ii * 8 60) (x + 3)2 , x e 2 , x2 + 4,_ x - 16 i2 + 6x + 9 x + I'— Put the following statements into the sign language of algebra: 61) The sum of x and y is z. ( __) 62) The prouuct of x and y is z. ( ) 63) The sum of the squares of x and y, increased by the cube of the difference of x and y lquals z. ( ) 64) If x is raised to the 4th power and decreased by the fifth power of y, the result is equivalent to the seventh root of z. 65) If a is divided by b the auotient is h plus k and the remainder is n. ( ) (45) In scoring this test one point is to be given for each answer that is to be put in parentheses with a possible total score of 79. It is expected that the scores on this test will be a good indication of the foundational knowledge of mathematics possessed by these engineering students. The various abilities tested are here listed with the problems in which they appear. There are some overlappings as several problems involve more than one algebraic process. Addition and subtraction of numbers 1 Multiplication of numbers 3,8,9 Division of numbers by inverting and then multiplying 4,5,8 Removal of parentheseéand then adding 7,15,16,29 L.C.D.: in addition or subtraction of ‘ fractions 6,56,57,58,59,60 in solving equations 52,35 Addition involving radicals 9 Multiplication involving radicals l4 Exponents: raising to a power 46,47 extracting a sQuare root 18 add when multiplying 39,41,43,45 subtract when dividing 40,42 zero exPonent equals 1 44 Reducing a fraction to lowest form 2,17 Factoring: common monomial 17,22,59 difference of two squares 25,27,30,58,60 trinomial, product of two binomials 24,25, 26,59,60 (46) Square root: of a perfect square fraction 11 of a two digit number divisible by a perfect square 10 of a negative number(imaginary)l2 Cube root 15 Solving equations for one unknown: by multiplication and division 28,29,51 by factoring first 50,54 by first reducing to L.C.D. and then clearing of fractions 52,55 by quadratic formula 55 by subtraction and then division56 Solving two simultaneous equations for two unknowns 57,58 Substituting given values in a given formula 20,21 Interpretation of graphs 49,50 Story problems 51,52,55,54 Ratio 55 Raising to the 2/5 power a number that has a cube root 48 The sign language of algebra 61,62,65,64,65 After this test has once been given to a group of students, the weaknesses of any specific student can be determined by checking the errors made on the test with the above list. This may also be consulted to discover the weak- nesses of a group when considered as a whole. More than just an understanding of how to perform a mathematical kkill is necessary for success in college D V n . . n e . s ' (47) mathematics. In performing each skill there are certain principles and concepts, notations, terminologies, and even other skills that should be thoroughly understood. The following chart, which shows an analysis of the skills that should be the tools of all students entering a college algebra class at Michigan State College, can be of help in finding exactly what is involved in the use of these math- ematiCal skills. OTHER PRINCIPLES NOTA- TERMIN- SKILLS SKILLS AND CONCEPTS#_ TIONS OLOGY INVOLVED I) Adding Multiplying or dividing NuESFEtEF'FEEtarifig and sub- both the numerator and Denomina- tracting denominator of a frac- tor fractions. tion does not change L.C.D. the value of the frac- tion. A fraction has three signs, the sign of the numerator, the sign of the denominator, and the sign of the frac- tion. Any two of these may be changed without changing the value of the fraction. 2) Multi- In division, invert the Quotient Factoring plying and divisor and proceed as Divisor dividing in multiplication. Inversion fractions 5)Removing In removing a paren- parenthe- thesis preceded by a sea negative sign, all 'the signs within the parentheses are changed. (48) 4) Solving If add, subtract, mul- Coeffi- Factoring of simple tiply, or divide one cient equations side of an equation by in one un- something, it is necessary known, and to do the same to the other a system side. of two si- multaneous linear equa- tions in two unknowns 5) Con- The x axis is hori- point: axis structing zontal and the y axis (1,0) and readingds vertical. etc. graphs 6) Factor- Difference of two monomial ing squares is equal to Binomial the product of the Trinomial sum and the difference Polynomial of their roots. Coefficient Square Cube 7) Handling Add exponents when exponents multiplying, subtract exponents when divi- ding. x0 = l 8) Taking A number has two square roots. roots, one positive and one negative. 9) Using V71 : 1 imaginary numbers (49) This chart can be used to help one to diagnose the specific cause or causes for failure in any one skill. If a student is unsuccessful in the use of any of these skills, it may be due to any one of these factors upon which each skill is based. All of these things should be investigated in helping a student to master a skill. In summarizing the findings of this study of the mathematics placement test for freshman engineers as given at Michigan State College in September, 1940, the following were determined: This test cannot be depended upon to predict success in mathematics courses at Mich- igan State College because insufficient time was allowed to test all the necessary fundamentals, becausd there were too many difficult problems that consumed too much time, and because the answers alone do not constitute a valid basis for such a prediction. Part I of this test proved to be more accurate in giving evidence of a student's capacities than the other two parts, and Part II was the least accurate in this respect. The test does not show any distinct difference between the three groups of engineering freshmen, those entering college algebra, trigonometry, and solid geometry; and it cannot be used in its present form to determine which students should be excused from college algebra and allowed to start anal- ytic geometry. Together with the high school record in mathematics this test could be used to determine which students should enroll in a review course without credit, (50) although it has not been set up ideally for this purpose. An easier test of fewer items would give approximately the same distribution of grades. After studying the errors made on this test by the F students and the A students and comparing them with errors made on similar tests at other colleges, it was learned that freshmen at different institutions make similar errors. The psychological test adds very little, if anything, to the knowledge of a student's ability in mathematics. The distribution of final grades is very similar in the three groups, those taking college algebra, trigonometry, and solid geometry. There is a higher correlation between this test and the first term's grade in the college algebra group than there is for the trigonometry and solid geometry groups. The test scores and the term's grades for the college algebra group have a linear regression. Benefiting from the results of the investigation of this placement test, a new test was formulated for freshman engineering students with the aim of testing for the fundamental knowledge considered nec- essary for college algebra, and to be used to determine Which.students are so poorly prepared that they should be given a review course, without credit. Before taking the test, the students should be told that it will be used for such a purpose. This will result in each student doing his best on the test. Thus if a student receives a relatively low score on this test, it is certain that he will be at a disadvantage when competing with students (51) possessing the proper mathematical background. and he will benefit greatly by enrolling in the review course. For aiding individual students, a list of the fundamental processes involved in this test was compiled with the ex- amples given in which they occur. This will be of use in determining Just where a student is weak. Also a list of the skills necessary in order to be able to do the work of college algebra is here given with the principles and concepts, notations, terminology, and other skills in- volved which lie back of each skill. Thus, although the test given in September, 1940 could be used for some purposes, this new test is expected to be a more accurate means of picking out students for a review course, without credit, and it would give more evidence of specific causes of failure. l. 2. 5. 4. 5. 6. BIBLIOGRAPHY Arnold, H.J. of Wittenberg College, "Abilities and Disabilities of College Students in Elementary Algebra". Journal of Educational Research, April, 1951. College Entrance Examination Board, General Report on the Mathematics Attainment Test, 1958. New York, New York, 1958. College Entrance Examination Ebard, Research Bulletin number 7, Report on the Mathematics Attainment Test of June 1956. New York, New York, December, 1956. COOperative Intermediate Algebra Test of the American Council on Education, form 1957, which includes quadratics and beyond. Dorwart, H.L. of Washington and Jefferson College, "Comments on the North Carolina Program in Freshman Mathematics". The American Mathematical Monthly, January, 1941. Keller, M.W., Shreve, D.R., and Remmers, H.H. of Purdue University, "Diagnostic Testing Program in Purdue University". The American Mathematical Monthly, January, 1941. APPENDIX In the appendix are included the various bits of statistics gathered in connection with this study and they are arranged in chart form. Although all the statistics were not utilized directly in the study, they are all included as they may be of interest to one who desires a more careful inspection of the results of the test. Scores on the Cooperative Intermediate Algebra Test, form 1957, Administered to the Freshman Engineering Students in September, 1940 of the Students Who Took College Algebra the Fall Term of 1940, Arranged in Groups Determined by First Term Grades A B C- D F 55' 55' IE' 12' IU’ 58 56 l9 15 10 46 57 21 l6 17 52 58 24 17 17 55 42 24 19 18 55 42 27 20 20 56 45 28 21 21 57 45 28 22 21 60 45 28 51 21 60 47 29 51 21 62 47 29 52 24 62 48 50 55 24 64 49 50 58 25 65 49 50 42 26 67 50 51 44 26 68 51 51 51 28 72 52 55 51 29 72 54 55 29 78 55 54 57 84 55 55 fig 55 56 56 57 47 59 59 50 59 so 40 61 40 51 4O 62 41 55 42 8'7 45 44 44 46 49 49 50 50 50 50 51 52 55 54 55 58 59 65 65 Scores on the Cooperative Intermediate Algebra Test, form 1957, Administered to the Ereshman Engineering Students in September, 1940 of the Students Who Took Solid Geometry the Fall Term of 1940, Arranged in Groups Determined by First Term Grades A B c D F 1'9 9'— 5" 7" B" 58 15 12 15 8 59 16 15 16 10 55 _ 19 17 22 11 72 25 17 24 12 28 22 55 18 28 22 59 29 50 25 51 26 51 28 52 28 58 29 4o 59 45 4o 82 4o 65 Mean Scores and Standard Deviations of the Three Groups of Students on the Cooperative Intermediate Algebra Test, form 1957. College Algebra Mean score of the whole group is 41.78 Means Standard Deviations A - 60.55 A -'ll.76‘ B - 51.57 B - 10.51 C - 40.20 C - 12.54 D " 29012 D " 12e48 F - 24.91 F - 8.94 Trigonometry Mean score of the whole group is 26.59 Means Standard Deviations A - 58.15 A - 14.87 B "' 35e11 B "' 7095 C - 25.75 C - 9.65 D - 22.60 D - 12.01 F - 20.56 F - 6.55 Solid Geometmy mean score of the whole group is 27.27 Means Standard Deviations A - 44.60 A'- 17.85 B "' 51.00 B - 16.555 C - 26.65 C - 15.56 D - 22.00 D - 10.59 F - 15.71 F - 6.98 (117) (V) This chart shows the number of college algebra students the first term that did each example correctly. They into groups according to the first term grades. Example A art I 10 1.9- Part II 2. 19 3. 2O 4. 19 5. ----18 6. 18 7. 16 8. 14 9. 16 10.--——16 11. 15 12. 17 13. 18 14. 15 15o """-11 16. 18 17. 12 18. 14 19. 12 20c “"""14 21. 17 22. 9 23. 7 24. 11 25.----18 26. 11 27. 6 28. 11 29. 15 30.----13 31. 12 .52. 0 33. 10 34. 3 35. ----- 2 36. O 57. O 38. O 39. 12 40. ----- 1 41. l l. 15 2a. 19 2b. 18 2c. 17 3.----l7 4ae 17 4b. 16 4c. 18 4d. 17 4e.----15 4f. 14 (20) B (30) C (49) D 56‘ '44 II 20 36 6 28 43 16 25 41 10 28 ------ 39 ------- 9 22 39 11 23 36 15 17 28 7 24 36 7 25 ------ 3O ------- 1 23 24 5 26 33 11 27 41 12 18 28 5 2O ------- 4 21 26 6 19 19 5 2O 25 2 16 36 5 ------ 20—------4 16 38 10 2O 17 1 4 5 1 13 15 1 19 ------ 22 ------- 8 7 6 1 2 5 1 12 2O 5 12 16 2 ------ 12-------4 14 12 5 1 O O 9 5 O 4 4 0 3 ------- 3 ------- O 1 O O 5 O 1 O O O 7 5 l 1 ------- O ------- O O O O 23 29 8 28 37 9 25 37 7 27 37 6 21 ------ 26 ------- 3 29 4O 12 28 38 9 28 39 14 28 38 11 25 ------ 53 ------- 8 18 2O 4 are divided (17) F (22) Total (158) I8 "I25- 10 91 14 121 15 110 ------- 7-------101 12 102 14 104 10 76 9 92 ------- 7--------78 6 75 8 95 15 111 0 66 ------- 5--------45 10 81 6 61 0 61 7 76 ------- 5-------—57 10 91 2 49 1 18 2 42 ------- 5-—-_-——-72 0 25 2 16 0 46 5 50 ------- 5——-—----44 2 45 0 1 0 24 0 11 ------- 0---------8 0 1 1 5 0 0 1 26 ------- 0----—-—--2 0 1 11 86 17 110 10 97 12 99 ....... 7—---—---74 15 115 12 105 14 115 14 108 ------ 11------—-90 2 58 Example 5. 6. 7a. 7b. 70o 7d. 8. 9a. 9b. 9c. 9d. 10. 11. Part III 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 52. 33. 34. 35. 36. 57. 58. 59. 40. 41. 42. A B C D F Total 11' 14‘ IT. '71" "6' 46 18 25 27 5 6 81 16 24 24 11 14 89 —---14 —————— 16 ------ 17 ------- 5 ------- 5 ........ 55 12 11 12 4 4 45 6 5 1 1 0 11 5 6 5 1 0 , 15 15 15 15 5 6 50 --——10 ——————— 9 ------ 10 ------- 2 ------- 7 ........ 58 9 9 7 2 2 29 8 5 5 1 1 18 4 5 2 2 1 12 9 6 5 5 1 24 19 50 48 15 17 129 20 27 45 15 14 119 20 28 20 15 15 96 19 27 58 12 14 110 ——-—20 ------ 50 —————— 55 ------ 12 ------- 8 ------- 105 15 24 51 6 6 80 7 11 18 2 0 58 17 25 56 7 10 95 15 21 52 9 10 87 ----15 ------ 25 ------ 15 ------- 5 ------- 4 -------- 56 16 25 15 7 7 66 14 28 59 8 10 99 17 20 52 5 9 85 12 24 55 4 10 85 ----16 ------ 17 ------ 25 ------- 5 ------- 2 -------- 65 14 15 18 8 7 60 7 9 15 8 8 45 7 11 5 5 0 26 12 21 19 8 12 72 ----- 7------l5------15-------6-------9-—-—----52 15 15 25 7 6 66 17 19 15 5 7 65 15 10 10 1 1 57 17 14 10 2 1 44 ----20- ----- 19 ------ 28 ------- 9 ------ 12 -------- ss 9 6 10 2 0 27 0 0 0 1 l 2 10 11 16 4 2 45 10 9 15 6 10 48 ----- 4——-----5-------5-—--—--0———---—1---—----11 5 2 4 1 1 11 0 1 1 1 0 5 5 5 6 0 0 14 1 2 2 0 0 5 ----- 1----——-5-------0-—--—--0-------0---------4 5 2 2 0 0 7 0 0 0 0 1 1 1 5 0 0 0 4 0 1 2 1 1 5 ----- 1------—1---——--0—-----~1-------1-------—-4 1 0 0 0 0 1 0 p 2 1 0 5 (v1) (vii) Rank Correlation Coefficients Between the Original Test and the Test Made of the Easy Ones, Namely Examples 1 Algebra Trigonometry A - . A - .89 B - .55 B - .69 C - .80 C - .82 D - .82 D - .88 F-OGU F-.39 Solid Geometry A B C D F 91.0 .67 .90 .83 .76 (vii) Rank Correlation Coefficients Between the Original Test and the Test Made of the Easy Ones, Namely Examples 1 Algebra A'. .55 .80 .82 06U anscaw Illl Through 15, 17, 18, 20, 24, and 25. Solid Geometry Trigonometry A - .89 B - .69 C - .82 D - .88 F - .59 A -‘1.0 B " .67 C " .90 D - .85 F "’ .76 AMERICAN COUNCIL ON EDUCATION c00PERATIVE INTERMEDIATE ALGEBRA TEST QUADRATICS AND BEYOND Form I937 by JOHN A. LONG. UniversiTy 0T ToronTo and L. P. SICELOFF. Columbia UniversiTy Please prinT: Name...) . .. .5 . .DaTe LasT FirsT Middle Grade or Class . ...... ., . . . Age ............................................ . ....... DaTe 0T BirTh ....................................... Yrs. Mos. School , .. ........... CiTy .............................................................. _ ....................... Sex ....................................... M. or F TiTle of algebra course you are now Taking ....................................................... InsTrucTor ............................................................ In whaT grade did you begin The sTudy 0T algebra? ....................................................... s ....................................................................... Number 0T years you have sTudied algebra (one semesTer = '/2 year: one quarTer = '/3 year): .................. General DirecTions: Do noT Turn This page unTil The examiner Tells you To do so. This examinaTion consis’rs of Three parTs. and requires 90 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 IT 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. No quesTions may be asked aTTer The examinaTion has begun. ParT MinuTes Row Score PercenTile l 40 || ’ 25 Ill 25 ToTaI 9O Scaled Score (See Table on key) Copyright I937, by The Cob‘perafive TesT Service. All Rights Reserved. Printed in U.S.A. 437 WesT 59Th STreeT, New York CiTy Part I Directions: Write the correct answer to each problem in the parentheses to the right of the problem. 1.51mp11ry: efi-11fi+10fi........................( 2. If §§1§_1.=.$§1%_1, then 8 equals. . . . . . . . . . . . . . . . . . . . . . ( . In the equation 5x - 5y = -9, when the value of x is 5, the value of y is. . ( 2 . When -4a2 + 5 is subtracted from a - 5a + 2, the remainder is . . . . . . . ( . Express V85 in simplest radical form . . . . . . . . . . . . . . . . . . . . ( .Thesumof2.41yand5.7yis........................I 3 4 5 6. In the equation N = 20 -'%r, when N = 8, r equals. . . . . . . . . . . . . . ( 7 8. The square root of 580689 is . . . . . . . . . . . . . . . . . . . . . . . . ( 9 . If 4X + y = 61, and X - y 3 4, than y equals 0 e e e e e e e e e e e e e e o ( r - 1 r + 5 10. If r - 4 = r - 1, then r equals. 0 O O O O O O O O I I O O O O O O 0 O O O O ( 11. The prOduCt Of ZVE-and svg 13. e e e o e e e o o O o o e e e e e e e o e e o ( 12. If 0.04x + 0.5 = 1.14, then x equals . . . . . . . . . . . . . . . . . . . . I 15. I varies directly as R. If M = 24 when R = 9, then when R = 6, M equals . . ( 16 is. . . . . . . . . . . . . . . . . . . . . . . . . ( 14. The square root of 56x 1 15. Simplify: 273 O I O O O O O O O O O O O O O O O O O O O O O I I O O I I O O ( 16. If a = 5 and b = -2, the numerical value of a3 + b 17. The negative root of the equation x2 + 2x - 24 = 0 is. . . . . . . . . . . . ( _ 2 18. Reduce l§§Z§——£1§ to lowest terms. . . . . . . . . . . . . . . . . . . . . . ( x ’ 7 19. If 108 76 = 1.8808, then 108 6:6 equals. 0 e e e e o e o 0 0 0 O O O O 0 O O ( 20. The larger of the two roots of 2x2 - 17x + 55 = 0 is . . . . . . . . . . . . ( ,” 21. Given sin 58° = .62, cos 58° = .79, ,I’ tan 58° = .78, the height, in feet, of the ,a’ flagpole in the accompanying diagram is . . . . ( {Ilaer (CH3 F”n 22. If the graph of the equation kx - 6y = 5 passes through the point (5,5), the value or k is O I O O C I O O I D O O O O O O C O O O O O O O O O O O O 0 ( 25. If a = 5, b = 2, n = 1, then the numerical value of abn+1 is . . . . . . . . ( 24. If g + %§ = 25, and %§ - g = 51, then x equals . . . . . . . . . . . . . . . ( 25. Express 6x2 + 17x - 10 as the product of two factors . . . . . . . . . . . .{( 26. If X2 + 5X - 6 ’ l = X, then X equals 0 e e o e e e e e e e e e s e e e e e ( Go on to the next page. . In the binomial expansion of (x + 5y The sum of 10 terms of an arithmetic series is 215. If the common difference is 5, the first term is. . . . . . . . . . . . . . . . . . . . . . . . 5 If log a = 1.5421 and log b = 0.9182, then log %7 equals . . . . . . . If log 4 = 0.602, then log 64 equals . . . . . . . . . . . . . . . . . .If log 2200 = 5.5424, and log 2210 = 5.5444, then 5.5458 is the log of . The roots of sz - 5x + 10 = 0 will be equal to each other if K equals The positive root of the equation x - V5x + 22 = 2 is. . . . . . . . . .What positive value of x will satisfy the pair of equations 5x2 - y2 = andx-y=-2? ooooooooooooooooooooooso... 30 - 22 = 11, and %§ - %? = 57, then y equals . . . . . . . . . . . If 7? y The sum of the infinite geometric series 5 + %% + Mlle (D .... is. O O O C 0 term 18 O O O I I O O O O O O I O O O C O I O O O O O O O 0 O O O 0 O O . In the binomial expansion of (a + b)12, the fourth term is . . . . . . . N varies as the square of c. If N = 48 when c = 4, then when c = 5, N equals 0 O O I C O O O O C O O O C 0 O O O O O O O I O I O O O O C 0 If 5x% = 1, then x-2 equals 0 O O O C O C O O C C O O O O O I C O O O O 2)7, the third term is . . . . . . Raw Score = Number right .If log 5.2 = 0.5051 and log 6.46 = 0.8102, then log (5200 x 646) equals. . . The second and fourth terms of a geometric series are 8 and 50. The third Part II Directions: Write the correct answer to each problem in the parentheses to the right of the problem. 1. The numerator of a given fraction is 5 less than the denominator. If the numerator is doubled and the denominator is increased by 7, the resulting fraction equals 1/5. What is the given fraction? . . . . . . . . . . . . . 2. Three sisters are left a bequest of £18,000. Mary is to receive 5 times as much as June, and June is to receive 31,000 more than Ruth. How much does June receive? Let x = the number of dollars June receives. Then: (a) The number of dollars Mary receives is, in terms of x, . . . . . ( (b) The number of dollars Ruth receives is, in terms of x, . . . . . ( (c) I would use the following equation to find x (solution not required) . . . . . . . . . . . . ( 5. Two positive numbers are in the ratio 5 to 2. If their difference is 48, the smaller number is. I I I I I I I I I I I I I I I I I I I I I I I I I I I 4. The length of a given rectangle is 7 inches greater than the width. Its area is doubled by increasing both its width and its length by 5 inches. What is the width, in inches, of the given rectangle? Let x = width, in inches, of given rectangle. Then: (a) The length, in inches, of the given rectangle is, in terms of x, ( (b) The area, in square inches, of the given rectangle is, in terms or x, I I I I I I I I I I I I I I I I I I I I I I I I I I I ( (c) The width, in inches, of the enlarged rectangle is, in terms of x,( (d) The length, in inches, of the enlarged rectangle is, in terms or x, I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ( (e) The area, in square inches, of the enlarged rectangle is, in terms or x, I I I I I I I I I I I I I I I I I I I I I I I (f) I would use the following equation to find x (solution not required) . . . . . . . . . . . . ( 5. I have $4.65 in my purse. There are 5 more quarters than dimes and twice as many nickels as quarters. How many dimes are there? . . . . . . . . . . . . ( 6. In a village of 800 people there are one-fourth as many children as there are adults. How many children are there? . . . . . . . . . . . . . . . . . ( 7. An express train and an airplane leave New York at the same time for a point 270 miles away. The airplane averages 90 miles an hour faster than the express and reaches its destination 4 hours sooner. What is the average speed, in miles per hour, of the express? Let x = average speed, in miles per hour, of the express. Then: (a) The average speed, in miles per hour, of the airplane is, in terms Of x, I I I I I I I I I I I I I I I I I I I I I I I I I I I ( (b) The number of hours for the express to make the trip is, in terms of x, I I I I I I I I I I I I I I I I I I I I I I I I I I I ( (c) The number of hours for the airplane to make the trip is, in terms of x, I I I I I I I I I I I I I I I I I I I I I I I I I I I ( (d) I would use the following equation to find x (solution not required) . . . . . . . . . . . . ( 8. Jim is one-fourth as old as his father. If Jim will be 0 years old in 8 years, what is the father's present age in years? . . . . . . . . . . . . ( Go on to the next page. lA grocer mixes candy worth 20¢ a pound with candy worth 55¢ a pound to make a nuxture of 100 pounds worth 25¢ a pound. How many pounds of the 20¢ candy does he use? Let x = the number of pounds of 20¢ candy used. Then: (a) The number of pounds of 55¢ candy used is, in terms of x, . . . . (b) The total value, in cents, of the 20¢ candy used is, in terms or x, I I I I I I I I I I I I I I I I I I I I I I I I I I I (c) The total value, in cents, of the 55¢ candy used is, in terms or x, I I I I I I I I I I I I I I I I I I I I I I I I I I I (d) I would use the following equation to find x (solution not required) . . . . . . . . . . . . ( LA large reservoir has two inlets. It can be filled through inlet A in 20 hours, or through inlet B in 50 hours. In how many hours can it be 2 filled through the two inlets working together? . . . . . . . . . . . . . . (.John and Arthur carry a 210-pound weight slung from a pole across their ‘ shoulders. If John is 5 feet from the weight, and Arthur is 7 feet from it, , how many pounds of the weight does John carry? (Neglect the weight of . the p016 I ) I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Raw Score = Number right - 6 Part III Directions: Each problem below is accompanied by several possible answers, only one of which is correct. In each problem find which is the correct answer, then write the number of that correct answer in the parentheses to the right of the problem. By exercising careful judgment and making shrewd guesses, you may profitably answer questions about which you are not absolutely sure; but since your score will be the number of correct answers diminished by a number proportional to the number of wrong answers, you should avoid answering questions about which you are totally ignorant. Shrewd guessing based on intelligent inference will improve your score, but wild guessing on questions that are entirely unknown to you will waste time which you could better put on other questions in the test, and may result in a large subtraction from the number of your correct answers. . 4 1 4b b 1. If % = %, then c equals (1) 7%, (2) % - z, (5)-E—%—Z, (4) TT’ (5) E - 4 . . . ( 2. If bx + = n, then x equals (1) % - a, (2) 2_%_E, (5) b(n - a), (4).£.%JE, 8. a (5)0-8....................._..................( N _ - 5 l - l N - l 5. If R — x, then K equals (1) N’ (2) x N’ (5) xN, (4) 3’ (5) x N' . . . . . ( 4. The product of r3 and r"12 is (l) r-SS’ (2) 2r'9, (5) r'4, (4) r'ls, (5) r'9. ( 6a - 14a6 ‘pa (5) -5 + 7a5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( equals (1) 6a + 7a5, (2) -5 - l4a6, (5) 6 + 7a6, (4) -5a - 14a5, r4 N LL _ 1 .J. _ LL 6. The product of 02 and c5 is (l) 010, (2) c7, (3) c , (4) clo, (5) c20 . . . . ( 7. 5- + 30 equals (l) g, (2) - mks , (3) 1%, (4) -4, (5) % . . . . . . . . . . . . ( l. i 8. Dws equals (1) w'z, (2) w55, (3) W7, (4) “12, (5) w . . . . . . . . . . . . . ( 3 3V 5 5 {— 3 9. If a = BEE, then n equals (1) —%M, (2) VaM - B, (3) %¥, (4) a - E: \ 3 (5) ($515) 0 o o o o o o o o o w o o o w o o o I o o 0 O o o o o o o o o o o o o o o o o ( 7a + g _ 3a :_5 9s + 32 19a + 1 4s + 9 9a - g (5) yflfi'gi o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I o o o o o o ( 7 - 11. If 7x + 5 = nx, then x equals (1) 3—3—3, (2) n § 7, (5) 5 - g, (4) 5 n, (5) 6’?‘? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 12. The product of two numbers is k. If one of the numbers is 35, the other is (1) 55 - k, (2) %§, (5) 55k, (4) k - 55, (5) §% . . . . . . . . . . . . . . . . ( E- R 2-1 a.-. 2-z 2.5. 13. If 2 - 5—1—3, then n equals (1) § R’ (2) 2R 5, (6) F _ R’ (4) F y; (5) 23 - Fy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 14. "When 3 is divided by r, the quotient is m + n and the remainder is c," may be s _ s _ 2 written algebraically as (1) F — r(m + n) + c, (2) F — m + n + r’ m + s s c (3) % = r n + c, (4) 3 = m + n + c, (5) F = m + n - F . . . . . . , , , . . , , , ( 18 q 15. Eta-equals (1)1142, (2) n4, (5) n“, (4) mm, (5) n‘“4. . . . . . . . . . .( n Go on to the next page. ).The quadratic equation whose roots are 7 and -3 is (l) x2 + 4x - 21 = O, (2) x2 - 4x + 21 = o, (5) x2 - 21x + 4 = o, (4) x2 - 4x - 21 = o, (5) x2 + 4x + 21 = 0. . . . C . C C O . C . . . . . O . O . . O .. O . C I O . . . U . . ( +. \r. 15 Questions 17-20 inclusive refer to the accompanying diagram. 17.x - 2y: 4 is the equation of line (1) AA, (2) BB, (3) CC, (4) DD, (5) EE . . . . . . A 18. 2x + y = -8 is the equation of line (1) AA, (2) BB, (3) cc, (4) DD, (5) as . . . . . . ( )(*' 19. The solution for the pair of equations repre- sented by lines AA and BB is (l) x = 2, y = 4, (2) x -3. y = 2, (3) x = 0. Y (4) x 4. y = 2. (5) x = -2. y 1 3. . . . . . ( 20. For line EE, when y equals -3, x equals (1) -32.! (2) 29 (3) ' s (4) 4p (5) '5 0 ( mud The product of 35'and 35'13 (l) %QE, (2) 3T3, (5) 1935, (4) 9T5, (5) @Z‘. . . ( "The sum of the cubes of two numbers, decreased by the square of the difference of the numbers," may be eXpressed algebraically as (l) (x + y)3 - (x - y)2, (2) 13+ y3 - (x2 - y 2), (3) ?———_§%§, (4) (x + y)3 - (x2 - ya), (5) x3 + y3 - (x - y)2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °,- . ( l P ). .1.=;4. l 22 xz 4“: X+z If K x + y’ then K equals (1) 4 + y, (2) x + 4y’ (3) , (4) 5 , i (5)4x+y ......O.......OOOOOOOOOOIOOOOI......I.( (Which2 one of the following is true? (1) (x3)4 = x7, (2) (x + %)(x - %) = x2 - l, (3) x—x2—:_l-_ = X " y, (4) 8(8X)2 = asxz, (5) Xs'xm = X31“ 0 e e e e e e e e e e e e o ( Log K2 equals (1) log K + log 2, (2) 2 log K, (5) (log x)2, (4) l2%_5, (5) K log 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( + - - i- - u- i- If 2x2 + 3x + 4 = 0, then x equals (1) EL:1%—3l, (2) _§__Z!.§§, (5) _§___!§1 (4) 9.3Litfléé, (5) :ELEEJfiii . . . . . . . . . . . . . . . . . . a .The product of the roots of cx2 - nx + s = O is (l) - g, (2) g, (3) %, (4) -%, (5)% e e e e e e e e e e e o e o e e o e e o e e e e o e e In the formula A = B C H, the symbols A, B, H, and C represent quantities which are always positive. We are sure to make A larger if we make (1) B and H larger, leaving C unchanged, (2) H smaller, leaving B and C unchanged, (3) B and C larger, leaving H unchanged, (4) H and C larger, leaving B unchanged, (5) C larger, leaving B and H mchanged O O O O O O O O I 0 O O O O O I O O O O O I O I O O O O O O O O O O O O ( B 29. The tangent of angle B in8 the right-triangle ABC is (1) LS— <2)15 (5) (4) 8 <5)” ( ’ TV’ 187' IE’ IE' ' ' ' ' 15‘ 'C Go on to the next page. 30. if ax + by = c and x - y = -n, then-y equals (1) 95:32a2, (2) H, (3) $13113, c + an be + n (4) m, (5) -_———a + b e e e o e e o e e e e e e e e e e e e e e e e e e o o o I e o 31. Which of the following points lies on the gra h of the equation 12 + 3y2 = 21? (l) (l, 3), (2) (-2, 3), (5) (Op 5), (4 (3, '2), (5) (’5) 1) 0 0 0 0 0 0 0 0 0 32. The sum of the roots of c 2 + nx + r = 0 is (l) - g, (2) g, (3) - %, (4) g. n . (5)-aeoeeoeoeooeeeeeeooeoeeoeoeeeoeee -4 4 33. (é) equals (1) 6T, (2) %, (3) 81, (4) - 33, (5) - 3%. . . . . . . . . . . 34. The first two terms of a geometric series are 3 and 12. The nth term is (1) 3(4)“, (2) 5(9)"‘1, (5) 5 + 4(n - l), (4) 5(4)“'1, (5) 5 + 9n. . . . . . . . . . . . . 35. "A certain two-digit number is 21 times as great as the amount by which the tens digit exceeds the units digit," may be written algebraically as (l) 10x + y = 21(10x - y), (2) ley = 21(x - y), (3) 10x + y = 21(x - y), (4) xy = 21(x - y), (5) ley = 21(10x - y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 V50 316 5 125 3 36. 3J2; equals (1) L59, (2) Tg—gQ/2, (3)) 3%, (4) 37. If a, b, and c are rational and b2 - 4ac = 49, then the roots of ax2 + bx + c = O are (l) rational and equal, (2) real, unequal, and irrational, (3) unequal and imaginary, (4) rational and unequal, (5) real, equal, and irrational. . . . . . . . My - gt 5 - v51 29 - v51 5 - v51 , 5 - v51 58.fi+v5equals (1)-———-—2 . (2)—--—2O . (3)_-—5+V§i' (4)-------—5 . (5 1 - -——§L—-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ’ W+xf3 39. In the accompanying diagram the graph OB may be interpreted as signifying that (l) x varies inversely as y, (2) the sum of x and y is constant, (3) x varies as the square of y, (4) the difference between x and y is constant, 01-11001»me 2 55 4 4o. (2 + st1)(5 - 4vCT) equals (1) 26 - 7V-1, (2) -14 + vvil, (5) 26 - 25vCl, (4) ~14 - 7v21, (5) 26 + TV- . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. The product of WE and 9; is (1) 33;5, (2) 9QE, (5) 29;, (4) 9i, (5) 23§7. . . 42. The geometric mean between 2 and 13 is (1) * V15, (2) %?, (3) t V26, (4) fig, (5) t\[%; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number wrong 4) Number right (5) x varies directly as y . . . . . . . . . . . ( Subtract ___1 20625-7 Raw score = Difference 1. w gum“. “I ,ww’u 'm-_- - .5. - - ‘T” "~— ———' rv—w.‘ v.“ w ‘7"_ 1T . _.”~.’" '5 _u.. .<_“.'-_ . ‘- ' a ‘ L'...,.~ , . .... r S . r1; .'» _ d - f! I .1 , 4 fruit: t’uy . 1) 1— a \. v :-";e'ises% '-. “s ”1.1: .' v r v- . '- A u . n ' . -. a. I ' ' - e r .- 0“. g ’ t}, J‘ I " ' «a _' , - n1? . ., . ,0}... . .‘l‘ ' .'_ '3, \‘e" , v34." =9 4 A v f '1 v. -:_ ' . ”:pi‘ i I‘ . " .. “g. h)... .( Q1395.- ;q_,._,§§, .- .§-;.”‘f_. 33.". z '5’:- .‘. \.'.o _ y- I. .' , (proyr Jr! 1 lllllllllllllll) 03056 05 l l l l 1' l' ' l l l l l l l l i l l l l l l 3 3 129 )IHIHWIUHIIH