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The purpose of renaming skills in fraction algorithms Carlson, Florine Kiyomi 1977

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THE PURPOSE OF RENAMING SKILLS IN FRACTION ALGORITHMS BY FLORINE KIYOMI CARLSON B.Ed . ,v University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF * THE REQUIREMENTS FOR THE DEGREE OF "* MASTER OF ARTS in 'THE FACULTY OF GRADUATE STUDIES Mathematics Department Faculty of Education University of British Columbia I We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 1977 ©Florine Kiyomi Carlson, 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department o r by his representatives. It is understood that copying o r publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics, Faculty of Education The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e September, 1977 ABSTRACT The study was designed in two parts. The purpose of the first part was to gather evidence to support a model hypothesizing relationships between the fraction algorithms, that is, specified algorithms for addition, subtraction, multip-lication, and division, and renaming skills, that is, specified skills for renaming for the algorithm (RAL) and renaming for the answer (RAN). The purpose of the second part was to investigate the effects in performance of the subtraction algo-rithm, of reviewing the five RAL skills hypothesized as relating directly to the subtraction of fractions algorithm (treatment one); and to investigate the effects in performance of each of the four algorithms, of reviewing the purpose of the five RAL skills within the subtraction algorithm (treatment two). A repeated measures, hierarchical design was employed in which six sixth-grade classes were assigned to one of three experimental groups, so that no treat-ment was replicated in any one school. Group A received both treatment one and treatment two. Group B received treatment one. While Group A was receiving treatment two, Group B was instructed in geometry. Group C (control group) re-ceived instruction in geometry and measurement during the treatment periods. Treat-ment one was administered on each of five consecutive days. Treatment two was administered in one day. The regular mathematics teacher for each classroom participated in the experiment. Lesson plans and worksheets for the treatments were provided by the experimenter. iii iii A pretest consisting of three parts: Part A on the seven RAL skills, Part B on the nine RAN skills, and Part C on the four algorithms, was adminis-tered on two consecutive days before the treatments, rbsttest one consisting of two parts: Part A on the five RAL skills, and Part B on subtraction, was ad-ministered in one day following treatment one. rbsttest two consisting of the same parts as the pretest, was administered on two consecutive days following treatment two. All tests were constructed by the experimenter. For Bart One of the problem, two computer programs, an item analysis and a frequency tabulation, were used. Evidence was found to support the pro-posed model. In other words, there appears to be a relationship between RAL skills and algorithm achievement, particularly for addition and subtraction; and between RAN skills and the ability to obtain a simplified answer for each of the four fraction algorithms. For Part Two of the problem, a random effects model, multivariate analysis of covariance was used to test the effects of the two treatments. Pretest scores were used as the covariate in the analysis. No statistical significance was found in performance of the subtraction algorithm following treatment one. No statisti-cal significance was found in performance of the four algorithms following treatment two. iv TABLE OF CONTENTS Chapter Page ABSTRACT ii ACKNOWLEDGEMENTS viii 1 THE PROBLEM 1 Background 1 Formation of the Model 5 Statement of the Problem 12 Definition of Terms 13 Hypotheses to be Tested 15 2 REVIEW OF THE RELATED LITERATURE 17 Introduction 17 Research of Specific Relevance to the Problem 17 Research of General Relevance to the Problem 19 Michigan Studies 19 Error Analysis Studies 24 Summary and Significance of the Research 29 3 DESIGN AND PROCEDURE 31 Experimental Design 31 Formation of the Group 32 The Population 32 The Sample 33 Assignment to Experimental Groups 33 Development of Measuring Instruments 35 Pilot Test 35 Pretest 38 Posttest One.. . 38 Posttest Two 38 Development of Instructional Materials 40 Procedure 40 Controls 42 Statistical Analysis 43 Data 43 Statistical Procedures 44 Statistical Assumptions 47 Table of Contents - continued Chapter Page 4 RESULTS OF THE STUDY 49 Results for Part One 49 Hypotheses One and Two 49 Preliminary Data Analysis 60 Hypothesis Four (H4) 62 5 CONCLUSIONS AND IMPLICATIONS 67 The Problem 67 Conclusions 67 Part One 67 Part Two 70 Limitations of the Study 71 Implications 71 Suggestions for Further Research 72 REFERENCES 74 APPENDICES A Pilot Test - Version One and Version Two 77 B Pretest 82 C Posttest One 87 D Posttest Two 90 E Lesson Plans and Worksheets 95 vi LIST OF TABLES Table Page 1 Frequency of Right, Wrong Answers and Omissions for Addition and Multiplication of Fractions 3 2 Renaming for the Algorithm (RAL) Skills with the Four Algorithms. 6 3 Renaming for the Answer (RAN) Skills with the Four Algorithms 7 4 RAL/ftAN PAIRS 8 5 Hoyt Estimate of Reliability on Subtests of the Pilot Test 37 6 Algorithm Items with the RAL and RAN Skills Required to Solve Them 39 7 Frequency and Percent of Students who have Attained Criterion Level on the Four Algorithms 49 8 Frequency and Percent of Students Who Have Attained Criterion Level on Each Renaming Skill 50 9 Frequency and Percent (in parenthesis) of Students Who Have or Have Not Attained Criterion Level on A ] , and RAL Skills 52 100 Frequency and Percent (in parentheses) of Students Who Have or Have Not Attained Criterion Level on A ] , A2, and RAN Skills 53 11 Correlations for the RAL/RAN Pairs 56 12 Means (X.) and Standard Deviations (s) for the Covariates 57 13 Means (X.) and Standard Deviations (s) for each Dependent Variable Measured in Posttest One 58 14 Means (X.) and Standard Deviations (s) for Each Dependent Variable Measured in Posttest Two 59 vii List of Tables - continued Table Page 15 Preliminary Data Analysis for Posttest One 60 16 Preliminary Data Analysis for Posttest Two 61 17 Summary of the Analysis of Covariance for Posttest One 62 18 F Values for RAL Skills on Posttest One 63 19 Summary of the Analysis of Covariance on Rjsttest Two 64 20 F Values for RAL and RAN Skills on Posttest Two 66 LIST OF FIGURES Figure 1 An Example of the "Area" model for the multiplication of fractions (Green, 1969) 21 2 An Example of the "Fractional forts of" Model for the Multiplication of Fractions (Green, 1969) 14 3 Assignment to Experimental Groups 34 4 Procedure 41 Acknowledgements The author would like to express her gratitude to the teachers of the classes participating in the study, to members of her thesis com-mittee—-Dr. Gail Spitler, Mr. Tom Bates, and Dr. Doug Owens—for their cooperation and encouragement, to Dr. Todd Rogers for consultation services regarding the statistical analysis, to fellow graduate students— Heather Kelleher, John Taylor, Lee Herberts and Tom O'Shea—for their constant support and willing advice, and to her husband, John, for his patience and his persistence in keeping things in perspective. Chapter 1 THE PROBLEM Background Positive rational numbers, or "fractions" as they are commonly referred to in the elementary school, form a considerable part of the mathematics content of the intermediate grade curriculum. By common usage, the word "fraction" sometimes means the number itself, and sometimes it means the symbol. The phrase "mixed number", in common usage, frequently means the symbol and not the number. Various writers have tried to avoid ambivalence by the use of such expressions as "fraction number" and "mixed numeral", but there is little, if any, evidence that when children are taught to make such distinctions, learning is en-hanced. In fact, the context usually makes clear whether number or numeral is intended, and most teachers probably make no verbal distinction. In accordance with common usage, the present study will allow the context to determine whether number or numeral is meant. (However, the reader is referred to page 13 for the definitions of terms used.) The intuitive notion of a unit fraction, for example, one-half, one-third, and one-fourth, that students are exposed to at the primary level, is expanded at the intermediate level to encompass not only other forms of fractions such as non-unit fraction, improper fraction, and mixed number, but also fundamental operations on fractions, such relations as ordering and equivalence, and renaming 1 2 skills. However, the emphasis at the intermediate level is on developing skill in adding, subtracting, multiplying, and dividing fractions. Recurring evidence from studies indicates that the level of pupil perfor-mance with fractions is discouragingly low. One of these studies is the National Assessment of Educational Progress (NAEP). The NAEP project (Carpenter & Cobum, 1975) tested approximately 2500 individuals in 1972-73 at each of four age levels: 9-year-olds, 13-year-olds, 17-year-olds, and adults between 26 and 35 years of age. For the purposes of the present study the performance of the 13-year-olds will be reported because they are most representative of students at the intermediate level. Their performance on whole number computation was much better than their performance on fraction computation. For example, 94 percent correctly answered a whole number addition item (38 + 19), whereas only 42 percent correctly answered an addition item involving two fractions (1/2+ 1/3). In multiplication, 83 percent multiplied two digits by one digit correctly (38 x 9), but only 62 per-cent multiplied two fractions correctly (1/2 x 1/4). Both fraction items used simple unit fractions, and the answers did not require simplifying. In another study, Lankford (1972) assessed the computational skills of 176 seventh-graders (average age of 13), and this was followed by an interview to determine the strategies that each student employed. The fraction items he used were more complex than the NAEP items. Table 1 shows the items along with the frequency of right answers, wrong answers, and omissions for the addition and multiplication of fractions. The table shows that a decreasing number of students answered the addition and multiplication items correctly as the complexity of the items increased. This 3 Table 1 Frequency of Right, Wrong Answers and Omissions for Addition and Multiplication of Fractions* Items Right Wrong Omitted Percent of items attempted with right answers Addition 3/4 + 5/2 = 75 84 17 47 3/8 + 7/8 = 97 48 31 67 5 7/8 + 2 1/2 = 71 56 49 56 2/3 + 1/2 = 68 49 59 58 Multiplication 2/3 x 3/5 = 96 57 23 63 2 1 / 2 x 6 = 48 62 66 44 5 1/2 x 3/4 = 38 56 82 40 *Lankford, 1972, p. 17. 4 was accompanied by an increase in the number of students who omitted the more complex items. Overall, Lankford found a lower percent of right answers, a higher percent of wrong answers, and a higher percent of omissions in fraction computations than with whole number computations. The figures were 35 percent right answers in fractions compared with 76 percent for whole numbers, 33 percent wrong answers for fractions compared with 21 percent for whole numbers, and 32 percent omissions for fractions compared with 3 percent omissions for whole numbers. Although both studies based their figures on student achievement as measured by a test on each of the four operations, it would appear that errors and incorrect strategies resulted from a misunderstanding of fundamental fraction concepts applied to the operations, rather than an inability to perform a particular operation. For example, in both studies a major source of error involving the addition of fractions was found to be the adding of numerators and denominators, as in 1/2 x 1/3 = 2/5. It would appear that students are viewing the two fractions as four separate whole numbers, rather than as two quantities. In multiplication, Lankford found many students first wrote equivalent fractions with a common denominator and then multi-plied the numerators only, placing the product over the common denominator, for example, 2/3 x 3/5 = 10/15 x 9/15 = 90/15. These students have been taught the process of finding common denominators, but it appears to have been learned as a strictly mechanical process. They do not seem to understand when and why it needs to be applied. In summary, fractions have traditionally posed greater difficulties for students at the intermediate level than whole numbers. One of the problems appears to be 5 the relationship between fundamental fraction concepts and the four basic computa-tional processes of addition, subtraction, multiplication, and division of fractions. Formation of the Model An analysis by the experimenter of the four basic computational operations with fractions revealed a common subskill, that of renaming fractions. In other words, in order to perform a particular algorithm for an operation some renaming skill (or skills) is (or are) required. The three kinds of fractions, proper, improper and mixed, were considered in the analysis. The algorithms chosen were the ones most prevalent in the three elementary school mathematics textbook series: Investigating School Mathematics (Eicholz, et al, 1974), Heath Elementary Mathe-matics (Dilley, et al, 1974), and Project Mathematics (Elliott, et al, 1975), re-commended for grades 4, 5 and 6 in the Mathematics Curriculum Guide (1977) for the British Columbia school system. The reader is referred to page 15 for examples of the four algorithms used. The necessary renaming skills are outlined in Tables 2, 3 and 4. Table 2 presents a detailed list of the RAL (renaming for the algorithm) skills in relation with the four algorithms. In addition and subtraction, renaming from lowest to higher terms for both proper fractions and mixed numbers is required. Most textbooks present this skill under the heading "equivalence of fractions". For example, in adding 3 2/5 + 4 3/4, both 3 2/5 and 4 3/4 are renamed to higher terms with a common denominator. Subtraction requires three more renaming skills 6 Table 2 Renaming for the Algorithm (RAL) Skills with the Four Algorithms 'Addi-tion Subtrac-tion Multip-lication Divi-sion 1. Lowest to higher terms - proper fractions example 1/3 = 3/9 * * 2. Lowest to higher terms - mixed numbers example 2 3/5 = 2 6/10 * * 3. One as an improper fraction example 1 = 3/3 * 4. Whole number as a mixed-improper example 4 = 3/3/3 * 5. Mixed number as a mixed-improper example 6 1/4 = 5 5/4 * * 6. Mixed number as an improper fraction example 6 1/4 = 25/4 * * 7. Whole number as an improper fraction * * example 3 = 3/1 - 6/2 7 Table 3 Renaming for the Answer (RAN) Skills with the Four Algorithms Addi-tion Subtrac- Multip-tion lication Divi-sion 8. Higher to lowest terms - proper fractions example 3/9 = 1/3 * * * * 9. Higher to lowest terms - mixed number example 2 4/8 = 2 1/2 * < * * * 10. Improper fraction as one example 4/4=1 * 11. Mixed-improper as a whole number example 2 7/7 = 3 * 12. Mixed-improper as a mixed number example 7 10/7 = 8 3/7 * 13. Improper fraction as a mixed number example 65/12 = 5 5/12 * * 14. Improper fraction as a whole number example 56/8 = 7 * * 15. Improper fraction as a mixed number to lowest terms - 2 renamings example 75/12 = 6 3/12 = 6 1/4 * * 16. Mixed-improper as a mixed number to lowest terms - 2 renamings example 3 6/4 = 4 2/4 = 4 1/2 * 8 Table 4 RAL/RAN Pairs RAL RAN 1. Lowest to higher-proper fractions 8. Higher to lowest-proper fractions 2. Lowest to higher-mixed numbers 9. 9. Higher to lowest-_imixed numbers m;«ed 3. One as an improper fraction 10. Improper fraction as one 4. Whole number as a improper mixed- 11. Mixed-improper as a whole number 5. Mixed number as a improper mixed- 12. Mixed-improper as a mixed number 6. Mixed number as an fraction i improper 13. Improper fraction as a mixed number 7. Whole number as an improper fraction 14. Improper fraction as a whole number 9 to account for regrouping in examples such as 6 1/3 - 2 2/3 and 4 - 1 3/4. These three skills are: renaming one as an improper fraction (1 = 3/3), renaming a whole number as a mixed improper number (4 = 3 3/3) and renaming a mixed*1 number as a mixed-improper number (6 1/4 = 5 5/4). Multiplication and division require two further renaming skills. They are: renaming a mixed number as an improper fraction (6 1/4 = 25/4), and renaming a whole number as an improper fraction (3= 3/1 = 6/2). Altogether, seven renaming skills were identified as essential for performing the usual algorithms for the addition, subtraction, multip-lication, and division of fractions. It should be noted that not all the renaming skills for a particular operation are needed to complete every example. As well as the seven RAL skills, nine more renaming skills were identified as necessary for simplifying the answer once an algorithm was performed. These nine skills will be referred to as RAN (renaming for the answer) skills. Table 3 presents a detailed list of the RAN skills in relation with the four algorithms. Re-naming from higher to lowest terms for both proper fractions and mixed numbers is required to simplify sums, differences, products and quotients. However, sums may also require four further renaming skills, namely: renaming an improper fraction as one (1/4 + 3/4 = 4/4 =1), renaming a mixed-improper number as a whole number (2 3/7 + 4/7 = 2 7/7 = 3), renaming a mixed-improper number as a mixed number (3 5/7 + 4 5/7 = 7 10/7 = 8 3/7), and renaming a mixed-improper number as a mixed number in lowest terms (1 3/4 + 2 3/4 = 3 6/4 = 4 2/4 = 4 1/2). Multiplica-tion and division answers require three additional and different renaming skills. They are: renaming an improper fraction as a mixed number (1 2/3 x 3 1/4 = 5/3 x 13/4 = 65/12 = 5 5/12), renaming an improper fraction as a whole number (8 -f- 1 1/7 = 8/1 x 7/8 = 56/8 = 7), renaming an improper fraction as a mixed number in lowest terms (1 2/3 x 3 3/4 = 5/3 x 15/4 = 75/12 = 6 3/12 = 6 1/4). As is evident from Tables 2 and 3, the renaming skills from lowest to higher (RAL 1 and 2) and higher to lowest (RAN 7 and 8) are separated for proper fractions and mixed numbers. The reasoning behind the separation is two-fold. First, students are exposed to proper fractions before mixed numbers. Consequently, they have more practice in computing with proper fractions than with mixed numbers, and, therefore, their ability to rename fromldowest to higher and higher to lowest may be different for proper fractions than for mixed numbers. Second, students are traditionally taught each of the four operations with proper fractions before being introduced to mixed number computations. Therefore, renaming skills for both proper fractions and mixed numbers are required to perform the operations,,as well as to simplify the answers. A comparison of the RAL and RAN skills revealed that each RAN skill except for categories 15 and 16 is the reverse of one of the RAL skills. For example, RAL skill 1 (renaming from lowest to higher terms for proper fractions) is the reverse of RAN skill 7 (renaming from higher to lowest terms for proper fractions). The reader is referred to Table 4 for a list of the seven RAL/RAN pairs"..;. RAN skills 15 and 16 require two consecutive renaming skills. RAN 15 is a combination of RAN 13,followed by RAN 9. RAN 16 is a combination of RAN 12, followed by RAN 9. Because there is no single matching RAL skill for either RAN 15 or RAN 16, they are not included in Table 4. It appears from the experimenter's analysis that renaming has two distinct purposes in fraction algorithms. One purpose is to perform a required step for an 11 algorithm - RAL. The second purpose is to simplify the answer once the algorithm is performed - RAN. Furthermore, for each of the two purposes (refer to Tables 2 and 3), the renaming categories related to addition and subtraction appear to be different from those related to multiplication and division, except for RAN cate-gories 8 and 9 (renaming from higher to lowest terms for proper fractions and mixed numbers), which appear to be related to all four algorithms. In other words, renaming ski I Is, not only appear to serve two purposes, but also appear to make distinctions among the four algorithms. This study, in attempting to distinguish between the RAL skills and the RAN skills, must also distinguish between two possible interpretations of the word "answer". These two interpretations are: the algorithm answer (referred to as A]) which is obtained solely by performing the algorithm correctly, and the simplified answer (referred to as A2) which is obtained by simplifying the algorithm answer. The current practice in the British Columbia school system, as judged by textbook presentations of the subject, is to teach fundamental fraction concepts and skills (including both RAL and RAN skills) prior to introducing the computation algorithms. However, the textbooks do not differentiate between the two renaming purposes, and therefore, teach RAL and RAN skills in the pairs outlined in Table 4. For each pair, the RAL skill is usually developed first follbwedtby'.the-corresponding RAN skill. Then, when algorithms are introduced, students are expected to have competency in both the necessary RAL and RAN skills, even though it appears that RAL skills only are needed to complete the algorithm. No recognition is given that there are two possible versions of "correct answer". The simplified answer is 12 usually the only one accepted as correct. Perhaps the acquisition of the fraction algorithms and related subskills is a more complex learning task than the one associated with whole number computa-tions. Is it possible that students can complete the algorithm, but cannot simplify the answer? Is it possible that students find the learning of the algorithm with accompanying RAL skills difficult enough without expecting concurrent mastery of the RAN skills needed to simplify the algorithm answer? Does the acquisition of the RAL and RAN skills occur simultaneously or independently? Statement of the Problem The purpose of this study is to examine the place of renaming skills in .fraction computations. The problem has two parts. The first part is to gather evi-dence to support the model hypothesizing the relationship between the sixteen re-naming categories (seven RAL and nine RAN skills) and the four algorithms. The questions to be answered in this part are: 1. Are there grade six students who do not attain criterion level on RAL skills, but who do attain criterion level on the corresponding algorithms when the correct algorithm answer is all that is required? To support the model proposed there should be no such students. 2. Are there grade six students who do not attain criterion level on RAN skills, but who do attain criterion level on the corresponding algorithms when only the correct simplified answer is accepted? To support the model proposed there should be no such students. 3. Is there a significant positive correlation between each of the seven RAL/kAN pairs outlined in Table 4? The second part of the problem is experimental in nature and is designed to investigate two effects. The first effect to be investigated will be performance on subtraction following a review of the five RAL skills related to the subtraction algorithm. The second effect to be investigated will be performance on all four algorithms following a review of the purpose of the five RAL skills in the sub-traction algorithm. The questions to be answered in this part are: 4. Is there a significant difference in performance on subtraction between grade six students who review the five RAL skills related to subtraction, and those who do not? 5. Is there a significant difference in performance on each of the four algorithms between grade six students who review the purpose of the five RAL skills within the subtraction algorithm, and those who do not? Definition of Terms For the purposes of this study, the following definitions of terms were adopted: A fraction refers to a positive rational number that can be written in the form a/b, where anand b are positive whole numbers and b/= 0. A proper fraction is one whose numerator is smaller than its denominator, for example, 3/5. A unit fraction is a proper fraction whose numerator is one, for example, 1/5. An improper fraction is one whose numerator is equal to or greater than its denominator, for example, 5/3 or 3/3. A mixed number: is a combination of a whole number and a proper fraction, for example, 1 3/5. A mixed-improper number is a combination of a whole number and an improper fraction, for example, 1 5/3. Lowest terms refers to a proper fraction or mixed number whose numerator and denominator have no common factor greater than one, for example, 3/4 or 2 .2/5. Simplifying means renaming a proper fraction or mixed number from higher to lowest terms, renaming an improper fraction as a whole number or mixed number in lowest terms, or renaming a mixed-improper number as a whole number or a mixed number in lowest terms. An algorithm is a procedure which, by correct sequential application to a computation example results in a correct answer. Within the context of the present study the algorithms referred to are the methods used to perform each of the four basic computations on fractions. They are as follows, with examples: addition of fractions 11 5/8=11 45/72 + 20 5/9 = 20 40/72 15 subtraction of fractions 16 2/6 = 16 14/42 = 15 56/42 - 12 3/7 = 12 18/42 = 12 18/42  3 38/42 =(3 19/21) multiplication of fractions 3 3/4 x 2 1/5 = 15/4 x 11/5 = 165/20= (8 1/4) division of fractions 2 2/7 4 3 1/9 = 16/7 x 9/28 = 144/196 = (36/49) The bracketed answer denotes the simplified or A2 answer. The underlined answer denotes the algorithm or A] answer. Hypotheses to be Tested Three major hypotheses are to be tested in Part One of this study. The first two are stated as predictions of the proposed model, independent of whether their forms are null, or otherwise. The third hypothesis, however, is stated in the null form, as it is based on a statistical test. H]: There arei no grade six students who do not attain criterion level on the seven RAL categories, but who do attain criterion level for the algorithm answer on the related computations, as measured by pretest scores. H2: There are no grade six students who do not attain criterion level on the nine RAN categories, but who do attain criterion level for the simplified answer on the related computations, as measured by pretest scores. 16 Hg: There is no significant correlation between each of the seven RAL/RAN pairs, as measured by pretest scores. As the hypotheses in Part Two are tested statistically, they are stated in null form, independent of whether they are consequences of the proposed model. H4: There is no significant difference in performance on the subtraction algorithm involving fractions, between grade six students who are given a review of the five RAL skills related to subtraction, and those who are not given the review, as measured by posttestst one scores. H5: There is no significant difference in performance on the four algorithms involving fractions between grade six students who are given a review of the purpose of the five RAL skills within the subtraction algorithm, and those who are not given the review, as measured by posttests two scores. Chapter 2 REVIEW OF THE RELATED LITERATURE Introduction Although specific research in the area of renaming in fraction work is lacking, several studies contributed towards or supported this study. They will be discussed under two headings: (1) research of specific relevance to the problem, and (2) research of general relevance to the problem. Research of Specific Relevance to the Problem The idea for the formation of a classification model to encompass the major renaming skills related to the four algorithms germinated from a study done by Up-richard and Phi Mips (1973). Their investigation was concerned with the generation and validation of a hypothesized learning hierarchy for rational number addition. Problems were divided into two levels: like -denominator, and unlike denominator. Classes within each level were identified by the nature of (prime or composite), and the relationship between (multiple or not) the denominators of two fractions. Of relevance to the present study are the five sum categories within each denomi-nator class that were generated and operationally defined in terms of renaming. However, the five sum categories do not exist in some classes. The five sum cate-gories in their hypothesized order of difficulty are: 17 18 Sum: II Sum: improper fraction renamed to one (1/4 + 3/4 = 4/4 = 1). Ill Sum: proper fraction renamed to simplest form (1/4 + 1/4 = 2/4 = 1/2). IV Sum: improper fraction renamed as a mixed numeral in simplest form, one renaming? (3/4 + 2/4 = 5/4 = 1 1/4). V Sum: improper fraction renamed as a mixed numeral in simplest form, two renamings, (3/4 + 3/4 = 6/4=1 2/4 = 1 1/2) (pp. 8-9). A test of the 45 tasks in the final hierarchy was constructed to assess mastery at each level. The test was then administered to 251 students in grades 4 through 8, with the majority in grades 5 and 6. "Pass" was defined as an expression of the correct sum in simplest form. The patterns of responses in the hierarchy were analyzed using two methods, the Walbasser method and pattern analysis. The following result in the empirically determined sequence is of interest in the present study. The sum categories in hierarchical order for like denominators were found to be I, II, IV, III, and V while for unlike denominators;they were I, II, III, IV, and V. In other words, sum category V (renaming an improper fraction as a mixed number with two renamings) was analyzed to be the most difficult of the five renaming categories generated. as a basis (category I was omitted as it did not require renaming), and analyzing the addition of fraction problems in current mathematics textbooks for grades six and seven, five renaming categories were decided upon as skills essential to renaming a sum in the present study. As in Uprichard and Phillips (1973) Using four of the categories from the Uprichard and Phillips (1973) study 19 study, the skills involving two renamings were hypothesized-to be the most difficult, and are therefore listed last. However, the rest of the renaming skills are not in a hypothesized order of difficulty. The skill of renaming-an improper fraction as a mixed number with one renaming (category IV) was excluded in the writer's classifi-cation for addition of fractions since it applies only to sums (and differences) less than 2. The addends traditionally used in textbooks are proper fractions (4/5 + 4/5 = 8/5=1 3/5), not improper (6/5 + 7/5 = 13/5 = 2 3/5), and there-fore do not exceed 2. Instead, this skill is more applicable to the renaming of products and quotients involving mixed numbers, and is therefore related to multi-plication and division. A further analysis of renaming skills required for differences, products, and quotients resulted in the RAN classification outlined in Table 3. The formation of the RAL categories, as outlined in Table 2, was a logical outgrowth of the RAN categories. Research of General Relevance to the Problem The research of general relevance to the problem can be organized into two groups. The first group will be referred to as the "Michigan studies". The second group will be referred to as error analysis studies. Michigan Studies The so-called "Michigan studies" (Payne, 1976, p. 145) are a series of experimental studies conducted at the University of Michigan beginning in 1968. The studies began as investigations of the efficacy of various alternative algorithms 20 for the four basic computations. They then shifted towards a more intensive exami-nation of what children learn while being taught a carefully developed algorithm. In particular, the focus was on when and how children learn equivalent fractions and how to generate them. It should be noted that the term "equivalent fractions" used by these studies refers to the four renaming skills numbered 1 (renaming from lowest to higher terms with proper fractions), 2 (renaming from lowest to higher terms with mixed numbers), 8 (renaming from higher to lowest terms with proper fractions), and 9 (renaming from higher to lowest terms with mixed numbers) as listed in Tables 2 and 3. The first in this series of "Michigan studies" was conducted by Bidwell (1968). He developed a learning sequence for three division of fraction algorithms, and then tested the efficacy of each algorithm with sixth graders. He found the inverse operation algorithm sequence to be consistently superior to either the complex frac-tion and common denominator sequences. The three algorithms he tested are: Common Denominator a/b - i - c/d = ad/bd -j- bc/fed = ad f be = ad/bc Complex Fraction a/b f c/d = a/b E aAx.d/c = a/b x d/c = a / b x d/<_ = c/d c/d x d/c 1 Inverse Operation a/b "T c/d = =^ Q x c/d = a/b ( |~] x c/d) x.d/c = a/b x d/c Q x (c/d x d/c) = a/b x d/c f~~| x 1 = a/b x d/c ^ = a/b x d/c =^ Q = ad/bc 21 Two of Bidwell's findings are of significance to the present study. He found major content areas that were poorly learned by the students. Equivalent fractions was one such content area and it is required to perform the complex fraction and com-mon denominator algorithms. The other finding showed the importance of careful development of the algorithm with the major subskills required in its performance. The premise in the present study is that renaming is a major subskill in the per-formance of the four basic computations. As Bidwell's study (1968) was concerned with the symbolic mode of repre-sentation, and not the concrete or pictorial modes, Green (1969) investigated the effects of concrete materials versus diagrams and two models for teaching multipli-cation of fractions on fifth graders. The two models were the "area" model and "finding fractional parts of 11 model. In the area model, a rectangular region was used to develop an algorithm for multiplication of fractions which extended an idea developed for multiplication of whole numbers. For example, 2/3 x 3/4 = n is illustrated by the shaded area in Figure 1. Since area = length x width, the area is 2/3 x 3/4. ,2 3 Area - — )C — 3 4 Figure 1. An example of the "Area" model for the multiplication of fractions (Green, 1969). The "fractional parts of" model is the more traditional model used where the first factor is used as a multiplier to find a fractional part of a region. To illustrate 2/3 x 3/4 = n using this approach, one begins with 3/4 of a region as in Figure 2a. Then the number 2/3 is used as an operator and two-thirds of the shaded region is found as in Figure 2b. Figure 2a Figure 2b Figure 2. An Example of the "Fractional Parts of" Model for the Multiplication of Fractions (Green, 1969). Green (1969) found that the "area" approach was better for learning multiplication of fractions than the "fractional parts of" model. Diagrams and manipulative mater-ials were equally effective in learning multiplication of fractions. Of importance to the present study is the difficulty all fifth graders had in Green's study with renaming a mixed number as an improper fraction and simplifying to lowest terms. Since both Bidwell and Green found that students had difficulty with re-naming to lowest terms, the subsequent study by Bohan (1970) was concerned with equivalent fractions. He designed and analyzed three sequences for teaching equi-valent fractions to fifth graders. One used diagrams to develop equivalent fractions, one used a paper folding technique, and a third introduced multiplication first and then the property of 1 (multiplicative identity). Each sequence included generalizing 23 the method for finding equivalent fractions, that is, multiplying and dividing the numerator and denominator by the same number. He found that no sequence was superior for changing equivalent fractions to lowest terms, but that the paper folding sequence was significantly better than the other two for generating equivalent frac-tions to higher terms. A further study on equivalent fractions was conducted by Coburn (1973), comparing a ratio model and a region model for generating equivalent fractions at the fourth grade level. He found both models to be equally effective with the re-naming generalization. However, the mean percentage correct for renaming to higher and lower terms for both treatments was just slightly better than 50. Therefore, he questioned equivalent fractions as a fourth grade topic. In fact, he recommended a delay in the verbalization of the rule for generating equivalent fractions until the need arose in the addition of unlike fractions. Coburn (1973) pointed towards the importance of a sound development of initial fraction work before learning the algorithms for the computations. Consequently, three pilot studies were conducted in 1974 by seven investigators: Payne, Coate, Ellerbruch, Greeno, Muangnapoe, Galloway & Williams, to design a sequence of initial fraction concepts and skills that fourth graders could be expected to master (Payne, 1976). The following topics were included in the sequence: identification of wholes and parts; partitioning units into equal-size parts; counting the number of parts to consider and parts in the whole unit and development of number pair ideas; introducing fraction symbols and word names; presenting mixed forms, fractions greater than one, and number lines; comparing fractions; and equivalent fractions. The generalization for developing equivalent fractions was not included, neither was 24 simplifying. Muangnapoe (1975) then investigated the initial fraction sequence (IFS) thus designed, to assess its effectiveness for the learning of the initial concepts and oral/written symbols for fractions in grades three and four. A major conclusion from his study was that third and fourth graders can achieve mastery at the 80 per-cent level of these initial fraction concepts and skills. Also, a strong suggestion was made for greater emphasis in instructional materials and methodology for begin-ning fraction work in the elementary schools. However, in the present study it was necessary to assume that the subjects had an appropriate background in beginning fraction work and could benefit by a review of renaming skills. Error analysis studies The second group of studies of general relevance to the present study ana-lyzed student errors. Many of the errors cited are attributable to deficiencies in the renaming skills outlined in Tables 2 and 3. In the 1920's two researchers, Morton (1924) and Bruekner (1928) conducted separate studies to analyze and categorize specific difficulties that students have with the four fraction computations. Morton (1924) collected data from the diagnostic test papers of three eighth graders. He identified nine types of errors in addition of fractions, seven in subtraction of fractions, nine in multiplication of fractions, and ten in division of fractions. It should be notedrthat only proper fractions were involved. A summary of the major causes of errors in each computation is as follows: A r s t l ^ ' o n (p. 119) 25 Addition (p. 119) percent a) wrong operation performed 40.2 b) error in computation 22.0 c) failures to convert addends to a common denominator 6.1 d) failure to reduce answers to mixed numbers 3.7 e) failure to reduce answers to lowest terms 2.4 Subtraction (p. 121) a) errors in computation 33.8 b) denominators subtracted 28.6 c) failure to reduce answers to lowest terms 26.0 d) failure to use a common denominator 11.7 Multiplication (p. 124) a) errors in computation 16.3 b) failure to reduce answers to lowest terms 16.3 c) wrong operation performed 9.3 d) denominators added 9.3 Division (p. 125) a) wrong operation performed 35.5 b) errors in computation 21.3 c) failure to reduce answers to lowest terms 1.4 d) failure to reduce answers; to mixed numbers 3.5 26 Most of the errors found were ascribed to three causes: (1) inadequate conception of the processes involved (mainly renaming difficulties), (2) confusion of the opera-tions (mainly using the wrong operation), and (3) lack of an adequate degree of skill in the fundamental operations (mainly computational errors). Of the renaming skills cited, simplifying answers, particularly in subtraction and multiplication, appears to present a difficulty. However, since mixed numbers were not included in the test items, the significance of the results have only limited value to the present study. Bruekner (1928) analyzed the different types of examples that students might encounter in the four operations involving both proper fractions and mixed numbers. Each type was different in some aspect which might cause difficulties. Forty dif-ferent types were identified in addition of fractions, forty-five in subtraction of fractions, forty-five in multiplication of fractions, and thirty-seven in division of fractions. Tests were then constructed for each operation and administered to 200 students in each of grades 5A, 6A, and 6B (A denotes the first half of the school year and B denotes the second half). Causes of errors were carefully studied and cate-gorized. A much more comprehensive list of errors was identified by Bruekner than by Morton. Bruekner summarized the major causes of errors in each operation as follows: Addition (p. 768) percent a) Lack of comprehension of process involved 20.2 b) Difficulty in reducing fractions to lowest terms 17.5 c) Difficulty with improper fractions 17.1 d) Computation errors 13.8 2,7 Subtraction (p. 768) percent a) Difficulty in borrowing 24.3 b) Used wrong process 20.3 c) Difficulty in reducing fractions to lowest terms 14.6 d) Lack of comprehension of process involved 14.6 e) Difficulty in changing fractions to common denominator 8.3 f) Computation errors 8.2 Multiplication (p. 769) a) Computation errors 28.7 b) Lack of comprehension of process involved 17.3 c) Difficulty in reducing fraction to lowest terms 17.3 d) Omitted example (probably lack of comprehension) 11.3 e) EaMure to change improper fractions to mixed numbers 8.8 f) Difficulty in changing mixed numbers to improper fractions 2.8 Division (p. 769) a) Used wrong process 31.1 b) Computation errors 13.8 c) Lack of comprehension of process involved 12.1 d) Difficulty in reducing fractions to lowest terms 8.9 e) Omitted example (lack of comprehension of process?) 8.3 Division (continued) percent f) Difficulty in changing mixed numbers to improper fractions 8.6 g) Failure to change improper fractions to mixed numbers 7.2 It appears that three major causes of errors for all four operations were: (1) lack of comprehension of the process involved (mainly performing one of the steps in the algorithm incorrectly, or not at all), (2) difficulty in simplifying fractions to lowest terms, and (3) difficulty in changing improper fractions to whole or mixed numbers. Of these three causes, the latter two are directly related to renaming skills. In a later study Anderson (1965), while analyzing errors made by 599 fifth grade pupils in addition and subtraction of fractions, found that the two categories with the highest percentages of errors related to equivalent fractions. The most frequent error found was incorrectly renaming fractions to lowest terms. The next most frequent error was incorrectly determining the numerator after the denominator of an equivalent fraction had been found. Multiplication and division of fractions were not analyzed. Once more, a frequent error in fractions appears to be related to renaming skills. In 1972, Lankford assessed and interviewed the computational capabilities of 176 seventh graders, and found simplification to lowest terms to be a major source of difficulty. In all the computations with fractions he found examples of correctly derived answers with errors occurring in converting to simpler form. This is the/only study found by the present writer that distinguished between the two types of answers, namely the algorithm answer and the simplified answer. Recurring problems with other renaming skills were also cited in Lankford's study. He found many errors were made in writing equivalent fractions with common denominators for addition and subtraction. Also, students had difficulty in renaming mixed numbers or mixed-improper numbers for regrouping in subtraction. The nature of wrong answers in multiplication and division appeared, not as a problem with renam-ing skills, but as lack of comprehension ofi the algorithm itself. For example, in multiplication many students first wrote equivalent fractions unnecessarily, and then incorrectly multiplied numerators, and placed the product over the common denomi-nator; Some pupils did not rename mixed numbers as improper fractions, but in-stead multiplied the fractions and affixed the whole number (5 1/2 x 3/4 = 5 3/8). In division, as in multiplication, a common practice was to divide the numerators and place the quotient over the common denominator as in the example, 9/10 — 3/10= 3/10, or to divide the numerators and denominators, ignoring the remainders, as in the example, 7/8 2/3 = 3/2. Some pupils forgot to use the reciprocal of the divisor before multiplying. From Lankford's study (1972), it appears that renaming poses more difficulty with addition and subtraction, rather than multip-lication and division. However, like Morton (1964) and Bruekner (1928), Lankford found simplification difficulties common to all four computations with fractions. Summary and Significance of the Research Although research on fractions is abundant, little attention has been paid to the various renaming skills required in fraction algorithms. If renaming has been examined at all, it is always with respect to the concept of equivalence as in Bohan's (1970), and Coburn's (1973) studies. In both these studies, renaming from 30 lowest to higher and higher to lowest terms was expected. A distinction was not made between the two renaming skills. However, the research is unanimous in citing simplifying to lowest terms, either as an added complication to their study, or as a major source of error. No experimental studies could be found by the present writer that distinguished between the algorithm answer and the simplified answer, when examining achievement scores on tests involving fraction computations. Either simplification was not re-quired, or the simplified answer was the only correct one accepted. Furthermore, no experimental studies could be found that examined the effects of teaching only the renaming skills required to perform a fraction computation algorithm. Therefore, it seems important that research be conducted to examine the purpose of renaming in fraction algorithms for computation. Chapter 3 DESIGN AND PROCEDURE Experimental Design The present study employed a repeated measures, hierarchical design in which six sixth-grade classes were nested within three experimental groups. Using the uniform code and graphic presentation employed by Campbell and Stanley (1963), the design of this study can be depicteded as follows: Group A: 0] X] O2 X 2 O3 Group B: Oj X J 0 2 Og Group C: 0| O2 Og The 0 refers to "some process of observation or measurement" (p. 6).' For the present study the 0 refers to the measuring instruments used: Oi is the pretest, 0 2 is post-test one, and O3 is iposttestittwoJwo. The X represents "the exposure of a group to an experimental variable or event, the effects of which are to be measured" (p. 6). Xj and X 2 refer to two different treatments. The O's and X's in a given row are applied to the same group, while the number of rows denotes the number of groups in the experiment. As indicated, the present study consists of three experimental groups designated A, B, and C. Experimental group A received two treatments (Xi which was a review of the five RAL skills related to subtraction, and X 2 which related these five RAL skills to subtraction). Experimental group B received only 31 32 the Xj treatment. On the day experimental group A was receiving the X 2 treat-ment, experimental group B received a lesson in geometry. Experimental group C, however, did not receive either the Xj or the X 2 treatments, but continued with their regular mathematics program, receiving instruction in geometry and measurement. In the design, it is also important to note that "the left-to-right dimension in-dicates the temporal order, and O's and X's vertical to one another are simultaneous" (p. 6). In the present study, "simultaneous" means on the same day during the regu-larly scheduled mathematics period. Therefore, the pretest (0]) was administered "simultaneously" to all three experimental groups, followed by treatment one (Xj) to experimental groups A and B only. Then, posttest one (02) was administered "simultan-eously" to all three experimental groups. Treatment two (X 2) was then given to experi-mental group A, followed by posttest two (Og) administered to all three experimental groups "simultaneously". For an illustration and full discussion of the hierarchical nature of the ex-perimental design, that is, classes nested within treatment groups, the reader is referred to Figure 3 and page 34. Formation of the Group The Population Sixth-graders were chosen as the population from which the experimental groups could be drawn. Two factors were involved in this decision. First, students of this grade could be expected to have had exposure to both RAL and RAN skills, as well as the addition and subtraction algorithms with mixed numbers. Therefore, their knowledge of the sixteen renaming categories, and performance on the four algorithms for computation could be measured. Second, it was felt that seventh 33 graders would have had too much instruction and review of the four basic computa-tions for the treatments to have any effect. The sample Six classes of heterogeneously grouped sixth-graders from three elementary schools were used in this study. The schools, located in a large suburban area, were selected on the basis of their size, that is, each school contained two intact grade six classes. No split classes of grades 5/6 or 6/7 were used. Although 157 students participated in the study, 114 were used in the final analysis. Two factors accounted for the decrease in the original number of students. First, 25 students were eliminated due to incomplete or missing data on one or more of the three tests. This left five classes, each containing 22 students, and one class with 19 students. Second, to facilitate the statistical analysis, the number of students in each class was equalized by randomly eliminating three students from each of the five classes containing 22 students. Although random selection of classes was not possible, it was felt that the classes were not originally selected on the basis of variables that are directly relevant to the study and, therefore, were considered as random samples from a common population. Assignment to Experimental Groups The experimenter was not at liberty to assign subjects at random to different treatments, but was required to use the groups already formed. Also, to avoid with-in school variance for any one treatment, no experimental group was replicated in the same school. Therefore, two classes from different schools were assigned to each experimental group. Within each school, random assignment of classes to experimental groups was done. The assignment of classes to experimental groups is summarized by Figure 3. Schools Experimental Group — — 1 2 3 Classes 1 2 23 4 5 6 A * * B * * C * * Figure 3. Assignment to Experimental Groups 35 Development of Measuring Instruments Pilot Test No standardized tests involving the renaming skills and fraction computa-tions were available'. Consequently, all tests were constructed by the experimenter. However, in order to determine total test reliability, subtest reliability, and ad-ministration time, a pilot test was constructed and administered to 46 students from two intact grade six classes. The classes were from one school located in the same suburban district as the experimental schools. The pilot test was constructed in three parts. Part A consisted of seven subtests of 35 items (five items per subtest), to measure performance on each of the RAL categories listed in Table 2. Part B consisted of nine subtests of 45 items (five items per subtest), to measure performance on each of the RAN categories listed in Table 3L Part C consisted of 24 items on the four algorithms (five items on addition, eight items on subtraction, five items on multiplication, and six items on division). Two versions of Part A and Part B were constructed to determine if differences in student performance occurred as a result of the magnitude of the mixed numbers. For example, one item used in version one was 5 7/8 = 5 E/72. The comparable item used in version two was 25 7/8 = 25 E/72. Comparing the two versions, twelve items were different in Part A and six items were different in Part B. The items in each version were randomly ordered using a table of random numbers (Glass & Stanley, 1970, pp. 510-512). In each class, every second student in the seating order received version one, and all other students received version two. 36 One regularly scheduled mathematics period was used to administer both versions. Part C of the pilot test was then administered to both classes on the following day. Copies of the two versions of the pilot test are presented in Appendix A. Total test reliability based on Hoyt's estimate was .98 for both version one and version two of Part A and Part B. The estimate for Part C was .88. Therefore, as the pilot test was considered to have a high reliability, the number of items per subtest on Parts A, B, and C was felt to be adequate. Based on subtest reliabilities, version one was chosen to form the core of items for Part A and Part B. Version one had a slightly higher Cronbach Alpha (r = .96), than version two (r = .95). Table 5 lists the reliability coefficients for the twenty subtests in the pilot test. Based on an item analysis, seven items on version one (two items in Part A, and five items in Part B) were revised. Four of the seven items were replaced with comparable items from version two, as the version two items had higher point-biserial correlations. The remaining three items were changed as they had lower point-biserial correlations (r ^ .7). The only revisions made to items in Part C were to ensure that the algorithm answer required simplifying. In this way, each item in Part C would have both an algorithm answer, and a simplified answer. Time to administer the pilot test was set at two, forty-minute periods, one period for Parts A and B, and one period for Part C. 37 Table 5 Hoyr Estimate of Reliability on Subtests of the Pilot Test Subtest Version One Version Two Part A - RAL Part B - RAN Part C 1 .84 .83 2 .81 .84 3 .88 .91 4 .93 .93 5 .91 .88 6 .85 .84 7 .92 .75 8 .77 .70 9 .84 .84 10 .87 .97 11 .86 .97 12 .84 .86 13 .90 .73 14 .78 .84 15 .84 .69 16 .90 ,83 Addition .35 Subtraction .84 Mu Implication .75 Division .73 38 Pretest The revised version of the pilot test was used as the pretest measure of the seven RAL skills, the nine RAN skills, and the four basic computations. Two forty-minute periods were used to administer the pretest. A copy of the pretest is presented in Appendix B. Table 6 shows a list of the algorithm items, together with the RAL skills required to obtain the algorithm answer, and the RAN skills required to simplify the answer. Posttest One This test consisted of two parts. Part A measured the five RAL skills re-lated to subtraction. Part B measured computational skill in subtracting fractions. The items used were identical to those used to measure the comparable skills in the pretest. The twenty-five items in Part A, and the eight items in Part B were placed in a random order using the same table of random numbers used for ordering the pilot test items. One forty-minute period was used to administer posttest one. Appendix C contains a copy of Posttest one. Posttest Two Like the pretest, posttest two consisted of three parts: Part A to measure the seven RAL skills (35 items), Part B to measure the nine RAN skills (45 items), and Part C to measure the four algorithms (24 items). The items used were identical to those used for the pretest. The only difference between the two tests was the use of a different random order for the items. Two forty-minute periods were used to administer posttest two. A copy of posttest two appears in Appendix D. "Table 6 Algorithm Items With the RAL and RAN Skills Required to Solve Them Addition Items RAL Skills* RAN Skills* 16 4/5 + 9 5/6 2 9 8 2/8 + 6 3/5 2 9 49 5/9 + 60 5/6 2 12 11 5/8 +20 5/9 2 12 34 12/20 + 32 21/25 2 16 Subtraction 84 3/10 - 14 7/10 5 9 31 7/36 - 16 13/36 5 9 1 9 - 4 7/21 4 9 40 -~22 32/40 4 9 16 2/6 - 12 3/7 2, 5 9 32 2/5 - 17 4/6 2, 5 9 9 3/16 - 4 5/10 2, 5 9 31 13/50 - 19 18/20 2, 5 9 Multiplication 8 X 4 3/4 6, 7 14 9 X 34/7 6, 7 13 2 2/9 X 3 3/5 6 14 3 3/4 X 2 1/5 6 15 3 1 / 4 X 2 1/5 6 13 Division 6 2/7 4 - H 6, 7 8 16 4/5 - f 4 6, 7 15 6-5- 2 2/5 6, 7 15 8 4 - 1 1 / 7 6, 7 14 3 3/4 4-1 2/3 6 15 2 2/7 - r 3 1/9 6 8 *refer to Tables 2 and 3 in Chapter 1 for the names of the skills 40 Development of Instructional Materials Treatment one (Xj) consisted of five forty-minute lessons, developed by the experimenter to review the five RAL skills related to subtraction. Treatment two (X 2) consisted of one forty-minute lesson to review the subtraction algorithm with reference to the five RAL skills. Each lesson was composed of a written lesson plan for the classroom teacher to follow, and a worksheet for each of the students to complete. Copies of the lesson plans with accompanying worksheets for both treat-ments are included in Appendix E. Three factors were involved in determining the type of, and use of, concrete and visual aids in the development of the lesson plans. First, Bohan(1970) observed that the paper folding technique appeared to make a logical connection between the concrete model of paper strips and the generalization for renaming fractions in higher terms. Second, Green (1969) found that the use of diagrams of regions was equally as effective as manipulating paper squares in ('earning to multiply fractions. Third, the lessons were considered to be a review of previously taught skills. Consequently, the experimenter decided that only the tepchers should use actual paper strips and paper folding to demonstrate the generation of equivalent fractions. Thereafter, diagrams of paper strips were used, followed by the symbolic representation of fractions. Procedure One month prior to the experiment, the pilot test was administered. Follow-ing the analysis of the pilot test, meetings were held with the principals of all of the participating schools, and the teachers of the participating classes to acquaint 41 them with the purpose of the study and the experimental procedure to be followed. At these meetings, the teachers were notified of their assignment to experimental groups, together with the treatments involved. Experimental group C teachers were asked to choose the geometry and measurement topics that they would be teaching during the experiment, and to inform the experimenter of the topics. Three days prior to the start of the experiment, a second set of meetings with the classroom teachers was held. The purpose of these meetings was to distri-bute the tests and the schedule for testing, together with the lesson plans, worksheets, and general instructions. A total of eleven consecutive school days was required for the experiment, five for testing and six for treatments. The pretest was administered on the first two days. The following five days were devoted to treatment one. On the eighth day, posttest one was administered. On the ninth day treatment two was given. Posttest two was administered on the tenth and eleventh days. Figure 4 provides a summary of the procedure followed. Days 1 22 3 3 4 45 6 76 8 9 10 11 P P P Experimental Group A R Treatment One O T. Two O E S S Experimental Group B T c Treatment One T Geometry T Experimental Group C c s Geometry and O Geometry T T Measurement N W E O Figure 4. Procedure 42 Each day of the experiment, the experimenter visited the participating classes to answer questions, and to collect tests or completed worksheets. . Controls Precautions were taken in attempting to control for several variables: initial class differences in achievement of fraction work, nested factor, testing effect, ex-perimenter bias, teacher effects, and Hawthorne-type effects. Each variable is dis-cussed in the following paragraphs. Pretest scores were used as covariates in an analysis of covariance to cont-rol for initial differences among the classes in fraction knowledge. As the three parts of the pretest (measuring RAL, RAN, and algorithmic skills, respectively) were considered to be conceptually different, it was decided to form three covariates: RALTOT (total scores on RAL skills), RANTOT (total scores on RAN skills), and ALGTOT (total scores on the four algorithms). Each covariate was used to adjust the scores onr tftenconre.spo;ndtngs'part For example, RALTOT was used to adjust the scores for Part A on posttest one, while ALGTOT was used to adjust the scores for Part B. For posttest two, RALTOT was used as the covariate for Part A, RANTOT was used as the covariate for Part B, and ALGTOT was used as the covariate for Part C. The factor of class differences nested within treatment groups was tested for significance. It was decided to employ a pooling strategy if no class differences were found. The practice effects of testing were controlled by Experimental Group C. As this group did not receive the experimental treatments, the sensitizing effects 43 of repeated testing could be detected by comparing their test results. Experimenter bias was minimized by having the regular mathematics teacher for each class participate in the experiment. The experimenter provided the instruc-tions, materials, and tests for each teacher. Teacher effects were controlled in three ways. First, the classes within treatment groups were randomly assigned. Second, the teachers in each treatment group were given the same written instructions to follow. Third, the experimenter visited each class to ensure, as much as possible, that the instructions were being followed. Hawthorne-type effects were controlled in two ways. First, all teachers were instructed to inform their classes that they were to take part in an experiment on fractions. Second, to ensure uniformity of the novelty effect of participating in an experiment, the experimenter visited all participating classes on a regular basis to collect completed tests, and worksheets. Statistical Analysis Data From the three test (pretest, posttest one, and posttest two) administrations, 50 different scores were obtained for each student. The pretest revealed 24 separate measures (7 for RAL skills, 9 for RAN skills, 4 for the algorithm answer on the four algorithms, and 4 for the simplified answer on the four algorithms). Posttest one resulted in 6 separate measures (5 for RAL skills related to subtraction, and 1 for the algorithm answer on subtraction), fosttest two resulted in 20 separate measures 44 (7 for RAL skills, 9 for RAN skills, and 4 for the algorithm answer on the four algorithms). Statistical Procedures To correspond with the two-part problem outlined in Chapter 1, the statisti-cal procedures will be presented under two headings, Part One and Part Two. Part One. This part of the study was designed to establish whether a relation-ship exists between renaming skills and the four fraction algorithms. Pretest data was used in this analysis. An item analysis (Nelson, 1974) was conducted on: pre-test results to determine subtest scores for each subject. The criterion level for each subtest of the seven RAL, and the nine RAN skills (five items per subtest) was set at four to ensure attainment of the renaming skills, while allowing for simple computational error. The criterion level for addition, which contained five items, was set at four to ensure that attainment meant adding two mixed numbers whose denominators have a common factor, for example, 9 and 6; or are relatively prime, for example, 5 and 6. Again, computational error was allowed for. Of the eight subtraction items, two involved subtracting like denominators, two involved subtracting denominators that have a common factor (16, 10), two involved subtracting denominators that were relatively prime (6, 7), and two involved subtracting a whole number from a mixe'd number. To ensure subtraction achievement, the criterion level was set at six. In this way, attainment of subtraction meant answering two of the four types of items correctly, allowing for computational errors in the other two types, or answering three of the four types correctly. 45 The criterion level for multiplication, which contained five items, was set at four to ensure that attainment meant multiplying a whole number by a mixed number, as well as two mixed numbers. Computational error was also allowed for. Of the six items for division of fractions, two involved dividing a whole number by a mixed number, two involved dividing a mixed number by a whole number, and two involved mixed numbers. To ensure division ability, the criterion level was set at five to ensure that attainment meant answering all three types of items, as well as allowing for computational error. The criterion level for the algorithm answer (A]) and the simplified answer (A 2) for each algorithm was the same. Therefore, a subject attained criterion on a particular skill or algorithm, by scoring at, or above the set criterion level. A computer program (Herberts, 1977) was used to tablulate the frequency of subjects who attained, or did not attain criterion level on each renaming skill (both RAL and RAN), together with the frequency of subjects that attained, or did not attain criterion level on the related computations with respect to the algorithm answer (Aj) and the simplified answer (A 2). To support the model hypothesizing a relationship between the renaming skills, and the four basic computations there should be virtually no students (less than three percent) who cannot attain criterion level on the necessary RAL skills, but who can attain criterion level for the algorithm answer. Also, there should be virtually no students (less than three percent) who" cannot attain criterion level on the necessary RAN skills, but who can attain criterion level for the simplified answer. Three percent was arbitrarily accepted, as it was felt that zero was too stringent. 46 The item analysis (Nelson, 1974) produced a correlation matrix for all the dependent variables measured. The critical value of the correlation was calculated to be .2185, or approximately .22, with 112 degrees of freedom (n - 2), at the .01 level of significance (Glass & Stanley, 1970, pp. 310 and 536). If the cor-relations between selected variables were found to be .22 higher, the null hypothesis of no correlation would be rejected. Part Two. Prior to testing for main treatment effects, a preliminary data ana-lysis was conducted to test for the nested factor. A multivariate analysis of covariance (Multivariance, 1974) was employed, in which classes within treatment levels were isolated. The covariate in each analysis was either RALTOT, RANTOT, of ALGTOT. If no differences were found at the .25 level of significance, it was decided to pool this source of variance with the subjects within treatments and classes, thereby increasing the power for testing the main effects. The .25 level was decided upon to lessen the probability of committing a Type 2 error, that is, of accepting the null hypothesis of zero interaction between classes within treatment levels, when it should be rejected (Kirk, 1968, p. 215; Winer, 1962, p. 203). As more than one dependent variable was involved in the second part of the study, a multivariate analysis of covariance (Multivariance, 1974) was em-ployed. The covariate in each analysis was either RALTOT, RANTOT, or ALGTOT. If the F ratios obtained in the analyses were found to be significant at the .05 level, a post hoc analysis (Winer, 1962, p. 586) for multiple comparisons was adopted to test the significance of the difference between adjusted means for each pair of groups. The effective error variances used in determining the F ratios required for the post hoc test were found from the formula: in which M S e r r o r ' is the adjusted mean square error for the dependent variable, n is the number of experimental units assigned to each of the experimental groups, "Txx is the sums of squares for the covariate, k is the number of experimental groups, and E ^ is the sums of squares within for the covariate. The formula used for computing the pairwise F ratios was: F X. j - X«2 M S error ' ' with k - 2 and 3 (n - 1) - 1 degrees of freedom. The numerator of the F ratio is obtained by subtracting the observed cell means for each pair of experimental groups on a specified dependent variable. The denominator is the effective mean square error for that dependent variable. The k refers to the number of experimental groups. The n refers to the number of experimental units assigned to each experimental group. Statistical Assumptions Three assumptions underlie the analysis of covariance. The first is that the distribution of scores is normal. The F distribution has been shown to be relatively insensitive to variations in normality (Campbell & Stanley, 1970, p. 372). With reference to the F test, Kirk (1968) concluded; In general, unless the departure from normality is so extreme that it can be readily detected by visual inspection of the data, the departure will have little effect on the probability associated with the test of significance (p. 61). 48 Histograms of the sets of scores to be compared in this study showed that the distri-bution of scores did not depart grossly from normal. Therefore, it was assumed that the distribution of scores would have little effect on the outcome of the analysis of covariance. The second assumption is the homogeneity of variances. As the test for significance involved an F ratio which is robust with respect to violation of this assumption when cell sizes are equal, no test was made (Kirk, 1968, p. 61). The third assumption is the homogeneity of within class regression coefficients. As little is known concerning the effect of violation of this assumption (Kirk, 1968, p. 469), no test was made. Chapter 4 RESULTS OF THE STUDY The results of the study are presented in two sections to correspond with Part One and Part Two of the problem. Results for Part One Hypotheses One and Two Table 7 presents a summary of the frequency and percentage of students who attained criterion level for the algorithm and simplified answers on the four algorithms. Table 8 presents a summary of the frequency and percentage of students who attained criterion level on each of the renaming skills. Table 7 Frequency and Percent of Students who have Attained Criterion Level on the Four Algorithms Computation Frequency*3 of A] Percentage of A, Frequency0 Percentage °<\ Addition 36 31.5 22 J9 .3 Subtraction 18 15.8 13 11.4 Multiplication 21 18.4 21 18.4 Division 14 12.3 14 12.3 a n =114 49 Table 8 Frequency and Percent of Students Who Have Attained Criterion Level on Each Renaming Skill Renaming Skill Frequency0 Percentage RAL 1 67 58.7 2 65 57.0 3 86 75.4 4 55 48.3 5 27 23.7 6 30 26.3 7 30 26.3 RAN 8 36 31.6 9 34 29.8 . 10 87 76.3 11 62 54.4 12 37 32.5 13 40 35.1 14 48 42.1 15 30 26.3 16 26 22.8 a n = 114 Tables 9 and 10 combine the data from Tables 7 and 8, and add another dimension, that of not attaining criterion level. This is denoted by the symbol r* used as a prefix. Therefore, the data in Tables 9 and 10 is tabulated according to six categories: 1. RA|A2 means attainment of criterion level for the renaming skills^RAL, see Table 9, or RAN, see Table 10), the algorithm answer (A]), and the simp-lified answer (A 2); 2. RAj«" 'A 2 means attainment of criterion level for the renaming skills (RAL or RAN), and the algorithm answer (Aj), but not the simplified answer (<* /A2); 3. R ^ A ^ A ^ i means attainment of criterion level for the renaming skills (RAL or RAN), but not for the algorithm answer ('•'Aj) .and not for the simplified answer ('*'A2); 4. r* RAjA 2 means attainment of criterion level for the algorithm (Aj) and simplified (A 2) answers, but not for the renaming skills (^ RAL or RAN); 5. R A i ^ A 2 means attainment of criterion level for the algorithm answer (Aj), but not for the renaming skills (r* RAL or**' RAN) and not for the simplified answer (** A 2 ) ; 6. R^Ajf 'Ar, means non-attainment of criterion level for the renaming skills (•^RAL or*> RAN), the algorithm answer ( ' "Aj), and the simplified answer (** A 2 ) . 52 Table 9 Frequency and Fercenr (in parentheses) of Students Who Have or Have Not Attained Criterion Level on A ] , A2 and RAL Skills ^Algorithms - Category ^Addition 'Subtraction * Multiplication - Division Required RAL Skills 1, 2 a 1 to 5 b 6, 7° 6, 7° 1. RAjA 2 42 (18.4) 62 (10.9) 14 (10.5) 17 ( 7.5) f 2. R A ] ^ A 2 26 (11.0) 21 ( 3.7) 0 ( 0.0) 0 ( 0.0) 3. R ^ A ^ A 2 64 (28.0) 217 (38.0) 25 (10.9) 32 (14.0) 4. ^ RA ] A 2 2 ( 0.9) 3 ( 0.5) 18 ( 7.9) 11 ( 4.8) 5. «" R A ^ A 2 2 ( 0.9) 4 ( 0.7) 0 ( 0.0) 0 ( 0.0) 6. r ' R ^ A j ^ A j 92 (40.4) 263 (46.0) 161 (70.6) 168 (73.7) a n = 228 b n = 570 53 Table 10 Frequency and Percent (in parentheses) of Students Who Have or Have Not Attained Criterion Level on A|, A 2 , and RAN Skills -Algorithms Category Addition Subtraction Multiplication Division Required RAN Skills 8 to 12, .16° 8, 9 b 8, 9, 13 to 15 c 8, 9, 13 Jo 15 c 1. RAjA 2 114 (16.7) 24 (10.5) 88 (15.4) 60 (10.5) 2. R A ^ A 2 44 ( 6.4) 6 ( 2.6) 0 ( 0.0) 0 ( 0.0) 3. R ^ A ^ A 2 124 (18.1) 40 (17.5) 100 (17.5) 128 (22.5) 4. e* RAjA 2 18 ( 2.6) 2 ( 0.9) 177((2.9) 10 ( 1.8) 5. ^ R A ^ A 2 40 ( 5.8) 4 ( 1.8) 0 ( 0.0) 0 ( 0.0) 6. / ^ R ^ A 1 ^ A 2 344 (50.3) 152 (66.7) 365 (64.0) 372 (65.3) a n = 684 b n = 228 c n = 570 54 To support the model hypothesizing a relationship between the renaming skills and the four algorithms, there should be virtually no students (less than 3 percent) in three of the categories: r-> RALA]A2, RAL k^, and / ^ R A N A ^ . On the basis of results from Table 9 hypothesis one (Hj) which predicted that there would be no students who do not attain criterion level on the RAL skills, but who do attain criterion level for the algorithm answer on the related computations, was confirmed for addition (US'/percent) and subtraction (1.2 percent), but not for mul-tiplication (7.9 percent) or division (4.8 percent). A further investigation of the students who were able to attain criterion level on multiplication and division, but not attain criterion level on the related RAL skills, revealed that they were all members of the three classes that had been exposed to the multiplication and division algorithm prior to the experiment. In other words, these students were able to obtain the correct algorithm answer for multiplication and division, but could not perform the necessary renaming skills when in isolation from the algorithm items. On the basis of results from Table 10, hypothesis two (H2) which predicted that there would be no students who do not attain criterion level on the RAN skills, but who do attain criterion level for the simplified answer on the related algorithms, was confirmed for all four algorithms (addition was 2.6 percent, subtraction was 0.9 percent, multiplication was 2.9 percent, and division was 1.8 percent). It is evident from Table 7 that the percentage of students attaining criterion level on renaming skills is low. In fact, only two renaming skills (RAL 3) which is renaming one as an improper fraction, and RAN 10 which is renaming an improper fraction as one), indicate knowledge at an acceptable level for sixth graders, that is, 55 75 percent and 76 percent respectively. The percentage of students attaining criterion level on the rest of the renaming skills varies from 23 percent to 59 percent. It is evident from Table 8 that the percentage of students attaining criterion level on the four algorithms is even lower than for renaming skills, with percentages ranging from 11 percent to 32;rpercent. Furthermore, a difference exists between the percentage of students who obtained the algorithm answer in addition and sub-traction, and the percentage who obtained the simplified answer (in favor of the former). However, no such differences were found for multiplication or division. Hypothesis Three (H3) Table 11 lists the correlations for the RAL/ftAN pairs. Since all the cor-relations are greater than the r c r i t ; c a | of .22 (p ^ .01), it was concluded that there is a significant positive correlation between each of the RAL/RAN pairs. Therefore, the null hypothesis of no correlation was rejected. It appears that the hypothesized correlations between the RAL/RAN pairs do exist. However, in analyzing the complete correlation matrix of all 16 renaming skills, the correlations obtained ranged between .22 and .90 (p K .01). Therefore, it appears that not only are the RAL/RAN pairs related as hypothesized, but that additional interrelation-ships between and among RAL and RAN skills may exist. Table 11 Correlations for the RAL/RAN Pairs RAL/RAN Pairs Correlations 1, 8 .565* 2, 9 .628* 3, 10 .415* 4, 11 .667* 5, 12 .709* 6, 13 .616* 7, 14 .517* *p < .0] Results for Part Two The means and standard deviations for the covariates, and the dependent variables measured on posttest one and posttest two are presented in Tables 12^  13 and 14 respectively. 57 Table 12 Means (X.) and Standard Deviations (s) for the Covariates Experimental Group , Covariate RALTOT RANTOT ALGTOT X. s X. s X. s A 12.25 8.72 10.34 11.41 1.53 3.02 B 17.50 10.36 18.89 15.52 3.54 5.19 C 22.74 9.73 30.03 12.96 12.45 7.95 'Table 13 Means (X.) and Standard Deviations((;s) ferreach Dependent Variable Measured in Posttest One Experimental Groups  s Dependent ' A " B C Variable 'Class 1 Class 2 -Class 3 'Class 4 -Class 5 Class 6 'X . ,ss Xr. s, X. s.. xX. s^ X. S-_ •RAL 1 3.7 1.9 3.1 2.1 3.7 1.9 4.2 1.6 4.7 1.2 4.1 1.6 2 3.9 1.8 2.9 2.3 3.5 2.1 4.3 1.7 4.5 1.4 4.3 1.6 3 5.0 0.0 4.9 0.2 4.4 1.4 4.6 1.0 4.5 1.4 4.9 0.3 4 4.7 1.1 3.9 2.1 4.4 1.6 4.7 1.1 4.2 1.9 4.1 1.8 5 3.1 2.1 2.2 2.2 2.7 2.3 3.8 1.9 3.8 2.1 3.0 2.3 Su Subtraction 0.6 1.9 0.6 1.7 1.3 2.0 2.2 3.1 5.6 2.7 3.3 2.9 Table 14 Means (X.) and Standard Deviations (s) for Each Dependent Variable Measured in Posttest -Two Experimental Groups Dependent A B C Variable Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 X. s X. s X. s X. s X . s X. s RAL 1 4Ui 1.2 3.3 2.2 3183 1.9? 4.2 1.6 4417 1.2 4.1 1.9 2 4.3 1.2 3.1 2.3 3.4 2.2 4.3 1.6 £4r5 1.2 4.0 1.0 3 4.8 0.5 4.8 0.9 4.6 1.1 4.7 0.9 4.5 1.6 4.6 1.2 4 4.7 0.7 3.9 2.0 4.0 2.0 4.7 1.2 4.2 1.9 3.7 2.0 5 1.7 2.0 1.5 2.0 3.3 . 2.2 3.7 1.9 3.8 1.9 2.9 2.3 6 1.3 1.8 0.7 1.8 if. 8 2.3 2.5 2.5 3.2 2.1 2.1 2.2 7 1.1 1.4 0.6 1.6 1.6 2.0 2.2 2.2 3.2 1.6 2.6 1.9 RAN 8 1.7 1.8 0.9 1.6 2.1 2.1 . 2.8 1.7 3.7 1.4 2.8 2.1 9 1.5 1.7 0.8 1.5 1.7 2.0 3.2 2.0 3.6 1.6 2.5 2.1 10 4.3 1.7 3.3 2.1 3.3 2.2 4.7 1.1 4.5 1.6 5.0 0.0 11 3.3 2.2 1.8 2.2 2.2 2.4 4.6 1.2 4.5 1.4 4.0 1.9 12 1.1 1.7 0.5 1.4 1.3 2.0 3.1 2.1 4.1 1.5 3.0 2.2 13 1.0 1.8 0.5 1.6 1.3 2.1 3.4 2.1 4.2 1.3 3.9 1.9 14 1.2 2.1 0.7 1.6 1.9 2.3 4.0 1.6 4.1 1.4 3.8 2.3 15 0.8 1.6 0.3 1.2 1.2 2.0 2.8 2.0 3.3 1.4 2.8 2.3 16 1.1 1.9 0.2 0.5 0.8 1.6 3.1 2.2 3.6 1.7 2.1 2.0 Addition 1.4 1.6 1.0 • 1.6 1.2 2.1 2.0 1.9 3.6 1.4 2.6 2.1 Subtraction 1.6 2.4 0.7 1.5 1.4 2.0 2.2 3.0 5.0 2.5 2.8 2.9 Multipli-cation 0.0 0.0 0.0 0.0 0.0 0.0 1.6 2.1 3.6 1.7 1.4 2.1 Division 0.0 0.0 0.2 0.5 0.2 0.5 0.7 1.4 4.2 2.0 1.2 2.2 Cn •o 60 Preliminary Data Analysis The F ratios for each dependent variable measured on the posttest one and posttest two, to determine the effects of the nested factor of class differences within treatment level, are given in Tables 15 and 16 respectively. Examination of the two tables indicates that 8 of the 26 obtained F values are less than the critical level set at .25, and therefore are significant. Thus, it was concluded that a signi-ficant difference between classes within treatment levels did exist, particularly with the dependent variables associated with the RAN skills, where three significant F values were found, and the algorithms where four significant F values were found. Consequently, it was decided not to pool the data of the two classes within each group. The experimental unit was now considered to be the class, not the individual student. The number of units within each treatment level was reduced from 38 to 2. Table 15 Preliminary Data Analysis for Posttest One Dependent Mean Square F ratio43 p less Variable Between Within than RAL , 1.1668 2.5153 0.4639 0.7082 2 2.7148 2.5728 1.0552 0.3714 3 0.7743 0.8069 0.9595 0.4148 4 1.8172 1.9989 0.9091 0.4394 5 1.6764 2.8628 0.5856 0.6258 Subtraction 6.8207 2.3021 2.9628 0.0355* *p < .25 a df = 3,107 61 Table 16 Preliminary Data Analysis for Posttest Two i Dependent Mean Square' F Ratioa p less Between Within than RAL j 2.7647 2.1423 1.2906 0.2815 2 3.3944 2.0969 1.6188 0.1893* 3 0.3895 0.9889 0.3939 0.7577 4 1.1317 1.8836 0.6008 0.6159 5 0.9706 2.6772 0.3625 0.7802 6 0.9545 2.6481 0.3604 0.7817 7 0.3074 1.8124 0.1696 0.9167 RAN 8 1.9086 1.6453 1.1613 0.3281 9 0.6067 1.4719 0.4122 0.7447 10 5.2973 2.3168 2.2865 0.0829* 11 6.3375 2.6835 2.3616 0.0755* 12 0.5531 1.37234 0.4030 0.7512 13 2.0505 1.5641 1.3110 0.2747 14 2.2821 1.7271 1.3213 0.2713 15 0.4269 1.6139 0.2645 0.8509 16 3.5395 1.3251 2.6710 0.0512* Addition 2.9191 2.1670 1.3471 0.2631 Subtraction 8.8325 2.7591 3.2012 0.0263* Multiplication 1.9830 1.1889 1.6679 0.1738* Division 5.9277 1.2805 4.6291 0.0044* *p < .25 a d f= 3, 107 62 Hypothesis Four (H^) Table 17 presents a summary of the results of the analysis of covariance used to determine the effects of reviewing the five RAL skills (treatment one) on performance of the subtraction algorithm. Table 17 Summary of the Analysis of Covariance for Posttest One Source of Variance df SS MS Univariate F Group 2 9.9396 4.6798 0.69 Classes 3 20.4621 6.8207 To be significant at the .05 level with 2 and 3 degrees of freedom, the F value must exceed 9.55. As the obtained F ratio was 0.69, the null hypothesis of no difference in performance on the subtraction algorithm involving fractions, between grade six students who are given a review of the five RAL skills related to subtraction, and those who are not given the review, was accepted. Since the omnibus F was not significant, a post hoc analysis, as described in Chapter 3, was not warranted. Therefore, it was concluded that treatment one did not significantly increase compu-tational achievement on subtraction of fractions. Table 18 summarizes the F values obtained for the RAL skills on posttest one. It is important to note that treatment one appears not to have had a signi-ficant effect on. performance of the five RAL skills. All the F values obtained were 63 less than the critical value of 9.55 (p .05), with 2 and 3 degrees of freedom. Table 18 F Values for RAL Skills on Posttest One Dependent Mean Square Univariate Variable Between Within B value 0 RAL 1 0.1799 1.1668 0.15 2 0.0638 2.7148 0.02 3 3.7773 0.7743 4.88 4 10.5177 1.8172 5.79 5 4.6061 12655644 2.75 a df = 2, 3 Hypothesis Five (H5) Table 19 presents a summary of the results of the analysis of covariance used to determine the effects of reviewing the purpose of the five RAL skills within the subtraction algorithm (treatment two), on the four algorithms. 64 Table 19 Summary of the Analysis of Covariance on Posttest Two Dependent Variable Source of Variance df SS MS Univar-iate F Addition Group 2 2.5160 1.2580 0.43 Classes 3 8.7573 2.9191 Subtraction Group 2 18.7336 9.3668 1.06 Classes 3 26.4975 8.8325 Multiplication Group 2 4.7444 2.3722 1.20 Classes 3 5.9490 1.9830 Division Group 2 7.0660 3.5330 3.81 Classes 3 2.7831 5.9277 The obtained F values: 0.43 for addition, 1.06 for subtraction, 1.20 for multi pi i— cation, and 3.81 for division, were less than the critical value of 9.55, at the .05 level of sign ificance. Therefore i, the nul 1 hypothesis of no difference in per-formance on the four algorithms was accepted. Consequently, the post ho c analysis, as described in Chapter 3, was not employed. It was concluded that treatment two did not significantly increase computational achievement on the four algorithms. Table 20 summarizes the F values obtained for the RAL and RAN skills on posttest two. It appears that treatment two had a significant effect on performance 65 of three of the seven RAL skills measured. The obtained F values (p ^ .05) were: 12.23 for RAL 3 which is renaming one as an improper fraction, 15.78 for RAL 4 which is renaming a whole number as a mixed-improper number, and 14.04 for RAL 5 which is renaming a mixed number as a mixed-improper number. Furthermore, it appears that performance on the nine RAL skills was not significantly affected. It appears that reviewing RAL skills does not facilitate performance of the RAN skills. On the basis of the results from Table 20, a post hoc analysis for multiple comparisons was applied to test the significance of the difference between each pair of means. However, no significance was obtained on the pairwise comparisons. Therefore, a complex comparison taking the average means of two groups with a third group was conducted (Winer, 1968, p. 586). Again, no significance was ob-tained. Further complex comparisons were not conducted as they were considered to be uninterpretable for the purposes of the present study. Therefore, although treat-ment two appears to have had a significant effect on three of the RAL skills, the results were not interpretable. 66 Table 20 F Values For RAL and RAN Skills on Posttest Two Dependent Mean Square Variable Between Within F value q R A L 1 1.4216 2.7647 0.51 2 2.6646 3.3944 0.78 3 4.7642 0.3895 12.23* 4 17.8569 1.1317 15.78* 5 13.6260 0.9706 14.04* 6 1.2179 0.9545 1.28 7 4.3055 0.3074 4.24 RAN 8 1.0514 1.9086 0.00 9 3.7464 0.6067 6.18 10 0.3889 5.2973 0.07 11 0.2145 6.3375 0.03 12 2.1164 0.5531 3.83 13 9.5472 2.0505 4.66 14 12.6844 2.2821 5.56 15 2.7589 0.4269 6.46 166 2.0220 3.5395 0.57 *p < .05 a df = 2, 3 Chapter 5 CONCLUSIONS AND IMPLICATIONS The Problem The present study had two purposes. The first purpose was to gather evi-dence to support the proposed model hypothesizing a relationship between the renaming skills, and the four algorithms. The second purpose was to determine the effect of reviewing five renaming for the algorithm (RAL) skills in performance of the subtraction algorithm, and the effect of reviewing the purpose of the five RAL skills within the subtraction algorithm1 in performance of the four algorithms. Conclusions The conclusions will be discussed in two parts, as in the previous chapters. Part One Two conclusions about the hypothesized relationship between renaming skills and the four algorithms, as outlined in the proposed model, can be.drawn from the results of the pretest analysis. First, there is evidence to support the hypothesis that a relationship exists between RAL skills, and the algorithm answer for addition and subtraction. Second, there is evidence to support the hypothesis that a re-lationship exists between RAN skills, and the simplified answer for all four algorithms. 67 68 Although the relationship between RAL skills and the algorithm answer for multiplication or division was not supported, the conclusion that there is no relationship cannot be made. As explained in Chapter 4, all students who were not able to attain criterion level on the RAL skills, but who were able to attain criterion level for the algorithm answer, were members of the three classrooms that had been exposed to the multiplication and division algorithms during the school year, prior to the experiment. Therefore, two alternatives may account for the lack of support. One explanation may be that the renaming skills related to the multiplication and division algorithms were performed mechanically, as one step in the algorithm. Consequently, although the students were able to obtain the algorithm answers, they were unable to perform the renaming skills in isolation from the algorithms. A second explanation may be that the relationship between RAL skills and the four algorithms is less critical for multiplication or division, than it is for addition or subtraction, as fewer renaming skills are required. For example, the same two renaming skills (RAL 6 and 7) are required to perform multiplication and division, whereas five other renaming skills are required to perform addition and subtraction. Of the five renaming skills, two are the same for both algorithms (RAL 1 and 2), while subtraction requires three more (RAL 3, 4 and 5). There-fore, it may be that a student can obtain the algorithm answer for multiplication or division more easily than for addition or subtraction. Although it would appear that simplification poses difficulties for addition and subtraction, but not for multiplication or division, the conclusion that simpli-fication is only a problem for addition or subtraction must be treated with caution. 69 The reason for the caution is the very low percentage of students attaining criterion level on each of the four algorithms. There appears to be greater difficulty asso-ciated with obtaining the algorithm answers for all four algorithms. The correspond-ing low percentages on attainment of criterion level on the RAL skills provide further evidence in support of the hypothesis that a relationship exists between RAL skills and each of the four algorithms. On the other hand, there is a possibility that the renaming skills for multiplication and division are taught, or learned, in a manner different from those for addition or subtraction. It may be that the acquisition of RAL 6 (renaming a mixed number as an improper fraction) and RAL 7 (renaming a whole number as an improper fraction) results in the simultaneous acquisition of RAL 13 (renaming an improper fraction as a mixed number) and RAL 14 (renaming an improper fraction as a whole number). Or, it may be that for multiplication and division, simplifying the algorithm answer is perceived as one more step in the algo-rithm, rather than as an isolated skill. However, further evidence, which is beyond the scope of the present study, is required to support the latter hypothesis. A third conclusion from the analysis of pretest data is that the hypothesized correlation between each of the RAL/kAN pairs does exist. In other words, the members of each pair appear to be dependent. Furthermore, inter-dependencies between, and among, the seven RAL and the nine RAN skills are^also in evidence. However, the nature of the dependencies could not be determined by the present study. In summary, the pretest analysis supports the proposed model which hypothe-sizes a relationship between renaming skills and the four algorithms. It would appear 70 that renaming skills have two distinct purposes: (1) to perform a required step for an algorithm, particularly for addition and subtraction, and (2) to simplify the algo-rithm answer for all four algorithms. Part Two From the results of the posttest analyses, it was concluded that neither treat-ment one, nor treatment two, was"' successful in increasing performance of the algo-rithms. The lack of significance in performance of the subtraction algorithm, fol-lowing treatment one, may be attributed to two factors. First, treatment one did not have a significant effect on performance of the RAL skills. Since a relationship between RAL skills and the subtraction algorithm was supported in Fbrt One, it may be that the lack of significance in performance of subtraction can be attributed, at least in part, to lack of significance in performance of the RAL skills. Second, it may be that the students were unable to apply the RAL skills reviewed, to the sub-traction algorithm, without direct instruction as to their application. Therefore, reviewing the RAL skills in ;isbfatibnin of the algorithm, did not affect performance of the algorithm. In treatment two, although the purpose of the five RAL skills within the subtraction algorithm was reviewed, performance on the four algorithms, including subtraction, was not significantly affected. Therefore, it appears that even with direct application of the RAL skills, performance on the algorithms is not affected. As both treatments were considered to be a review of previously taught skills, the experimenter assumed that sixth graders would be familiar with the beginning fraction concepts and skills, as well as the subtraction algorithm. The low perfor-mance? on the pretest do not support this assumption. Therefore, it may be that other factors, such as comprehension of initial fraction concepts and skills, apart from re-naming skills, are confounding the treatment effect. Limitations of the Study A major limitation of the study concerned the selection and assignment of classes to treatment groups. The experimenter was informed, prior to the start of the study, that three of the classes had already been exposed to the four algorithms as part of their mathematics curriculum that year, and that the other three classes had not. Furthermore, although assignment of classes within treatment groups was done randomly, the result was one experimental group containing two classes that had been exposed to fractions, one experimental group containing two classes that had not been exposed to fractions, and one experimental group containing one class that had been exposed to fractions and one class that had not. Consequently, the class differences within treatment groups, and between treatment groups was significantly different, prior to the start of the study. Implications As a consequence of the findings of this study, two implications for the classroom were drawn. First, as renaming skills appear to serve two distinct purposes in fraction algorithms, it would seem reasonable to attempt to teach the algorithms with a 72 separate emphasis on the renaming skills. As teaching the RAL skills in isolation from the algorithm does not appear to improve either the RAL skills themselves, or algorithmic ability, an attempt should be made to emphasize specifically the RAL skills required to perform the algorithm. Additionally, as teaching the RAL skills does not appear to facilitate performance of RAN skills, an attempt should be made to emphasize specifically the RAN skills required to simplify the algorithm answer. Second, the low level of performance on both renaming skills, and each of the algorithms, suggests poor understanding of beginning fraction concepts and skills. Consequently, more emphasis needs to be placed on fraction understanding, prior to work on algorithms. Suggestions for Further Research Since results of Part Two of the study must remain inconclusive because of systematic class differences beyond the control of the experimenter, a replication study might well be conducted, employing controls to minimize these differences. For example, it might be possible to select classes that have either all received, or-all'not' received, instruction on fractions, prior to conducting the experiment. However, if replication studies are conducted, it is recommended that the number of dependent variables be limited, perhaps by focussing on fewer renaming skills and algorithms, in order to facilitate the statistical analysis. As a result of the present research, three problems deserve further study. One problem is to determine the nature of the dependencies between, and among, renaming skills. Another problem is to investigate the relationship of the renaming skills to the multiplication and division algorithms. A third problem is to compare the relationship of renaming skills with the four algorithms, as it appears that the relationship between the renaming skills and the multiplication and division algo-rithms may be different from the relationship between the renaming skills and the addition and subtraction algorithms. 74 REFERENCES Anderson, R. C. A Comparison to two procedures for finding the least common denominator in the addition of unlike, unrelated fractions (Doctoral dissertation, University of Iowa, 1965). Dissertation Abstracts, 1966, 26, 5901. Bidwell, J . K. A Comparative study of the learning structures of three algorithms for the division of fractional numbers (Doctoral dissertation, University of Michigan, 1968). Dissertation Abstracts International, 1968, 29A, 830. Bohan, H. J . A Study of the effectiveness of three learning sequences for equi-valent fractions (Doctoral dissertation, University of Michigan, 1970). Dissertation Abstracts International, 1971, 31 A, 6270. Bruekner, L. J . Analysis of errors in fractions. Elementary School Journal, June 1928 (a), 28, 760-770. Campbell, D. T., & Stanley, J . C. Experimental and quasi-experimental designs  for research. Chicago : Rand McNally College Publishing Company, 1963. Carpenter, T. P., & Coburn, T., Reys, R., & Wilson, J . Results and implications of the NAEP mathematics assessment : elementary school. The Arithmetic Teacher, October, 1975, 22, 438-450. Carpenter, T. P., & Coburn, T. Notes from the national assessment : addition and multiplication with fractions. The Arithmetic Teacher, February, 1976, 23, 137-142. Coburn, T. G . The Effect of a ratio approach and a region approach in equi-valent fractions and addition/subtraction for pupils in grade four (Doctoral dissertation, University of Michigan). Dissertation Abstracts International, 1974, 34, 4688A - 4689A. Dilley, C. A. , Rucker, W. E., & Jackson, A. E. Heath Elementary Mathematics (Canadian Edition), Teachers' annotated edition, levels 4, 5 & 6. Toronto : D. C. Heath Canada Ltd., 1974. Eicholz, R. E., O'Daffer, P., & Fleenor, C. Investigating School Mathematics, Teachers' Edition, Books 4, 5 & 6. Don Mills, Ontario : Addison-Wesley (Canada) Ltd., 1974. Elliott, H. A. , Bye, M. P., Hanwell, A. P., Hay, R., Jorden, J . M. , MacLean, J . R., Neufeld, K. A. Project Mathematics, Teacher's Guide-book, 4, 5, 6. Toronto : Holt, Rinehard and Winston of Canada, Limited, \775. 75 Glass, Gene V., and Stanley, Julian C. Statistical Methods in Education and  Psychology. New Jersey : Prentice Hall Inc., 1970. Green, G . A. A Comparison of two approaches, area and finding a part of, and two instructional materials, diagrams and manipulative aids, on multiplication of fractional numbers in grade five (Doctoral dissertation, University of Michigan, 1969). Dissertation Abstracts International, 1970, 31 A, 676-77. Herberts, Lee. Frequency Tabulation Program. Vancouver : University of British Columbia, June, 1977. Kirk, Rodger E. Experimental Design : Procedures for the Behavioral Sciences. California : Brooks/Cole Publishing Company, 1968. Lankford, F. G . , Jr. Some computational strategies of seventh grade pupils. Washington, D.C. : U.S. Office of Education Froject no. 2-C-013, Government Printing Office, October, 1972. Mathematics Curriculum Guide Years One to Twelve. Victoria : Ministry of Education, 1977. Morton, R. L. An Analysis of pupils' errors in fractions. Journal of Educational Research, 1924, 9, 117-125. Muangnapoe, C. An Investigation of the learning of the initial concept and oral/ written symbols for fractional numbers in grades three and four. Unpublished doctoral dissertation, University of Michigan, 1975. Multivariance, User's Guide. National Educational Resources, Inc., Chicago, Illinois. Version V, Release 2, June, 1974. Nelson, Larry Richard. Guide to Lertap Use and Interpretation. New Zealand: University of Otajo. 1974. Payne, Joseph N. Review of Research on Fractions. In R. Lesh (Ed.), Number  and Measurement, papers from a Research Workshop. Georgia : University of Georgia, 1976. Uprichard, A. E., & Phillips, E. R.; An Intraconcept analysis of rational number addition : A Validation Study.. Journal of Research in Mathematics Education, January 1977, 8, 7-16. Winer, B. J . Statistical Principles in Experimental Design. New York : McGraw Hill, 1962. 76 APPENDIX A Pilot Test - Version One and Version Two Pilot Test - Version One Part A: DIRECTIONS: Fill in the missing numbers 77 * ' * 8 • • 9 - = 9 — * 9 27 8 = • 32 30 § = ° 7 7 • 23 = 22 g * * 6 • 7 35 = 5 • 72 19 O 9 24 " 8 24 • i i = io § 1 = 24 • 10 5 6 10 • 42 5 ° 7 - = -9 . 9 10 a 23 — = 23 — 11 55 * 2 Q 1 6 ~3 = 3 3 4 • 16 15 = 14 ^ • x _ 121 • * 8 7 _ £ 12 12 13 24 • 48 4 ° 4 0 f 0 = 4 0 60 • 7 = 6 12 * * 11 = 1 2 1 • 2 5 • 60 9 = 9 • • 5 = I * 16 g = 15 • 12 1 = • 7 9 D 21 - = 20 — • 15 = - 3 * * * • 1 = n 1 = 9 • 5 D 6 6 " 6 5 • 5 - = 4 -° 8 8 20 2 D 0 • 49 = 48 ^ • 35 | - 34 , & - J J 78 Part B: DIRECTIONS: Simplify the following to lowest terms. 20 ^ = U 60 60 72 12 12 = 22 3 100 10 i $ = 11 31 = 27 24 36 = 92 24 180, 60 40 ^ = 96 5 § = 32 19 ^ -• 11 43 36 " 4 8 100 56 12 30 30 _ 6 126 8 * i o f . 29 12 11 132 11 72 75 — = / D 72 * 2 2 = 7 47 12 = 36 36 80 42 19 — = 24 16 80 12, 24 25 75 5 " 8 = 65 104 64 10 10 * 6 8 » ¥ -* 12 15 .9 246 10 36 66 11 32 * 1 3 7 40 1 * 9 23 - = ^ 5 110 110 48 18 18 111 20 20 =i = U 18 *denotes the items that are different from Version Two. Pilot Test - Version Two Part A: DIRECTIONS: Fill in the missing answers. 14 19 - = ]9 8 = 30 - = oo _ oo 9 J 27 32 7 7 2 3 - 2 2 § - — 7 _ „ a 1 9 • Q 7 " 35 2 5 8 " 2 5 I T 9 24 = 8 — 11 = 10 — 1 = -21. 10 I = 10 • 7- 5- D 23 10 • 11 2 3 55 • 42 7 9 9 * 2 • 1 6 3 * — 3 _ 4 • 28 15 - - S 2 -• 1 = • * 8 3; = - 5 . 12 ]2 13 24 • 48 40 i =40 • 60 17 • = 1 6 12 * * i i . - i 2 i • 9 12 a 36 • 15 • 1 * 11 n 16 == - 15 1 - -• 2 1 A = 20 a 15 • 12 12 21 11 3 * * * • 1 = — 1 = -19 • 5 • 1 0 6 = IT" 15 5 • 8" 1 4 8 49 = 48 20 35 - . - 34 — 2 5 • 60 80 Part B: DIRECTIONS: Simplify the following to lowest terms. 20 I S -60 60 72 12 12 223 100 10 i§ 10 27 24 36 92 24 180 _ 60 72 40 ^  = 96 19 i i -11 43 _ 36 48 100 30 6 126 11 * 29 i 2 = 11 132 11 72 75 ^ = = 72 * 12 'i3 -13 47 _ 12 36 36 il OICN CO i-t 42 16 24 80 25 75 _ 5 65 ± 2 * . . 64 10 10 * 20 12 i§ = 15 246 _ 10 36 3 66 _ 7 32 1 1 40 = 25 25 * 23 12 -15 110 110 48 ±2 = 18 H I _ 20 20 2§ -18 * denotes the items that are- different from Version One Pretest 82 PRETEST - PART A: DIRECTIONS: Fill in the missing numbers. 27 8 32 3 0 7 7 23 22 9 (2) (7) (6) (4) s • 7 • 1 9 • • ^ = TT - 25 ^ = 25 - — 9 ^ = 8 — — 11 = 10 — — 7 35 8 64 24 24 8 (1) (2) (5) (4) 24 5 1—1 5 L_l i n I 1 1 = = 10 § = 10 — — 7 5 _ ___ 23 i£ » 23 — — j — i 6 92 9 9 11 55 ' (3) (2) (6) (2) • „ • 2 9 1 1 60 121 16 f = - E^  = —— 15 = 14 - 2 — 1 = -±- i -3 3 1 6 • • • (6) (1) 1 ' (4) 1 1 (3) 8 12" ]2 1 3 . • 40 • *7 • 24 48 60 7 = 6 ~ (6) (1) (2) (4) 121. 11 = • a, ,_• 9 = 9 5 = 5 5 60 (1) " • (7) • (7) n • 1 6 12 - 1 5 I T • 21 • 15 = 2_ ______ 11 3 (5) (3) (5) (7) x _ 9 10 c 5 • — A (3) • (3) 6 6 (6) 5 8 ~ 4 ~ (5) 20 49 = 48 35 | = 34 • 9 . • 7 (5) 12 36 (1) Note: the bracketed number refers to the number of the RAL category in Table 2. 83 PRETEST : PART B DIRECTIONS: Simplify Hie following to lowest terms. 60 _ 12 - 223 (11) 72 (8) 12 (10) , 100 (13) (16) 27 (12) 24 36 = (8) 92 .6 .. (15) 180 60 (14) „~ 72 4 0 96 = (9) 5 § = 32 (9) 19 ^  = 11 (11) 43 _ 48 56 12 30 _ 36 (13) 100 (8) 30 (9) 6 (14) 126 = 8 ' = (15) » s - (12) 29 12 11 (12) 84 7 (14) 75 Zi = 72 2 Z _ 47 36 _ (11) 2 7 " (11) 12 (13) 36 (10) 80 _ 42 24 16 80 12 (15) (1.6) '24 (8). 25 (15) 75 5 (14) 5 6 42 " (12) 65 104 _ 64 (16) 10_ '10 (10) 8 12 (8) 29 i° = 18 (9) 12 !5 9 (16) 246 10 (15) 36 66 11 32 25 3 (14) 7 (13) 40 (9) 25 (10) "| - (12). 110 _ 110 (10) 48 18 18 (ID 111 20 (13) 28 20 — = 18 (16) Note: the bracketed number refers to the number of the RAN category in Table 3. PRETEST - PART C: DIRECTIONS: Answer each question. Show all your work in the space provided. Simplify each answer to lowest terms if possible + 9 I « I X 2 49 I 3 3 - - l 2 -4 * 3 +60 34 +32 12 20 21 25 - I •"•I 16 -12 31 36 19 2 ? - 3 i 7 • 9 4 -2-' 21 2 3 2 - X 3 - = 9 5 4 5 9 f e 10 - 1 8 X +20 | 85 PART C - continued 40 32 i - 3 50 22 31 40 •19 £ 20 4 9 X 3 - = 6 - 2 | -« 5 APPENDIX C Posttest One 87 POSTTEST 1 : PART A DIRECTIONS: Fill in the missing numbers •11 • 1 6 12 = 1 5 " I T ,9 12 _ • 36 *1 = - i ± -• (3) X' 23 = 22 • 9 (5) (1) (4) i = i 2 i 3/_ • 9 • 2 1 n =20 — 60 • 4 ; 16 (3) (1) (5) 12) • i 9 27 2 • 35 - = 34 — 7 7 2 _ n 5 60 (3) (2) (5) (1) s • 10 - = 10 — — 6 42 15 = 14 -|g-g 11 = 10 • 5 8 - 4 — 8 (2) (5) (4) , • 2 5 e > 2 5 " 6 T n • s . • • 1 = 7 7 35 1 = 1 T (2) (3) (1) (3) 13 _ • io • n =23 — 49 = 48 * L j • U) 9 ^ = 8 24 • 24 48 24 -(-ir (2) (5) • Note: the bracketed number refers to the number of the RAL category in Table 2. 88 POSTTEST 1 DIRECTIONS: PART B Answer each question Show all your work Simplify each answer to lowest terms if possible 31 13 ' 50 "19 In5 20 •» fo 1 6 1 40 32 -22 32 40 -4 10 19 - f e — 4 21 -16 ±3 36 APPENDIX D Posrtesr Two 90 POSTTEST 2 - PART A : DIRECTIONS: Fill in the missing numbers i l • 1 6 1 2 = — 23 • = 22 9 6 _ _ • 7 35 1 = 7 (5) (4) (1) (3) 9 9 9 12 _ • 36 11 121 23 • 55 (6) (1) (2) 7 • 40 — = —==-10 60 (2) 7 = • 6 T T (4) 30 7 7 (6) 13 24 _ • 48 (1) 5 • 10 7 = 10 — -6 42 9 = 35 2 • 7 = 3 4 — 21 • 11 (2) (5) (5) L—I 60 1 I 7 I 1 15 = — " ' " [ J 1 1 = 1 ° — (7) . ' (4) -(4) • (3) 2 5 • 60 (l) • (7) • 24 (5) i = 2 _ 3 _ • 7 • 2 5 8 = 2 5 -64" (2) S . D • (3) 4 16 (l) (7) 5 • 1 0 6 = — q 8 Q • 1 - 2 4 ^ 5 A • (6) 9 9 — y 27 (2) • ,3, 5 8 4 • 8 (5) 1 = I 2 i _1Q — - /I Q 20 a • 16 3 . — (6) • (3) ^3 — • 4iO • (4) Note: the bracketed number refers to the number of the RAL category in Table 2. 91 POSTTEST 2 - PART B: DIRECTIONS: Simplify the following to lowest terms. 180 92 223 12 60 (14) 6 (15) 100 (13) 12 (10) 36 48 32 11 = 40 110 3 (14) 100 (8) (9) 110 " (10) (9) 111 20 (13) 72 75 ^  = 72 (ID 5 6 ^ = 30 (9) 23 |- 24 25 246 (12) 36 (8) 25 (10) 10 (15) (ID 20 |2 = 60 (11) 42 (12) 30 _ 6 (14) 10 1§ = 10 (16) 80 12 (15) 42 19 ^  = 24 (16) 75 5 (14) 36 _ 47 84 36 (10) 12 (13) 7 (14) (16) 27 (12) 66 7 (13) 29 i° = 18 (9) (16) 43 _ 80 16 126 _ 36 (13) 25 (15) 24 (8) 8 (15) 8 10 _ 60 12 (8) 10 (10) 72 (8) (12) 19 ^  = 11 (ID « M - (ID 29 i 2 = 11 (12) (16) 40 — = 96 (9) Note: the bracketed number refers to the number of RAN category in Table 3. 92 POSTTEST 2 - PART C DIRECTION: Answer each question. Show all your work. Simplify each answer to lowest terms 2 • 12 19 6 2- - 11 = 34 J „ 7 -^21 • 4' 21 + 3 2 25 84 10 6 — 2 - = 5 4 9| +60 8 - 1 7 8 X 4 -4 X 3 - = 7 • 9 10 5 . 6 + 9 6 16 — 4 2 3 2 5 x 3 5 93 PART'C - continued 40 -22 32 40 +20 | APPENDIX E Lesson Plans and Worksheets APPENDIX E General instructions for using the Lesson Plans 1. Please follow the lesson plans as closely as possible. Do not add new words or concepts that are not in the plans even if the addition appears logical to you. 2. Teacher directions are in regular type. Statements and questions to be directed to students are in capital type and enclosed in a box. Expected student responses are in brackets. 3. The lesson plans are yours to keep. Feel free to underline and make notes on the sheets as you use them. 4. If students complete the worksheets assigned before the period is over, do not assign any more math work. Perhaps they can catch up on other work or read a book. 5. If yo'u don't have enough time to mark the worksheets in the same class period as the lesson, mark them together as a review the next day before starting the next lesson. 6. Please collect the worksheets after they are marked. I will collect them afterwards. 7. READ THE LESSONS THOROUGHLY BEFORE TEACHING. Lesson 1  Objectives 1. To give proper fraction names, both oral and written, for the shaded regions of a paper strip. 2. To give fraction names, both oral and written, for one whole. 3. To give equivalent fraction names, both oral and written, for the shaded region of identicalppaper strips. Materials Four shaded paper strips numbered 1 to 4 (provided) Blank paper strips (provided) Crayon to shade in the regions Introduction (10 minutes) Hold up the paper strip numbered 1. Demonstrate how it was folded. The black line has been drawn in to emphasize the fold line. WHAT IS THE FRACTION NAME FOR THE SHADED REGION? (one-half) HOW DO WE WRITE IT? (one over two) Write the fraction symbol beside the picture. WHAT; DOES THE BOTTOM NUMBER OR DENOMINATOR MEAN? (the number of equal parts) WHAT DOES THE TOP NUMBER OR NUMERATOR MEAN? (the number of shaded parts) 97, The blackboard lay-out for this part of the development should look like the following, but place each drawing below the other: /// //// 3 <3 EACH OF THESE PICTURES AND FRACTION SYMBOLS DESCRIBES ONE WHOLE. 4 3 8 In front of each fraction symbol write "1 =", i .e., 1 = ^, 1 = ^, and 1 = - . WHAT ARE SOME OTHER FRACTION NAMES FOR ONE WHOLE? (Accept three others and write them on the board as 1 = .) WHAT ABOUT ONE OVER ONE? IS THIS A FRACTION NAME FOR ONE? (yes) HOW DO WE WRITE IT? (one over one) HOW CAN WE SHOW IT WITH A PAPER STRIP? (Take a SHADED paper strip that is not folded.) THE DENOMINATOR 1 MEANS THE NUMBER OF EQUAL PARTS. THE NUMERATOR 1 MEANS THE NUMBER OF SHADED PARTS. Draw a picture of | above the other three pictures. 2. Take a new paper strip and fold it in half. Unfold it and shade in one part. Blacken the fold line. Show the students. Draw a picture of this on the board. Write - beside the picture on the left-hand side. 2 98 I AM NOW GO ING TO FOLD THE STRIP A L O N G THE OTHER SIDE INTO TWO EQUAL PARTS. Unfold and blacken the fold line. HOW MANY EQUAL PARTS HAVE I MADE? (four) HOW MANY ARE SHADED? (two) WHAT FRACTION NAME DESCRIBES THE SHADED REGION? (two-fourths) Complete the picture drawn earlier by drawing in the second fold line. Beside 1 2 the picture on the right hand side write - = - . ONE-HALF IS EQUIVALENT TO TWO-FOURTHS. Take a second new paper strip and fold it in half. Unfold and shade in one part. Blacken the fold line. Draw a picture of it underneath the one for two-fourths. Write - on the left hand side of it. 2 AGAIN I HAVE ONE-HALF SHADED. I AM NOW GOING TO FOLD THE OTHER SIDE OF THE STRIP INTO THREE EQUAL PARTS, unfold and blacken the fold line. WHAT FRACTION IS SHADED NOW? (three-sixths) ONE-HALF IS EQUIVALENT TO THREE-SIXTHS. Complete the picture by drawing in the other fold lines. 1 3 Beside the picture on the right hand side write - - = g . Follow the above sequence to develop two equivalent fractions for two-thirds. You will have to use two new strips to do this. The blackboard lay-out for this part of the development should look like the following: THESE TWO PICTURES (point to the appropriate ones) SHOW^ EQUIVALENT FRACTIONS FOR ONE-HALF. THESE OTHER TWO SHOW EQUIVALENT FRACTIONS FOR TWO-THIRDS. Seatwork (10 minutes) Hand out Worksheet 1. Erase the boardwork. Collect and mark them together the next day as a review before starting Lesson 2. 100 WORKSHEET 1 NAME DATE TEACHER SCHOOL 1. Look at each picture. Fill in the missing numbers in the columns. The first one is done for you. PICTURE NUMBER OF SHADED PARTS TOTAL NUMBER OF EQUAL PARTS FRACTION SYMBOL (a) ///// ///// 2 4 ((c) Z (d) <»> 7. mm (0 m 2. Draw in the fold lines and shade in the appropriate amount to show the following fraction names for one whole. WORKSHEET 1 continued a) two-halves b) six-sixths c) d) four-fourths , 10 e) — 10 2. Look at each picture. Write the fraction name for the shaded region beside each picture. Then fill in the missing numbers. E05 • • In the following two pictures, just the fold lines are drawn. Shade in the appropriate amount to show an equivalent amount for . Write the fraction name beside each picture. 102 WORKSHEET 1 continued 4. Look at each picture. Write the fraction name for the shaded region beside each picture. Then fill in the missing numbers. In the two pictures below shade in the appropriate amount to show equivalent 2 fractions for - . 3 Write the fraction name beside each picture. Lesson 2  Objectives 1. To generalize a method for finding a fraction that is equivalent to a given fraction. 2. To use this method fo find the numerator or denominator of a fraction given two equivalent fractions. Introduction (10 minutes) Mark Lesson 1 worksheet together. Collect. Development (20 minutes) 1. Draw a picture of one-half on the board, write •[ on the left hand side of IN YESTERDAY'S LESSON WE STARTED WITH A PAPER STRIP FOLDED INTO ONE-HALF. WE THEN FOLDED A L O N G THE OTHER SIDE TO MAKE TWO EQUAL PARTS. WE THEREFORE DOUBLED THE TOTAL NUMBER OF EQUAL PARTS, I.E., THE DENOMINATOR DOUBLED. Draw in the fold line on the picture above. On the right hand side of it write 2 WE HAVE ALSO DOUBLED THE NUMBER OF SHADED PARTS, I.E., THE NUMERATOR HAS DOUBLED. Write 1 x 2 above the 2 x 2, e.g., -Then write " = - beside it. 4 THE RESULT IS TWO-FOURTHS. WE CAN SAY THEN THAT ONE-HALF IS EQUIVALENT TO TWO-FOURTHS. Write - = — on the far right hand side of the picture. 2 4 Draw another picture of one-half on the board below the one showing two-fourths. Write - on the left hand side of it. 2 HOW DID WE FOLD THE PAPER STRIP TO MAKE THREE-SIXTHS? (We folded along the other side to make three equal parts.) WE THEREFORE TRIPLED THE TOTAL NUMBER OF EQUAL PARTS, I.E., THE DENOMI-NATOR TRIPLED. Draw in the fold lines on the picture you drew. On the right hand side of it write 2 x 3 , e.g., ~ 1 x 3 = 3 2 x 3 6 THE RESULT IS THREE-SIXTHS. WE CAN SAY THEN THAT ONE-HALF IS EQUIVALENT TO THREE-SIXTHS. 1 3 Write - = - on the far riqht hand side of the picture. 2 6 105 HOW DO WE FIND EQUIVALENT FRACTIONS FOR A GIVEN FRACTION? (We multiply the numerator and the denominator of the given fraction by the same number.) THIS NUMBER IS SOMETIMES CALLED THE MULTIPLIER. If you feel your students have not fully grasped the concept of how to find equi-valent fractions from a given one, follow the above sequence to find equivalent fractions for two-thirds. In t.hisscgse yourclb'lackbo'ardclay-out should look like the following: 2 3 2x2 = 4 3 x 2 6 2 x 3 6 3 x 3 2 3 2 3 4 6 6 9 2. WE MULTIPLY THE NUMERATOR* AND DENOMINATOR OF A GIVEN FRACTION BY THE SAME NUMBER TO MAKE A FRACTION THAT IS EQUI-VALENT TO IT. LET US USE THIS METHOD TO FIND A MISSING NUM-ERATOR OR DENOMINATOR WHEN WE ARE GIVEN TWO FRACTIONS THAT ARE ALREADY EQUIVALENT. 1 • Write - = — on the board. 3 12 WE WANT TO FIND HOW MANY .TWELFTHS ARE EQUIVALENT TO ONE-HALF? WHAT MUST TWO BE MULTIPLIED BY TO MAKE IT EQUIVALENT TO TWELVE? (six) THEREFORE, THE SAME NUMBER MUST BE MULTIPLIED 106 WITH THE ONE TO FIND THE MISSING NUMERATOR. WHAT NUMBER SHOULD G O IN THE BOX? (six) Write " x 6" beside the 2 and the 1. Write 6 in the box. ONE-HALF IS EQUIVALENT TO SIX-TWELFTHS. Write - = on the board. 3 24 WE WANT TO FIND HOW MANY TWENTY-FOURTHS ARE EQUIVALENT TO TWO-THIRDS. WHAT MUST THE THREE BE MULTIPLIED BY TO MAKE , IT EQUIVALENT TO TWENTY-FOUR? (eight) THEREFORE, THE SAME NUMBER MUST BE MULTIPLIED WITH THE TWO TO FIND THE MISSING NUMERATOR. WHAT NUMBER SHOULD G O IN THE BOX? (sixteen) 2 x 8 1 6 Write " x 8" beside the 3 and the 2, e.g., = 3 x 8 24 TWO-THIRDS IS EQUIVALENT TO SIXTEEN TWENTY-FOURTHS. HOW WOULD YOU FIND THE MISSING DENOMINATOR IN THIS QUESTION? Write - = — on the board. (5 x 3 6 • Write 18 in the box. Seatwork (10 minutes) Hand out Worksheet 2. Erase the Collect. = 15, therefore 6 x 3 = | | .) boardwork. Mark the worksheet togeth* WORKSHEET 2 NAME DATE 108 TEACHER SCHOOL 1. Fill in the missing numbers. The first one is done for you. -A I i (a) (b) (c) 4 * A 3_x 4 x 5 x 20 • 24 30 (d) (e) (f) 8 x Q 2. Fill in the missing numbers. 5 x 40 6 x • 1 X • 3 x 27 10 x 60 11 X • 4 _ • 7 28 1 10 9 3 5 - -35 (d) 1 = <f) n 12 • 3 18 • 44 • (g) (h) (0 4 x (a) i 8 to it. 1 «> 5 • 7 x 56 3.x • 10 x 100 6 x 54 7 x • 3 - • 4 28 4 20 ,T = • 7 49 5 " • equivalent Lesson 3  Objectives 1. To write mixed numbers given a picture. 2. To rename the fraction part of a mixed number to an equivalent fraction. 3. To use the method of finding equivalent fractions to find the numerator or denominator of two equivalent mixed numbers. Materials Five paper strips totally shaded (prepare before lesson) One paper strip with one-third shaded (prepare before lesson) Review of Lesson 2 (5 minutes) 1. WHO CAN GIVE ME A N EQUIVALENT FRACTION FOR THREE-FIFTHS? (accept three) Write them on the Board. HOW DID YOU FIND THEM? (multiplied the numerator and denominator of three-fifths by the same number) On the board show how each equivalent fraction 3 x 2 6 3 x 3 9 was made, e.g., 5 x 2 10 ' 5 x 3 15 2. On the board write * = • and 2 - 2 7 35 4 [—] HOW DO WE FIND THE MISSING NUMBERS? ( 7 x 5 =35, therefore 4 x 5 = ; 3 x 9 = 27, therefore 4 x 9 = ) Fill in the missing numbers. Erase the boardwork. Development (15 minutes) [ 1. TODAY WE WILL FIND OUT ABOUT MIXED NUMBERS. Hold up four and one-third shaded paper strips. HOW MANY WHOLE SHADED STRIF5 DO I HAVE? (four) WHAT PART OF A STRIP IS SHADED? (one-third) HOW MANY STRIPS DO I HAVE ALTOGETHER? (four and one-third) Draw a picture of the strips on the board. I HAVE ONE PLUS ONE PLUS ONE PLUS ONE PLUS ONE-THIRD SHADED PAPER STRIPS. Write 1 + 1 + 1 + 1 + 1 underneath the picture. WE CAN 3 COMBINE THE ONES RESULTING IN 4 + 1. Write this underneath the 3 previous line. FINALLY WE CAN SHORTEN THIS TO 4 ^ . Write this on the board. THIS IS CALLED A MIXED NUMBER BECAUSE IT HAS A WHOLE NUMBER ./PART /AND A FRACTION PART. 2. Draw a picture of three and three-fourths shaded paper strips. HOW MANY WHOLE SHADED STRIPS ARE THERE? (three) WHAT PART OF A STRIP IS SHADED? (three-fourths) WHAT MIXED NUMBER DESCRIBES THIS PICTURE? (three and three-fourths) HOW DO WE WRITE IT? ( 3 J ) I l l The boardwork for 1 and 2 should look like the following: 1. m m m m m m m m 4 + 5 4 l 3 + -3 3 -3. WHO CAN GIVE ME A N EQUIVALENT FRACTION FOR THE ONE-HALF IN 4 i . ? (Accept two different ones and write them on the board, e.g., 4 1 2 - I and 4 - = 4 - .) 2 8 Draw a picture of 4 ^ and the two equivalent fractions. IS THE SHADED PART OF EACH PICTURE THE SAME? (yes) WHAT HAS CHANGED THEN? (the name for one-half) The boardwork should look like this: VZZ\ ZZ2 1Z22 LZ22 ED S" 4 • 3-' m V Z A s 4 \ 4 4 4. WHO CAN GIVE ME A N EQUIVALENT FRACTION FOR THE ONE -FOURTH IN 2 | ? (Accept two different ones and write them on the board, e.g., 2 ^ = 2 | and 2 ^ = 2 — - . ) Draw a picture of 2 - and the two equivalent fractions. i IS THE SHADED PART OF EACH PICTURE THE SAME? (yes) WHAT HAS CHANGED THEN? (the name for one-fourth) The boardwork should look like this: ezTJ tzza m YZZA xrm WZk WZA Write the following on the board: 2 3 • HOW e 5 i = 5*=" 18 • 16 2= 16 -5 30 2 ! 22 8 4 9 2 = 9 24 4 • 11 Z— 11 2§ 10 • HOW CAN WE FIND THE MISSING NUMERATORS AND DENOMINATORS AND KEEP THE MIXED NUMBERS EQUIVALENT? (Use the method we learned in Lesson 2 for making a fraction equivalent to a given one, i .e., find the number we multiply the numerator and denominator by to make an equivalent fraction.) 113 For each pair of mixed numbers on the board write in the multiplier, then fil in the missing number, e.g., 5 = 5 3 x 6 18 Seatwprk (10 minutes) Hand out Worksheet 3. Erase boardwork. Mark worksheet 3 together. Collect. 114 WORKSHEET 3 NAME DATE TEACHER SCHOOL  1. Write mixed numbers in the blank spaces provided for the shaded part of each picture. 2. Shade in the strips to show the given mixed numbers (a> '» " I II (b) 2 i = ( I I I I b (c) 3 1 I | I ~| | I I 4 1 I I 1 \ 1 J (d) i • • (a) 2 I I j 1 WORKSHEET 3 continued 3. Write mixed numbers for each of the following: (a) 2 + | = (b) 6+ \ = (c) 1 + 1 + 1 + 1+1 + 1 = (d) 1 + 1 + 1 + - = 4 (e) 8 + n CNI CO Follow the exam e.g. 62- = 3 (a) (a) 4 3 ^ = 5 (*b) (c) ' f o " (e) 2 = 1 + 1 + 1 + 1 + 1 + 1 + 1 3 3 5. Fill in the missing numbers. WORKSHEET 3 continued 117 Lesson 4  Objectives 1. To rename a mixed number as a mixed-improper using drawings of paper strips. 2. To generalize a method for renaming a mixed number as a mixed-improper. Materials Use the paper strips prepared from Lesson 3. Development (20 minutes) TODAY WE ARE GO ING TO LOOK AT ANOTHER WAY TO RENAME A MIXED NUMBER.  Hold up>4 - shaded paper strips. 3 HOW MANY PAPER STRIPS DO I HAVE? (four wholes and one-third of 4 -) Draw a picture of the strips on the board. Beside it write 4 3 Underneath it write 1 + 1 + 1 + 1 + - . 3 I AM NOW TAKING ONE OF THE WHOLE STRIPS AND FOLDING IT INTO THIRDS LIKE THE FRACTION PART. Do this with one of the whole shaded strips. Then draw in the fold lines on the drawing. HOW MANY THIRDS IN ONE WHOLE? (three) WE NOW HAVE ONE PLUS ONE PLUS ONE PLUS THREE-THIRDS PLUS ONE-THIRD. 3 1 Write 1 + 1 + 1 + - + - underneath the last line on the board. ADDING UP THE WHOLES WE HAVE 3. ADDING UP THE FRACTIONS WE HAVE 4 . ALTOGETHER WE HAVE 3 PLUS i WHICH WE SHORTEN TO 3 - . 3 3 3 THIS IS CALLED A MIXED-IMPROPER NUMBER BECAUSE IT HAS A WHOLE NUMBER PART AND A N IMPROPER FRACTION PART. WE MADE 3 | FROM 4 | . As you say the numbers write them on the board. Hold up 3 - shaded paper strips. HOW MANY PAPER STRIPS DO I HAVE? (three and three-fourths) 3 Draw a picture of 3 - shaded paper strips on the board. Underneath it 4 write 1 + 1 + 1 + - . Beside it write 3 - . Take one of the whole strips and fold it into fourths. Draw in the fold lines on the picture. WHY DID I USE FOURTHS? (We want the same denominator as the fraction part in order to combine them. 4 3 Write 1 + 1 + - + - underneath the last line 4 4 HOW MANY WHOLES ARE THERE? (two) HOW MANY FOURTHS ARE THERE? (seven) WHAT MIXED-IMPROPER NUMBER HAVE WE MADE? ( 2 - ) HAVE WE CHANGED THE AMOUNT OF SHADED STRIPS? (no) 4 WHAT HAS CHANGED THEN? (changed one whole into a fraction) WE MADE 2 \ from 3 \ . 4 4 The blackboard lay-out should look like the following: 4 i 1 4 - 1 + 3 3 ! HERE ARE 5 EASY STEPS TO FOLLOW IN RENAMING A MIXED NUMBER TO A MIXED IMPROPER WITHOUT USING STRIPS OR PICTURES. Write the following sequence on the board explaining and writing each step along the way. 5?. 7 5 + 3-7 4 + l + | 4 + Z - ; + 2 7 7 4 + 7 4-° 7 Step 1 Separate whole number and fraction Step 2 Separate one whole from the rest Step 3 Rename one whole to a fraction Step 4 Add the fractions Step 5 Shorten step 4 4 9 Following Steps 1 to 5 show how 4 -=can be renamed to 3 - and ,5,.,!! 5 Write the following questions on the board. 10 HOW DO WE FIND THE MISSING NUMBERS? (follow Steps 1 to 5) IN THE FIRST QUESTION ONE WHOLE HAS BEEN SEPARATED FROM 6 LEAVING 5 AND RENAMED TO TWO-HALVES. THE TWO-HALVES HAS THEN BEEN ADDED TO ONE-HALF GIVING HOW MANY HALVES? (three) WH/AT IS THE MISSING NUMBER? (3) Write the missing number in the box. The boardwork should look like the followin Hand out Worksheet 4. Erase boardwork. ' Mark Worksheet 5 together. Collect. WORKSHEET 4 NAME DATE. 121 TEACHER SCHOOL. 1. Follow steps 1 to 5 to show how each mixed number is renamed to a mixed improper. An example is done for you. Do not write the words of the steps in the questions you answer. Just follow the steps. e.g., 3 = 3 + ^ Step 1 Separate whole number and fraction. = 2 + 1 + ~ Sf eP 2 Separate whole numbers. 4 1 = 1 + - + —- Step 3 Rename one whole to a fraction. 4 4 5 = 1 + — Step 4 Add fractions. 4 - 1 5 = 1 - Step 5 Shorten step 4. 4 a) 6 f = c) 4 | = b) 8 - = c) 3 WORKSHEET 4 continued 2. Fill in the missing numbers. Lesson 5  Objectives 1. To rename a whole number greater than one as a whole number and fraction. 2. To generalize the renaming of a whole number greater than one as a whole number and fraction. TODAY WE ARE GO ING TO LOOK AT A WAY TO RENAME A WHOLE NUMBER AS A WHOLE NUMBER AND FRACTION.  Hold up four whole shaded strips of paper. Draw a picture of them on the board, e.g.: I AM NOW TAKING ONE OF THE SHADED STRIPS AND FOLDING IT INTO FIFTHS. Draw in the fold lines on your picture. HOW MANY FIFTHS IN ONE WHOLE? (five) HOW MANY WHOLE STRIPS ARE LEFT? (3) WE NOW HAVE FOUR AND FIVE-FIFTHS SHADED STRIPS. HERE IS WHAT WE DID USING NUMBERS INSTEAD OF STRIPS AND PICTURES: Materials Five shaded paper strips (prepare before class) Development (20 minutes) SEPARATED ONE WHOLE FROM THE REST RENAMED ONE WHOLE AS A FRACTION COMBINED THE WHOLE NUMBER AND FRACTION Follow the same development with pictures and numbers to rename 6 as 5 - and 4 3 as 2 - . 2 i o - D f »-•§ • iFIND THE MISSING NUMBERS. Write them in the boxes? Make sure your students can answer these six questions before handing out the worksheet. Use paper strips and pictures if necessary. Seatwork (10 minutes) Hand out worksheet 5. Erase boardwork. Mark worksheet 5 together. Collect. Write 4 = 3 + 1 Write 4 = 3 + -5 Write 4 = 3 -5 Write the following questions on the board. • 4 = 3 7 = 6 n 10 WORKSHEET 5 NAME DATE TEACHER SCHOOL 1. Follow the example to fill in the missing numbers. e.g. 11 = 10 + 1 = 10 + - = 10 -3 3 (a) 8 = 7 + 1 = 7 + — = (b) 22 = 21 + • = 21 + I=L ( C ' r - , r - i • (c) 16 = Q + 1 = •*+J=i. (d) 7 -= • +f I > = 6 + 5 13 • 2. Fill in the missing numbers. • • (a) 5 = 4 - g - (e) 2 = 1 • 9 (b) 21 = 20 (f) 16 = 15 y 10 14 . . . 4 _ O . • WORKSHEET 5 continued 3. Match the following by drawing lines to each pair of equivalent fractions. A. 4 l 7 (D 3-° 7 B. 7 (2) 7 C. 4< 7 (3) 7 D. 40? 7 (4) 12 3 — 7 E. 45 7 (5) 3? 7 Lesson 6  Objective To relate the renaming skills from Lessons one to five to sample subtraction questions. Introduction (5 minutes) 4 2 Write 6 - - 1 - = on the board vertically. 5 5 HOW DO WE FIND THE ANSWER? (subtract the fraction parts, then subtract the whole numbers) WHAT IS THE ANSWER? ( 5 \ ) \ 2 Development (20 minutes) TODAY WE ARE GO ING TO USE WHAT WE LEARNED LAST WEEK ABOUT RENAMING FRACTIONS, WHOLE NUMBERS AND MIXED NUMBERS TO THE SUBTRACTION OF FRACTIONS.  1. Subtracting mixed numbers with unlike denominators. 2 On the board write 8 -WE CANNOT SUBTRACT FIFTHS FROM THIRDS. WE NEED A COMMON DENOMINATOR. HOW ABOUT 15? Write " = M 15 beside the first mixed number and " = 4 — 1 1 beside the second one. 15 NOW HOW DO WE FIND THE MISSING NUMERATORS? ( 3 x 5 = 15, therefore, 2x5 = | { and 5 x 3 = 15, therefore 1 x 3 =| | ) Write in the numerators and subtract. On the board write 6 ^ 3 WE CANNOT SUBTRACT FOURTHS FROM SEVENTHS. WE NEED A COMMON DENOMINATOR. HOW ABOUT 28? Write " = 6 — " beside the first mixed 28 number and " = 1 — " beside the other. NOW HOW DO WE FIND THE 1 28 MISSING NUMERATORS? ( 7 x 4 = 28, therefore 6 x 4 = Q and 4 x _7_ = 28, therefore 3 x 7 = | [ ) Write in the numerators and subtract. 2. Subtracting mixed numbers involving ^borrowing'1 ©n the board write 6 -5 - 2 2-5 WE CANNOT SUBTRACT THREE-FIFTHS FROM ONE-FIFTH. HOW CAN WE INCREASE THE NUMBER OF FIFTHS IN 6 \? (rename 6 \ to a mixed 5 o improper) HOW DO WE DO THAT? (Take one whole from 6 leaving 5 and rename one whole to five-fifths. Add five-fifths to one-fifth resulting in six-fifths.) Write " = 5 t " beside 6 \ . 5 5 129 WE DO NOT HAVE TO RENAME 2 f SO REWRITE IT UNDERNEATH 5 -5 5 NOW WE CAN SUBTRACT. 2 Write 8 - on the board. WHAT IS THE FIRST THING WE HAVE TO DO IN ORDER TO SUBTRACT? (find a common denominator) WHAT SHOULD WE USE? (42) Write " = 8 — " and " = 4 — " beside the appropriate mixed number, e.g., 8 1 = 8 4 2 _45 = 4 -6 42 HOW DO WE FIND THE NUMERATORS? ( 7 x 6 = 42, therefore 2 x 6 = | | and 6 x 7 = 42, therefore 5 x 7 = Q ) Write in the numerators. NOW mn WE SUBTRACT? (no) WHY NOT? (?| jis too small) WHAT CAN WE 2 DO? (rename 8 - to a mixed-improper number) HOW DO WE DO THAT? 42 (Take one whole from 8 leaving 7, rename one whole to — and add it to 42 1 2 TI, i f 7 5 4 ^ — The result is 7 — 42 42 Wrif e " = 7 — " and " = 4 — " beside the appropriate mixed number, e.g., 8 2- = 8 12 = 7 * 7 42 42 5 35 35 - 4 - = 4 — = 4 — 6 42 42 NOW WE CAN SUBTRACT. WHAT IS THE ANSWER? (3 — ) 42 3. Subtracting a mixed number from a whole number. On the board write 2 7 SEVENTHS MUST BE SUBTRACTED FROM SEVENTHS. WHERE CAN WE GET SOME SEVENTHS IN ORDER TO SUBTRACT THREE SEVENTHS FROM IT? (Rename two wholes to one whole and a fraction. The result is one and seven-sevenths.) 7 3 Write " = 1 - " beside the 2 and rewrite 1 - . Your boardwork should now look like this: " ' 7 -I3- = 1 3 -7 7 NOW WE CAN SUBTRACT. WHAT IS THE ANSWER? (1 ? ) On the board write 16 -42-5 Follow the same sequence as above to rename 16 to 15 - . 5 Then subtract. Seatwork (15 minutes) Hand out Worksheet 6. Erase boardwork. Mark worksheet 6 together. Collect. WORKSHEET 6 NAME DATE 131 TEACHER SCHOOL Find the differences. Show all your work. 4 ' 1 1 •2 I -4 | -8 | 20 10 | 18 | -4 4 -4 - -11 1 9 5 1 2 2 "3 13 | 14 21 7 5 4 -9 4 _p 2 6 9 5 8 7 _ 1 1 7 2 3 f 2 3 5 TO 2 0 " 6 f2 -1 11 "10 1 

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