division with common fraction and decimal divisors

556

Division with Common Fraction andDecimal DivisorsFoster E. GrossnickleLeiand M. Perry

Division of whole numbers, when thedivisor is a two-or-more place numeral,is the most difficult of the four opera-tions on whole numbers. Similarly,when the divisor is either a commonfraction or a decimal, division is themost difficult of the four operations.

Many pupils find it difficult to understand division when the divisor is either ofthese two types, but the rule governing each operation is relatively simple to ap-ply. Consequently, division by a common fraction is often introduced with therote learned rule, "invert the divisor and multiply." Similarly, division by a deci-mal is often taught by the rote learned rule, "move the decimal point to the rightin the dividend as many places as there are decimal places in the divisor." Theserote learned procedures are unsatisfactory in a modern program of learning.

This paper aims to find what procedures have been used in dealing with frac-tional and decimal divisors within the current century. The following four ques-tions will be investigated:

1. How did writers of mathematics books within the past three or four centuriesdeal with these divisors?

2. How did early writers of methods books deal with these divisors?3. What method does educational research show to be superior for dealing with

these divisors?4. What method is used in current pupil’s books for dealing with these divisors?

In the light of answers to these four questions, the writers will make recom-mendations for dealing with fractional and decimal divisors.

Division of Common Fractions

Early Rules and Definitions

According to Smith, the first printed books in the 1500s gave three methods ofdividing by common fractions: 1

School Science and MathematicsVolume 85 (7) November 1985

Division . . . Fraction . . . Divisors 557

1. Change the dividend and divisor to like fractions and divide the numerators(Common denominator method).

2. Cross multiply as is 2/3 - 3/4� - � = �

a ka c3. Follow a formula: � - � = � as in:

b c kb

3 I2 - 2- 3x4 13 13

5~

13~

5~

13 ~4x5~20

Historically, division has been defined in three different ways.2 First, Fibo-nacci, in 1202, defined division as "finding how often a smaller number is con-tained in the larger number."2 This type of division answers the ques-tion: "How many times is 2 contained in 6?" This idea is one of measuring 6units by 2 units to find how many 2’s in 6 and represents the measurement con-cept of division.A second definition of division given in the early 1500s was known as parti-

tioning a given number into a whole number of parts to find how many in eachpart. Partitive division furnishes a valuable approach in helping a pupil discoverhow division by a whole number is related to multiplying by a fraction because 6- 2 is easily related to Vi of 6.A third historical definition of division, given in the late 1500s, consisted in

"finding a number which has the same ratio as the dividend has to the divisor."This ratio concept is useful in solving a rate problem.

Approaches Used in Early Mathematics Books

Writers of mathematics books over the years have warned against rote learning.Colburn, in 1863, criticized rote learning because it "entirely reverses the naturalprocess for the pupil is expected to learn general principles before he has ob-tained the particular ideas from which they are composed." Colburn believedthat every number situation should be introduced with a practical problem.3

Goff’s text for pupils, published in 1889, presented division of fractions withoral examples such as: "When % bushel of oats are divided equally among 5horses, how much will each receive?" The solution given was: "Each horse willreceive Ys of 5 eighths of a bushel, or I/eighth of a bushel." Then a rule wasstated: "A fraction may be divided by a whole number either by dividing the nu-merator or multiplying its denominator by the divisor." 4

Brooks, in 1904, advocated the abolition of rote procedures. Arithmetic toooften "is taught by a collection of rules to be committed by memory and appliesmechanically to solutions of problems." 5 He advocated an inductive reasoningapproach to help pupils understand the derived rule. Pupils derived the rule fordivision of fractions by studying particular examples. The rule discovered was,


558 Division . . . Fraction . . . Divisors

’ ’To divide by a fraction either multiply the multiplicand by the reciprocal of thedivisor or divide the numerator of the multiplicand by the divisor." 6

Although early writers of arithmetic books emphasized relating learning topractical problems, the process of dividing of fractions was predominately "bythe rule" method. McClymonds and Jones, in then Advanced Arithmetics, pub-lished in 1910, provided a two-page introduction to division of fractions whichconsisted of 12 measurement type problems.7 One problem directed students tomeasure a 1-foot line with a measure of Vi foot and find that Vi foot measures a1-foot line 2 times. Then the students were told that "2 is the reciprocal of 1/2.When the product of two numbers is 1, the numbers are reciprocals of eachother." The introduction concluded with the rule: "To divide a fraction, multi-ply the reciprocal of the divisor by the dividend."

Early authorities deplored learning to divide by a fraction by rote and advo-cated introducing the topics with practical problems. Division was defined inthree ways: (1) a measurement situation; (2) a partitive situation; (3) as a ratio.Each of these definitions provided a meaningful approach to division of com-mon fractions.

Approaches used by Early Writers of Methods Books

Thorndike, in 1924, wrote one of the first popular methods books in which hedealt with division of fractions. He stated that division "is still approached withelaborate caution and with various means of showing why one must invert andmultiply or multiply by the reciprocal." 8 The difficulty with division, Thorndikemaintained, was not with the "invert and multiply" rule, but with the fact thatyoungsters have had drilled into their minds that "divide makes smaller." Howmany pupils believe that 8 divided by Vi is four rather than the correct answer16?Morton, in 1939, recommended using the partition concept of division to in-

troduce dividing a fraction by a whole number and the measurement concept tointroduce dividing a whole number by a fraction.9 To divide a fraction by a frac-tion, Morton would rename the fractions with common (like) denominators, andthen divide the numerators. The division of the like denominators gives a quo-tient of one, so the quotient of two like fractions is the quotient of the numera-tors. After the pupils understood this procedure, they divide by a fractional divi-sor by multiplying by its reciprocal.Buckingham, in 1947, presented two methods of division of fractions.10 The

first approach was the common denominator method and the second approachinvolved multiplying by the reciprocal of the divisor. He endorsed both methods,but recommended the common denominator method for initial instruction andthe reciprocal approach as a shortcut procedure.

Brueckner and Grossnickle, in 1947, presented the rationale for division by afraction, such as 6 divided by Vs which may be expressed either as 6 - V3 or ¥3or VsVTas follows: n



-1 \/i = -3- x -2- /-^x 6 = 1 \/T- 9.

The following three mathematical principles are involved in the given solution:

1. Muliplying both divisor and dividend by the same nonzero number does notchange the quotient.

2. The product of a number multiplied by its reciprocal is 1.3. The quotient of any number divided by 1 is that number.

Spitzer, in 1948, suggested that division of fractions be introduced in problemsituations involving dividing a whole number by a fraction and solved as a meas-urement process as 4 - !/2, which asks how many Vi’s in 4.12 This problem canalso be solved by the common denominator method as shown:

Spitzer noted that prior to 1948 the common denominator method proved tobe effective in small experimental studies but had never been tested on a largescale.

Approaches used in SMSG "New Math" Programs

In 1963, the School Mathematics Study Group (SMSG), introduced division offractions by finding how many objects are in Vz of a set of 12 objects.13 The solu-tion involved taking 12 objects and separating them into 3 equal parts to showthat Vs x 12 and 12-3 give the same result. It was noted that ’/3 is the recipro-cal of 3. Then the rule was expressed as "Multiply by the reciprocal of the divi-sor."

What Research Says About Division of Fractions

Brownell, in 1938, reported a field study by a teacher who introduced division offractions using the common denominator method.14 The last week of the study,the teacher introduced the inversion method. The more able pupils adopted thismethod at once, but the other pupils preferred to use the common denominatormethod.

Both Brooke, in 1954, and Capps, in 1960, compared the common denomina-tor method and the inversion method.15116 The results in each case showed no sig-nificant difference between the two procedures.

Kirch, in 1964, studied two methods of teaching division of a fraction by afraction.17 One method used a meaningful approach to build an understandingof the process. The other method was a mechanical procedure of inverting andmultiplying. The results showed no significant difference between the twogroups for low and normal range students on the test given immediately aftercompletion of the program. However, on a later test, there was a significant dif-ference in the achievement of the two procedures in favor of the meaningful ap-proach, especially for middle and high ability students.



Bergen, in 1966, compared the complex fraction method of dividing fractionswith the common denominator and inversion approach.18 The complex fractionapproach involved writing a division example such as ^ � Va as a complex frac-tion: .

~6__2_*3

The example is then solved by multiplying both the numerator and denominatorby V2, the reciprocal of the denominator, making the divisor one, and the quo-tient the product of the numerator Y(, x %. The results showed that the commondenominator method was better for division of a whole number by a proper frac-tion and for division of a mixed number by a proper fraction. On other types ofexamples, the complex fraction and inversion methods were significantly betterthan the common denominator method.

"... the teacher introduced the inversion method.^

Bidwell, in 1968, completed his doctoral study comparing three procedures fordivision of fractions; namely, the common denominator approach, the inversionmethod, and the complex fraction approach.19 The results showed that the groupthat used the inversion method had the highest mean score and the lowest percentof transfer error. The complex fraction method was the next most effective ap-proach, and the common denominator method was found to be the least effec-tive.

"Frequent errors occur in division of decimals when pupilsuse a mechanical procedure . . . .??

Division of Fractions in Recent Textbooks for Pupils

The authors examined six recently published mathematics textbooks for pupils inthe elementary school to determine the procedure used to introduce division by afractional divisor.20 All six texts presented the topic at the sixth-grade level. Fiveof the six books presented division by a decimal fraction before division by acommon fraction. All six books introduced division of a fraction with a meas-urement situation, such as 3 - Vz, read "How many Vz’s are in 3?" All six seriesgeneralized in the first lesson that dividing by a fraction is the same as multiply-ing by the reciprocal (or multiplicative inverse) of the divisor. Only one of thetextbooks series used the term "multiplicative inverse" instead of the term "re-ciprocal." The common denominator approach was not introduced in any of thesix series, nor was the complex fraction approach.



Summary of Division by Common Fractions

The results of an analysis of the writings of early authors of elementary mathe-matics books and of research dealing with division by a fractional divisor may besummarized as follows:

1. The early writers dealing with the topic disapproved of rote learning of therule, "invert and multiply," but teachers were given almost no help to presentthe topic meaningfully.

2. There was no agreement in methods books whether to introduce the topicby dividing a whole number by a fraction or vice versa.

3. Manipulative materials were not widely used to objectify the work until af-ter WW II when the meaning theory of learning was widely adopted.

4. Some early research studies favored the common denominator method fordivision by a fractional divisor, but this method is not used in today’s textbooksfor learners.

5. Today’s textbooks for pupils in the elementary school find the quotient fora fractional divisor by multiplying by the reciprocal of the divisor.

Division of Decimals

Methods Used in Early Books

Simon Stevin wrote the first book dealing with decimal fractions in 1585 butdecimals were not widely used until about 200 years later.21 Goff, in 1889, pre-sented division of decimals with an example involving dividing a decimal by awhole number as shown: 22

17.28

9T9^Detailed directions were given to the learner for working the example startingwith, "Place divisor and dividend as in division of integers .... It is evident thatthe number of decimal places in the quotient equals the excess of the number inthe dividend over the number in the divisor." This method of finding the place-ment of the decimal point in the quotient is known as the subtraction method.

Brooks, in 1904, indicated that two methods may be used to deal with a deci-mal divisor.23 They are: (1) Use the principles of dividing by a common fraction,and (2) use the method derived from the pure decimal conception, which is thesimpler and more practical method. This method is illustrated as follows:

.2\4862486-2243101010-10 1

McClymonds and Jones, in 1910, began their presentation of division of deci-mals with questions such as: 24 "How many $2 in $4?" "How many 2 tenths in4 tenths?" What is the quotient of:

4VT A\TJ .04\/~08" .004\/~W8



Then they concluded: "If the divisor contains tenths, tenths of the dividendmay give a whole number quotient." The rule to follow: "Place the decimalpoint in the quotient above and after the figure in the dividend occupying thesame order as the lowest order in the divisor."

Thorndike, in 1923, suggested that initial work with division of decimalsshould be verified by common fractions.25 The placement of the decimal pointshould be verified in exercises such as:

1.23\/^65

The trial quotient cannot be 2 because 2 x 1.23 is less than 24.6. It cannot be 200because 200 x 1.23 is more than 24.6. So it must be 20 because 20 x 1.23 =

24.6." Thorndike emphasized the use of estimation in determining the placementof the decimal point in the quotient.Morton, in 1939, indicated that when the divisor is a whole number, the num-

ber of decimal places in the quotient is the same as the number of those places inthe dividend.26 The pupil checks the correctness of this process by use of com-mon fractions.Morton suggested that another approach to the placement of the decimal point

in the quotient is the inverse of the way the decimal point is determined in theproduct. Stated another way: ’ The number of decimal places in the divident mi-nus the number of places in the divisor equals the number of places in the quo-tient."Buckingham, in 1947, stated that the location of the point in the quotient can

be determined by three procedures: (1) Estimation, (2) Divide by a whole num-ber, then the point in the quotient will be directly above the point in the dividend,and (3) If the divisor is a decimal, multiply both the divisor and dividend by thesmallest power of 10 that will make the divisor a whole number.27

Brueckner and Grossnickel, in 1947, recommended that an example involvingdivision of decimals be changed so that the divisor is a whole number by multi-plying both divisor and dividend by a power of 10.28 The power is the same as thenumber of decimal places in the divisor. Then every example in division of deci-mals will involve either dividing a decimal by a whole number or dividing twowhole numbers.

Frequent errors occur in division of decimals when pupils use a mechanicalprocedure, such as the use of carets to shift the decimal point in the divisor anddividend without understanding the operation. This is especially true in an exam-ple of the type, .5 \/~2. The pupil does not know if the starting place in shifiingthe decimal point should be to the left or to the right of the numeral 2 in the divi-dend.

Spitzer, in 1948, recommended that division by a whole number be introducedfirst to develop the rule that, ’The quotient decimal point is to be placed imme-diately above the dividend decimal point . . . ." 29 This means the divisor mustbe a whole number.



Approaches Used in SMSG "New Math" Programs

The first method for division of decimals suggested by the School MathematicsStudy Group in 1963 consisted of writing the example in common fractionalform and then multiplying numerator and denominator by a power of 10 so thatboth terms of the fraction are whole numbers as shown:

53.75 53.75 x 100 537553.75 - .5 = �� = �� = �� = 107/2

.5 .5 x 100 50

The second method involved moving the decimal point in both divisor anddividend the necessary number of places to make the divisor a whole number(The caret procedure).

Research Findings Regarding Division of Decimals

Linquist, in 1983, reported the results of The Third National Assessment ofEducational Progress in Mathematics and found that achievement increased for13-year-olds on almost every computational skill except for division of decimalsand concluded: "The performance on fractional computation was low and stu-dents seem to have done their computation with little understanding." 3I Thefollowing percent of correct results for three types of examples were:

(a).04\/TT 39 percent

(b).3yT^J 50 percent(c) 4 \/8.4 84 percent

Very probably, if the results for examples of the type .4 N/T? were given, thepercent of correct responses would have been lower than for any of the threetypes given.

The senior writer conducted a study to determine the cause of errors in divi-sion of decimals. He attempted to determine whether errors resulted from divi-sion, from the placement of the point in the quotient, from the placement of thequotient and the placement of the point, or from a combination of all three ofthe alternatives. An analysis of the test results justified the following conclu-sions.

1. The division process was not a dominant factor in the number of errors made.2. The placement of the decimal point was the chief source of error in division of

decimals.32Students used two methods to place the decimal point in the quotient: (1)

Shifting the decimal point by using carets which is a mechanical procedure, and(2) changing the divisor to a whole number by multiplying both divisor and divi-dend by a power of 10. Pupils who used the first method made more errors thanby the second method. If pupils were taught the second method meaningfully,most of the errors in division of decimals could be eliminated.



Flournoy studied the success that pupils had in placing the decimal point in thequotient with two different methods.33 Group I learned the method of makingthe divisor a whole number. Group II learned the subtractive method for theplacement of the point in the quotient. The results of the tests were expressed interms of pupil ability. Group I excelled for pupils of average or below averageability. Group II excelled for pupils above average in ability.

Faires studied two equated groups of pupils in the fifth grade.34 One groupstudied common fractions first and the other group studied decimals first. Thegroup that studied decimals first performed significantly better than the groupthat began with common fractions.

Approaches Used by Elementary Textbooks

The writers examined the same six series of elementary mathematics textbooksmentioned before to determine the approach in division of decimals.35The presentation in all series was similar. These texts introduce division of a

decimal by a whole number first, and then make the decimal divisor a wholenumber by multiplying the divisor and dividend by a power of 10.

Summary of Division by Decimals

Although computation with decimal fractions is only about 200 years old, theprocedures for dealing with them have been quite uniform. Early writers dealingwith division by a decimal divisor recommended making the divisor a wholenumber by shifting the point in both the divisor and dividend. This procedureusually resulted in a mechanical operation and as a consequence many errorswere made.Modern mathematics textbooks for elementary school recommend making the

divisor a whole number by multiplying both divisor and dividend by a power of10. This is an application of the identity element of multiplication. When thedivisor is a whole number, a decimal point may be in both the dividend and thequotient or in the quotient only.One approach of great importance in division by decimals is the approxima-

tion of the quotient to see if the answer given is sensible. Of the references citedin this paper, few gave emphasis to this topic. Yet, approximation of the quo-tient for checking the answer is highly important in division of decimals.The achievement of pupils in division of decimals as shown by The National

Assessment tests is very low, indicating the work is not fully understood by thepupils.

Recommendations

After reviewing the literature dealing with division of common and decimal frac-tions, the writers make the following recommendations pertaining to that opera-tion.



1. Present a problem involving the measurement concept of division to introduce divisionof a whole number by either a common fraction or a decimal. The measurement conceptshows that division is a shortened form of subtraction. To find how many VA yard in 6yards, have the pupil measure by ¥4 yard to find the answer is 8 pieces.

2. Use manipulative and visual aids in introducing division by both common and deci-mal fractions. The learner uses these aids until he/she is able to operate meaningfully withsymbols.

3. Have the class discover that division by a fraction, such as 14, follows the multiplica-tion-division pattern for whole numbers such as 3 - Vi = 3 x 2.

4. Introduce the concept of the reciprocal to give the mathematical basis for changing adivision example to one in multiplication when the divisor is a common fraction.

5. When a class learns to divide by a decimal before learning to divide by a commonfraction, change the common fraction divisor to a decimal and then solve the example. Thesame pattern applies when the sequence is reversed. Have pupils use rounding off tech-niques for repeating decimal divisors. Make extensive use of hand-held calculators.

6. When the divisor is a decimal, transform the example so that the divisor is a wholenumber by multiplying both divisor and dividend by the power of 10 which is the same asthe number of decimal places in the divisor.

7. After the class understands the work in item 6, introduce the short-cut of sliding thedecimal point in the divisor and dividend by use of carets.

8. Have pupils approximate the answers in division of decimals to see if the quotients aresensible. Use a calculator to check the accuracy of the answers.

9. The final and most important recommendation is to be certain the pupil understandsthe procedure before developing computational skill. With the increased use of calculatorsand computers in the classroom, computational skill is relegated to a minor role comparedwith estimation of the answer.

References

1. Smith, David Eugene. A History of Mathematics. Vol. 11. New York: Dover Publica-tions, 1925.

2. Smith, David Eugene. A History of Mathematics. Vol. II, 128.3. Colburn, Warren. Intellectual Arithmetic Upon the Inductive Method of Instruction.

New York: Hurd and Houghton, 1863.9-13.4. Goff, Milton B. Practical Arithmetic. New York: Taintor Brothers and Company,

1889. 141-142.5. Brooks, Edward. The Philosophy of Arithmetic. Philadelphia: Normal Publishing

Company, 1904.6. Brooks, Edward. The Philosophy of Arithmetic. 428-429.7. McClymonds, J. W. and D. R. Jones. Advanced Arithmetic. Sacramento: California

State Series, 1910. 124-125.8. Thorndike, Edward L. The Psychology of Arithmetic. New York: The Macmillan

Company, 1924. 72.9. Morion, Robert Lee Teaching Arithmetic in the Elementary School, Vol. Ill, Upper

Grades. New York: Silver Burdett Company, 1939. 134-139.10. Buckingham, Burdette R. Elementary Arithmetic: Its Meaning and Practice. New

York: Ginn and Company, 1947. 270-271.11. Brueckner, Leo J. and Foster E. Grossnickle. How to Make Arithmetic Meaningful.

Philadelphia: The John C. Winston Company, 1947. 339-47.12. Spitzer, Herbert F. The Teaching of Arithmetic. Cambridge, Mass: The Riverside

Press, 1948.



13. Bell, Max S., cd. Studies in Mathematics, Vol. IX, Brie/Course in Mathematics/orElementary School Teachers. PaloAlto, Calif.: Stanford University, 1963. 275-291.

14. Brownell, Wiliiam A. "Two Kinds of Learning in Arithmetic." Journal of Educa-tional Research XXX, May 1938. 341-347.

15. Brooke, George Milo. "The Common Denominator Method in Division of Frac-tions," Dissertation Abstracts. 14:2290-2291; 1954.

16. Capps, Leion Roger. "A Comparison of the Common Denominator and InversionMethods of Teaching Division of Fractions." Dissertation Abstracts. 21:819, October1960.

17. Kirch, Percy. "Meaningful Vs Mechanical Methods of Teaching Division of Frac-tions." School Science and Mathematics, XIII. December 1964, 697-708.

18. Bergen, Patricia M. "Action Research on Division of Fractions." The ArithmeticTeacher, XIII. April 1966. 293-295.

19. Bidwell, James King. "A Comparative Study of the Learning Structures of ThreeAlgorithms for the Division of Fractions." Dissertation Abstracts. 29:830A,September 1968.

20. Elementary Textbook Series: Addison-Wesley, 1981; Harcourt Brace Jovanovich(1981); Open Court, 1981; Silver Burden, 1981; Macmillan 1982; and Scotl-Foresman,1980.

21. Smith, Vol. II, 1925,240-241.22. Goff, 94-96.23. Brooks, 459.24. McClymonds and Jones, 68-70.25. Thorndike, 81-82.26. Morton, 158-162.27. Buckingham, 331-335.28. Bruecknerand Grossnickle, 361-373.29. Spitzer.297.30. Bell, 293-314.31. Lindquist, Mary Montgomery. "The Third National Mathematics Assess-

ment: Results and Implications for Elementary and Middle Schools," The ArithmeticTeacher, XXXI December 1983. 16.

32. Grossnickle, Foster E. "Types of Errors in Division of Decimals," Elementary SchoolJournal, XLII, November 1941. 184-194.

33. Flournoy, Frances. "A Consideration of Pupils’ Success with Two Methods for Plac-ing the Decimal Point in the Quotient," School Science and Mathematics, LIX, June1959,445-455.

34. F’aires, DanoMiller. "Computation with Decimal Fractions in the Sequence of NumberDevelopment," Dissertation Abstracts 23’A\83, 1963.

35. Elementary Textbook Series: Addison-Wesley, 1981; Harcourt Brace Jovanovich,1981; Open Court, 1981; Silver Burdett, 1981; Macmillan, 1982; and Scott-Foresman,1980.

Foster E. GrossnickleLeIandM. PerryJersey City State CollegeCalifornia State University,Jersey City, New Jersey 07305Long Beach, California 90808


division with common fraction and decimal divisors

Documents