understanding partitive division of fractions

UNDERSTANDING PARTITIVE DIVISION OF FRACTIONSAuthor(s): Jack M. Ott, Daniel L. Snook and Diana L. GibsonSource: The Arithmetic Teacher, Vol. 39, No. 2 (OCTOBER 1991), pp. 7-11Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194943 .

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UNDERSTANDING PARTITIVI« DIVISION OF FRACTIONS Jack M. Ott, Daniel L Snook, and Diana L Gibson

first-grade students start addition of whole numbers, they are given sets of ob-

jects to combine and count to determine the sum. Most educators agree that begin- ning students need experience combining groups of objects, such as combining two pencils and three pencils to get a group of five pencils. Such activities are needed to give students an understanding of what addition means before moving on to faster - but less meaningful - ways to sum numbers, such as the use of an algorithm or calculator. This approach is based on the learning principle that meaning comes from knowing the things signified by the symbols. Thus, familiar, concrete experience - actual or recalled - should be a first step in the development of new abstract concepts and their symbolization. Though this principle of moving from the concrete to the abstract is extensively used in teaching the addition of whole numbers, it is used much less in teaching the division of fractions. The inevitable result in that a student may be able to use an algorithm by rote to divide fractions, but the skill is useless because it is devoid of meaning.

Consider the following problem:

1 + I = ^x^ = * = 2- 4 3 4 14 4

What does the answer "2 1/4" mean? How can the answer be larger than the number being divided? If one cannot inter-

pret or make sense out of the answer, then even an efficient means of getting the answer is useless. Inability to interpret the results of division problems such as the foregoing is usually not the result of an inability to decode the meaning of the fraction symbols (3/4, 1/3, 2 1/4) but, instead, is the result of a lack of understanding of what division by a fraction means. In other words, the cause of problem is decoding the meaning of the sym- bol -r in the context of fractions. What are the meanings of division-of-fraction problems?

The meanings of division-of-fraction exercises are the same as those for division of whole numbers. Consider the following statement:

2x3 =6

(Number (Number in of groups) x each group) = Total

By mathematical convention, the first num-

ber of a multiplication sentence indicates the number of equal groups and the second number, the size of each group. Related to each such multiplication sentence are two division sentences.

The first division sentence results when the first factor in the multiplication sentence is missing. It is represented by the equation x3 = 6or6-=-3 = . One is given the total, 6, and the number in each group, 3, and asked to find the number of groups of size 3 in 6. This approach is called the measurement meaning of division. A problem illustrating this meaning of division is the following:

Samantha has 6 cookies and is going to give some friends 3 cookies each. How many friends will get cookies?

The second division sentence results when the second factor in the multiplication sentence is missing. It is represented by the equation 2 x = 6 or 6 -r 2 = . In this approach, the partitive meaning of division, one knows the total,

Familiar concrete experiences, such as those involving money, are a first step in the development of the abstract concept of division of fractions.

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Jack Ott teaches at the University of South Carolina, Columbia, SC 2920X. He is interested in teaching mathematics for understanding. Daniel Snook is the Exit Examination Coordinator for Lower Richland High School, Hopkins, SC2906 1 . His work deals with students at remedial levels in mathematics, reading, and writing. Diana Gibson is an educational assis- tant at the University of South Carolina. Her interest is media-assisted instruction.

OCTOBER 1991 7



6, and the number of groups, 2, and is asked to find the number in each group when 6 is distributed into 2 equal groups. An example of the partitive meaning of division is the following:

Samantha has 6 cookies to be given to 2 friends. How many cookies does each friend get?

Since these meanings are not obvious, students need experience dividing numbers in a more concrete and meaningful manner before moving on to more abstract means of dividing.

Such concrete experiences are easy to devise and are relatively easy for a student to follow as long as the numbers are whole numbers. However, meaningful concrete experiences related to division of fractions are much more difficult for teachers to devise and for students to follow.

In dealing with common and decimal fractions, the easiest of the meanings of division to demonstrate - and hence for a student to comprehend - is measurement. Several excellent articles in the Arithmetic Teacher discuss how the measurement meaning of division of fractions can be demonstrated concretely. One such article suggests the use of egg cartons (Hyde and Nelson 1967); another, the use of money (Steiner 1987); and yet another, the use of fraction bars (Trafton and Zawojewski 1984). Although the measurement meaning of division of fractions has received considerable attention, a review of the literature indicates that the partitive meaning for division of fractions has been almost totally ignored. The authors of this article found no explanation in the Arith- metic Teacher of the partitive meaning of division by a fraction or mixed number. Similarly, they found that elementary textbooks typically give careful explanations of the measurement meaning of division of fractions but seldom treat the partitive meaning at all.

The partitive meaning of division of fractions has been very resistant to clear, concrete explanations. In fact, one writer offering an explanation for measurement division of fractions went so far as to say that partitive division of fractions does not make sense. A more accurate observation would have been that no sensible explanation has yet been offered. This void in the literature must be filled, as the partitive meaning of division of fractions is no less

^^^^^^^^^^^H Partitive division of a fraction by a whole number

If - of an egg carton is divided into 2 equivalent sets, how large is each set?

5 + 0 2 = 5 1 5 - + 0 2 = - x- = - 6 6 2 12

Partitioning means to distribute. How can - be distributed into 2 sets? 6

5 10 5 ~" - - Put - in each set. 6 12 12

How many are in each set?

Answer: Each set is _ of an egg carton. 1 2 * Number of sets into which to distribute

^^^^^^^^a| Application of partitive division of a fraction by a whole number

A 2-kg bag of potting soil costs- dollar. How much does 1 kg cost?

- dollar +2 = - x- = - dollar, or a quarter, per kilogram

Partitioning means to distribute. How can - dollar be distributed over 2 kg?

^^ l kg 1 kg - dollar = 2 quarters. One quarter is assigned to each 2 kilogram.

How much does one kilogram cost?

Answer: One kilogram of potting soil costs - dollar, or a quarter. 4

important than the measurement meaning.

Hyde and Nelson's work with egg cartons and Steiner's work with money can be extended to give students an intuitive understanding of the partitive meaning of division of fractions. The easiest example to demonstrate is that of a fraction divided by a whole number. See figure 1. Since

5/6 of an &gg carton is 1 0 of 1 2 sections, if 5/6 of an egg carton is distributed into two equivalent sets or parts, each set will be 5/1 2 of an egg carton. As with whole numbers, partitive division names the size of each set, which in this example is 5/12. A monetary application of this idea is given in figure 2. When the standard algorithm for dividing a fraction by a whole number

8 ARITHMETIC TEACHER



H^^Vj^Q^^I Partitive division of a fraction by a proper fraction

1 2 If - of an egg carton is to form - of a set, what is the size of one set?

6 3 1

2^2_2 ^_1 6

' 3

" Jb 2

" 4

2 1 2

Partitioning means to distribute. How can - be distributed into - set?

JET 1

^e T 2 - set 1 set 3

12 n 1 ,1 - = - n Put - into each - set. 6 12 12 3

How much is in one whole set?

Answer: Each whole set is - of an egg carton. 4

* Number of sets into which to distribute

^^^^^^^^^^^^^H Application of partitive division of a fraction by a proper fraction

3 2 If flower seeds cost 60<t ( - dollar) for - oz., what is the price per ounce?

3 2 3 3 9 -j dollars - oz. = - x - = - dollar, or 904 peroz. -j 5 3 5 2 10

3 2 Partitioning means to distribute. How can - dollar be distributed over- oz.?

5 3

| dollar 60<t (-^dollar) ' for- oz. 90<t (^dollar) 10 for 1 oz.

5 5 ' 3 10

1 Each -oz. costs 30$.

How much does one ounce cost? ç Answer: One ounce of flower seeds costs 90<t, or ttt dollar.

is introduced, the results can be checked with results gotten using egg cartons or money.

A more difficult instance of partitive division to comprehend is division by a proper fraction. See figure 3. This problem indicates that 1/6 of an egg carton - 2 sections - is to be distributed into 2/3 of a whole set or part; one section constitutes each third set. Solvers are then asked to find the size of one whole set. Since 3 thirds compose one whole, then 3 sections - or 1/4 of an egg carton - compose a whole set. This problem demonstrates that when the divisor is less than 1, the quo- tient is larger than the dividend. A monetary application of this idea is given in figure 4.

The situation in which a fraction is divided by a mixed number combines the foregoing concepts. See figure 5. Here the task is to distribute 5/6 of an egg carton - 10 sections - into 2 1/2 sets or parts. As shown in figure 5, each whole set contains 4 sectionsand the 1/2 set contains 2 sections. Since the answer to a partitive division problem gives the amount in one whole set, the answer is 4/12, or 1/3, of an egg carton. An application of partitive division by a mixed number is given in figure 6. Such exercises drive home to students the fact that in partitive division one divides the product (total) by the number of sets or parts into which it is to be partitioned to find the amount in one whole set or the amount per unit. With this understanding firmly in mind, problems such as the following are greatly simplified:

Mrs. Foster's will states that her estate of $3000 should be shared by her children as follows:

William 1 share

Judy 1 - shares 2

Martha 1 - shares 4 3

Total 3 - shares 4

How can her attorney determine how much each of her children should receive? Since $3000 is to be divided into a total of 3 3/4 shares, the attorney can divide $3000 by 3 3/4 to find the amount in one share (partitive division) as follows:

OCTOBER 1991 9



^^^^^^^^H Partitive division of a fraction by a mixed number

If - of an egg carton is divided into 2 - sets, what is the size of each set? 6 2

1 1 5 „ 1 5 5 ^5 * 1 - + „ 2- = - + - = - x- = - 6 2 6 2 *> 3 3

3 ] 5 1

Partitioning means to distribute. How can - be distributed over 2 - sets? 6 2

(®@@®©® * ol = f©@' /®©' /®P' i®@®©@©J * 2

= i@@y tëXS/ '@L; I - ^J

1 set 1 set - set 2

- = Put - in each whole set and - into the half set. 6 12 12 12

How much is in one whole set?

Answer: Each whole set is - of an egg carton. 3 * Number of sets into which to distribute

^^^^^^^^^M Application of partitive division of a fraction by a mixed number

1 3 If 2 - oz. of pumpkin seeds costs 75<t ( - dollar), how much does 1 oz. cost?

2 4

3 1 3 2 3 - dollar +2 - oz. = - x - = - dollar, or 30<t per oz. 4 2 4 5 10

3 1 Partitioning means to distribute. How can - dollar be distributed over 2 - oz.?

4 2

-dollar 30* for 30* for 15<tforloz. 4 1 oz. 1 oz. 2

3 - dollar = 75* (6 dimes and 3 nickels). Each oz costs 3 dimes, and the half- 4 oz costs 3 nickels. How much does 1 ounce cost? ~ Answer: One ounce of pumpkin seeds costs 30*, or - dollar.

3 15 $3000 + 3 - shares = 3000 + -

4 4

200 = x - = $800 per share

1 AS 1

Thus, William, who is to get one share, will receive $800. Judy's and Martha's shares are, respectively, $ 1 200 and $ 1 000.

A special instance of partitive division is finding an average. Consider the following problem:

Debbie drove 125 km in 2 1/2 hours. What was her average speed?

Since rates are amount per unit, the answer is found by partitive division:

125km -h 2 - h = 125 + - 2 2

25

= - x - = 50km/h 1 JS

1

Yet another use of partitive division is in finding unit prices, as demonstrated in figures 2, 4, and 6. Thus, partitive division by fractional numbers is a very useful concept and is at least as important to modeling problem situations as the measurement concept of division of fractions. Division of fractions is only as meaningful as the number of meanings one can give to the symbols. Thus, it is very important to give all students a complete understanding of the meaning of division as applied to fractions by also teaching partitive division. Understanding is usually an excellent antidote for fear and anxiety. We feel that students need early concrete experiences that clearly demonstrate the meanings of division of fractions; this belief is supported by the NCTM's (1989) Curriculum and Evaluation Standards for School Mathematics, which recommends that new content be explored empirically to help students understand mathematical abstractions. The document states that "concepts are the substance of mathematical knowledge. Students can make sense of mathematics only if they understand its concepts and their meanings and interpretations" (NCTM 1989, 223).

To summarize, the problem 3/4 ■=■ 1 1/2 has but one answer, 1/2, which can be found rotely by the traditional invert-the-

I o ARITHMETIC TEACHER



This student uses cut-up egg cartons to develop an intuitive understanding of the partitive meaning of division of fractions.

I 2>

divisor algorithm or by using one of the new fraction calculators. But the result is meaningless until someone gives it a meaningful interpretation. The result may have either of two interpretations depend- ing on the type of situation being mod- eled. One type of situation makes use of the measurement meaning of division and results in the interpretation that 1/2 group of size 1 1/2 is contained in 3/4. A second type of situation makes use of the partitive meaning of division; the division sentence is interpreted to indicate that if 3/4 is distributed into 1 1/2 groups, a whole group is of size 1/2.

References

Hyde, David, and Marvin N. Nelson. "Save Those Egg Cartons!" Arithmetic Teacher 14 (November 1967):578-79.

National Council of Teachers of Mathematics, Com- mission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.

Steiner, Evelyn E. "Division of Fractions: Develop- ing Conceptual Sense with Dollars and Cents." Arithmetic Teacher 34 (May 1987):36-42.

Trafton, Paul R., and Judith S. Zawojewski. "Teach- ing Rational Number Division: A Special Problem." Arithmetic Teacher 31 (February 1984):20-22.

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understanding partitive division of fractions

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