Sinicrope, Rose, Harold W. Mick, and John R. Kolb. “Interpretations of Fraction Division.” In Making Sense of Fractions, Ratios, and
Proportions, 2002 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Bonnie Litwiller, pp. 153–61.
Reston, Va.: NCTM, 2002.
17
Interpretations of Fraction Division
Rose Sinicrope
Harold W. Mick
John R. Kolb
IF OUR students are to construct a rich, relational understanding of fraction
division, we as teachers need a framework for fraction-division situations
that will help us select problem types and to design tasks. What is fraction
division? What is entailed? To answer these questions from a teaching and
learning perspective, we need to know what kinds of situations are fractiondivision situations, what reasoning occurs within these situations, and what
mathematical generalizations can be made.
Fraction-division algorithms can arise as abstractions of procedures used
to reason out the solutions to different problem situations. By exploring different algorithms, problem situations, and instructional models, we can categorize fraction-division situations, here called “interpretations.”
For whole-number division, problem situations can be categorized as
measurement division (determining the number of groups); partitive division (determining the size of each group); or the inverse of a Cartesian product (determining a dimension of a rectangular array). Fraction division can
be explained by extensions of all three of these whole-number interpretations. But these extensions are not sufficient; division as the determination
of a unit rate and division as the inverse of multiplication are also important
fraction-division interpretations.
MEASUREMENT DIVISION
When Warrington (1997) asked her fifth- and sixth-grade students to
describe what 4 ÷ 2 meant to them, the most common response was, “It
153
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electronically or in other formats without written permission from NCTM.
154
MAKING SENSE OF FRACTIONS, RATIOS, AND PROPORTIONS
means how many times does 2 fit into 4 or how many groups of two fit into
four?” This measurement interpretation is a common instructional focus for
whole-number division. As Warrington’s students demonstrated, it is also a
meaningful interpretation for fraction division. Her students reasoned that 2
divided by 1/2 is 4 because “one-half goes into 2 four times” and “ if you had
two candy bars and you divided them into halves, you’d have four pieces.”
Instructional models like pattern blocks also use this measurement interpretation. Pattern blocks—yellow hexagons, red trapezoids, blue parallelograms, and green triangles—are used in fraction-division instruction. The
pattern block pieces are constructed in such a way that two green triangles
will cover a blue parallelogram, for example, and six green triangles or two
red trapezoids will cover a yellow hexagon. For the convenience of fractiondivision instruction, the unit is often defined as the region formed by two
adjacent hexagons. This unit makes it easy to show halves, thirds, fourths,
sixths, and twelfths. The problems are presented using shapes. Students may
be asked, for example, to determine how many red trapezoids, 1/4 of the
whole, it will take to cover 11 green triangles, 11/12 of the whole, that is
11/12 ÷ 1/4. Since 1 red trapezoid will cover 3 green triangles (1/4 = 3/12), it
will take 11 ÷ 3, or 3 2/3, trapezoids to cover the 11 triangles (See fig. 17.1.).
The algorithm that represents the procedural reasoning in this type of
division is the common-denominator algorithm for the division of fractions:
a
c
ad
bc
ad
÷
=
÷
=
b
d
bd
bd
bc
The first step in the common-denominator algorithm is to express both the
divisor and the dividend as fractions with like denominators. Once the
denominators (the units of measure) are the same, the numerators are divided as in figure 17.1.
It is possible to relate the procedural reasoning used in the solution of
measurement divisions to the invert-and-multiply algorithm. The instructional process follows the pattern of first dividing a whole number by a unit
fraction, for example, 4 ÷ 1/3. Here the reasoning is that there are 3 thirds in
1, and hence four times as many in 4, or 12 one-thirds in 4. Next, the question asked is, “How many two-thirds are in 4, or 4 ÷ 2/3?” The reasoning is
that since there are 12 one-thirds in 4, then there are half as many two-thirds
in 4, or 1/2 of 12 = 6; that is, dividing by 2/3 is equivalent to multiplying by 3
and then multiplying by 1/2, or 3 × 1/2 = 3/2, or multiplying by the reciprocal of 2/3.
FRACTION DIVISION INTERPRETATIONS
Fig. 17.1. 11/12 ÷ 1/4
155
156
MAKING SENSE OF FRACTIONS, RATIOS, AND PROPORTIONS
PARTITIVE DIVISION
Another meaning of whole-number division that can be extended to fraction division is partitive division. The following is an example of partitive
division with whole numbers that is a sharing situation.
Jo has 12 pies. She will share them equally among 3 friends. How many
pies will each friend receive?
If we use the same division meaning and extend it to fractions, then the
problem takes the following form:
Jo has 12 sixteenths of a pie. She will share the pie equally among 3
friends. How much of the pie will each friend receive?
Solution:
12 wholes ÷ 3 = 4 wholes
12 sixteenths ÷ 3 = 4 sixteenths
Although this is a contrived “real world” problem, understanding the situation enables us to translate real-world problems of a similar type into a
fraction-division expression with an algorithmic solution that is meaningful. Perhaps in our haste to get to the invert-and-multiply algorithm, we
often skip the division of a fraction by a whole number. These divisions can
take two forms:
a
a÷c
when c divides a .
÷c=
b
b
a
a
when c does not divide a .
÷c=
b
b×c
In her study of prospective teachers’ knowledge of fraction division,
Tirosh (2000) used the following partitive division in which the divisor does
not divide the numerator (p. 9):
Four friends bought 1/4 kilogram of chocolate and shared it equally. How
much chocolate did each person get?
The solution requires dividing 1/4 by 4. A strip of paper folded into four
equal parts with one part shaded can represent the chocolate. With three
sections folded back in accordion-fold style to show only the 1/4 kg of
chocolate, fold the 1/4 section into four equal parts and mark the amount
one person will get. Unfold the strip to name the amount of chocolate, 1/16
kg.
FRACTION DIVISION INTERPRETATIONS
157
DETERMINATION OF A UNIT RATE
A different aspect of partitive division occurs if the situation is not about
sharing but focuses on the size of one group. The following problem is an
example of fraction division as the determination of a unit rate.
A printer can print 20 pages in two and one-half minutes. How many
pages does it print per minute?
A possible solution:
20 pages in 2 1/2 minutes
40 pages in 5 minutes
8 pages in 1 minute
20
20 × 2
40
40 ÷ 5
8
20
=
=
=
=
=
1
5
5
5
5 ÷ 5
1
× 2
2
2
2
2
Note that the process of multiplying by 2 and dividing by 5 is equivalent to
multiplication by the fraction 2/5, which provides another rationale for the
invert-and-multiply algorithm.
Ott, Snook, and Gibson (1991) use the term partitive division for this
division interpretation. For them, the determination of the unit rate is
equivalent to determining the size of one set; it is the same question that is
asked in the partitive division of whole numbers. Ott, Snook, and Gibson
(1991, p. 9) used the following question in their explanation of the partitive division of fractions: “If 1/6 of an egg carton is to form 2/3 of a set,
what is the size of one set?” They used the opposite order of dividing by
the numerator and then multiplying by the denominator to determine the
size of one set. Summarizing the actions, we find that dividing by 2/3 is
accomplished by multiplying by 3/2 (see fig. 17.2.). With other examples, it
becomes possible to generalize that dividing by a fraction is equivalent to
multiplying by the reciprocal of the fraction, which is the invert-and-multiply algorithm:
ad
a
a
d
×
b
bc
ad
b
c
=
=
=
c
1
bc
c
d
×
d
d
c
158
MAKING SENSE OF FRACTIONS, RATIOS, AND PROPORTIONS
Fig. 17.2. 1/6 ÷ 2/3
FRACTION DIVISION INTERPRETATIONS
159
DIVISION AS THE INVERSE OF MULTIPLICATION
The following problem is an example of the inverse of a fraction multiplication in which one of the fractions is an operator:
In a seventh-grade survey of lunch preferences, 48 students said they prefer pizza. This is one and one-half times the number of students who
prefers the salad bar! How many prefer the salad bar?
A Possible Solution:
11/2 × number for salad = 48
3/2 × number for salad = 48
If you multiply [the number for salad] by 3 and divide by 2, you get 48.
To undo, multiply by 2 and divide by 3, or multiply by the fraction 2/3.
48 ÷ 1 1/2 = 48 3/2 = 48 2/3
32 prefer salad.
Although different strategies can be used to solve this problem, a general
approach is reversing the procedure of the original multiplication. Multiplication by 1 1/2 is equivalent to multiplication by 3/2. To multiply by 3/2,
multiply by 3 and divide by 2. The reverse is to multiply by 2 and divide by
3, or to multiply by 2/3. The symbolic representation becomes
a
c
a
d
÷
= × .
b
d
b
c
DIVISION AS THE INVERSE OF A CARTESIAN
PRODUCT
Another interpretation for the division of fractions is the inverse of a
Cartesian product. The problem situation may be one of area, where the
total area and one dimension of a rectangular region are known. The problem is to determine the other dimension.
For example, to determine the width of a rectangle that has a length of 3/4
units and an area of 6/20 square units, first form a corner of a rectangle with
two sides (see fig. 17.3.). Along one side, mark off a length of 3/4 units. The
unit length is arbitrary, but put tick marks at each fourth. In order to deal
with the area, which is measured in twentieths, mark off the other side in
fifths, so that the area will be represented in little rectangles representing
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MAKING SENSE OF FRACTIONS, RATIOS, AND PROPORTIONS
Fig. 17.3. 6/20 ÷ 3/4
FRACTION DIVISION INTERPRETATIONS
161
1/20 of a square unit. Next, determine the width by marking off one-fifths
until you use all 6/20 square units of area. Since in our rectangle each onefifth in width uses three 1/20 square units of area, the width is 2/5 units. An
algorithm that describes this process is an algorithm in which numerator
divides numerator and denominator divides denominator—for example,
6
3
6÷3
2
÷ =
= .
20
4
20 ÷ 4
5
SUMMARY
Fraction division has many different interpretations. For instance, we
divide to determine how many times one quantity is contained in a given
quantity, to share, to determine what the unit is, to determine the original
amount, and to determine a dimension for an array. For the teacher of
mathematics, an exploration of different interpretations of fraction division
forms a framework for designing instruction through posing problems. As
students solve the teacher-posed problems and similar problems, they can
eventually generate algorithms for solving even “larger sets” of problems.
This connection between the problem context and the fraction-division
algorithm is a link that is often missing in students’ understanding and performance (Piel and Green 1994). These problem contexts for fraction division are varied in ways deeper than whether the problem is about pizza or
gallons of milk. These contexts encourage different procedural reasoning to
solve the problems. These situations or interpretations can be categorized as
measurement division, partitive division, determination of a unit rate, the
inverse of multiplication, and the inverse of a Cartesian product.
REFERENCES
Ott, Jack M., Daniel L. Snook, and Diana L. Gibson. “Understanding Partitive Division of Fractions.” Arithmetic Teacher 39 (October 1991): 7–11.
Piel, John A., and Michael Green. “De-Mystifying Division of Fractions: The Convergence of Quantitative and Referential Meaning.” Focus on Learning Problems in
Mathematics 16 (Winter 1994): 44–50.
Tirosh, Dina. “Enhancing Prospective Teachers’ Knowledge of Children’s Conceptions: The Case of Division of Fractions.” Journal for Research in Mathematics Education 31 (January 2000): 5–25.
Warrington, Mary Ann. “How Children Think about Division of Fractions.” Mathematics Teaching in the Middle School 2 (May 1997): 390–94.