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1
0.1. DIVIDING FRACTIONS
Excerpt from: Mathematics for Elementary Teachers, First Edition, by
c 2003, by Addison-Wesley
Sybilla Beckmann. Copyright 0.1
Dividing Fractions
In this section, we will discuss the two interpretations of division for fractions,
and we will see why the standard “invert and multiply” procedure for dividing
fractions gives answers to fraction division problems that agree with what
we expect from the meaning of division.
The Two Interpretations of Division for Fractions
Let’s review the meaning of division for whole numbers, and see how to
interpret division for fractions.
The “how many groups?” interpretation
With the “how many groups?” interpretation of division, 12 ÷ 3 means the
number of groups we can make when we divide 12 objects into groups with
3 objects in each group. In other words, 12 ÷ 3 tells us how many groups of
3 we can make from 12.
Similarly, with the “how many groups?” interpretation of division,
5 2
÷
2 3
means the number of groups we can make when we divide 25 of an object
into groups with 23 of an object in each group. In other words, 25 ÷ 32 tells us
how many groups of 32 we can make from 52 . For example, suppose you are
making popcorn balls and each popcorn ball requires 23 of a cup of popcorn.
If you have 2 21 = 52 of a cup of popcorn, then how many popcorn balls can
you make? In this case you want to divide 25 of a cup of popcorn into groups
(balls) with 32 of a cup of popcorn in each group. According to the “how
many groups?” interpretation of division, you can make
5 2
÷
2 3
popcorn balls.
2
The “how many in one (each) group?” interpretation
With the “how many in each group?” interpretation of division, 12÷3 means
the number of objects in each group when we distribute 12 objects equally
among 3 groups. In other words, 12 ÷ 3 is the number of objects in one group
if we use 12 objects to evenly fill 3 groups. When we work with fractions, it
often helps to think of “how many in each group?” division story problems as
asking “how many are in one whole group?”, and it helps to think of filling
groups or part of a group. So in the context of fractions, we will usually
refer to the “how many in each group?” interpretation as “how many in one
group?”.
With the “how many in one group?” interpretation of division,
3 1
÷
4 2
is the number of objects in one group when we distribute 43 of an object
equally among 12 of a group. A clearer way to say this is: 43 ÷ 21 is the
number of objects (or fraction of an object) in one whole group when 34 of
an object fills 21 of a group. For example, suppose you pour 43 of a pint of
blueberries into a container and this fills 12 of the container. How many pints
of blueberries will it take to fill the whole container? In this case, 43 of a
pint of blueberries fills (i.e., is distributed equally among) 21 of a group (a
container). So according to the “how many in one group?” interpretation
of division, the number of pints of blueberries in one whole group (one full
container) is
3 1
÷
4 2
One way to better understand fraction division story problems is to think
about replacing the fractions in the problem with whole numbers. For example, if you have 3 pints of blueberries and they fill 2 containers, then how
many pints of blueberries are in each container? We solve this problem by
dividing 3 ÷ 2, according to the “how many in each group?” interpretation.
Therefore if we replace the 3 pints with 43 of a pint, and the 2 containers with
1
of a container, we solve the problem in the same way as before: 3 ÷ 2 now
2
becomes 34 ÷ 12 .
Here is another way to think about the problem. Because 21 of the container is filled, and because this amount is 34 of a pint, therefore 21 of the
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0.1. DIVIDING FRACTIONS
number of pints in a full container is
3
4
of a pint. In other words:
3
1
× number of pints in full container =
2
4
Therefore
number of pints in full container =
Dividing by
1
2
Versus Dividing in
3 1
÷
4 2
1
2
In mathematics, language is used much more precisely and carefully than in
everyday conversation. This is one source of difficulty in learning mathematics. For example, consider the two phrases:
dividing by 21 ,
dividing in 12 .
You may feel that these two phrases mean the same thing, however, mathematically, they do not. To divide a number, say 5, by 21 means to calculate
5 ÷ 12 . Remember that we read A ÷ B as A divided by B. We would divide
5 by 12 if we wanted to know how many half cups of flour are in 5 cups of
flour, for example. (Notice that there are 10 half-cups of flour in 5 cups of
flour, not 2 21 .)
On the other hand, to divide a number in half means to find half of that
number. So to divide 5 in half means to find 12 of 5. One half of 5 means
1
× 5. So dividing in 21 is the same as dividing by 2.
2
The “Invert and Multiply” Procedure for Fraction Division
Although division with fractions can be difficult to interpret, the procedure
for dividing fractions is quite easy. To divide fractions, such as
3 2
2
÷
and 6 ÷
4 3
5
we can use the familiar “invert and multiply” method in which we invert the
divisor and multiply by it:
3 2
3 3
3·3
9
÷ = · =
=
4 3
4 2
4·2
8
4
and
6÷
reciprocal
2
6 2
6 5
6·5
30
= ÷ = · =
=
= 15
5
1 5
1 2
1·2
2
Another way to describe this “invert and multiply” method for dividing
fractions is in terms of the reciprocal of the divisor. The reciprocal of a
C
is the fraction D
. In order to divide fractions, we should multiply
fraction D
C
by the reciprocal of the divisor. So in general,
A C
A D
A·D
÷
= ·
=
B D
B C
B·C
Explaining Why “Invert and Multiply” is Valid by Relating Division to Multiplication
The procedure for dividing fractions is easy enough to carry out, but why is it
a valid method? Before we answer this question in general, consider a special
case. Recall that every whole number is equal to a fraction (for example,
6 = 16 ). Therefore we can apply the “invert and multiply” procedure to
whole numbers as well as to fractions. According to this procedure,
2÷3=
2 3
2 1
2·1
2
÷ = · =
=
1 1
1 3
1·3
3
Notice that this result, that 2 ÷ 3 = 32 , agrees with our findings earlier in
this chapter: that we can describe fractions in terms of division, namely that
A
= A ÷ B.
B
In general, why is the “invert and multiply” procedure a valid way to
divide fractions? One way to explain this is to relate fraction division to
fraction multiplication. Recall that every division problem is equivalent to a
multiplication problem (actually two multiplication problems):
A ÷ B =?
is equivalent to
?·B =A
(or B·? = A). So
3 2
÷ =?
4 3
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0.1. DIVIDING FRACTIONS
is equivalent to
3
2
= .
(1)
3
4
Now remember that we want to explain why the “invert and multiply” rule
for fraction division is valid. This rule says that 43 ÷ 23 ought to be equal to
?·
3·3
4·2
Let’s check that this fraction works in the place of the ? in Equation 1. In
3·3
other words, let’s check that if we multiply 4·2
times 23 , then we really do get
3
:
4
3·3·2
3 · (3 · 2)
3 · (3 · 2)
3
3·3 2
· =
=
=
=
4·2 3
4·2·3
4 · (2 · 3)
4 · (3 · 2)
4
Therefore the answer we get from the “invert and multiply” procedure really
is the answer to the original division problem 43 ÷ 32 . Notice that the line
of reasoning above applies in the same way when other fractions replace the
fractions 32 and 43 used above.
It will still be valuable to explore fraction division further, interpreting
fraction division directly rather than through multiplication.
Class Activity 0A: Explaining “Invert and Multiply” by
Relating Division to Multiplication
Using the “How Many Groups?” Interpretation to Explain Why “Invert And Multiply” Is Valid
Above, we explained why the “invert and multiply” procedure for dividing
fractions is valid by considering fraction division in terms of fraction multiplication. Now we will explain why the “invert and multiply” procedure is
valid by working with the “how many groups?” interpretation of division .
Consider the division problem
2 1
÷
3 2
The following is a story problem for this division problem:
How many
1
2
cups of water are in
2
3
of a cup of water?
6
Or, said another way:
How many times will we need to pour 21 cup of water into a
container that holds 23 cup of water in order to fill the container?
From the diagram in Figure 1 we can say right away that the answer to this
problem is “one and a little more” because one half cup clearly fits in two
thirds of a cup, but then a little more is still needed to fill the two thirds of
a cup. But what is this “little more”? Remember the original question: we
want to know how many 12 cups of water are in 32 of a cup of water. So the
answer should be of the form “so and so many 21 cups of water.” This means
that we need to express this “little more” as a fraction of 21 cup of water.
How can we do that? By subdividing both the 21 and the 32 into common
parts, namely by using common denominators.
1/2 cup
2/3 cup
1/2 cup =
3/6 cup
2/3 cup =
4/6 cup
Figure 1: How Many 1/2 Cups of Water Are in 2/3 Cup?
When we give 21 and 23 the common denominator of 6, then, as on the
right of Figure 1, the 21 cup of water is made out of 3 parts (3 sixths of a
cup of water), and the 32 cup of water is made out of 4 parts (4 sixths of a
cup of water), so the “little more” we were discussing above is just one of
those parts. Since 21 cup is 3 parts, and the “little more” is 1 part, the “little
more” is 13 of the 12 cup of water. This explains why 23 ÷ 12 = 1 13 : there’s a
whole 21 cup plus another 13 of the 21 cup in 23 of a cup of water.
To recap: we are considering the fraction division problem 32 ÷ 21 in terms
of the story problem “how many 12 cups of water are in 23 of a cup of water?”
If we give 21 and 23 the common denominator of 6, then we can rephrase
the problem as “how many 63 of a cup are in 64 of a cup?” But in terms of
Figure ??, this is equivalent to the problem “how many 3s are in 4?” which
is the problem 4 ÷ 3, whose answer is 34 = 1 31 . Notice that 34 is exactly the
same answer we get from the “invert and multiply” procedure for fraction
division:
2 2
2·2
4
2 1
÷ = · =
=
3 2
3 1
3·1
3
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0.1. DIVIDING FRACTIONS
So the “invert and multiply” procedure gives the same answer to 23 ÷ 21 that
we arrive at by using the “how many groups?” interpretation of division.
The same line of reasoning will work for any fraction division problem
A C
÷
B D
A
C
Thinking logically, as above, and interpreting B
÷D
as “how many
A
cups of water?”, we can conclude that
of water are in B
C
D
cups
A C
A·D
B ·C
A·D
÷
=
÷
= (A · D) ÷ (B · C) =
B D
B·D B·D
B·C
The final expression, A·D
, is the answer provided by the “invert and multiply”
B·C
procedure for dividing fractions. Therefore we know that the “invert and
multiply” procedure gives answers to division problems that agree with what
we expect from the meaning of division.
Class Activity 0B: “How Many Groups?” Fraction Division Problems
Using the “How Many in One Group?” Interpretation
to Explain Why “Invert And Multiply” Is Valid
Above, we saw how to use the “how many groups?” interpretation of division
to explain why the “invert and multiply” procedure for fraction division is
valid. We can also use the “how many in one group?” interpretation for
the same purpose. This interpretation, although perhaps more difficult to
understand, has the advantage of showing us directly why we can multiply
by the reciprocal of the divisor in order to divide fractions.
Consider the following “how many in one group?” story problem for 21 ÷ 35 :
You used 12 of can of paint to paint 53 of a wall. How many cans
of paint will it take to paint the whole wall?
This is a “how many in one group?” problem because we can think of the
paint as “filling” 53 of the wall. We can also see that this is a division problem
by writing the corresponding number sentence:
3
1
· (amount to paint the whole wall) =
5
2
8
Therefore
1 3
÷
2 5
We will now see why it makes sense to solve this problem by multiplying
1
by the reciprocal of 53 , namely by 53 . Let’s focus on the wall to be painted,
2
as shown in Figure 2. Think of dividing the wall into 5 equal sections, 3 of
amount to paint the whole wall =
the 1/2 can of paint is divided equally
among 3 parts
the amount of paint for the full wall is
5 times the amount in one part
Figure 2: The Amount of Paint Needed for the Whole Wall is
Used to Cover 53 of the Wall
5
3
of the
1
2
Can
which you painted with the 21 can of paint. If you used 12 a can of paint to
paint 3 sections, then each of the 3 sections required 21 ÷ 3 or 21 × 31 cans
of paint. To determine how much paint you will need for the whole wall,
multiply the amount you need for one section by 5. So you can determine
the amount of paint you need for the whole wall by multiplying the 12 can of
paint by 31 and then multiplying that result by 5, as summarized in Table 1.
But to multiply a number by 31 and then multiply it by 5 is the same as
multiplying the number by 35 . Therefore we can determine the number of
cans of paint you need for the whole wall by multiplying 21 by 53 :
5
1 5
· =
2 3
6
This is exactly the “invert and multiply” procedure for dividing
shows that you will need 56 of a can of paint for the whole wall.
1
2
÷ 35 . It
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0.1. DIVIDING FRACTIONS
use
1
2
can paint
for
↓ ÷3 or × 31
use
1
6
can paint
↓ ×5
3
5
of the wall
↓ ÷3 or × 31
for
1
5
of the wall
↓ ×5
use
5
6
can paint
use
1
2
in one step:
can paint for 35 of the wall
↓ × 35
use
5
6
can paint
for 1 whole wall
↓ × 35
for 1 whole wall
Table 1: Determining How Much Paint to Use for a Whole Wall if
Paint Covers 53 of the Wall
1
2
Can of
10
The argument above works when other fractions replace 21 and 35 , thereby
explaining why
A C
A D
÷
= ·
B D
B C
In other words, to divide fractions, multiply the dividend by the reciprocal
of the divisor.
Class Activity 0C: “How Many in One Group?” Fraction Division Problems
Class Activity 0D: Are These Division Problems?
Exercises for Section 0.1 on Dividing Fractions
1. Write a “how many groups?” story problem for 1 ÷ 75 . Use the story
problem and a diagram to help you solve the problem.
2. Write a “how many in one group?” story problem for 1 ÷ 34 . Use the
situation of the story problem to help you explain why the answer is
4
= 1 13 .
3
3. Annie wants to solve the division problem
story problem:
3
4
÷ 12 by using the following
I need 21 cup of chocolate chips to make a batch of cookies.
How many batches of cookies can I make with 43 of a cup of
chocolate chips?
Annie draws a diagram like the one in Figure 3. Explain why it would
be easy for Annie to misinterpret her diagram as showing that 43 ÷ 12 =
1 14 . How should Annie interpret her diagram so as to conclude that
3
÷ 12 = 1 12 ?
4
4. Which of the following are solved by the division problem 43 ÷ 12 ? For
those that are, which interpretation of division is used? For those that
are not, determine how to solve the problem, if it can be solved.
(a)
3
4
of a bag of jelly worms make
worms are in one bag?
1
2
a cup. How many cups of jelly
11
0.1. DIVIDING FRACTIONS
1/4 cup left
1/2 cup
makes
one batch
Figure 3: How Batches of Cookies Can We Make With 43 of a Cup of Chocolate Chips if 1 Batch Requires 21 Cup of Chocolate Chips?
(b)
of a bag of jelly worms make 21 a cup. How many bags of jelly
worms does it take to make one cup?
3
4
(c) You have 43 of a bag of jelly worms and a recipe that calls for 21 of
a cup of jelly worms. How many batches of your recipe can you
make?
(d) You have 34 of a cup of jelly worms and a recipe that calls for 21 of
a cup of jelly worms. How many batches of your recipe can you
make?
(e) If 34 of a pound of candy costs 21 of a dollar, then how many pounds
of candy should you be able to buy for 1 dollar?
(f) If you have 43 of a pound of candy and you divide the candy in 21 ,
then how much candy will you have in each portion?
(g) If 21 of a pound of candy costs $1, then how many dollars should
you expect to pay for 34 of a pound of candy?
5. Frank, John, and David earned $14 together. They want to divide it
equally, except that David should only get a half share, since he did half
as much work as either Frank or John did (and Frank and John worked
equal amounts). Write a division problem to find out how much Frank
should get. Which interpretation of division does this story problem
use?
6. Bill leaves a tip of $4.50 for a meal. If the tip is 15% of the cost of
the meal, then how much did the meal cost? Write a division problem
to solve this. Which interpretation of division does this story problem
use?
12
7. Compare the arithmetic needed to solve the following problems.
(a) What fraction of a
of water?
1
3
cup measure is filled when we pour in
(b) What is one quarter of
(c) How much more is
1
4
1
3
1
3
1
4
cup
cup?
cup than
1
4
cup?
1
3
(d) If cup of water fills of a plastic container, then how much
water will the full container hold?
8. Use the meanings of multiplication and division to solve the following
problems.
(a) Suppose you drive 4500 miles every half year in your car. At the
end of 3 34 years, how many miles will you have driven?
(b) Mo used 128 ounces of liquid laundry detergent in 6 12 weeks. If
Mo continues to use laundry detergent at this rate, how much will
he use in a year?
(c) Suppose you have a 32 ounce bottle of weed killer concentrate.
The directions say to mix two and a half ounces of weed killer
concentrate with enough water to make a gallon. How many gallons of weed killer will you be able to make from this bottle?
9. The line segment below is 23 of a unit long. Show a line segment that
is 25 of a unit long. Explain how this problem is related to fraction
division.
2
3
unit
Answers To Exercises For Section 0.1 on Dividing Fractions
1. A simple “how many groups?” story problem for 1 ÷ 75 is “how many 57
of a cup of water are in 1 cup of water?” Figure 4 shows 1 cup of water
and shows 57 of a cup of water shaded. The shaded portion is divided
into 5 equal parts and the full cup is 7 of those parts. So the full cup
is 75 of the shaded part. Thus there are 57 of 75 of a cup of water in 1
cup of water, so 1 ÷ 75 = 57 .
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0.1. DIVIDING FRACTIONS
1 cup
5
7
1
5
of a cup
Figure 4: Showing Why 1 ÷ 57 =
Water are in 1 Cup of Water
7
5
by Considering How Many
5
7
each piece is
of the shaded
portion
of a Cup of
2. A “how many in one group?” story problem for 1 ÷ 43 is “if 1 ton of
dirt fills a truck 34 full, then how many tons of dirt will be needed to
fill the truck completely full?”. We can see that this is a “how many
in one group?” type of problem because the 1 ton of dirt fills 43 of a
group (the truck) and we want to know the amount of dirt in 1 whole
group. Figure 5 shows a truck bed divided into 4 equal parts with 3
of those parts filled with dirt. Since the 3 parts are filled with 1 ton
of dirt, each of the 3 parts must contain 31 of a ton of dirt. To fill the
truck completely, 4 parts, each containing 31 of a ton of dirt are needed.
Therefore the truck takes 34 = 1 31 tons of dirt to fill it completely, and
so 1 ÷ 43 = 34 .
the 1 ton of dirt
is divided equally
among 3 parts
truck bed
4 parts are needed
to fill the truck;
each part is 1/3 of
a ton, so 4/3 tons
of dirt are needed
to fill the truck
Figure 5: Showing Why 1 ÷ 34 = 34 by Considering How Many Tons of Dirt
it Takes to Fill a Truck if 1 Ton Fills it 34 Full
3. Annie’s diagram shows that she can make 1 full batch of cookies from
14
her 34 of a cup of chocolate chips and that 41 cup of chocolate chips will
be left over. Because 14 cups of chocolate chips are left over, it would
be easy for Annie to misinterpret her picture as showing 43 ÷ 12 = 1 41 .
But the answer to the problem is supposed to be the number of batches
Annie can make. In terms of batches, the remaining 41 cup of chocolate
chips makes 12 of a batch of cookies. We can see this because 2 quartercup sections make a full batch, so each quarter-cup section makes 12 of
a batch of cookies. So by interpreting the remaining 14 cup of chocolate
chips in terms of batches, we see that Annie can make 1 12 batches of
chocolate chips, thereby showing that 34 ÷ 21 = 1 21 , not 1 14 .
4. (a) This problem can be rephrased as “if 21 of a cup of jelly worms
fill 43 of a bag, then how many cups fill a whole bag?”, therefore
this is a “how many in one group?” division problem illustrating
1
÷ 34 , not 43 ÷ 12 . Since 21 ÷ 34 = 21 · 43 = 23 , there are 32 of a cup of
2
jelly worms in a whole bag.
(b) This problem is solved by 43 ÷ 21 , according to the “how many in
each group?” interpretation. A group is a cup and each object is
a bag of jelly worms.
(c) This problem can’t be solved because you don’t know how many
cups of jelly worms are in 43 of a bag.
(d) This problem is solved by 43 ÷ 12 , according to the “how many
groups?” interpretation. Each group consists of 12 of a cup of jelly
worms.
(e) This problem is solved by 34 ÷ 21 , according to the “how many in
one group?” interpretation. This is because you can think of the
problem as saying that 43 of a pound of candy fills 21 of a group
and you want to know how many pounds fills 1 whole group.
(f) This problem is solved by
not dividing by 12 .
3
4
× 21 , not
3
4
÷ 21 . It is dividing in half,
(g) This problem is solved by 34 × 12 , according to the “how many
groups?” interpretation because you want to know how many 12
pounds are in 34 of a pound. Each group consists of 21 of a pound
of candy.
5. If we consider Frank and John as each representing one group, and
David as representing half of a group, then the $14 should be dis-
15
0.1. DIVIDING FRACTIONS
tributed equally among 2 12 groups. Therefore, this is a “how many in
one group” division problem. Each group should get
1
5
2
28
3
6
14 ÷ 2 = 14 ÷ = 14 · =
= 5 = 5 = 5.60
2
2
5
5
5
10
dollars. Therefore Frank and John should each get $5.60 and David
should get half of that, which is $2.80.
6. According to the “how many in one group?” interpretation, the problem is solved by $4.50 ÷ 0.15 because $4.50 fills 0.15 of a group and we
want to know how much is in 1 whole group. So the meal cost
$4.50 ÷ 0.15 = $4.50 ÷
100
$450
15
= $4.50 ·
=
= $30
100
15
15
7. Each problem, except for the first and last, requires different arithmetic
to solve it.
(a) This is asking: 14 equals what times 31 ? We solve this by calculating
1
÷ 13 , which is 43 . We can also think of this as a division problem
4
with the “how many groups?” interpretation because we want to
know how many 31 of a cup are in 41 of a cup. According to the
meaning of division, this is 41 ÷ 31 .
(b) This is asking: what is
1
1
× 31 = 12
.
4
1
4
of
1
?
3
We solve this by calculating
1
(c) This is asking: what is 13 − 41 ? The answer is 12
which happens to
be the same answer as in part (b), but the arithmetic to solve it
is different.
(d) Since 14 cup of water fills 31 of a plastic container, the full container
will hold 3 times as much water, or 3 × 41 = 34 of a cup. We can
also think of this as a division problem with the “how many in one
group?” interpretation. 41 cup of water is put into 31 of a group.
We want to know how much is in one group. According to the
meaning of division it’s 41 ÷ 13 , which again is equal to 43 .
8. (a) The number of 21 years in 3 34 years is 3 34 ÷ 12 . There will be that
many groups of 4500 miles driven. So after 3 34 years you will have
16
driven
3 1
15 1
(3 ÷ ) × 4500 = ( ÷ ) × 4500
4 2
4
2
15
× 4500
=
2
= 33, 750
miles.
(b) Since one year is 52 weeks there are 52 ÷ 6 12 groups of 6 12 weeks
in a year. Mo will use 128 ounces for each of those groups, so Mo
will use
1
13
(52 ÷ 6 ) × 128 = (52 ÷ ) × 128
2
2
104
=
× 128
13
= 1, 024
ounces of detergent in a year.
(c) There are 32 ÷ 2 12 groups of 2 21 ounces in 32 ounces. Each of those
groups makes 1 gallon. So the bottle makes 32 ÷ 2 12 = 12 54 gallons
of weed killer.
9. One way to solve the problem is to determine how many 32 units are in 52
units. This will tell us how many of the 32 unit long segments to lay end
to end in order to get the 52 unit long segment. Since 52 ÷ 23 = 15
= 3 34 ,
4
there are 3 43 segments of length 23 units in a segment of length 52 units.
So you need to form a line segment that is 3 times as long as the one
pictured, plus another 34 as long:
Problems for Section 0.1 on Dividing Fractions
1. A bread problem: If one loaf of bread requires 1 41 cups of flour, then
how many loaves of bread can you make with 10 cups of flour? (Assume
that you have enough of all other ingredients on hand.)
(a) Solve the bread problem by drawing a diagram. Explain your
reasoning.
17
0.1. DIVIDING FRACTIONS
(b) Write a division problem that corresponds to the bread problem.
Solve the division problem by “inverting and multiplying.” Verify
that your solution agrees with your solution in part (a).
2. A measuring problem: You are making a recipe that calls for 23 cup
of water, but you can’t find your 31 cup measure. You can, however,
find your 41 cup measure. How many times should you fill your 41 cup
measure in order to measure 23 of a cup of water?
(a) Solve the measuring problem by drawing a diagram. Explain your
reasoning.
(b) Write a division problem that corresponds to the measuring problem. Solve the division problem by “inverting and multiplying.”
Verify that your solution agrees with your solution in part (a).
3. Write a “how many groups?” story problem for 4 ÷ 32 and solve your
problem in a simple and concrete way without using the “invert and
multiply” procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the “invert and multiply”
procedure.
4. Write a “how many groups?” story problem for 5 41 ÷ 1 34 and solve your
problem in a simple and concrete way without using the “invert and
multiply” procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the “invert and multiply”
procedure.
5. Jose and Mark are making cookies for a bake sale. Their recipe calls
for 2 41 cups of flour for each batch. They have 5 cups of flour. Jose and
Mark realize that they can make two batches of cookies and that there
will be some flour left. Since the recipe doesn’t call for eggs, and since
they have plenty of the other ingredients on hand, they decide they can
make a fraction of a batch in addition to the two whole batches. But
Jose and Mark have a difference of opinion. Jose says that
2
1
5÷2 =2
4
9
and so he says that they can make 2 92 batches of cookies. Mark says
that two batches of cookies will use up 4 12 cups of flour, leaving 12 left,
18
so they should be able to make 2 12 batches. Mark draws the picture in
Figure 6 to explain his thinking to Jose. Discuss the boys’ mathematics:
Figure 6: Representing 5 ÷ 2 41 by Considering How Many 2 41 Cups of Flour
are in 5 Cups of Flour
what’s right, what’s not right, and why? If anything is incorrect, how
could you modify it to make it correct?
6. Marvin has 11 yards of cloth to makes costumes for a play. Each
costume requires 1 12 yards of cloth.
(a) Solve the following two problems:
i. How many costumes can Marvin make and how much cloth
will be left over?
ii. What is 11 ÷ 1 21 ?
(b) Compare and contrast your answers in part (a).
7. A laundry problem: You need 43 of a cup of laundry detergent to wash
one full load of laundry. How many loads of laundry can you wash with
5 cups of laundry detergent? (Assume that you can wash fractional
loads of laundry.)
(a) Solve the laundry problem by drawing a diagram. Explain your
reasoning.
(b) Write a division problem that corresponds to the laundry problem.
Solve the division problem by “inverting and multiplying.” Verify
that your solution agrees with your solution in part (a).
8. Write a “how many groups?” story problem for 2 ÷ 43 and solve your
problem in a simple and concrete way without using the “invert and
multiply” procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the “invert and multiply”
procedure.
0.1. DIVIDING FRACTIONS
19
9. Write a “how many groups?” story problem for 13 ÷ 14 and solve your
problem in a simple and concrete way without using the “invert and
multiply” procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the “invert and multiply”
procedure.
10. Write a “how many groups?” story problem for 21 ÷ 23 and solve your
problem in a simple and concrete way without using the “invert and
multiply” procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the “invert and multiply”
procedure.
11. Fraction division story problems involve the simultaneous use of different wholes. Solve the following paint problem in a simple and concrete
way without using the “invert and multiply” procedure. Describe how
you must work simultaneously with different wholes in solving the problem.
A paint problem: You need 43 of a bottle of paint to paint
a poster board. You have 3 21 bottles of paint. How many
poster boards can you paint?
12. An article by Dina Tirosh, [?], discusses some common errors in division. The following problems are based on some of the findings of this
article.
(a) Tyrone says that 12 ÷5 doesn’t make sense because 5 is bigger than
1
and you can’t divide a smaller number by a bigger number. Give
2
Tyrone an example of a sensible story problem for 21 ÷ 5. Solve
your problem and explain your solution.
(b) Kim says that 4 ÷ 13 can’t be equal to 12 because when you divide,
the answer should be smaller. Kim thinks the answer should be
1
because that is less than 4. Give Kim an example of a story
12
problem for 4 ÷ 13 and explain why it makes sense that the answer
1
really is 12, not 12
.
13. Write a story problem for 34 × 12 and another story problem for 34 ÷ 12
(make clear which is which). In each case, use elementary reasoning
about the story situation to solve your problem. Explain your reasoning.
20
14. Sam picked 12 of a gallon of blueberries. Sam poured the blueberries
into one of his plastic containers and noticed that the berries filled
the container 32 full. Solve the following problems in any way you like
without using a calculator. Explain your reasoning in detail.
(a) How many of Sam’s containers will 1 gallon of blueberries fill?
(Assume Sam has a number of containers of the same size.)
(b) How many gallons of blueberries does it take to fill Sam’s container
completely full?
15. A road crew is building a road. So far, 32 of the road has been completed
and this portion of the road is 34 of a mile long. Solve the following
problems in any way you like without using a calculator. Explain your
reasoning in detail.
(a) How long will the road be when it is completed?
(b) When the road is 1 mile long, what fraction of the road will be
completed?
16. Will has mowed 32 of his lawn and so far it’s taken him 45 minutes.
For each of the following problems, solve the problem in two ways:
1) by using elementary reasoning about the story situation and 2) by
interpreting the problem as a division problem (say whether it is a
“how many groups?” or a “how many in one group?” type of problem)
and by solving the division problem using standard paper and pencil
methods. Do not use a calculator. Verify that you get the same answer
both ways.
(a) How long will it take Will to mow the entire lawn (all together)?
(b) What fraction of the lawn can Will mow in an hour?
17. Grandma’s favorite muffin recipe uses 1 34 cups of flour for one batch of
12 muffins. For each of the following problems, solve the problem in
two ways: 1) by using elementary reasoning about the story situation
and 2) by interpreting the problem as a division problem (say whether
it is a “how many groups?” or a “how many in one group?” type
of problem) and by solving the division problem using standard paper
and pencil methods. Do not use a calculator. Verify that you get the
same answer both ways.
0.1. DIVIDING FRACTIONS
21
(a) How many cups of flour are in one muffin?
(b) How many muffins does 1 cup of flour make?
(c) If you have 3 cups of flour, then how many batches of muffins
can you make? (Assume that you can make fractional batches of
muffins and that you have enough of all the ingredients.)
18. Write a “how many in one group?” story problem for 4 ÷ 31 and use
your story problem to explain why it makes sense to solve 4 ÷ 31 by
“inverting and multiplying,” in other words by multiplying 4 by 31 .
19. Write a “how many in one group?” story problem for 4 ÷ 32 and use
your story problem to explain why it makes sense to solve 4 ÷ 32 by
“inverting and multiplying,” in other words by multiplying 4 by 32 .
20. Write a “how many in one group?” story problem for 9 ÷ 43 and use
your story problem to explain why it makes sense to solve 9 ÷ 43 by
“inverting and multiplying,” in other words by multiplying 9 by 43 .
21. Write a “how many in one group?” story problem for 21 ÷ 34 and use
your story problem to explain why it makes sense to solve 21 ÷ 43 by
“inverting and multiplying,” in other words by multiplying 12 by 34 .
22. Write a “how many in one group?” story problem for 1 ÷ 2 21 and use
your story problem to explain why it makes sense to solve 1 ÷ 2 21 by
“inverting and multiplying”.
23. Give an example of either a hands-on activity or a story problem for
elementary school children that is related to a fraction division problem (even if the children wouldn’t think of the activity or problem as
fraction division). Write the fraction division problem that is related to
your activity or story problem. Describe how the children could solve
the problem by using logical thinking aided by physical actions or by
drawing pictures.
24. Buttercup the gerbil drank 23 of a bottle of water in 1 21 days. Assuming
Buttercup continues to drink water at the same rate, how many bottles of water will Buttercup drink in 5 days? Use multiplication and
division to solve this problem, explaining in detail why you can use
multiplication when you do and why you can use division when you do.
22
25. If you used 2 21 truck loads of mulch for a garden that covers 34 of an acre,
then how many truck loads of mulch should you order for a garden that
covers 3 21 acres? (Assume that you will spread the mulch at the same
rate as before.) Use multiplication and division to solve this problem,
explaining in detail why you can use multiplication when you do and
why you can use division when you do.
26. If 2 12 pints of jelly filled 3 21 jars, then how many jars will you need
for 12 pints of jelly? Will the last jar of jelly be completely full? If
not, how full will it be? (Assume that all jars are the same size.) Use
multiplication and division to solve this problem, explaining in detail
why you can use multiplication when you do and why you can use
division when you do.