0.1 Dividing Fractions - University of Georgiamath.uga.edu/~sybilla/5020Fa03/FractionDivision.pdfClass Activity 0A: Explaining \Invert and Multiply" by Relating Division to Multiplication

$Page 1: 0.1 Dividing Fractions - University of Georgiamath.uga.edu/~sybilla/5020Fa03/FractionDivision.pdfClass Activity 0A: Explaining \Invert and Multiply" by Relating Division to Multiplication$
0.1. DIVIDING FRACTIONS 1

Excerpt from: Mathematics for Elementary Teachers, First Edition, bySybilla Beckmann. Copyright c© 2003, by Addison-Wesley

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 dividingfractions gives answers to fraction division problems that agree with whatwe 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 tointerpret division for fractions.

The “how many groups?” interpretation

With the “how many groups?” interpretation of division, 12 ÷ 3 means thenumber of groups we can make when we divide 12 objects into groups with3 objects in each group. In other words, 12 ÷ 3 tells us how many groups of3 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 5

2of an object

into groups with 2

3of an object in each group. In other words, 5

2÷ 2

3tells us

how many groups of 2

3we can make from 5

2. For example, suppose you are

making popcorn balls and each popcorn ball requires 2

3of a cup of popcorn.

If you have 2 1

2= 5

2of a cup of popcorn, then how many popcorn balls can

you make? In this case you want to divide 5

2of a cup of popcorn into groups

(balls) with 2

3of 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 meansthe number of objects in each group when we distribute 12 objects equallyamong 3 groups. In other words, 12÷3 is the number of objects in one groupif we use 12 objects to evenly fill 3 groups. When we work with fractions, itoften helps to think of “how many in each group?” division story problems asasking “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 usuallyrefer to the “how many in each group?” interpretation as “how many in onegroup?”.

With the “how many in one group?” interpretation of division,

3

4÷

1

2

is the number of objects in one group when we distribute 3

4of an object

equally among 1

2of a group. A clearer way to say this is: 3

4÷ 1

2is the

number of objects (or fraction of an object) in one whole group when 3

4of

an object fills 1

2of a group. For example, suppose you pour 3

4of a pint of

blueberries into a container and this fills 1

2of the container. How many pints

of blueberries will it take to fill the whole container? In this case, 3

4of a

pint of blueberries fills (i.e., is distributed equally among) 1

2of a group (a

container). So according to the “how many in one group?” interpretationof division, the number of pints of blueberries in one whole group (one fullcontainer) is

3

4÷

1

2

One way to better understand fraction division story problems is to thinkabout replacing the fractions in the problem with whole numbers. For ex-ample, if you have 3 pints of blueberries and they fill 2 containers, then howmany pints of blueberries are in each container? We solve this problem bydividing 3 ÷ 2, according to the “how many in each group?” interpretation.Therefore if we replace the 3 pints with 3

4of a pint, and the 2 containers with

1

2of a container, we solve the problem in the same way as before: 3÷ 2 now

becomes 3

4÷ 1

2.

Here is another way to think about the problem. Because 1

2of the con-

tainer is filled, and because this amount is 3

4of a pint, therefore 1

2of the


number of pints in a full container is 3

4of a pint. In other words:

1

2× number of pints in full container =

3

4

Therefore

number of pints in full container =3

4÷

1

2

Dividing by 1

2Versus Dividing in 1

2

In mathematics, language is used much more precisely and carefully than ineveryday conversation. This is one source of difficulty in learning mathemat-ics. For example, consider the two phrases:

dividing by 1

2,

dividing in 1

2.

You may feel that these two phrases mean the same thing, however, mathe-matically, they do not. To divide a number, say 5, by 1

2means to calculate

5 ÷ 1

2. Remember that we read A ÷ B as A divided by B. We would divide

5 by 1

2if 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 offlour, not 21

2.)

On the other hand, to divide a number in half means to find half of thatnumber. So to divide 5 in half means to find 1

2of 5. One half of 5 means

1

2× 5. So dividing in 1

2is the same as dividing by 2.

The “Invert and Multiply” Procedure for Fraction Di-vision

Although division with fractions can be difficult to interpret, the procedurefor dividing fractions is quite easy. To divide fractions, such as

3

4÷

2

3and 6 ÷

2

5

we can use the familiar “invert and multiply” method in which we invert thedivisor and multiply by it:

3

4÷

2

3=

3

4·3

2=

3 · 3

4 · 2=

9

8

4

and

6 ÷2

5=

6

1÷

2

5=

6

1·5

2=

6 · 5

1 · 2=

30

2= 15

Another way to describe this “invert and multiply” method for dividingfractions is in terms of the reciprocal of the divisor. The reciprocal of areciprocalfraction C

Dis the fraction D

C. In order to divide fractions, we should multiply

by the reciprocal of the divisor. So in general,

A

B÷

C

D=

A

B·D

C=

A · D

B · C

Explaining Why “Invert and Multiply” is Valid by Re-lating Division to Multiplication

The procedure for dividing fractions is easy enough to carry out, but why is ita valid method? Before we answer this question in general, consider a specialcase. Recall that every whole number is equal to a fraction (for example,6 = 6

1). Therefore we can apply the “invert and multiply” procedure to

whole numbers as well as to fractions. According to this procedure,

2 ÷ 3 =2

1÷

3

1=

2

1·1

3=

2 · 1

1 · 3=

2

3

Notice that this result, that 2 ÷ 3 = 2

3, agrees with our findings earlier in

this chapter: that we can describe fractions in terms of division, namely thatA

B= A ÷ 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 tofraction multiplication. Recall that every division problem is equivalent to amultiplication problem (actually two multiplication problems):

A ÷ B =?

is equivalent to

? · B = A

(or B·? = A). So3

4÷

2

3=?


is equivalent to

? ·2

3=

3

4. (1)

Now remember that we want to explain why the “invert and multiply” rulefor fraction division is valid. This rule says that 3

4÷ 2

3ought to be equal to

3 · 3

4 · 2

Let’s check that this fraction works in the place of the ? in Equation 1. Inother words, let’s check that if we multiply 3·3

4·2times 2

3, then we really do get

3

4:

3 · 3

4 · 2·2

3=

3 · 3 · 2

4 · 2 · 3=

3 · (3 · 2)

4 · (2 · 3)=

3 · (3 · 2)

4 · (3 · 2)=

3

4

Therefore the answer we get from the “invert and multiply” procedure reallyis the answer to the original division problem 3

4÷ 2

3. Notice that the line

of reasoning above applies in the same way when other fractions replace thefractions 2

3and 3

4used above.

It will still be valuable to explore fraction division further, interpretingfraction 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 Ex-plain Why “Invert And Multiply” Is Valid

Above, we explained why the “invert and multiply” procedure for dividingfractions is valid by considering fraction division in terms of fraction multi-plication. Now we will explain why the “invert and multiply” procedure isvalid by working with the “how many groups?” interpretation of division .

Consider the division problem

2

3÷

1

2

The following is a story problem for this division problem:

How many 1

2cups of water are in 2

3of a cup of water?

6

Or, said another way:

How many times will we need to pour 1

2cup of water into a

container that holds 2

3cup of water in order to fill the container?

From the diagram in Figure 1 we can say right away that the answer to thisproblem is “one and a little more” because one half cup clearly fits in twothirds of a cup, but then a little more is still needed to fill the two thirds ofa cup. But what is this “little more”? Remember the original question: wewant to know how many 1


3of a cup of water. So the

answer should be of the form “so and so many 1

2cups of water.” This means

that we need to express this “little more” as a fraction of 1

2cup of water.

How can we do that? By subdividing both the 1

2and the 2

3into 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 1

2and 2

3the common denominator of 6, then, as on the

right of Figure 1, the 1

2cup of water is made out of 3 parts (3 sixths of a

cup of water), and the 2

3cup 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 ofthose parts. Since 1

2cup is 3 parts, and the “little more” is 1 part, the “little

more” is 1

3of the 1

2cup of water. This explains why 2

3÷ 1

2= 11

3: there’s a

whole 1

2cup plus another 1

3of the 1

2cup in 2

3of a cup of water.

To recap: we are considering the fraction division problem 2

3÷ 1

2in terms

of the story problem “how many 1


3of a cup of water?”

If we give 1

2and 2

3the common denominator of 6, then we can rephrase

the problem as “how many 3

6of a cup are in 4

6of a cup?” But in terms of

Figure ??, this is equivalent to the problem “how many 3s are in 4?” whichis the problem 4 ÷ 3, whose answer is 4

3= 11

3. Notice that 4

3is exactly the

same answer we get from the “invert and multiply” procedure for fractiondivision:

2

3÷

1

2=

2

3·2

1=

2 · 2

3 · 1=

4

3


So the “invert and multiply” procedure gives the same answer to 2

3÷ 1

2that

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

B÷

C

D

Thinking logically, as above, and interpreting A

B÷ C

Das “how many C

Dcups

of water are in A

Bcups of water?”, we can conclude that

A

B÷

C

D=

A · D

B · D÷

B · C

B · D= (A · D) ÷ (B · C) =

A · D

B · C

The final expression, A·D

B·C, is the answer provided by the “invert and multiply”

procedure for dividing fractions. Therefore we know that the “invert andmultiply” procedure gives answers to division problems that agree with whatwe expect from the meaning of division.

Class Activity 0B: “How Many Groups?” Fraction Di-vision Problems

Using the “How Many in One Group?” Interpretationto Explain Why “Invert And Multiply” Is Valid

Above, we saw how to use the “how many groups?” interpretation of divisionto explain why the “invert and multiply” procedure for fraction division isvalid. We can also use the “how many in one group?” interpretation forthe same purpose. This interpretation, although perhaps more difficult tounderstand, has the advantage of showing us directly why we can multiplyby the reciprocal of the divisor in order to divide fractions.

Consider the following “how many in one group?” story problem for 1

2÷ 3

5:

You used 1

2of can of paint to paint 3

5of 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 thepaint as “filling” 3

5of the wall. We can also see that this is a division problem

by writing the corresponding number sentence:

3

5· (amount to paint the whole wall) =

1

2

8

Therefore

amount to paint the whole wall =1

2÷

3

5

We will now see why it makes sense to solve this problem by multiplying1

2by the reciprocal of 3

5, namely by 5

3. Let’s focus on the wall to be painted,

as shown in Figure 2. Think of dividing the wall into 5 equal sections, 3 of

the 1/2 can of paint is divided equallyamong 3 parts

the amount of paint for the full wall is5 times the amount in one part

Figure 2: The Amount of Paint Needed for the Whole Wall is 5

3of the 1

2Can

Used to Cover 3

5of the Wall

which you painted with the 1

2can of paint. If you used 1

2a can of paint to

paint 3 sections, then each of the 3 sections required 1

2÷ 3 or 1

2× 1

3cans

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 determinethe amount of paint you need for the whole wall by multiplying the 1

2can of

paint by 1

3and then multiplying that result by 5, as summarized in Table 1.

But to multiply a number by 1

3and then multiply it by 5 is the same as

multiplying the number by 5

3. Therefore we can determine the number of

cans of paint you need for the whole wall by multiplying 1

2by 5

3:

1

2·5

3=

5

6

This is exactly the “invert and multiply” procedure for dividing 1

2÷ 3

5. It

shows that you will need 5

6of a can of paint for the whole wall.


use 1

2can paint for 3

5of the wall

↓ ÷3 or ×1

3↓ ÷3 or ×1

3

use 1

6can paint for 1

5of the wall

↓ ×5 ↓ ×5

use 5

6can paint for 1 whole wall

in one step:use 1

2can paint for 3

5of the wall

↓ ×5

3↓ ×5

3

use 5

6can paint for 1 whole wall

Table 1: Determining How Much Paint to Use for a Whole Wall if 1

2Can of

Paint Covers 3

5of the Wall

10

The argument above works when other fractions replace 1

2and 3

5, thereby

explaining whyA

B÷

C

D=

A

B·D

C

In other words, to divide fractions, multiply the dividend by the reciprocalof the divisor.

Class Activity 0C: “How Many in One Group?” Frac-tion 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 ÷ 5

7. 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 ÷ 3

4. Use the

situation of the story problem to help you explain why the answer is4

3= 11

3.

3. Annie wants to solve the division problem 3

4÷ 1

2by using the following

story problem:

I need 1

2cup of chocolate chips to make a batch of cookies.

How many batches of cookies can I make with 3

4of a cup of

chocolate chips?

Annie draws a diagram like the one in Figure 3. Explain why it wouldbe easy for Annie to misinterpret her diagram as showing that 3

4÷ 1

2=

11

4. How should Annie interpret her diagram so as to conclude that

3

4÷ 1

2= 11

2?

4. Which of the following are solved by the division problem 3

4÷ 1

2? For

those that are, which interpretation of division is used? For those thatare not, determine how to solve the problem, if it can be solved.

(a) 3

4of a bag of jelly worms make 1

2a cup. How many cups of jelly

worms are in one bag?


1/2 cupmakesone batch

1/4 cup left

Figure 3: How Batches of Cookies Can We Make With 3

4of a Cup of Choco-

late Chips if 1 Batch Requires 1

2Cup of Chocolate Chips?

(b) 3

4of a bag of jelly worms make 1

2a cup. How many bags of jelly

worms does it take to make one cup?

(c) You have 3

4of a bag of jelly worms and a recipe that calls for 1

2of

a cup of jelly worms. How many batches of your recipe can youmake?

(d) You have 3

4of a cup of jelly worms and a recipe that calls for 1

2of

a cup of jelly worms. How many batches of your recipe can youmake?

(e) If 3

4of a pound of candy costs 1

2of a dollar, then how many pounds

of candy should you be able to buy for 1 dollar?

(f) If you have 3

4of a pound of candy and you divide the candy in 1

2,

then how much candy will you have in each portion?

(g) If 1

2of a pound of candy costs $1, then how many dollars should

you expect to pay for 3

4of a pound of candy?

5. Frank, John, and David earned $14 together. They want to divide itequally, except that David should only get a half share, since he did halfas much work as either Frank or John did (and Frank and John workedequal amounts). Write a division problem to find out how much Frankshould get. Which interpretation of division does this story problemuse?

6. Bill leaves a tip of $4.50 for a meal. If the tip is 15% of the cost ofthe meal, then how much did the meal cost? Write a division problemto solve this. Which interpretation of division does this story problemuse?

12

7. Compare the arithmetic needed to solve the following problems.

(a) What fraction of a 1

3cup measure is filled when we pour in 1

4cup

of water?

(b) What is one quarter of 1

3cup?

(c) How much more is 1

3cup than 1

4cup?

(d) If 1

4cup of water fills 1

3of a plastic container, then how much

water will the full container hold?

8. Use the meanings of multiplication and division to solve the followingproblems.

(a) Suppose you drive 4500 miles every half year in your car. At theend of 33

4years, how many miles will you have driven?

(b) Mo used 128 ounces of liquid laundry detergent in 6 1

2weeks. If

Mo continues to use laundry detergent at this rate, how much willhe 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 killerconcentrate with enough water to make a gallon. How many gal-lons of weed killer will you be able to make from this bottle?

9. The line segment below is 2

3of a unit long. Show a line segment that

is 5

2of a unit long. Explain how this problem is related to fraction

division.

2

3unit

Answers To Exercises For Section 0.1 on Dividing Frac-

tions

1. A simple “how many groups?” story problem for 1÷ 5

7is “how many 5

7

of a cup of water are in 1 cup of water?” Figure 4 shows 1 cup of waterand shows 5

7of 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 cupis 7

5of the shaded part. Thus there are 7

5of 5

7of a cup of water in 1

cup of water, so 1 ÷ 5

7= 7

5.


5

7

1

5of a cup

1 cup each piece is

of the shaded

portion

Figure 4: Showing Why 1 ÷ 5

7= 7

5by Considering How Many 5

7of a Cup of

Water are in 1 Cup of Water

2. A “how many in one group?” story problem for 1 ÷ 3

4is “if 1 ton of

dirt fills a truck 3

4full, then how many tons of dirt will be needed to

fill the truck completely full?”. We can see that this is a “how manyin one group?” type of problem because the 1 ton of dirt fills 3

4of a

group (the truck) and we want to know the amount of dirt in 1 wholegroup. Figure 5 shows a truck bed divided into 4 equal parts with 3of those parts filled with dirt. Since the 3 parts are filled with 1 tonof dirt, each of the 3 parts must contain 1

3of a ton of dirt. To fill the

truck completely, 4 parts, each containing 1

3of a ton of dirt are needed.

Therefore the truck takes 4

3= 11

3tons of dirt to fill it completely, and

so 1 ÷ 3

4= 4

3.

the 1 ton of dirt is divided equallyamong 3 parts

4 parts are neededto fill the truck;each part is 1/3 ofa ton, so 4/3 tonsof dirt are needed to fill the trucktruck bed

Figure 5: Showing Why 1 ÷ 3

4= 4

3by Considering How Many Tons of Dirt

it Takes to Fill a Truck if 1 Ton Fills it 3

4Full

3. Annie’s diagram shows that she can make 1 full batch of cookies from

14

her 3

4of a cup of chocolate chips and that 1

4cup of chocolate chips will

be left over. Because 1

4cups of chocolate chips are left over, it would

be easy for Annie to misinterpret her picture as showing 3

4÷ 1

2= 11

4.

But the answer to the problem is supposed to be the number of batches

Annie can make. In terms of batches, the remaining 1

4cup of chocolate

chips makes 1

2of a batch of cookies. We can see this because 2 quarter-

cup sections make a full batch, so each quarter-cup section makes 1

2of

a batch of cookies. So by interpreting the remaining 1

4cup of chocolate

chips in terms of batches, we see that Annie can make 1 1

2batches of

chocolate chips, thereby showing that 3

4÷ 1

2= 11

2, not 11

4.

4. (a) This problem can be rephrased as “if 1

2of a cup of jelly worms

fill 3

4of a bag, then how many cups fill a whole bag?”, therefore

this is a “how many in one group?” division problem illustrating1

2÷ 3

4, not 3

4÷ 1

2. Since 1

2÷ 3

4= 1

2· 4

3= 2

3, there are 2

3of a cup of

jelly worms in a whole bag.

(b) This problem is solved by 3

4÷ 1

2, according to the “how many in

each group?” interpretation. A group is a cup and each object isa bag of jelly worms.

(c) This problem can’t be solved because you don’t know how manycups of jelly worms are in 3

4of a bag.

(d) This problem is solved by 3

4÷ 1

2, according to the “how many

groups?” interpretation. Each group consists of 1

2of a cup of jelly

worms.

(e) This problem is solved by 3

4÷ 1

2, according to the “how many in

one group?” interpretation. This is because you can think of theproblem as saying that 3

4of a pound of candy fills 1

2of a group

and you want to know how many pounds fills 1 whole group.

(f) This problem is solved by 3

4× 1

2, not 3

4÷ 1

2. It is dividing in half,

not dividing by 1

2.

(g) This problem is solved by 3

4× 1

2, according to the “how many

groups?” interpretation because you want to know how many 1

2

pounds are in 3

4of a pound. Each group consists of 1

2of a pound

of candy.

5. If we consider Frank and John as each representing one group, andDavid as representing half of a group, then the $14 should be dis-


tributed equally among 2 1

2groups. Therefore, this is a “how many in

one group” division problem. Each group should get

14 ÷ 21

2= 14 ÷

5

2= 14 ·

2

5=

28

5= 5

3

5= 5

6

10= 5.60

dollars. Therefore Frank and John should each get $5.60 and Davidshould get half of that, which is $2.80.

6. According to the “how many in one group?” interpretation, the prob-lem is solved by $4.50÷ 0.15 because $4.50 fills 0.15 of a group and wewant to know how much is in 1 whole group. So the meal cost

$4.50 ÷ 0.15 = $4.50 ÷15

100= $4.50 ·

100

15=

$450

15= $30

7. Each problem, except for the first and last, requires different arithmeticto solve it.

(a) This is asking: 1

4equals what times 1

3? We solve this by calculating

1

4÷ 1

3, which is 3

4. We can also think of this as a division problem

with the “how many groups?” interpretation because we want toknow how many 1

3of a cup are in 1

4of a cup. According to the

meaning of division, this is 1

4÷ 1

3.

(b) This is asking: what is 1

4of 1

3? We solve this by calculating

1

4× 1

3= 1

12.

(c) This is asking: what is 1

3− 1

4? The answer is 1

12which happens to

be the same answer as in part (b), but the arithmetic to solve itis different.

(d) Since 1

4cup of water fills 1

3of a plastic container, the full container

will hold 3 times as much water, or 3 × 1

4= 3

4of a cup. We can

also think of this as a division problem with the “how many in onegroup?” interpretation. 1

4cup of water is put into 1

3of a group.

We want to know how much is in one group. According to themeaning of division it’s 1

4÷ 1

3, which again is equal to 3

4.

8. (a) The number of 1

2years in 33

4years is 33

4÷ 1

2. There will be that

many groups of 4500 miles driven. So after 3 3

4years you will have

16

driven

(33

4÷

1

2) × 4500 = (

15

4÷

1

2) × 4500

=15

2× 4500

= 33, 750

miles.

(b) Since one year is 52 weeks there are 52 ÷ 6 1

2groups of 61

2weeks

in a year. Mo will use 128 ounces for each of those groups, so Mowill use

(52 ÷ 61

2) × 128 = (52 ÷

13

2) × 128

=104

13× 128

= 1, 024

ounces of detergent in a year.

(c) There are 32÷2 1

2groups of 21

2ounces in 32 ounces. Each of those

groups makes 1 gallon. So the bottle makes 32÷2 1

2= 124

5gallons

of weed killer.

9. One way to solve the problem is to determine how many 2

3units are in 5

2

units. This will tell us how many of the 2

3unit long segments to lay end

to end in order to get the 5

2unit long segment. Since 5

2÷ 2

3= 15

4= 33

4,

there are 33

4segments of length 2

3units in a segment of length 5

2units.

So you need to form a line segment that is 3 times as long as the onepictured, plus another 3

4as long:

Problems for Section 0.1 on Dividing Fractions

1. A bread problem: If one loaf of bread requires 1 1

4cups of flour, then

how many loaves of bread can you make with 10 cups of flour? (Assumethat you have enough of all other ingredients on hand.)

(a) Solve the bread problem by drawing a diagram. Explain yourreasoning.


(b) Write a division problem that corresponds to the bread problem.Solve the division problem by “inverting and multiplying.” Verifythat your solution agrees with your solution in part (a).

2. A measuring problem: You are making a recipe that calls for 2

3cup

of water, but you can’t find your 1

3cup measure. You can, however,

find your 1

4cup measure. How many times should you fill your 1

4cup

measure in order to measure 2

3of a cup of water?

(a) Solve the measuring problem by drawing a diagram. Explain yourreasoning.

(b) Write a division problem that corresponds to the measuring prob-lem. Solve the division problem by “inverting and multiplying.”Verify that your solution agrees with your solution in part (a).


3and solve your

problem in a simple and concrete way without using the “invert andmultiply” procedure. Explain your reasoning. Verify that your solutionagrees with the solution you obtain by using the “invert and multiply”procedure.

4. Write a “how many groups?” story problem for 5 1

4÷ 13

4and solve your


5. Jose and Mark are making cookies for a bake sale. Their recipe callsfor 21

4cups 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 therewill be some flour left. Since the recipe doesn’t call for eggs, and sincethey have plenty of the other ingredients on hand, they decide they canmake a fraction of a batch in addition to the two whole batches. ButJose and Mark have a difference of opinion. Jose says that

5 ÷ 21

4= 2

2

9

and so he says that they can make 2 2

9batches of cookies. Mark says

that two batches of cookies will use up 4 1

2cups of flour, leaving 1

2left,

18

so they should be able to make 2 1

2batches. Mark draws the picture in

Figure 6 to explain his thinking to Jose. Discuss the boys’ mathematics:

Figure 6: Representing 5 ÷ 2 1

4by Considering How Many 2 1

4Cups of Flour

are in 5 Cups of Flour

what’s right, what’s not right, and why? If anything is incorrect, howcould you modify it to make it correct?

6. Marvin has 11 yards of cloth to makes costumes for a play. Eachcostume requires 1 1

2yards of cloth.

(a) Solve the following two problems:

i. How many costumes can Marvin make and how much clothwill be left over?

ii. What is 11 ÷ 1 1

2?

(b) Compare and contrast your answers in part (a).

7. A laundry problem: You need 3

4of a cup of laundry detergent to wash

one full load of laundry. How many loads of laundry can you wash with5 cups of laundry detergent? (Assume that you can wash fractionalloads of laundry.)

(a) Solve the laundry problem by drawing a diagram. Explain yourreasoning.

(b) Write a division problem that corresponds to the laundry problem.Solve the division problem by “inverting and multiplying.” Verifythat your solution agrees with your solution in part (a).


4and solve your



9. Write a “how many groups?” story problem for 1

3÷ 1

4and solve your


10. Write a “how many groups?” story problem for 1

2÷ 2

3and solve your


11. Fraction division story problems involve the simultaneous use of differ-ent wholes. Solve the following paint problem in a simple and concreteway without using the “invert and multiply” procedure. Describe howyou must work simultaneously with different wholes in solving the prob-lem.

A paint problem: You need 3

4of a bottle of paint to paint

a poster board. You have 3 1

2bottles of paint. How many

poster boards can you paint?

12. An article by Dina Tirosh, [?], discusses some common errors in divi-sion. The following problems are based on some of the findings of thisarticle.

(a) Tyrone says that 1

2÷5 doesn’t make sense because 5 is bigger than

1

2and you can’t divide a smaller number by a bigger number. Give

Tyrone an example of a sensible story problem for 1

2÷ 5. Solve

your problem and explain your solution.

(b) Kim says that 4÷ 1

3can’t be equal to 12 because when you divide,

the answer should be smaller. Kim thinks the answer should be1

12because that is less than 4. Give Kim an example of a story

problem for 4÷ 1

3and explain why it makes sense that the answer

really is 12, not 1

12.

13. Write a story problem for 3

4× 1

2and another story problem for 3

4÷ 1

2

(make clear which is which). In each case, use elementary reasoningabout the story situation to solve your problem. Explain your reason-ing.

20

14. Sam picked 1

2of a gallon of blueberries. Sam poured the blueberries

into one of his plastic containers and noticed that the berries filledthe container 2

3full. 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 containercompletely full?

15. A road crew is building a road. So far, 2

3of the road has been completed

and this portion of the road is 3

4of a mile long. Solve the following

problems in any way you like without using a calculator. Explain yourreasoning 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 becompleted?

16. Will has mowed 2

3of 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) byinterpreting 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 pencilmethods. Do not use a calculator. Verify that you get the same answerboth 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 3

4cups of flour for one batch of

12 muffins. For each of the following problems, solve the problem intwo ways: 1) by using elementary reasoning about the story situationand 2) by interpreting the problem as a division problem (say whetherit is a “how many groups?” or a “how many in one group?” typeof problem) and by solving the division problem using standard paperand pencil methods. Do not use a calculator. Verify that you get thesame answer both ways.


(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 muffinscan you make? (Assume that you can make fractional batches ofmuffins and that you have enough of all the ingredients.)


3and use

your story problem to explain why it makes sense to solve 4 ÷ 1

3by

“inverting and multiplying,” in other words by multiplying 4 by 3

1.


3and use


3by


2.


4and use


4by


3.

21. Write a “how many in one group?” story problem for 1

2÷ 3

4and use

your story problem to explain why it makes sense to solve 1

2÷ 3

4by

“inverting and multiplying,” in other words by multiplying 1

2by 4

3.

22. Write a “how many in one group?” story problem for 1 ÷ 2 1

2and use

your story problem to explain why it makes sense to solve 1 ÷ 2 1

2by

“inverting and multiplying”.

23. Give an example of either a hands-on activity or a story problem forelementary school children that is related to a fraction division prob-lem (even if the children wouldn’t think of the activity or problem asfraction division). Write the fraction division problem that is related toyour activity or story problem. Describe how the children could solvethe problem by using logical thinking aided by physical actions or bydrawing pictures.

24. Buttercup the gerbil drank 2

3of a bottle of water in 1 1

2days. Assuming

Buttercup continues to drink water at the same rate, how many bot-tles of water will Buttercup drink in 5 days? Use multiplication anddivision to solve this problem, explaining in detail why you can usemultiplication when you do and why you can use division when you do.

22

25. If you used 2 1

2truck loads of mulch for a garden that covers 3

4of an acre,

then how many truck loads of mulch should you order for a garden thatcovers 31

2acres? (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 andwhy you can use division when you do.

26. If 21

2pints of jelly filled 3 1

2jars, then how many jars will you need

for 12 pints of jelly? Will the last jar of jelly be completely full? Ifnot, how full will it be? (Assume that all jars are the same size.) Usemultiplication and division to solve this problem, explaining in detailwhy you can use multiplication when you do and why you can usedivision when you do.

Documents

0.1 Dividing Fractions - University of Georgiamath.uga.edu/~sybilla/5020Fa03/FractionDivision.pdfClass Activity 0A: Explaining \Invert and Multiply" by Relating Division to Multiplication