Excerpt from: Mathematics for Elementary Teachers, First Edition, by Sybilla 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 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 52 of an object into groups with 23 of an object in each group. In other words, 52÷ 23 tells us how many groups of 23 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 212 = 52 of a cup of popcorn, then how many popcorn balls can you make? In this case you want to divide 52 of a cup of popcorn into groups (balls) with 23 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.
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
4÷ 1 2
is the number of objects in one group when we distribute 34 of an object equally among 12 of a group. A clearer way to say this is: 34 ÷ 12 is the number of objects (or fraction of an object) in one whole group when 34 of an object fills 12 of a group. For example, suppose you pour 34 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, 34 of a pint of blueberries fills (i.e., is distributed equally among) 12 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 4÷ 1
2
One way to better understand fraction division story problems is to think about 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 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 34 of a pint, and the 2 containers with
1
2 of a container, we solve the problem in the same way as before: 3 ÷ 2 now becomes 34 ÷ 12.
Here is another way to think about the problem. Because 12 of the con- tainer is filled, and because this amount is 34 of a pint, therefore 12 of the
number of pints in a full container is 34 of 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
12Versus Dividing in
12In mathematics, language is used much more precisely and carefully than in everyday conversation. This is one source of difficulty in learning mathemat- ics. For example, consider the two phrases:
dividing by 12, dividing in 12.
You may feel that these two phrases mean the same thing, however, mathe- matically, they do not. To divide a number, say 5, by 12 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 212.)
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
2 ×5. So dividing in 12 is 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 procedure for dividing fractions is quite easy. To divide fractions, such as
3 4÷ 2
3 and 6 ÷ 2 5
we can use the familiar “invert and multiply” method in which we invert the divisor and multiply by it:
3 4÷ 2
3 = 3 4 ·3
2 = 3 · 3 4 · 2 = 9
8
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 dividing fractions is in terms of the reciprocal of the divisor. The reciprocal of a reciprocal
fraction CD is the fraction DC. 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 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 = 61). 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 = 23, agrees with our findings earlier in this chapter: that we can describe fractions in terms of division, namely that
A
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 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 4 ÷2
3 =?
is equivalent to
? · 2 3 = 3
4. (1)
Now remember that we want to explain why the “invert and multiply” rule for fraction division is valid. This rule says that 34 ÷ 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 other words, let’s check that if we multiply 34·3
·2 times 23, 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 really is the answer to the original division problem 34 ÷ 23. Notice that the line of reasoning above applies in the same way when other fractions replace the fractions 23 and 34 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 Ex- plain 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 multi- plication. 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 3 ÷ 1
2
The following is a story problem for this division problem:
How many 12 cups of water are in 23 of a cup of water?
Or, said another way:
How many times will we need to pour 12 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 23 of a cup of water. So the answer should be of the form “so and so many 12 cups of water.” This means that we need to express this “little more” as a fraction of 12 cup of water.
How can we do that? By subdividing both the 12 and the 23 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 12 and 23 the common denominator of 6, then, as on the right of Figure 1, the 12 cup of water is made out of 3 parts (3 sixths of a cup of water), and the 23 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 12 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 = 113: there’s a whole 12 cup plus another 13 of the 12 cup in 23 of a cup of water.
To recap: we are considering the fraction division problem 23÷12 in terms of the story problem “how many 12 cups of water are in 23 of a cup of water?”
If we give 12 and 23 the common denominator of 6, then we can rephrase the problem as “how many 36 of a cup are in 46 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 43 = 113. Notice that 43 is exactly the same answer we get from the “invert and multiply” procedure for fraction division:
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 23 ÷12 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
B ÷ C D
Thinking logically, as above, and interpreting BA÷ DC as “how many DC cups of water are in BA cups 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, AB·D
·C, is the answer provided by the “invert and multiply”
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 Di- vision 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 12÷35: You used 12 of can of paint to paint 35 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” 35 of 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
Therefore
amount to paint the whole wall = 1 2 ÷3
5
We will now see why it makes sense to solve this problem by multiplying
1
2 by the reciprocal of 35, namely by 53. 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 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 53 of the 12 Can Used to Cover 35 of the Wall
which you painted with the 12 can of paint. If you used 12 a can of paint to paint 3 sections, then each of the 3 sections required 12 ÷3 or 12 × 13 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 13 and then multiplying that result by 5, as summarized in Table 1.
But to multiply a number by 13 and then multiply it by 5 is the same as multiplying the number by 53. Therefore we can determine the number of cans of paint you need for the whole wall by multiplying 12 by 53:
1 2 ·5
3 = 5 6
This is exactly the “invert and multiply” procedure for dividing 12 ÷ 35. It shows that you will need 56 of a can of paint for the whole wall.
use 12 can paint for 35 of the wall
↓ ÷3 or ×13 ↓ ÷3 or ×13
use 16 can paint for 15 of the wall
↓ ×5 ↓ ×5
use 56 can paint for 1 whole wall in one step:
use 12 can paint for 35 of the wall
↓ ×53 ↓ ×53
use 56 can paint for 1 whole wall
Table 1: Determining How Much Paint to Use for a Whole Wall if 12 Can of Paint Covers 35 of the Wall
The argument above works when other fractions replace 12 and 35, thereby explaining why
A B ÷ C
D = A B · D
C
In other words, to divide fractions, multiply the dividend by the reciprocal of 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 ÷ 57. 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 3 = 113.
3. Annie wants to solve the division problem 34 ÷12 by using the following story problem:
I need 12 cup of chocolate chips to make a batch of cookies.
How many batches of cookies can I make with 34 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 34÷12 = 114. How should Annie interpret her diagram so as to conclude that
3
4 ÷ 12 = 112?
4. Which of the following are solved by the division problem 34 ÷ 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) 34 of a bag of jelly worms make 12 a cup. How many cups of jelly worms are in one bag?
1/2 cup makes one batch
1/4 cup left
Figure 3: How Batches of Cookies Can We Make With 34 of a Cup of Choco- late Chips if 1 Batch Requires 12 Cup of Chocolate Chips?
(b) 34 of a bag of jelly worms make 12 a cup. How many bags of jelly worms does it take to make one cup?
(c) You have 34 of a bag of jelly worms and a recipe that calls for 12 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 12 of a cup of jelly worms. How many batches of your recipe can you make?
(e) If 34 of a pound of candy costs 12 of a dollar, then how many pounds of candy should you be able to buy for 1 dollar?
(f) If you have 34 of a pound of candy and you divide the candy in 12, then how much candy will you have in each portion?
(g) If 12 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?
7. Compare the arithmetic needed to solve the following problems.
(a) What fraction of a 13 cup measure is filled when we pour in 14 cup of water?
(b) What is one quarter of 13 cup?
(c) How much more is 13 cup than 14 cup?
(d) If 14 cup of water fills 13 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 334 years, how many miles will you have driven?
(b) Mo used 128 ounces of liquid laundry detergent in 612 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 gal- lons 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 52 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 Frac- tions
1. A simple “how many groups?” story problem for 1 ÷57 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 75 of 57 of a cup of water in 1 cup of water, so 1 ÷ 57 = 75.
5 7
1
of a cup 5
1 cup each piece is
of the shaded portion
Figure 4: Showing Why 1 ÷ 57 = 75 by Considering How Many 57 of a Cup of Water are in 1 Cup of Water
2. A “how many in one group?” story problem for 1 ÷ 34 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 34 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 13 of a ton of dirt. To fill the truck completely, 4 parts, each containing 13 of a ton of dirt are needed.
Therefore the truck takes 43 = 113 tons of dirt to fill it completely, and so 1 ÷ 34 = 43.
the 1 ton of dirt is divided equally among 3 parts
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 truck bed
Figure 5: Showing Why 1 ÷ 34 = 43 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
her 34 of a cup of chocolate chips and that 14 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 34 ÷ 12 = 114. But the answer to the problem is supposed to be the number of batches Annie can make. In terms of batches, the remaining 14 cup of chocolate chips makes 12 of a batch of cookies. We can see this because 2 quarter- cup 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 112 batches of chocolate chips, thereby showing that 34 ÷ 12 = 112, not 114.
4. (a) This problem can be rephrased as “if 12 of a cup of jelly worms fill 34 of a bag, then how many cups fill a whole bag?”, therefore this is a “how many in one group?” division problem illustrating
1
2 ÷ 34, not 34 ÷ 12. Since 12 ÷ 34 = 12 ·43 = 23, there are 23 of a cup of jelly worms in a whole bag.
(b) This problem is solved by 34 ÷ 12, 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 34 of a bag.
(d) This problem is solved by 34 ÷ 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 ÷ 12, according to the “how many in one group?” interpretation. This is because you can think of the problem as saying that 34 of a pound of candy fills 12 of a group and you want to know how many pounds fills 1 whole group.
(f) This problem is solved by 34 × 12, not 34 ÷ 12. It is dividing in half, not dividing by 12.
(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 12 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-
tributed equally among 212 groups. 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 = 53 5 = 56
10 = 5.60
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 prob- lem 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 ÷ 15
100 = $4.50 ·100
15 = $450
15 = $30
7. Each problem, except for the first and last, requires different arithmetic to solve it.
(a) This is asking: 14 equals what times 13? We solve this by calculating
1
4÷13, which is 34. We can also think of this as a division problem with the “how many groups?” interpretation because we want to know how many 13 of a cup are in 14 of a cup. According to the meaning of division, this is 14 ÷ 13.
(b) This is asking: what is 14 of 13? We solve this by calculating
1
4 × 13 = 121 .
(c) This is asking: what is 13−14? The answer is 121 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 13 of a plastic container, the full container will hold 3 times as much water, or 3 × 14 = 34 of a cup. We can also think of this as a division problem with the “how many in one group?” interpretation. 14 cup of water is put into 13 of a group.
We want to know how much is in one group. According to the meaning of division it’s 14 ÷ 13, which again is equal to 34.
8. (a) The number of 12 years in 334 years is 334 ÷ 12. There will be that many groups of 4500 miles driven. So after 334 years you will have
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 ÷ 612 groups of 612 weeks in a year. Mo will use 128 ounces for each of those groups, so Mo will use
(52 ÷ 61
2) × 128 = (52 ÷ 13
2 ) × 128
= 104 13 ×128
= 1, 024 ounces of detergent in a year.
(c) There are 32 ÷ 212 groups of 212 ounces in 32 ounces. Each of those groups makes 1 gallon. So the bottle makes 32 ÷ 212 = 1245 gallons of weed killer.
9. One way to solve the problem is to determine how many 23 units are in 52 units. This will tell us how many of the 23 unit long segments to lay end to end in order to get the 52 unit long segment. Since 52 ÷23 = 154 = 334, there are 334 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 114 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.
(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 13 cup measure. You can, however, find your 14 cup measure. How many times should you fill your 14 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 prob- lem. 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 ÷ 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.
4. Write a “how many groups?” story problem for 514÷134 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 214 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
5 ÷ 21 4 = 22
9
and so he says that they can make 229 batches of cookies. Mark says that two batches of cookies will use up 412 cups of flour, leaving 12 left,
so they should be able to make 212 batches. Mark draws the picture in Figure 6 to explain his thinking to Jose. Discuss the boys’ mathematics:
Figure 6: Representing 5 ÷ 214 by Considering How Many 214 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 112 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 ÷ 112?
(b) Compare and contrast your answers in part (a).
7. A laundry problem: You need 34 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 ÷ 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.
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 12 ÷ 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 differ- ent 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 prob- lem.
A paint problem: You need 34 of a bottle of paint to paint a poster board. You have 312 bottles 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 this article.
(a) Tyrone says that 12÷5 doesn’t make sense because 5 is bigger than
1
2 and you can’t divide a smaller number by a bigger number. Give Tyrone an example of a sensible story problem for 12 ÷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
12 because that is less than 4. Give Kim an example of a story problem for 4 ÷13 and explain why it makes sense that the answer really is 12, not 121 .
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 reason- ing.
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 23 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, 23 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 23 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 134 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.
(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 ÷ 13 and use your story problem to explain why it makes sense to solve 4 ÷ 13 by
“inverting and multiplying,” in other words by multiplying 4 by 31. 19. Write a “how many in one group?” story problem for 4 ÷ 23 and use
your story problem to explain why it makes sense to solve 4 ÷ 23 by
“inverting and multiplying,” in other words by multiplying 4 by 32. 20. Write a “how many in one group?” story problem for 9 ÷ 34 and use
your story problem to explain why it makes sense to solve 9 ÷ 34 by
“inverting and multiplying,” in other words by multiplying 9 by 43. 21. Write a “how many in one group?” story problem for 12 ÷ 34 and use
your story problem to explain why it makes sense to solve 12 ÷ 34 by
“inverting and multiplying,” in other words by multiplying 12 by 43. 22. Write a “how many in one group?” story problem for 1 ÷ 212 and use
your story problem to explain why it makes sense to solve 1 ÷ 212 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 prob- lem (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 112 days. Assuming Buttercup continues to drink water at the same rate, how many bot- tles 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.
25. If you used 212 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 312 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 212 pints of jelly filled 312 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.
