section 1: fraction addition - mathed – mathematics...

Fraction Operations Unit

Copyright © 2007 by Daniel Siebert. All rights reserved.

Section 1: Fraction Addition Goal To help students learn to add fractions by reasoning about pictures and to record their explanations in writing. Big Ideas Addition of fractions often involves finding a common partitioning that can be used to count the resulting sum. For example, we can draw the following picture to illustrate the sum of 2/5 + 1/3: While we get a qualitative sense for how much this might be, we don't know exactly how much of a whole the combined shaded area is. We can't call it 3/5, because the shaded part on the right is much larger than 1/5. Likewise, it isn't 3/3, either. We need to have some common partitioning that will allow us to count up the total amount of area. Being able to "count up" the pieces is a critical idea in performing fraction operations. This common partitioning, also known as the common denominator, needs to yield pieces that are just the right size so that we can still make our two fractions 2/5 and 1/3. To know whether a partitioning will work as a common partitioning, we must show two things: that the way we have partitioned both regions leads to pieces that are equal in size (hence are common), and that we can form both fractions using that partitioning. Depending on the method and model we use, one of these two conditions may be easier to show than the other. There are many different methods for determining what partitionings would work as a common partitioning. Strategy 1: One method for finding a common partitioning is to increment the number of partitions by 1 until we find a partitioning that works. For example, in the problem above, we know that anything smaller than fifths won't work, because we won't be able to make 2/5 from that partitioning. Sixths won't work, because we can't make fifths from sixths. Similarly any partitioning with pieces larger than 1/15 won't work, because we won't be able to make both 2/5 and 1/3 from those partitionings. Fifteenths will work, because we can take the 15 pieces and make five equal groups of 3/15. This corresponds to fifths. So 2/5 would be two 3/15s, or 6/15. We could also take the 15 pieces and make three equal groups of 5/15. This corresponds to thirds.



Strategy 2: A second method is to increase by a multiple of one of the denominators until we find a partitioning that we can use to make the other fraction. So for the problem above, we could try increasingly larger multiples of 5 until we find one that we can make thirds from. Fifths and tenths don't work, but fifteenths will for the same reasoning as in Method 1. Strategy 3: We can make a common denominator by cutting each of the five 1/5s into three equal parts and each of the three 1/3s into five equal parts. In other words, we are making thirds in our fifths and fifths in our thirds. We know this yields a common partitioning because we get 15 parts in each whole, and we make both 2/5 and 1/3 from those fifteen parts. In fact, the way we cut the parts, we never lose sight of the 2/5 and the 1/3 in our picture. Strategy 4: Find a whole number that is divisible by both denominators, and make a partitioning with that many equal-sized pieces. For example, 15 is divisible by both 5 and 3. Since it is divisible by 5, that means we can make 5 equal groups (of three 1/15s from the fifteen 1/15s in 1), or fifths, from this partitioning. Thus, we can make 2/5. Likewise, because 15 is divisible by 3, we can make 3 equal groups (of five 1/15s from the fifteen 1/15s in 1), or thirds, from this partitioning. Therefore, 15 will work because we can make both fractions with this partitioning. Once we have a common partitioning, we can then count how much we have in the sum, because the pieces are now the same size in both wholes. In the first whole we have six 1/15s, and in the second whole we have five 1/15s. This gives us a total of eleven 1/15s, or 11/15.



Section 2: Fraction Subtraction Goal To help students write story problems for fraction subtraction number sentences and to solve fraction subtraction problems with a picture and a written explanation. Big Ideas The first important idea is to realize that when we are subtracting one fraction from another, both fractions are fractions of one. For example, given the subtraction number sentence 1/2 – 1/3, the 1/3 is 1/3 of 1, not 1/2. Thus, the following story problem would not be correct for 1/2 – 1/3:

If 1/2 of a pizza is left after a pizza party, and I have to leave 1/3 of it for my brother, how much pizza is left for me to eat?

In this story problem, the 1/3 is 1/3 of 1/2 of a pizza. Thus, the number sentence for the story problem would be 1/2 – (1/3 of 1/2) or 1/2 – (1/3 x 1/2). The following is a correct story problem for 1/2 – 1/3:

If 1/2 of a pizza is left after a pizza party, and I have to leave 1/3 of a pizza for my brother to eat, how much of a pizza is left for me to eat?

In this problem, 1/2 and 1/3 are 1/2 and 1/3 of a whole pizza. Also, the answer is in terms of a whole pizza. The following story problem would be incorrect, because although the 1/2 and 1/3 are both of a whole pizza, the difference is not:

If 1/2 of a pizza is left after a pizza party, and I have to leave 1/3 of a pizza for my brother to eat, what fraction of the remaining pizza can I eat?

The reasoning involved in subtracting fractions is similar to the reasoning we use to add fractions. For example, if I wish to compute 2/5 – 1/3, I might draw both of these quantities, as shown below: The question I ask is, How much of a whole would I have left if I took 1/3 of a whole away from 2/5? Just like addition, I can’t answer this question immediately because the partitions in each whole are different and do not permit an accurate comparison of the two quantities. I must first find a common partitioning before I can remove 1/3 from 2/5 and know what fraction of a whole is left. To find a common denominator, I could use



any of the strategies from the last section. Once I find that 15 works as a common denominator, I can partition my two wholes further to get the following picture: From this picture, I can see that 2/5 is equivalent to 6/15, and 1/3 is equivalent to 5/15. Then when I remove 1/3 from 2/5, I will have 1/15 left.



Section 3: Justifying Addition and Subtraction Algorithms Goal To help students make connections between their pictures of fraction addition and subtraction and the traditional algorithms for adding and subtracting fractions, and to be able to justify the traditional algorithms using pictures and written explanations. Big Ideas To make sense of the traditional algorithms, it is necessary to explain the following: 1. Why do we need to find a common denominator when we add or subtract fractions

with unlike denominators?

We find a common denominator, because that is equivalent to finding a common partitioning. We need a common partitioning (common denominator) so we can count to find the sum or the difference of the number of (equal-sized) pieces we have in the two fractions.

2. Why can we multiply the numerator and denominator of a fraction by the same

whole number?

When we multiply the numerator and denominator of a fraction by the same number, we are making smaller partitions in the original partitions. For example, if we have 2/3 and we multiply the numerator by 5, then we are creating 5 partitions in each of our two shaded thirds, which creates two groups of five shaded parts from the two existing part, or 2 x 5 shaded parts.

Now we have 10 partitions of a particular size, but what fraction of the whole are they? To find out, we have to partition the remaining third into fifths as well. When we do this, we create 15 parts of equal size.



We can see from our picture that each third yields a group of 5 smaller parts, so that altogether, we have 3 groups of 5 smaller parts. By multiplying the denominator, 3, by 5, we get the total number of new parts in our whole. Simply put, multiplying the denominator by the whole number gives us the total number of equal parts in the whole after we create the new partitioning, and multiplying the numerator by the whole number gives us the number of shaded parts in the whole after it has been repartitioned. 3. Why would we want to multiply the numerator and denominator of a fraction by the

same whole number?

We don't multiply the numerator and denominator by a randomly selected whole number. We choose the whole number carefully so that it leads to a partitioning that will be of the same size as the partitioning of the other fraction (albeit we may have to change the size of the other fraction's partitioning, too, so that the two partitionings are of the same size). In other words, we multiply the numerator and denominator by a carefully selected whole number to get a common denominator.

4. How do we know which whole number to multiply the numerator and denominator

by?

We have to multiply by a number a that will give us a common denominator, or a common partitioning. We can use many of the same methods from Section 1, or we can merely multiply the numerator and denominator by the denominator of the other fraction. If we do this to both fractions, we will get a common partitioning.

For example, suppose our problem is 1/3 – 1/5. To find a common denominator, we might multiply the numerator and denominator of the first fraction by 5, and the numerator and denominator of the second fraction by 3. This has the effect of creating fifths in the thirds and thirds in the fifths. In the first partitioning we have 3 groups of 5, and in the second we have 5 groups of 3. We know this gives us the same number of pieces because 3 groups of 5 is 15, as is 5 groups of 3. But we don't have to compute what 3 groups of 5 and 5 groups of 3 are to know that they are equal; we can reason about it with the following picture:

In this picture, we can see from the way that the parts are aligned and grouped that 3 groups of 5 is equivalent to 5 groups of 3. To change from one to the other, all we have to do is group in a different direction. In general, this way of reasoning and



grouping will always work to show that multiplying each fraction's numerator and denominator by the other fraction's denominator will result in a common denominator. Of course, this might not be the smallest possible common denominator for the two fraction, but it is nonetheless a common denominator, and thus will allow us to add or subtract the two fractions.

5. Why can we add or subtract numerators when we have achieved a common

denominator?

Once we have a common denominator, then we have created a common partitioning between our two fractions, and thus have created pieces of the same size. Now it is possible to count the pieces to compute a sum or difference.

6. Why don't we add the denominators when adding fractions?

We do not add the denominators, because we want to talk about the pieces in relation to a single whole, not two wholes. For example, when we added 2/5 + 1/3 in Section 1, we found a common denominator and drew the following picture:

In the rectangle on the left, we can see that 2/5 is equivalent to 6/15. On the right, we see that 1/3 is equivalent to 5/15. If we add both the numerators and denominators together, we get 11/30. This is wrong, because the shaded part is not 11/30 of 1. It is in fact 11/30 of 2. Thus, an answer of 11/30 without noting that 11/30 is of 2, and not 1, is incorrect. On the other hand, an answer of 11/30 of 2 is technically correct for 6/15 + 5/15. However, it doesn't really answer the fundamental question we are asking when we add 6/15 and 5/15, namely, How many wholes do we get when we add 6/15 and 5/15? In order to answer this question, we choose not to add the denominators so that the size of the pieces continue to be measured in relation to a single whole.



Section 4: Fraction Multiplication Goal To help students learn to multiply fractions by reasoning about pictures and to record their explanations in writing. Big Ideas One way to think of multiplication is in terms of repeated groups of the same size. For example, 3 x 4 could be interpreted to mean 3 groups of 4 or 4 groups of 3. In the US, the convention is for the first number to represent the number of groups and the second number to represent the number in each group. Thus, for this unit we will assume that 3 x 4 means three 4s, or 4 + 4 + 4. An important idea in multiplication is that the two numbers in the multiplication problem do not mean the same thing. For 3 x 4, the 4 means four 1s. In contrast, the 3 means three 4s, not three 1s. This seems like a trivial distinction until we start thinking about multiplying fractions, such as 1/4 x 4/7. How can we make sense of this number sentence? If we apply our understanding of whole number multiplication to this number sentence, we realize that the 4/7 is 4/7 of 1, while the 1/4 is 1/4 of 4/7. The task then becomes to determine just how much a fourth of 4/7 is. If we look at a picture of 4/7, we notice that it is already divided into four equal parts. Thus, each of those parts must be a 1/4 of 4/7. We can check this using an iterating image of fractions. If we take one of those four (shaded) parts and iterate it another 3 times, we recreate the 4/7. Thus, each piece must be a 1/4 of 4/7. Once we determine what size of a piece is equivalent to 1/4 of 4/7, the last task is to find the appropriate name for that piece. In particular, we would like to know the size of that piece in relation to a whole. We notice that what we called 1/4 of 4/7, when iterated another 6 times, gives us a whole. This means that the piece must be 1/7 of a whole, because 7 copies of it makes a whole. Thus, our answer to 1/4 x 4/7 is 1/7. Sometimes deciding how to draw a fraction of another fraction is not so obvious. For example, suppose we wanted to compute 2/5 x 3/8. This problem is asking us to find 2/5 of 3/8 of 1. We can start by drawing a picture of 3/8.



There is no obvious way of dividing 3 parts into 5 parts so that we can find what 2/5 is. One possible solution is that instead of taking 2/5 of the whole 3/8, we take 2/5 of each of the three 1/8s that make up 3/8, and then add the 2/5 of each 1/8 together to find out how much 2/5 of 3/8 is. Thus, I might take the first 1/8 on the left and divide it into 5 equal pieces, with one of those pieces being 1/5 of 1/8: Once I know what 1/5 of 1/8 is, I can take two of those pieces to get 2/5 of 1/8. Then I can repeat this process for the other two 1/8s in 3/8. To find out what I should call this amount, I continue to partition the other non-shaded eighths in my picture so that I can see what size this smaller piece is in relation to the whole.



Once I am finished, I can see that there are 40 pieces of equal size in my picture, suggesting that each piece is 1/40. I have six of these 1/40s shaded, so my answer is 6/40. I could simplify this in my picture by grouping my 1/40s into groups of two. This would give me a total of 20 equal pieces, or 1/20s, with three of the 1/20s shaded in. Thus, my answer could also be called 3/20.



Section 5: Explaining Multiplication Algorithms Goal To help students be able to explain why the algorithm for multiplying fractions works using pictures and written explanations. Big Ideas The algorithm for multiplying fractions can be closely tied to reasoning with pictures of fraction multiplication. For example, suppose we wanted to show the following:

2

3!

4

5=

2! 4

3! 5

One way to show that this is true is by considering three different multiplication problems:

1. Show

1

3!

1

5=

1

3! 5.

Explanation: To compute 1/3 × 1/5 using pictures, we would first start with a picture of 1/5, and then divide the 1/5 into 3 equal parts, taking one of those parts to represent 1/3 of 1/5. Then to find out what part of a whole the 1/3 of 1/5 is, we would divide the other fifths into thirds to create pieces of equal size throughout the entire whole. Thus, in each fifth we get three pieces. This is the same as five groups of 3, or 5 × 3 pieces. However, we could also group horizontally, which would give us three groups of 5, or 3 × 5 pieces. Thus, each piece would be 1/(3 ×5), because it takes 3 × 5 of those pieces to make a whole.



2. Show

1

3!

4

5=

4

3! 5 using the fact that

1

3!

1

5=

1

3! 5.

Explanation: We know that 1/3 of 1/5 is 1/(3 × 5) from Step 1. Then what is 1/3 of 4/5? From the picture below, we can see that 1/3 of 4/5 is the same as four copies of 1/3 of 1/5. Thus, 1/3 of 4/5 is four pieces of size 1/(3 × 5), or 4/(3 × 5).

3. Show that

2

3!

4

5=

2! 4

3! 5using the fact that

1

3!

4

5=

4

3! 5.

Explanation: 2/3 of 4/5 is the same as two 1/3s of 4/5: Thus, since 1/3 of 4/5 is 4/(3 × 5), then 2/3 of 4/5, or two 1/3s of 4/5, would be two

groups of 4/(3 × 5). So we have 2 × 4 pieces of size 1/(3 × 5), or (2 × 4)/(3 × 5). In general, the multiplication algorithm can be explained by first looking at what size pieces result from multiplying unit fractions. Then we can use our understanding of what non-unit fractions mean, such as 4/5 means four 1/5s, to compute how many pieces of that particular size result from multiplying the two fractions. Optional: Explaining Cross-Canceling Students may want to know why cross-canceling works when multiplying fractions. The following is a brief explanation. You will have to supplement the ideas if you wish to discuss them with your students. 1. Canceling the denominator of the first fraction with the numerator of the second

fraction.

1/3 of 1/5 1/3 of 4/5

2/3 of 4/5 1/3 of 4/5



Example: Show that

5

/ 6 !

/ 3

4=

5

2!

1

4.

Explanation: When we can cross cancel between the denominator of the first fraction and the numerator of the second fraction, it means that there is an efficient way to partition the second factor so that we can make the requisite number of copies we need to make. For example, we can take 5/6 of 3/4 by taking each 1/4 in 3/4 and splitting it in half to make sixths in the 3/4, and then take 5 of those 1/6 of 3/4, as illustrated below: This method requires much fewer partitions than a more standard approach to computing the product, such as the one illustrated below. It is no accident that cross canceling with the denominator of the first fraction and the numerator of the second fraction corresponds to the existence of a more efficient way of partitioning our fractions than a standard method would suggests (try, for example, the problem 5/6 × 8/9). Furthermore, it is the first picture, where we made as few partitions as possible, that helps us see that 5/6 × 3/4 is equivalent to 5/2 × 1/4. Notice that in this picture, the piece that is 1/2 of 1/4 is also 1/6 of 3/4. Thus, five of those pieces could be called either 5/2 of 1/4 or 5/6 of 3/4.

2. Canceling the numerator of the first fraction with the denominator of the second fraction.

Example: Show that

/ 3

4!

5

/ 6 =

1

4!

5

2.

Explanation: Taking 3/4 of 5/6 is the same as taking 1/4 of 5/6 three times, because that's what 3/4 of 5/6 means — three 1/4s of 5/6. To see the connection between 3/4 × 5/6 and 1/4 × 5/2, it is helpful to take 1/4 of 5/6 from three copies of 5/6 rather than taking 1/4 three times from the same copy of 5/6. In the picture below, the



heavily shaded portion represents 1/4 of three copies of 5/6, which is equivalent to 3/4 of 5/6: By rearranging the shaded portions, we get the following picture: Notice that this is actually 1/4 of 5/2. We can do this same process for any two fractions. However, we will not be able to perform the last step, where we simplified the resultant fraction, unless we can cross-cancel between the numerator of the first fraction and the denominator of the second fraction.



Section 6: Fraction Division from a Measurement Model Goal To help students learn to divide fractions from a measurement perspective by reasoning about pictures, and to record their explanations in writing. Big Ideas Because there are two interpretations for whole number division, namely measurement and sharing, there are also two interpretations for fraction division. In this section, we discuss measurement division. Fortunately, the measurement model can be applied directly to fraction division without modification. For example, suppose our division problem is 3/4 ÷ 1/3. Using the measurement model of division, we would interpret this number sentence as asking the question, How many 1/3s are in 3/4? The following picture, where we overlap 3/4 with a picture of a whole divided into thirds, suggests that there are at least two 1/3s in 3/4, with a small amount leftover: How much is left over? By finding a common partitioning of twelfths, we get a better idea of how much is leftover:



The heavy lines show how the thirds match up with the smaller partitions in the 3/4. Note that there is one small piece leftover. There are two ways of viewing this leftover part: we can either interpret it in terms of the whole, or in terms of 1/3. In terms of the whole, the leftover part is 1/12, because it takes twelve pieces of this size to make a whole. However, this leftover part is also 1/4 of a 1/3, because it takes four pieces of this size to make a 1/3. The problem, then, is knowing which name to use for the leftover part. Because the question we are trying to answer is, How many 1/3s are in 3/4, we usually write the answer as 2 1/4, meaning that there are 2 1/4 thirds in 3/4. However, we could have also written the answer as 2 remainder 1/12. This means that there are two 1/3s in 3/4, with a remainder of 1/12 (of 1). On the other hand, to write the answer as 2 1/12 is incorrect, because in this case we are mixing units — the 2 is 2 one-thirds, while the 1/12 is 1/12 of 1. In summary, when using the measurement model of division, we are asking how many of the divisor (the second fraction) are in the dividend (the first fraction). The quotient tells us the number of copies of the divisor that are in the dividend.



Section 7: Fraction Division from a Sharing Model Goal To help students learn to divide fractions from a sharing perspective by reasoning about pictures, and to record their explanations in writing. Big Ideas The sharing interpretation does not lend itself so easily to division of fractions. To see this, let's start with a whole number sharing problem for the division problem 6 ÷ 3:

Michael wants to give an equal number of suckers to three friends. If he has six suckers, how many suckers should he give to each friend?

Now try changing the numbers in the story problem to include fractions. For example, suppose we use the division problem 3/4 ÷ 1/3:

Michael wants to give an equal number of suckers to 1/3 friends. If he has 3/4 suckers, how many suckers should he give to each friend?

This story problem now makes no sense. After all, how can you split something among a fractional number of groups, particularly if that fraction is less than one? However, consider a different context for 6 ÷ 3:

If Anna can knit 6 socks in three hours, how many socks can she knit in one hour? This story problem also involves division from a sharing perspective, because to find the answer we would split the number of socks equally among three groups, each group corresponding to one hour. How does this context work for the division problem 3/4 ÷ 1/3?

If Anna can knit 3/4 socks in 1/3 hours, how many socks can she knit in an hour? In fact, this context does make sense, and 3/4 ÷ 1/3 yields the correct answer to the story problem. This suggests that perhaps we have found a legitimate context for fraction division from a sharing perspective. The above problem can help us rethink the template for sharing so that it extends meaningfully to fractional numbers. When we dealt with a ÷ b by sharing from a whole number perspective, we always asked the question, "If b groups get a amount, how much does each group get if a were split evenly among the b groups?" This question doesn't make much sense if b is a fraction. However, by following the pattern in the story problem above, we can change the template for a sharing problem and still have it make sense for a fractional divisor. The new template looks like the following:



If b groups get a amount, then how much does one group get? In the sock problem, we have 1/3 of an hour corresponding to 3/4 of a sock. The question we are asking is how many socks correspond to one whole hour. Note that underlying these types of contexts is the restriction that the rate of amount per group remains constant as the number of groups changes. Without this restriction, the problem can no longer be solved by division. For example, if Anna changes the speed at which she knits socks during the last 2/3 of an hour, then our prediction of how much she knits per hour will be wrong. Note that this new way of phrasing a sharing problem also makes sense for whole numbers. For example, if we have 6 ÷ 3, the new template asks, If 3 groups get 6, then how much does one group get? Given the underlying restriction that the rate of amount per group must be constant, the answer is 2. Solving Problems with Pictures Now that we have a template for the meaning of sharing division problems, we can attempt to solve them. Suppose that we want to solve 2/3 ÷ 4/5. Using the template above, we get the following statement:

If 4/5 of a group gets 2/3, how much does a whole group get? One way to solve this problem is to figure out how much 1/5 of a group gets. If we know that, then we just take five copies of that amount and we have how much a whole group gets. Since 4/5 of a group corresponds to 2/3, I need to split 2/3 into four equal parts to find how much each 1/5 of a group gets. To do this, I split each of the two 1/3s into two equal part, which yields 4 equal parts in the 2/3 segment. If I split the remaining third into two equal parts, too, then I can see that each part is 1/6 of the whole, because it takes 6 copies of them to make a whole.

4/5 of a group corresponds to 2/3

1/5 of a group corresponds to 1/4 of 2/3



Each of these 1/6 parts now corresponds to 1/5 of a group. To find how much a whole group gets, I take 5 of these 1/6 parts. So a whole group gets five 1/6s, or 5/6. So 2/3 ÷ 4/5 = 5/6. Is This Really Sharing? To recap, we've found that we can extend one type of sharing story problem from whole numbers to fractions, and we used this to rephrase our template question for sharing division. Then we used this to solve a division of fractions problem from a sharing perspective. But exactly what about this new way of thinking about fraction relationships is sharing? It doesn't seem like we're doing any division at all! If we look at the first part of the problem above, however, we get back to the roots of sharing. Note that in the above problem, the first thing we did was split the 2/3 evenly into four groups, each group corresponding to one of the four one-fifths in the divisor. In division of fraction problems involving sharing, this will commonly be the first step. For example, if we have the problem a/b ÷ c/d, we will split the a/b into c equal groups, each corresponding to what 1/d of a group gets. So we actually are still sharing. The only difference between whole number and fraction sharing problems is that in the fraction problems, once we find how much 1/d of a group gets, we have to iterate that amount d times to see how much a whole group gets.




a whole group corresponds to 5/6



Writing Story Problems Two things are important to keep in mind when writing story problems for a sharing division of fractions problem. 1. Make sure you choose a context that involves quantities that can be meaningfully

cut into fractional parts. Without this flexibility, your story problem could make little sense.

2. The question you ask at the end determines whether your story problem is a ÷ b or

b ÷ a. The divisor, or second number in the division problem always corresponds to whatever you are asking to be the whole. For example, take another look at the knitting story problem above for the division problem 3/4 ÷ 1/3. We could change this problem to 1/3 ÷ 3/4 by changing the question we ask:

If Anna can knit 3/4 socks in 1/3 hours, how many hours does it take her to knit one sock?

Note that the first part of the story problem remains the same. However, because our goal now is to complete one sock and see how many hours that corresponds to, 3/4 becomes the divisor, and 1/3 of an hour corresponds to that fraction of a sock. We must now split the 1/3 evenly among the 3 one-fourths to see how much time corresponds to a 1/4 of a sock, and then iterate this amount of time four times to see how much time corresponds to one sock. So the division problem is 1/3 ÷ 3/4.



Section 8: Explaining the Invert and Multiply Rule Goal To help students draw pictures to explain why the invert and multiply rule works, and to record their explanations in writing. Big Ideas Because we have two models of division, we need to develop two explanations for why the invert and multiply rule works. Surprisingly, the explanation for measurement is typically harder to construct than the explanation for sharing. The Measurement Perspective There are two parts to explaining why the invert and multiply rule works from a measurement perspective: 1. Why invert the divisor?

Explanation: The reason we invert the divisor is connected to what the reciprocal of the divisor means. Recall that the reciprocal of a fraction can be found by dividing 1 by that fraction. For example, to find the reciprocal of 2/3, we could compute 1 ÷ 2/3. In our picture below, we note that there are 3/2 two-thirds in 1, because each third of one corresponds to 1/2 of 2/3, and because there are 3 thirds in 1, there are three 1/2 of 2/3 in 1, or 3/2 two-thirds in 1.

From our picture, we can tell that the reciprocal of 2/3 is 3/2. But what does the 3/2 mean? It tells us how many 2/3 are in 1. In general, the reciprocal of a fraction tells us how many of that fraction are in 1. Thus, we invert the divisor, because by doing so, we get the reciprocal of the divisor, which tells us how many of that divisor are in 1.

2/3 of 1

1/2 of 2/3 of 1



2. Why multiply the reciprocal of the divisor times the dividend?

Explanation: Once we know how many of our divisor are in 1, then we can figure out how many of that divisor are in the dividend. For example, suppose we were interested in computing 3 ÷ 2/3. We know that there are 3/2 two-thirds in 1, so for each 1 in 3, we have 3/2 two-thirds. This gives us 3/2 + 3/2 + 3/2 two-thirds in 3, or 3 x 3/2 two-thirds in 3. Thus, 3 ÷ 2/3 = 3 x 3/2. This works for non-whole number dividends as well. For example, suppose I wanted to compute 1 1/2 ÷ 2/3. I know that each 1 in the divisor gives me 3/2 two-thirds. So for 1 1/2 ones, I'm going to get 1 1/2 groups of 3/2 two-thirds, or 1 1/2 x 3/2 two-thirds. Thus, 1 1/2 ÷ 2/3 = 1 1/2 x 3/2.

The Sharing Perspective To see why the IM rule works for sharing, lets first solve a sharing division problem. Consider the division problem 3/4 ÷ 2/3. From a sharing perspective, this problem asks, If 2/3 of a group gets 3/4, how much does a whole group get? To find the answer, we first find out how much 1/3 of a group gets. Since 2/3 gets 3/4, we need to split 3/4 evenly in half. Splitting each of the three one-fourths in half and shading in half of each fourth gives us the amount that 1/3 of a group gets. To find how much of the whole this is, we partition the remaining fourth into halves as well, finding that we have created eight equal pieces in our whole. So each piece must be 1/8. Since three are shaded, the amount that 1/3 of a group gets is 3/8. Now to find out how much a whole group gets, we make two more copies of the amount that 1/3 of a group gets. Thus, we get three copies of 3/8, or 9/8.



The process that we have just gone through is actually the same process we would have used for calculating 3/2 x 3/4. To calculate this multiplication problem, we would have first found what half of 3/4 is, which is the first thing we did when we were solving the sharing division problem. Then we would find 3/2 by taking three copies of 1/2 of 3/4, which is the second part of the sharing division problem above. Thus, not only does 3/2 x 3/4 give us the same answer as 3/4 ÷ 2/3, the multiplication process also produces the exact same actions on 3/4 as the division problem. Thus, 3/4 ÷ 2/3 = 3/2 × 3/4, a variation of the IM rule.



Section 9: Complex Fractions Goal To be able to make meaning for complex fractions using either an iterating or partitioning perspective, and to explain why a complex fraction is equivalent to the fraction in the numerator multiplied by the reciprocal of the fraction in the denominator. Big Ideas So far we have only dealt with fractions that have whole number numerators and denominators. What happens if we have fractions in the numerator and/or denominator? It turns out that if the fraction in the denominator is greater than 1, it's not too hard to figure out what it means. We consider this case first. Denominator Greater Than 1

Consider the following fraction:

!

21

2

32

3

We can try to interpret this fraction using either an iterating or partitioning perspective. From an iterating perspective, the fraction means that it is 2 1/2 copies of 1/(3 2/3), where 1/(3 2/3) is a piece such that 3 2/3 copies of it makes a whole. Below is an example of a whole partitioned in terms of 1/(3 2/3), where the solid lines separate the 1/(3 2/3) pieces except for the piece on the right, which is 2/3 of a 1/(3 2/3) piece. I can tell that the marked pieces are 1/(3 2/3) because it takes 3 2/3 copies of one of these pieces to make up a whole. In order to draw a (2 1/2)/(3 2/3), I need to shade in 2 1/2 of the 1/(3 2/3) pieces. My picture looks like the following:

1/(3 2/3) 2/3 of 1/(3 2/3)



To figure out what quantity we have without using complex fractions, we need to find a common denominator. We need a partitioning that allows us to make halves of 1/(3 2/3) so we can draw 2 1/2 copies of 1/(3 2/3), and also a partitioning that allows us to make 2/3 of a copy of 1/(3 2/3) so that we can divide the piece on the far right of our whole into equal pieces, too. The partitioning that will work is sixths of 1/(3 2/3). This yields the following picture: From the picture, we can see that we have partitioned our whole into 22 equal pieces, so each piece is 1/22. Furthermore, we have 15 of those pieces shaded, so (2 1/2)/(3 2/3) = 15/22. From a partitioning point of view, the complex fraction (2 1/2)/(3 2/3) means 2 1/2 copies of 1/(3 2/3), where 1/(3 2/3) is the part we get from chopping the whole into four pieces, three of which are the same size, and the fourth of which is 2/3 the size of one of the first three pieces. If we were to construct a whole with these characteristics, we would end up constructing the whole just as we did above and shading the same amount of the whole. Thus, after we created a common partitioning, we would have found the same equivalent fraction. Denominator less that 1

Consider the following fraction:

!

2

3

3

4

From an iterating point of view, this fraction means 2/3 copies of 1/(3/4), where 1/(3/4) is the amount such that when it is iterated 3/4 of a time, or in other words, if we make 3/4 of a copy of it, we get 1. To create a 1/(3/4), I first start with 1. I know that 1 is 3/4 of 1/(3/4), because as noted above, 3/4 of a copy of 1/(3/4) is 1. If I partition 1 into three

2 1/2 copies of 1/(3 2/3)



equal parts, or thirds, each should correspond to a fourth of 1/(3/4). So four copies of 1/3 should yield a 1/(3/4). Now I need 2/3 of a 1/(3/4) to yield (2/3)/(3/4). I have already partitioned the 1/(3/4) into four equal pieces. I want to keep these original partitions, because it will help me compare 2/3 of 1/(3/4) with my original 1, and thus enable me to see the complex fraction in simplified form. So I need to create a common partitioning in my 1/(3/4). Since I already have fourths, and I also need to be able to make thirds, I will choose 1/12s as my common partitioning, because I can make both fractions with 1/12s. I know that 1/3 of 1/(3/4) is 4/12 of 1/(3/4) because three groups of 4/12 of 1/(3/4) makes a whole 1/(3/4). So 2/3 of 1/(3/4) is 8/12 of 1/(3/4). Now when I compare this amount with the original whole, I take the original partitions in my whole and cut them into three equal pieces so that the size of the pieces match up with the size of pieces in the 1/(3/4). Although these pieces are 1/12 of 1/(3/4), they are 1/9 of the whole, because 9 of them make up the whole. 2/3 of 1/(3/4) was 8 of these pieces, so (2/3)/(3/4) is 8/9 of a whole.

1 whole

1/(3/4) of a whole

1/4 of 1/(3/4) of a whole




8/9 of a whole



From a partitioning perspective, (2/3)/(3/4) doesn't make much sense. Using our typical way of phrasing fractions from a partitioning perspective yields the following description: (2/3)/(3/4) means 2/3 of a copy of 1/(3/4), where 1/(3/4) is the amount we get by splitting our whole into 3/4 equal pieces. What we can do to make sense of it, though, is to modify the way we think of partitioning, much the same way we did when we modified our definition of sharing so that we could do sharing division with fractions. The partitioning perspective can be modified to allow for fractional denominators less than one by thinking of partitioning the whole into the same number of parts as the numerator of the fraction in the denominator, but recognizing these as fractional parts of the piece, not the whole piece itself. For example, we can split 1 into three equal pieces, or thirds of 1, each corresponding to 1/4 of the piece 1/(3/4). Then to create the piece 1/(3/4), we iterate the 1/3 of 1 four times to create 1/(3/4). We would end up with the following picture. To figure out how much (2/3)/(3/4) is, we would proceed in the same way we did with the iterating perspective. Looking at the pictures for iterating and partitioning, we might be tempted to say that there really is no difference between the two ways of thinking. To point out the differences, however, consider what someone would do if they only had the iterating perspective for the complex fraction. Because 1/(3/4) from this perspective means finding an amount that 3/4 of it makes 1, the way to proceed would be to guess an amount, take 3/4 of it, and check to see if 3/4 of the amount was equal to 1. If 3/4 of the amount was too big, start with a smaller amount; if it was too small, start with a bigger amount. Continue to guess and check until you find the right amount. From a partitioning perspective, however, we have a strategy for creating the original amount. We partition the whole appropriately and iterate. Once again, these two strategies complement each other. We can use the partitioning perspective to create the amount that corresponds to our complex fraction, and then

1 whole

1/(3/4) of a whole

1/3 of a whole corresponds to 1/4 of a 1/(3/4)



use iterating to check and make sure that we are right. For example, we can create 1/(3/4) using the partitioning strategy above, and then check to make sure we have the right amount by using the iterating perspective, or taking 3/4 of the amount and checking to make sure it equals 1.

section 1: fraction addition - mathed – mathematics...

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