Fractions Section 1: Iterating and Partitioning

Fraction Unit Page 1

Goal

To help students develop iterating and partitioning images for fractions that they can use to (1) reason about part-whole relationships, and (2) describe, both orally and in writing, what a fraction means.

Big Ideas

Fractions are often taught using whole number ideas and language, which leads children to think about fractions as two whole numbers located vertically in space, not as a single, meaningful entity. In this section, I present two different images for fractions that are uniquely different from whole number reasoning, namely iterating and partitioning.

Iterating

One image for thinking about fractions is iteration. To conceive of a fraction from an iteration perspective, first start with a unit fraction, such as 1/5. How can you tell if something is 1/5? An amount is 1/5 if five copies of it equals 1. The image here is of taking 1/5 and iterating it 4 more times to make a whole.

1/5

The power in this image is that it gives a way to check to see if something is 1/5. Students can use this image when reasoning about fractions to make sure that they are attributing the correct size to fractional pieces. It also emphasizes that fractions are always a fraction of something. 1/5 is 1/5 of what? Of 1. How do you know? Because five copies of 1/5 is 1.

Once we understand what unit fractions are from an iterative perspective, then we can talk about the meaning of non-unit fractions, such as 4/5. Four-fifths means four one- fifths, where one-fifth is understood to be the amount such that five copies of that amount, when combined, equal a whole.

Iterating vs. "Out of" Thinking

You may have difficulty thinking about fractions in terms of iterations because you may have thought about fractions mostly from an "out of" perspective. For example, you may

Copyright © 2007 by Daniel Siebert. All rights reserved. Fraction Unit Page 2 think of 1/5 as 1 out of 5 things or parts. The problem with this image of fractions is that it is really whole number thinking. You are thinking of the fraction as two whole numbers, 1 and 5. It doesn't require you to use any fraction ideas. The "out of" image of fractions leads to several breakdowns, including some of the following:

• Lack of uniformity in size. All of the pieces below are 1/5, because they are each 1 out of 5 pieces.

• It obscures that fractions are always fractions of something. 1/5 thought of as 1 out of 5 is one-fifth of what? 1? 5?

• Complex fractions don't make sense. What does 3/2 mean? 3 out of 2 things? How can you have 3 out of 2 things?

• Lack of relative sizes. 2/8 is bigger than 1/3, because 2 candies out of 8 candies is more than 1 candy out of 3 candies.

• Obscures the actions of iterating and partitioning (partitioning discussed below). When children say something like "3 out of 5 equal parts" to explain 3/5, they are not thinking of performing an action, like cutting up a whole. They are missing the actions that are used to create the fraction, and that are an essential part of understanding what the fraction means.

Partitioning

A second image for thinking about fractions is partitioning. To conceive of a fraction from a partitioning perspective, first start with a unit fraction, such as 1/5. How can you tell if something is 1/5? It is 1/5 if it is the size of a piece you would get by taking a whole and splitting it into 5 equal parts.

1/5 1/5 1/5 1/5 1/5

The power in this image is that it gives a way to generate 1/5. We get 1/5 by taking a whole and splitting it into 5 equal parts. Students can use this image when reasoning about fractions to create a fractional piece of the correct size. Like the iterating image, this image also emphasizes that fractions are always a fraction of something. 1/5 is 1/5

Once we understand what unit fractions are from an partitioning perspective, then we can talk about the meaning of non-unit fractions, such as 4/5. Four-fifths means four one-fifths, where one-fifth is understood to be the amount that one gets from partitioning 1 into 5 equal pieces.

The image of partitioning is a welcome addition in reasoning about fractions. For example, if a student only has the iteration image to work with and is asked to determine how much 1/5 of a particular amount would be, they would have to use the guess and check method to find the amount. In other words, they would have to guess an amount, iterate it to see if it worked, and if it didn't, continue to modify the guess and iterate to check until a close enough approximation is achieved. With partitioning, the student has a direct method for creating 1/5: divide the amount into 5 equal parts.

The images of partitioning and iterating are very compatible. In fact, to function well with fractions, both images are necessary. Partitioning helps us be able to create fractions of different sizes, and iteration can be used to check whether we have achieved the right fraction by iterating and comparing with the whole. For example, we create 1/5 by partitioning 1 into 5 equal parts, and then check that it is 1/5 by iterating one of the parts 5 times to get 1. In this unit on fractions, one image should not be stressed over the other. You should become flexible in using both perspectives.

Partitioning vs. "Out Of"

You may have difficulty distinguishing between fractions from a partitioning perspective and the more common (but problematic) "out of" perspective. Because of this difficulty, I purposely started our discussion of what fractions mean with the iteration image, because this is so different from the "out of" perspective so as not to be easily confused. Partitioning and iterating images are fundamentally different ways of conceiving of fractions from the "out of" perspective.

The key difference between the partitioning and "out of" perspectives is that partitioning always keeps the partitioning process in the foreground. For example, when making fifths, we start with a whole and partition it into 5 equal parts to yield fifths. From the "out of" perspective, we don't really care where the five parts came from, what they were five parts of, and maybe not even that they are five equal parts. We are merely interested in taking a whole number amount from a whole number total. In fact, we never have to conceive of the five things as a whole whatsoever. They can remain five things. Because the partitioning image never loses sight of the partitioning process, it also never loses sight of the part-whole relationship, and thus represents fraction reasoning, not whole number reasoning.

Fractions Section 2: Fractions as Quotients

Goals

Help students to

• Distinguish between sharing and measurement situations;

• Write story problems for both the sharing and measurement interpretations of division;

• Draw pictures and write explanations for why quotients can be thought of as fractions, using both the sharing and measurement interpretation of division;

• Draw pictures and write explanations for why fractions can be thought of as quotients, using both the sharing and measurement interpretations.

Big Ideas

Two Models for Whole Number Division

There are two ways of conceiving of whole number division: sharing and measurement. The difference between these two interpretations is best illustrated using story problems:

Problem 1: Edward has eight cookies to give to his four friends. How many cookies will each friend get if they each get the same amount?

Problem 2: Edward has eight cookies and wants to give each of his friends four cookies. How many friends can he give cookies to?

The first problem is an example of a sharing situation. Edward is sharing his eight cookies among his four friends. To find out how many cookies each person gets, Edward can give each friend a cookie, and then continue to divvy out the cookies until all the cookies are gone. Thus, the action of sharing is divvying out cookies one at a time. Notice that in this problem we know how many total cookies we have, and how many people we want to give cookies to, but not how many cookies each person gets. That’s why we divide, to find out how many cookies each person gets. In general, for sharing situations, we know how many objects we have, and how many groups we have to make, but not how many should be in each group:

! Total $ ! Number $ ! Amount $ # & ÷ # & = # & " amount% " of groups% " per group%

The second problem is an example of a measurement situation. Edward is seeing how many groups of four cookies he can make from eight cookies. To find how many friends

Edward can give cookies to, Edward starts making groups of four cookies until all the cookies get used up. Thus, the action of measurement is grouping the cookies together to make as many groups of the specified amount as possible. Notice that in this problem, we know how many total cookies we have and how many cookies each person gets. What we don’t know is how many people we can give cookies to. In general, for measurement situations, we know how many objects we have and how many of those objects should be in each group, but not how many groups we can make:

Total Amount Number ÷ = amount per group of groups

The following chart highlights the differences between the two interpretations of division:

Sharing Measurement Action Divvying out equal amounts to Making groups of a certain size each group Unknown The amount each group gets The number of groups Guiding If I have eight objects and want If I have eight objects and want to Question to split them into four groups, make groups that contain four then how much will be in each objects, how many groups can I group? make?

Connecting Whole Number Division to Fractions

We can use these two interpretations of division to help us understand why fractions can be thought of as quotients. To illustrate this, consider 3/4 = 3 ÷ 4.

Sharing Perspective: To conceive of 3 ÷ 4 from a sharing perspective, we ask the question, If we have three and we want to split it evenly among four groups, how much would each group get? We know right away that there is not enough for each group to get 1. One way to solve this problem is to divide each of the three 1s that make up three into four parts. Once we do this, we can divvy one part of each 1 to each of the four groups, so that each group gets 1/4 of each of the three 1’s. Thus, each group gets 3/4, where 3/4 is understood to be 3/4 of 1.

Measurement Perspective: To conceive of 3 ÷ 4 from a measurement perspective, we ask the question, If I want to make groups of 4 things from a total of 3 things, how many groups can I make? Obviously, there is not enough to make a whole group of 4, but we can make part of a group of 4. We can see what fraction of a group of 4 there is in 3 by comparing the two amounts. We see from the picture below that each 1 in 4 is 1/4 of 4, because it takes four 1s to make a 4. Three has three of these 1s, or three 1/4s of 4. So there are 3/4 of a 4 in 3.

One important difference between the sharing and measurement interpretations for 3/4 is that when interpreting division from the sharing perspective, 3/4 is 3/4 of 1. In contrast, from the measurement perspective, 3/4 is 3/4 of 4.

Starting with Fractions and Ending Up With Division

What we've done above is show that any whole number division problem can be easily transformed into a fraction in two different ways. So in essence, we've shown that "division is just fractions," at least for whole numbers.

What we have left to show is that "fractions are division." To do this, we need to start with pictures of fractions and somehow see division in them. Consider, for example, the fraction 7/4. How do we see 7/4 as a division problem? First, consider the following picture of 7/4:

Note that this is 7/4. Each of the small sections are 1/4 , because when they are replicated four times, they produce 1. Furthermore, I have 7 of those 1/4s, so that's 7/4. However, I can also look at it from a measurement perspective. If I think about the division problem 7 ÷ 4 from a measurement interpretation, I am trying to find how many groups of 4 I can make from 7. To do this, I would group the 7 objects into 4 groups, and then count how many groups I have made. But this looks exactly like what has been done in the picture above. Note that in my picture of 7/4, I have 7 things (the small sections) grouped into groups of 4. Thus, this picture also represents 7 ÷ 4 from a measurement perspective.

To see 7 ÷ 4 from a sharing perspective, I need to start with a different picture of 7/4:

Note that this picture is also 7/4, because I have seven copies of 1/4, each small section a 1/4 of the whole because four copies of a small section make a whole. But with a little imagination, I can also see this picture as being 7 ÷ 4 from a sharing perspective. Recall that if I were dividing 7 by 4 using a sharing interpretation, I would be trying to share 7 things among 4 groups. To do this, I might take each of the 7 things and split them into fourths. Then I would divvy one-fourth of each thing to each group, so that every group got 1/4 of each thing. Then the shaded part I have above would merely be the amount that one group got.

Fractions Section 3: Fractions as Ratios

Goals

To help students conceive of ratios as fractions, and to be able to see and describe how they see these fractions in pictures they draw to represent the ratios.

Big Ideas

We often think of ratios one of two ways. The first way is in terms of the relationship between two numbers. For example, we might say something like the ratio of girls to boys at the school dance was 3 to 2. Another way of saying this is that for every 3 girls at the dance, there were 2 boys. We can symbolize the ratio of girls to boys by writing 3:2. However, we can also write the ratio of girls to boys as 3/2. What does 3/2 mean in this situation? The 3/2 is 3/2 of what? These questions push us to consider ratios in a second way, namely, as a single number. To help us see what the 3/2 means, consider the picture below, which represents the composition of the population at the school dance:

 

The entire rectangle represents the total population at the dance. Notice that in order to draw the ratio of girls to boys as 3:2, we needed to divide our population into 5 equal parts. Then 3 parts would be girls and 2 parts would be boys.

Where is the 3/2 in our picture? To be able to see the three halves, we need to be able to see the five equal parts that make up our picture in different ways. Some of the ways we can think about them include

• as fifths of the whole population, because it takes five copies of one of the parts to equal the number of people at the dance;

• as thirds of the girl population, because it takes three copies of one of the parts to equal the number of girls;

• as halves of the boy population, because it takes two copies of one of the parts to equal the number of boys.

Using the last interpretation of what one of the small parts represents, we can see that three of those parts equals the number of girls at the dance, and each part is 1/2 of the number of boys at the dance, so the number of girls at the dance is 3/2 times as many as the number of boys at the dance. In other words, there are 3/2 times as many girls at the dance as boys.

The reason that 3/2 is not immediately obvious from the picture above is that 3/2 comes from a part-part comparison, whereas the fractions we considered in the previous sections were all a result of part-whole comparisons. Naturally, we could also make part-whole comparisons in our picture above, such as concluding that girls represented 3/5 of the whole population. You should be able to see all of these comparisons in the picture.

Notice that 3/2 times as many girls as boys is different from 3/2 times more girls than boys. In order to have 3/2 more girls than boys at the dance, our picture would have to look like the following:

 

In this picture, the part of the girls population that is more than the boys population, namely three parts, is 3/2 as many as the number of boys. So the number of girls would be 3/2 times more than the number of boys.

Fraction Section 4: Equivalent Fractions

Goal

Students will be able to simplify fractions by drawing pictures and reasoning about those pictures. They will be able to write written explanations to communicate their reasoning. They will be able to explain what dividing the numerator and denominator by a whole number greater than one accomplishes in their picture, and why a fraction can be simplified if and only if both the numerator and denominator can be divided by a whole number larger than 1.

(Note: We will talk about subdividing fractions to yield equivalent fractions when we discuss common denominators in the section on fraction addition.)

Big Ideas

Equivalent Fractions Using the Measurement Interpretation of Division

Consider the picture below of 20/24:

We can simplify this picture by grouping the 1/24s into larger groups. Below are two different ways of grouping the 1/24s.

(a) (b)

In (a), the 1/24s were grouped into groups of 2. We can see twelve equal parts in our whole, so each part is 1/12. Ten of those parts are shaded, so we have 10/12. Because 10/12 and 20/24 represent the same amount of a whole, they are equivalent fractions. This shows that we can simplify the fraction 20/24 to 10 /12. By doing so, we not only have a fraction with smaller numbers in the numerator and denominator, but we also have a picture that is easier to read. This suggests that in terms of pictures, simplifying fractions means grouping smaller partitions into larger, equal-sized partitions, such that each new piece is either entirely shaded or entirely unshaded.

Typically we are taught to simplify fractions by dividing the numerator and denominator by the same whole number. To get from 20/24 to 10/12, we would divide 20 by 2 and 24 by 2. How can we see this in our picture? If we use the measurement interpretation of division, this would lead to the questions, How many twos are in 20 and 24? We answer this question in our picture by grouping the 1/24s into groups of 2 so we can see just how many groups of 2 we made. We found that we got 12 groups of 2, the answer to 24 ÷ 2. We were careful, though, to group the shaded 1/24s with other shaded 1/24s (i.e., we didn’t try to form a group of 2 with one shaded and one unshaded 1/24). We created 10 shaded groups of 2, the answer to 20 ÷ 2. Simply put, dividing the numerator and denominator by a particular whole number results in creating larger groups by fusing together that particular number of smaller parts. In the above example, dividing by 2 led to the creation of new, larger groups by repeatedly fusing two 1/24s together.

As we see in (b), 20/24 can be simplified even more. In this picture, we see that 20/24 is equivalent to 5/6. Furthermore, we notice that to get this picture, we grouped the 1/24s into groups of 4. This corresponds to dividing 20 by 4 (to get the number of shaded larger parts) and 24 by 4 (to get the number of larger parts in the whole).

We can see that (b) can no longer be simplified any further. To simplify it, we would have to be able to group the remaining pieces evenly into larger groups, with all the groups consisting of all shaded or all unshaded pieces. This is not possible for 5/6. For example, if we tried to group the six parts into three equal groups, we would end up with a group that would be half shaded. This violates the condition that the simplified fraction consists of parts that are either entirely shaded or entirely unshaded. The symbolic representation, namely (2 1/2)/3, further suggests that trying to simplify the fraction this way leads to a more complex representation than the original 5/6.

Once we understand what dividing the numerator and denominator by a whole number corresponds to in a picture of the fraction, we can make sense of the following statements:

1. If a whole number greater than 1 divides both the numerator and denominator of a fraction, then the fraction can be simplified: If a whole number divides the numerator, that means we can make groups of that size from the shaded parts without having any leftover. If the number also divides the denominator, that means we can make groups of that number from the parts that makeup the whole, with none left over. Thus, the fraction can be simplified.

2. If the only whole number that evenly divides the numerator and denominator is 1, then the fraction is in lowest terms: If a whole number does not divide the numerator of a fraction, then it is not possible to make groups of that size from the shaded region without having some left over. If a whole number does not divide the denominator, then it is not possible to make groups of that size from the parts that make up the whole without having some left over. Therefore, in order to simplify a fraction, a whole number must divide both the numerator and

denominator. If there are no whole numbers greater than 1 that divide both the numerator and denominator evenly, then the fraction cannot be simplified further.

Equivalent Fractions Using the Sharing Interpretation of Division

It is possible to explain the algorithm for simplifying fractions using the sharing interpretation of division, although it may not be as straightforward as the measurement interpretation. To illustrate how the sharing interpretation might be used, consider again the fraction 20/24. We can divide the numerator and denominator by 4. Dividing 20 by 4 is equivalent to asking, If I take the 20 shaded 1/24s and divide them evenly into four groups, how many will be in each group? Dividing 24 by 4 asks a similar question, namely, If I take the twenty-four 1/24s that make up a whole and divide them evenly into four groups, how many will be in each group? This leads to the following picture for 20/24:

From this picture, we see the 20/24 is divided into four equal groups, each of which is 5/6 shaded. When we use the sharing interpretation, we are not making larger groups made up of all shaded or unshaded parts, like we do when using the measurement interpretation. Instead, we are splitting the shaded and unshaded parts equally among four groups, with the understanding that once we know what fraction of each of these smaller groups is shaded, we will also know what fraction of the whole is shaded. In this case, we see that each of the four equal parts is 5/6 shaded. And because the whole is comprised of these four groups, it makes sense that the entire whole is also 5/6 shaded.

Fractions Section 5: Fractions and Decimals

Goal

To help students to use pictures and their reasoning about the meaning of fractions to (1) convert decimals to fractions and (2) convert fractions to decimals using both a part- whole and a quotient interpretation of fractions.

Big Ideas

The ideas we learned about simplifying fractions from the last section can be used to reason about the process of converting fractions to decimals and vice versa.

Converting Decimals to Fractions

To begin this process, we need to first be able to interpret what decimals mean. For the decimal .65, we can think of it one of two ways: as six-tenths and five-hundredths, or as sixty-five hundredths. If we use the second way of thinking about it, then we have in fact converted it to a fraction, albeit a fraction that could still be simplified further. If we think of .65 as 6/10 and 5/100, then we can use pictures to help us think about what this might be in terms of a single fraction. The picture below represents 6/10 and 5/100:

We can turn the block with the tenths into hundredths by drawing 9 equally spaced horizontal lines. Then we can combine the shaded regions into one block, yielding 65/100:

We can further simplify this fraction by grouping the hundredths into groups of five- hundredths. Doing this yields the following picture:

In this picture, we can see that each part that contains five-hundredths is 1/20 of the whole, because it takes 20 of them to make a whole. Since .65 makes 13 of those pieces, we know that .65 is equivalent to thirteen 1/20's, or 13/20. Note that there is no other way of grouping the hundredths into larger size pieces such that the larger pieces both evenly partition the shaded region and the whole. Thus, this is the fraction in simplest form.

Converting Fractions to Decimals

Suppose that we wanted to convert 3/4 into its decimal representation. The goal then would be to find some fraction that is equivalent to 3/4 and whose denominator is some power of ten.

Strategy 1: One way to do this is to start with tenths, and continue with increasingly smaller sized pieces (1/100's, 1/1000's, etc.) until we finally get a partitioning that can be divided evenly into fourths. Tenths will not work, because there is no way to group 10 tenths into four equal groups evenly. But hundredths will work:

Note that in each of the four parts, there are 25/100. Then three of the fourths would be 75/100, or .75.

Strategy 2: A second method that builds on a part-whole understanding of fractions is really partitioning with added constraint that we have to respect the division into tenths, hundredths, etc. We start by divvying out tenths into four groups to make fourths. Of course, if we do this, we will have two tenths leftover. But we know that each fourth will be more than 2/10 and less than 3/10.

We can take the remaining two-tenths and divide them into ten pieces each, so that each of the smaller pieces would be hundredths. Doing this yields 20/100, which can be evenly distributed among the four groups.

2/10 2/10 2/10 2/10

When the 20/100 are evenly distributed among the 4 groups, each group will get 5/100. This will give each group a total of 2/10 and 5/100, or .25. Then 3/4, which is three of the groups, would be 6/10 and 15/100, which is equivalent to 7/10 and 5/100, or .75.

Strategy 3: A third method starts with 3/4 being interpreted as a quotient from 3 ÷ 4. Using a sharing interpretation of division, we ask the question, If 3 were split among 4 groups evenly, how much would each group get? To answer this question in terms of a decimal representation, we use the same technique as above, but instead of starting with 1, we start with 3. Then we partition each of the 3 ones into tenths, and try to distribute them evenly among the four parts. If we do this, each group will get 7/10, with 2/10 leftover. Then we would divide each of the 1/10s into ten parts to get hundredths, and distribute the resulting 20/100 evenly among the four groups so that each group had 7/10 and 5/100, or .75. This would be the amount that each of the four groups would get if 3 were evenly split among them.

(We can also use a measurement interpretation by asking how many 4s are in 3. We could divide the 4 into 100 equal pieces so that each resultant piece was 1/100 of 4. This would give us 25 pieces in each of the 4 ones. Then we would partition the 3 ones in three into 25 pieces so that we could make a comparison between the 3 and the 4. We would find that 3 contains 75 pieces of the 100 pieces in 4. So 3 is 75/100, or .75, of 4.)