To explain three major meanings of fractions.
This packet is aimed at prospective teachers and explains three major meanings of fractions: (1) part-whole relationships, (2) quantities and (3) ratios.
The differences among these are subtle, and most people don't need to worry very much about these distinctions. But teachers need to understand these ideas because they can help explain students' understanding of fractions, including alternate correct strategies as well as misconceptions.
Who cares what a fraction is? If you are going to be teaching students mathematics at pretty much any level, you do, bunky!
What makes this a question worth reading about is that a fraction isn't just one thing. Don't believe me? Ask a sixth-grader and a mathematician. Typical responses will be these:
Don't worry if you don't understand number 2. If you are reading this, you probably don't. The point is that most of us are closer to the sixth-grader's idea of fraction than to the mathematician's idea. That's OK. But even the sixth-grader's idea can be picked apart. That is what this packet aims to do.
Math educators have identified many different interpretations of fractions. Those wanting more detail can get started with some of the very interesting work done at the Rational Number Project at the University of Minnesota. I will focus on three of these:
Imagine that you are sharing 2 cookies among 3 people. How much does each person get? You might begin to answer this question by cutting each cookie into 3 pieces. Then each person can have two of these pieces, as in the picture below.
The first person gets the two pieces numbered "1", the second person gets those numbered "2" and the third person gets those numbered "3". Each piece is one-third of a cookie, so each person gets two-thirds of a cookie.
The fraction 2/3 means "two out of three parts that make up the whole".
We say that two fractions are equivalent when we can partition or group the parts of one to make the other. So 2/3 is equivalent to 4/6 because I can cut each of my thirds into two equal pieces, making sixths. And on doing so, each of the 2 pieces each child received has become 4 pieces. This is shown in the picture below.
In this case, we are thinking of the fraction 2/3 as a single amount. The difference is subtle but important.
Consider the number 37. We tend to think of this as an amount; it is more than 36, less than 38 and a lot less than 100. We don't tend to think of of 37 as being 3 tens plus 7 units. We can think of it this way, but we tend to think of 37 as a number rather than as a something built out of numbers.
The part-whole meaning of fractions focuses our attention on the number (2 and 3) from which 2/3 is built, rather than on an expression of quantity. If we think of 2/3 as a quantity, we think of the fraction as a single amount. 2/3 is more than 1/2 but less than 3/4 and it quite a bit less than 2. When we think of a fraction as a quantity, it is helping to answer the question "how much?" without needing to picture a whole cut into parts.
In the picture below, the first person gets 2/3 of the first cookie, the third person gets 2/3 of the other cookie and the second person gets 2/3 of a cookie, but takes some of each cookie.
The emphasis here is on how much cookie each person gets, rather than on the number of pieces. In this case we are thinking about 2/3 as a number, rather than as two separate numbers.
In this view, two fractions are equivalent if they represent the same amount. 2/3 is equivalent to 4/6 because we can see that they would represent equal amount of cookie. We might check this visually by drawing 2/3 of a cookie and 4/6 of a cookie carefully (but even careful drawings can be deceiving).
In the third case, we don't think of part-whole or quantity; we think of the fraction as representing the relationship between the number of cookies and the number of children. We might say to ourselves, "For every two cookies, there are three children". In this case, we are not naming the result of the sharing (2/3 of a cookie).
But notice that it is the multiplicative relationship between the numbers that matters with a ratio. The additive relationship would be there is one more child than there are cookies, and this is called the difference. One thing that makes ratios tricky is that we write them in the same form that we write the results. So the same expression: 2/3 might refer to the ratio (2 cookies to 3 children) and it might refer to the result (2/3 of a cookie per child).
We don't do that with differences. We don't write the result of subtracting 3-2 as "3-2", we write "1". But a ratio and the result of the operation are written identically: 2/3.
Two ratios are equivalent if the same relationship holds between the two numbers. Another way to describe this is For every two cookies there are three children. In the drawing below, both ratios show this same relationship, so they are equivalent.
When we have two equivalent ratios, we talk about "scaling" one up or down to the other. In this example, 2/3 is equivalent to 4/6 because I can show the 2 to 3 relationship twice in the 4 to 6 relationship. The 4 to 6 relationship represents having scaled the 2 to 3 relationship by a factor of 2.
The key difference here is that 4 to 6 represents twice as many cookies and twice as many children. When we are thinking about a part-to-whole relationship, we cut things into smaller pieces to show equivalence. When we are thinking about ratios, we scale to show equivalence.
Mathematically knowledgable people can move swiftly and seamlessly among these ideas-usually without noticing.
Two people are sharing three cookies? we think, If everyone is going to get the same amount, then each gets 2/3 of a cookie... Let's see...how can I do this? I'll cut each cookie into three pieces... etc.
But if we are teaching children about fractions (or otherwise working with people who are not as knowledgable about or adept with fractions), we need to be aware of these different ways of thinking. We need to be aware so that we can evaluate their ideas (as right or wrong), and so that we can provide help and explanations that build on their ideas.
The following are some examples of questions that are challenging to answer if you are still figuring out the relationships among these different meanings of fractions.