Three Meanings

One reason for the difficulty in teaching fractions is that, IMO, they have three meanings.

First, they simply mean divide. So, "84 divided by 4" can be represented as 84/4.

The second meaning is as a number. There are numbers between 1 and 2, and we can represent these numbers are fractions. For example, 1 2/3.

Of course, there is overlap between these two definitions. You can even try to argue that they are the same. But I think you will get in trouble if you do, because they are different.

In the first, we are moreso talking about a process than a result. For example, we could say that (x*x + 2*x + 1)/(x+1) = x + 1. We are not talking about numbers here. In the second, the concern is with the result of the process, not the process itself. Here, 1 1/2 is the same as 1.5, and both are the same as 3/2, 6/4, and the number midway between 1 and 2.

In the first, we could have numerators bigger than demoninators, or as noted, variable expressions. In the second we are primarily going to be using mixed fractions, where the fraction is proper (numerator smaller than the denominator). We would rarely use 3/2 as indicating a length in inches.

The Third Meaning

There is a third meaning to fractions, as proportions. Suppose we find out that 2/3 of the children in a class are boys. This tells us something. But it does not tell us how many boys are in the class.

In this third sense, fractions are just like percentages or probabilities. The lowest value is zero and means none; the highest value is one and that means all.

Of course, this is the form we need for making decisions. If you want to decide whether to get a flu shot, you need to the probability of getting the flu without the shot, which is (roughly) the fraction/proportion of people getting the flu out of all the people not getting a flu shot. The actual number of people who get the flu without getting a flu shot is irrelevant.

I ask my class to differentiate these two sentences. "I ate half of a banana" and "I ate half of the bananas". In the first, half is being used as a number; in the second, half is being used as a proportion. "I ate one and a half bananas" makes sense; "I ate one and a half of the bananas" doesn't make sense.

I also happened to ask if you could add fractions, and the answer from some students was no. From the third definition, this is actually somewhat reasonable. Suppose you knew that 1/2 the people in a class were male, and 1/3 had their birthdays in the summer. There is no question for which these two numbers can be meaningfully added. Of course, as numbers or lengths, it makes sense to add fractions.

Are the Fractions Numbers?

I ask classes to list numbers. They are prone to just mention the counting numbers. On a bad day, they won't remember zero; on a good day, you might even get the negative numbers. You are unlikly to get fractions.

Then I ask if there are numbers between 1 and 2. Then I am likely to get fractions. Then I take a vote -- are the fractions numbers?

Usually the fractions lose the vote. One time they won and then the teacher later told the class that they weren't numbers. Actually, with no definition of "number", there is no right or wrong answer. And I am sympathetic to the claim that proportions are not numbers, though I think the fractions do point to real numbers and hence "are" true numbers.

If the class votes that fractions are not numbers, then you are back to the question of whether they are numbers between 1 and 2. Or to ask the question in a more concrete way, are there lengths between 1 inch and 2 inches?

One third-grade class voted overwhelmingly that fractions were numbers. I asked what a fraction was, which is a good exercise by itself, and they said, I think correctly, something to the effect of one number in the numerator and one number in the demoninator. (This left out the concept of division, which is central to the first definition of fraction but not essentially to the other definitions; as will be discussed below, students are exposed to fractions as part of a circle.)

So I asked if 1/2 / 6 was a fraction. They really didn't like calling that a fraction. But the standard way to get out of it was to say that 1/2 wasn't a number, and they had voted that fractions were numbers. (I gave them the word "integer", but they didn't use it in their definition construction.)

The Ubiquitous Circle

Fractions apparently are first defined as parts of a circle. For example, the fraction 1/3 means to divide the circle into 3 parts, then color in one of the parts. As noted above, while the circle is divided into three parts to represent the fraction 2/3, two is never divided by three. So this representation does not fit the first meaning of fraction.

If the circle is taken as representing the number 1, then the circle fits the second definition. But I sincerely doubt that it how the circle is meant. The circle perfectly fits the third definition. If there are six bananas and I eat six of them, then we divide the circle into six parts, color in three, and see that I have eating 3/6 of the bananas. In this representation, the circle in a way is six bananas, and it has no connection to the number 1.

The circle concept can be bent to fill the other definitions. One third-grader could see that 1/2 / 3 was a number, using the circle concept -- she divided her circle into three parts and then imagined coloring in 1/2 of a part. Similarly, when faced with 84/4, she asked how that was possible -- you can't divide a circle up into 4 parts and then color in 84 of them. I said you would need 21 circles, and she seemed to understand that.

A Beginning Exercise

Have some unit of measure. The "paper", equal to the width of a piece of standard paper, is very convenient. (You can make a lot of them very quickly.)

Now, there is a problem whenever you measure something, in that it won't be exactly some whole number of papers. You can have them measure.

Then once they see this problem, try to communicate a length to them. Tell them to draw a line that is this long: Divide the paper into 3 equal parts, then draw the line equal to two of these parts. You can use this to draw lines of any length.

I sometimes say that in the imaginary country of Riki-Tiki, they would say that something was 14 + 5 parts out of 7 papers long. This was then shorted to 14 5/7 and everyone knew what that meant.

Other Meanings

The beginning exercise communicates the second meaning of fractions, as representing numbers. Ways of trying to communicate the third meaning are discussed elsewhere.