Fractions are answers to division problems. Consider 7/3. We can draw a picture of 7 divided into 3 equal pieces
The answer is the point on the number line at the right edge of the first piece. We see that it is between 2 and 3. Exactly where is it?
We really need to answer some easier questions first. What is 1/2? Take 1 unit and divide it into 2 pieces.
What is 1/3? Take 1 unit and divide it into 3 equal pieces.
The first division point is called 1/3, the second division point is called 2/3 which is the answer to the candy bar problem.
For instance consider the candy bar problem
Elena, Hortencia, and Toshawna have two candy bars. They want to split them equally among themselves. How many candy bars does each one get?
Each person gets 2/3 of a candy bar. The middle person gets their candy bar in 2 pieces, but everyone gets the same amount of candy.
If we plot the thirds between the whole numbers in the original problem we see
The answer is 1/3 more than 2.
At this point there is a discussion of proper and improper fractions and mixed numbers. We see in the picture that 7/3 = 21/3. Since the whole number 2 tells us how many groups of 3 there are in 7 it can be found by dividing 3 into 7 and obtaining a quotient of 2. The remainder of 1 will be the number of thirds which remain to get us to our number of 7/3.
At this point the students are asked to plot the halves, thirds, fourths, and sixths between 0 and 1. The result looks something like this
At this point the students can see that
The conclusion is that if you multiply or divide the top and bottom of a fraction by the same thing, you wind up with what we will call a different name for the same number. To see a picture of the relationship between say 2/3 and 4/6 we can use the following picture.
If you want to divide something into 6 pieces it helps if it has already been divided into 3 pieces because then all you have to do is to divide each of the thirds in half. This process will give is 3 groups of 2 which is an illustration of a multiplying 2x3 = 6. But in the process of dividing all 3 thirds in half we have divided each of the two thirds in half giving us a total of four sixths.
At this point the idea of reducing or simplifying fractions is discussed and terminology like "expressed in lowest terms" is introduced. Students are given exercises in converting improper fractions to mixed numbers and back, and reducing fractions to lowest terms. They are also asked to construct number lines and plot fractional points on them.
Number lines are only one way of illustrating fractions. They are useful in that they are among the easiest to draw, and because of the fact that other representations such as pies and candy bars are often number lines disguised by either wrapping them in a circle or giving them some depth in their drawings to turn them into candy bars for example. There are, however other two and three dimensional illustrations of fractions which do not translate to number lines as readily. Here is a two dimensional illustration of changing 2/3 into 4/6..