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. 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. 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. 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.