We already know how to lay numbers out in a row on
the number line, and we
know that a fraction is a whole
cut into equal parts. Put the two ideas together and you can find exactly where a
fraction lives on the line.
The trick is to look at the gap from 0 to
1 — one whole unit — and split it into
n equal steps, where n is the
denominator. Then the numerator
m simply counts how many of those steps to take, starting from
0:
\frac{m}{n}\ =\ m \text{ steps of size } \tfrac{1}{n}
So \frac{3}{4} means: split 0 to
1 into 4 equal steps, then count
3 of them. The fraction sits three of the four steps along — closer
to 1 than to 0.