Fractions on the number line
A fraction began as pieces of a pie. Where does it sit among the other numbers, and is there always room for one more?
Where you are 2 inside
What this area is about
Here a fraction stops being pieces of a thing and becomes a number with a place of its own. On a number line the stretch from 0 to 1 is one whole. Cut it into b equal parts, for a whole number b \neq 0, and each part has length \displaystyle \frac{1}{b}; the point \displaystyle \frac{a}{b} is a of those jumps from 0, past 1 when a > b.
How these fit together
Placing a fraction. Locating a fraction is the procedure: mark equal parts between whole numbers, count unit fraction jumps from 0, and read a point someone else has marked.
What lies between. Finding a fraction between two fractions shows that between any two different fractions there is always another, found by cutting the pieces smaller. So no two fractions are neighbors, and the line never fills up.
A path through it
Locate a fraction first; finding one between two others uses those points.
The number line is where the next ideas live: equivalent fractions are two names for one point, comparing fractions reads which point is further right, benchmark fractions are landmarks on it, and negative fractions extend it to the left of 0.