A fraction as part of a whole
A sandwich is cut and shared. When does a piece really count as a half or a third, and how does one number say how much you have?
Where you are 4 inside
What this area is about
This is where fractions begin. A whole is cut into equal parts, and a fraction \displaystyle \frac{a}{b} says that a of those b equal parts are taken. Two rules hold it up: the parts must be the same size, and the whole must be named, since half of a cupcake is not half of a cake.
How these fit together
The ground rules. Equal shares and naming the whole defines halves, thirds and fourths as pieces of the same size that fill the whole exactly. Equal shares that look different adds that same size does not mean same shape: two pieces are equal when they hold the same amount.
Three pictures of a fraction. Area and length models cut a shape or a length into b equal parts and shade a of them, with a at most b here. A fraction of a set counts objects in a group instead, each object standing for one equal part.
A path through it
Equal shares first: everything else depends on it. The other three follow in any order.
From here come unit fractions, the single pieces every fraction is built from, then fractions on the number line, where the length picture becomes a point, and later a fraction as division.