Unit fractions

What this area is about

A unit fraction is one piece when a whole is cut into equal parts: \displaystyle \frac{1}{b}, with numerator 1 and a whole number b \neq 0 as denominator. Every other fraction is a count of these pieces: \displaystyle \frac{a}{b} is a copies of \displaystyle \frac{1}{b}. That single fact runs underneath counting, adding and placing fractions on a line.

How these fit together

Both children are procedures built on that one fact, used in two directions.

Building up. Counting by unit fractions takes the copies one at a time, \displaystyle \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, and keeps going past the whole, so \displaystyle \frac{4}{3} is reached the same way as \displaystyle \frac{2}{3}. Only the numerator changes. It also counts backward.

Taking apart. Decomposing a fraction writes one fraction as a sum of smaller fractions with the same denominator, such as \displaystyle \frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} or \displaystyle \frac{2}{6} + \frac{3}{6}.

A path through it

The two children can be taken in either order.

Next come adding fractions with like denominators, which reads a decomposition from the parts back to the total, fractions greater than one, a fraction times a whole number, and sharing a fraction into equal parts.