In Grade 3 you built fractions from unit fractions: $\dfrac{3}{4}$ is three pieces of size $\dfrac{1}{4}$. Turn that around and you can break a fraction into parts in many ways.
Key idea
A fraction can be split into a sum of fractions with the same denominator, as long as the numerators add up: $\dfrac{5}{8} = \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{3}{8} = \dfrac{2}{8} + \dfrac{3}{8}$. Mixed numbers split the same way: $2\dfrac{1}{8} = 1 + 1 + \dfrac{1}{8} = \dfrac{8}{8} + \dfrac{8}{8} + \dfrac{1}{8}$.
Why it works
$\dfrac{5}{8}$ is just 5 eighth-pieces. You can sort 5 pieces into piles any way you like (1 and 4, or 2 and 3, or 1, 1 and 3) and you still have 5 eighths.
Worked example: Missing part
Fill in: $\dfrac{5}{8} = \dfrac{2}{8} + \dfrac{?}{8}$
- Count eighths: 2 plus what makes 5?
- $5 - 2 = 3$.
Answer: $\dfrac{3}{8}$
Worked example: Split a mixed number
Write $2\dfrac{1}{4}$ as a sum of fourths.
- Each whole is $\dfrac{4}{4}$.
- $2\dfrac{1}{4} = \dfrac{4}{4} + \dfrac{4}{4} + \dfrac{1}{4}$.
Answer: $\dfrac{4}{4} + \dfrac{4}{4} + \dfrac{1}{4}$
Common mistake
Changing the denominators while splitting, like writing $\dfrac{5}{8} = \dfrac{2}{4} + \dfrac{3}{4}$. The pieces must stay eighths. Check by adding the numerators back: they should make 5, over the same 8.