When a whole is cut into equal parts, one of those parts is a unit fraction. One of 4 equal parts is written $\frac{1}{4}$: the bottom number says how many equal parts make the whole, and the top number says we have 1 of them.
Why it works
Cutting the same whole into more pieces makes each piece smaller. A pizza for 8 people gives smaller slices than the same pizza for 4 people. So $\frac{1}{8}$ is less than $\frac{1}{4}$, even though 8 is more than 4, and $\frac{1}{4}$ is less than $\frac{1}{3}$: 4 equal parts of a whole are smaller than 3 equal parts. The bigger number counts more parts, and more parts means smaller parts.
Key idea
$\frac{1}{b}$ is one of $b$ equal parts of a whole. The larger $b$ is, the smaller $\frac{1}{b}$ is. It takes $b$ of them to make the whole.
Worked example: Name it
A bar is cut into 6 equal parts and 1 part is shaded. What fraction is shaded?
- 6 equal parts: each part is $\frac{1}{6}$.
- 1 part is shaded.
Answer: $\frac{1}{6}$
Worked example: Which is bigger?
Which is greater, $\frac{1}{3}$ or $\frac{1}{8}$?
- Both are one piece of the same whole.
- Thirds are 3 parts; eighths are 8 parts, so eighths are smaller pieces.
Answer: $\frac{1}{3}$
Worked example: Make a whole
How many pieces of size $\frac{1}{5}$ make one whole?
- $\frac{1}{5}$ is one of 5 equal parts.
- All 5 parts make the whole.
Answer: $5$
Common mistake
Thinking $\frac{1}{8}$ is bigger than $\frac{1}{4}$ because 8 is bigger than 4. That habit works for whole numbers, but here the 8 means the whole is cut into more, smaller parts. Check with the bars: the $\frac{1}{8}$ piece is half as long as the $\frac{1}{4}$ piece.