Count by fractions
Theory
Counting by fractions is a Year 4 maths skill: taking equal fraction-sized steps such as one quarter, one third or one half. This page shows how to count on a number line, reach a whole number like \(\dfrac{3}{3}=1\), keep counting past one whole into mixed numbers such as \(1\dfrac{1}{5}\), count back, and rename a mixed number as an improper fraction. In Western Australia this is taught in Year 5. Curriculum: VC2M4N04 / AC9M4N04.
To count by fractions you take steps that are all the same size. Counting by sixths, each step is one sixth: \(\dfrac{1}{6},\dfrac{2}{6},\dfrac{3}{6},\dfrac{4}{6},\ldots\) The top number counts how many parts you have, and the bottom number (how many parts make a whole) stays the same.
After enough parts you reach a whole. Three thirds make one whole, so \(\dfrac{3}{3}=1\). In the same way \(\dfrac{6}{6}=1\) and \(\dfrac{8}{8}=1\). The count does not stop there.
Keep going and you pass one whole. \(\dfrac{6}{5}\) is one whole and one more fifth, written as the mixed number \(1\dfrac{1}{5}\). A fraction like \(\dfrac{6}{5}\), with a top number bigger than the bottom, is an improper fraction. Both name the same point on the number line.
You can also count back: take one part away each time, so the top number goes down by one, until you reach \(0\). A fraction is greater than one whole when its top number is bigger than its bottom number, and it is a whole number when the top number is a multiple of the bottom, like \(\dfrac{6}{3}=2\).
Counting by fractions has no formula to memorise, but these facts help you know when the count reaches a whole and how to name the parts past one.
How to count by fractions
- Find the step. The bottom number tells the step size. Counting by eighths, each step is \(\dfrac{1}{8}\).
- Add one part. Keep the bottom number the same and add one to the top number: \(\dfrac{5}{8}\) then \(\dfrac{6}{8}\) then \(\dfrac{7}{8}\).
- Pass the whole. When the top and bottom match you have one whole. Do not stop. Keep counting into mixed numbers, like \(\dfrac{9}{8}=1\dfrac{1}{8}\).
How to count back
- Take one part away. Keep the bottom number and take one from the top number each step.
- Watch the wholes. Going back from \(1\dfrac{1}{5}\), the next step is \(1\), which is \(\dfrac{5}{5}\), then \(\dfrac{4}{5}\).
How to change a mixed number into an improper fraction
- Change the wholes into parts. Each whole is as many parts as the bottom number. In fifths, 1 whole is 5 fifths and 2 wholes are 10 fifths.
- Add the extra parts. Add the fraction part on top. The bottom number stays the same.
| \(\text{one more fifth}\) | \(\to\) | \(\dfrac{10}{5}\) |
| \(10\text{ fifths}\) | \(=\) | \(5\text{ fifths} + 5\text{ fifths}\) |
| \(\dfrac{10}{5}\) | \(=\) | \(2\) |
The next fraction is \(\dfrac{10}{5}\), which is exactly \(2\) wholes.
| \(\text{step}\) | \(=\) | \(\dfrac{1}{6}\) |
| \(\text{after }\dfrac{6}{6}\) | \(\to\) | \(\dfrac{7}{6}\) |
| \(\dfrac{7}{6}\) | \(=\) | \(1\dfrac{1}{6}\) |
The missing fraction is \(\dfrac{7}{6}\).
| \(\text{arrow}\) | \(=\) | \(5\text{ thirds from }0\) |
| \(\dfrac{5}{3}\) | \(=\) | \(\dfrac{3}{3}+\dfrac{2}{3}\) |
| \(\dfrac{5}{3}\) | \(=\) | \(1\dfrac{2}{3}\) |
The arrow is on \(\dfrac{5}{3}\), which is \(1\dfrac{2}{3}\).
| \(7\text{ third-cups}\) | \(=\) | \(\dfrac{7}{3}\) |
| \(\dfrac{7}{3}\) | \(=\) | \(\dfrac{6}{3}+\dfrac{1}{3}\) |
| \(\dfrac{7}{3}\) | \(=\) | \(2\dfrac{1}{3}\) |
That is \(2\dfrac{1}{3}\) cups of oats.
| \(\text{step}\) | \(=\) | \(\text{take away }\dfrac{1}{3}\) |
| \(\dfrac{2}{3}\) | \(\to\) | \(\dfrac{1}{3}\) |
| \(\dfrac{1}{3}\) | \(\to\) | \(0\) |
The next two numbers are \(\dfrac{1}{3}\) and \(0\).
| \(1\text{ whole}\) | \(=\) | \(5\text{ fifths}\) |
| \(2\text{ wholes}\) | \(=\) | \(2 \times 5 = 10\text{ fifths}\) |
| \(2\dfrac{3}{5}\) | \(=\) | \(\dfrac{10}{5}+\dfrac{3}{5} = \dfrac{13}{5}\) |
\(2\dfrac{3}{5}\) is the same as \(\dfrac{13}{5}\): counting by fifths from \(0\) takes 13 steps.
Common pitfalls
Frequently asked questions
How do you count by fractions?
Take equal steps that are all the same size. Counting by eighths you add one eighth each time: \(\dfrac{1}{8},\dfrac{2}{8},\dfrac{3}{8},\dfrac{4}{8}\). The top number goes up by one and the bottom number stays the same.
How many sixths make a whole?
Six sixths make one whole, because \(\dfrac{6}{6}=1\). A whole is always made when the top and bottom numbers match.
What comes after one whole when counting by fifths?
One whole is \(\dfrac{5}{5}\), and the count keeps going. The next fifth is \(\dfrac{6}{5}\), which is the same as \(1\dfrac{1}{5}\). You do not go back to \(\dfrac{1}{5}\).
What is a mixed number?
A mixed number has a whole-number part and a fraction part, like \(1\dfrac{2}{5}\). It names an amount that is more than one whole but not a whole number of parts.
How do you write 11/5 as a mixed number?
Take out the wholes first. \(\dfrac{11}{5}=\dfrac{10}{5}+\dfrac{1}{5}\), and \(\dfrac{10}{5}=2\), so \(\dfrac{11}{5}=2\dfrac{1}{5}\).
Do you count by fractions past one whole?
Yes. The count does not stop at one whole. After you reach a whole you keep adding the same fraction, moving into mixed numbers such as \(1\dfrac{1}{5}\) and \(1\dfrac{2}{5}\).