Fractions on a number line
Theory
Fractions on a number line is a Year 4 maths skill: cut the line from 0 to 1 into equal parts, then count parts from 0 to name or place a fraction. This page explains how the equal parts give the bottom number and the parts from 0 give the top number, how to show fractions greater than 1, how equal fractions such as one half and two quarters land on the same point, and how to compare and order fractions by their place on the line. In Western Australia this is taught in Year 5. Curriculum: VC2M4N04 / AC9M4N04.
A number line is a straight line with numbers marked in order. To show fractions, cut the length from \(0\) to \(1\) into equal parts. The parts must be the same size.
The number of equal parts is the bottom number (the denominator). The number of parts counted from \(0\) is the top number (the numerator). If the line from \(0\) to \(1\) has \(4\) equal parts, each part is \(\dfrac{1}{4}\), and the tick that is \(3\) parts from \(0\) is \(\dfrac{3}{4}\).
A number line does not stop at \(1\). It can keep going to \(2\) and beyond, so a point can be a fraction greater than 1, like \(\dfrac{5}{2}\). Two fractions that land on the same point, such as \(\dfrac{1}{2}\) and \(\dfrac{2}{4}\), are equal fractions.
The further right a point is, the bigger the fraction. A fraction is closer to 1 when fewer parts are left between it and \(1\). To compare points on two different lines, the lines must be the same length from \(0\) to \(1\).
Fractions on a number line do not need formulas, but these facts help you name and match points.
How to read a fraction on a number line
- Count the equal parts. Count the gaps between \(0\) and \(1\). This number is the bottom number.
- Count from 0. Count how many parts the point is from \(0\). This number is the top number.
- Write the fraction. Put the parts from \(0\) over the number of equal parts, like \(\dfrac{3}{4}\). Keep counting past \(1\) for fractions greater than one.
How to compare and order fractions on a number line
- Mark each fraction. Use lines of the same length, or rename the fractions into the same parts, such as eighths.
- Read from left to right. The point furthest left is the smallest and the point furthest right is the largest.
- Find the whole numbers either side. A fraction with a top number bigger than its bottom number sits past \(1\). Check which whole numbers it lies between.
| \(\text{equal parts}\) | \(=\) | \(5\) |
| \(\text{parts from }0\) | \(=\) | \(3\) |
| \(\text{fraction}\) | \(=\) | \(\dfrac{3}{5}\) |
The arrow points to \(\dfrac{3}{5}\).
| \(\text{each part}\) | \(=\) | \(\dfrac{1}{6}\) |
| \(\dfrac{5}{6}\) | \(=\) | \(5\text{ parts from }0\) |
\(\dfrac{5}{6}\) is the fifth tick after \(0\), just one part before \(1\).
| \(1\) | \(=\) | \(3\text{ thirds}\) |
| \(\text{arrow}\) | \(=\) | \(4\text{ thirds}\) |
| \(\text{fraction}\) | \(=\) | \(\dfrac{4}{3}\) |
\(\dfrac{4}{3}\) is just past \(1\), between \(1\) and \(2\).
| \(1\text{ quarter}\) | \(=\) | \(2\text{ eighths}\) |
| \(6\text{ eighths}\) | \(=\) | \(6 \div 2 = 3\text{ quarters}\) |
| \(\dfrac{6}{8}\) | \(=\) | \(\dfrac{3}{4}\) |
\(\dfrac{6}{8}\) and \(\dfrac{3}{4}\) mark the same spot.
| \(\dfrac{1}{2}\) | \(=\) | \(\dfrac{4}{8}\) |
| \(3,\ 4,\ 7\text{ eighths}\) | \(\to\) | \(\dfrac{3}{8},\ \dfrac{1}{2},\ \dfrac{7}{8}\) |
The order is \(\dfrac{3}{8}\), \(\dfrac{1}{2}\), \(\dfrac{7}{8}\). \(\dfrac{7}{8}\) is closest to \(1\), only one eighth away.
(i) What is her mistake?
(ii) What fraction is the point?
| \(\text{ticks counted}\) | \(=\) | \(5\text{ (including }0\text{)}\) |
| \(\text{gaps from }0\) | \(=\) | \(5 - 1 = 4\) |
| \(\text{point}\) | \(=\) | \(\dfrac{4}{6}\) |
(i) She counted the tick at \(0\). A fraction counts the equal gaps from \(0\), not the ticks. (ii) The point is \(\dfrac{4}{6}\), four sixths along the line.
Common pitfalls
Frequently asked questions
How do you find a fraction on a number line?
Count the equal parts between \(0\) and \(1\) to get the bottom number, then count how many parts the point is from \(0\) to get the top number. A point \(3\) parts along a line cut into \(5\) equal parts is \(\dfrac{3}{5}\).
What is the bottom number of a fraction on a number line?
The bottom number, called the denominator, is the number of equal parts the line from \(0\) to \(1\) is cut into. Count the gaps between ticks, not the tick marks.
How do you show a fraction bigger than 1 on a number line?
Keep counting the parts past the \(1\) mark. On a line from \(0\) to \(2\) cut into fifths, the tick just past \(1\) is \(\dfrac{6}{5}\), because it is \(6\) fifths from \(0\). It lies between \(1\) and \(2\).
Are two thirds and four sixths the same point on a number line?
Yes. \(\dfrac{2}{3}\) and \(\dfrac{4}{6}\) land on exactly the same point, because each third is 2 sixths. Fractions that mark the same spot are equal fractions.
How many parts is seven tenths from 0?
\(\dfrac{7}{10}\) is \(7\) parts from \(0\) on a line cut into \(10\) equal parts. It sits closer to \(1\) than to \(0\).
What fraction is exactly halfway between 0 and 1?
\(\dfrac{1}{2}\) is exactly halfway between \(0\) and \(1\). The same point can also be written as \(\dfrac{2}{4}\) or \(\dfrac{4}{8}\).