Fractions on a Number Line

Look at a ruler or a measuring jug: between the big whole-number marks sit smaller ones, and each says exactly how far along you are. Those in-between marks are fractions, sitting at precise points on a line.

You can already lay numbers out in a row on the number line, and you know that a fraction is a whole cut into equal parts. Put the two ideas together and something lovely happens: a fraction stops being a piece of pizza and becomes an exact place — a single point — on the line.

Picture the gap from 0 to 1. That gap is one whole. A fraction lives inside that gap (or just past it). To find it, you do exactly what you do when you share a chocolate bar: you cut the whole into equal pieces, then you count.

The denominator (the bottom number) says how many equal parts to split the whole into.
The numerator (the top number) says how many of those parts to count, walking from 0.

\frac{m}{n}\ =\ m \text{ steps, each of size } \tfrac{1}{n}

Take \frac{3}{4}. The denominator is 4, so split the gap from 0 to 1 into four equal steps. The numerator is 3, so count three steps from 0. You land three-quarters of the way along — much closer to 1 than to 0. It is like cutting a pizza into four slices and eating three:

pizza cut into quarters

Press play, then replay it: each time the unit from 0 to 1 is split into a different number of equal steps, and a marker hops step by step to land exactly on a fraction. Watch how the denominator decides how fine the steps are, and the numerator decides how many you take.

See it: cut, then count

Here is the same idea, frozen so you can study it. The line below is split into a number of equal steps (the denominator), and the dot has hopped a number of steps from 0 (the numerator). Read the labels under the whole numbers, count the gaps to the dot, and check the fraction it shows. Press Refresh for a fresh one.

Two worked examples

Finding \frac{2}{5}. The denominator is 5, so chop 0 to 1 into five equal steps. The numerator is 2, so count two steps. The point sits a little less than halfway — two of five steps along.

Finding \frac{1}{3}. Three equal steps from 0 to 1; count one. The point sits one-third of the way along, to the left of \frac{1}{2} — because cutting the whole into three pieces makes each piece smaller than cutting it into two. The more pieces you cut, the shorter each step. That is the single most important idea on this whole page, and the "Watch out!" below says it again.

What about fractions bigger than 1?

The line does not stop at 1 — it keeps going. So fractions can too! Take \frac{5}{4}: split each whole into 4 equal steps and count five of them. The first four steps carry you all the way to 1; the fifth step carries you one quarter past 1. So \frac{5}{4} sits just beyond 1, a quarter of the way toward 2:

Whenever the numerator is bigger than the denominator, the fraction lands past 1. When they are equal — like \frac{4}{4} — you land exactly on 1.

Different names, same spot

Here is a surprise. Split the whole into two and take one step: you reach \frac{1}{2}. Now split the same whole into four and take two steps: you reach \frac{2}{4}. Look where you stop — it is the very same point, slap in the middle of the line. Fractions that land on the same place are equivalent:

\frac{1}{2}\ =\ \frac{2}{4}\ =\ \frac{3}{6}\ =\ \frac{4}{8}

They all name the halfway point — different cuts, same destination. The number line is what makes this obvious: two fractions are equal exactly when their dots sit on top of each other.

It feels like it should be — 8 is more than 2, after all. But no! A bigger denominator means you cut the cake into more slices, so each slice is smaller. One slice of a cake cut into eight (\frac{1}{8}) is a thin sliver; one slice of a cake cut into two (\frac{1}{2}) is a giant half. So \frac{1}{8} sits much closer to 0 than \frac{1}{2} does.

a cake to share

The two classic number-line traps:

Split the whole into equal parts first, then count the gaps — not the tick marks. A line from 0 to 1 cut into quarters has five ticks (counting both ends) but only four equal steps. The denominator counts the steps between ticks, never the ticks themselves.
A bigger denominator makes smaller pieces. Cutting into more parts makes each part shorter, so \frac{1}{8} is less than \frac{1}{2}, even though 8 > 2.

Sal Khan places fractions on a number line here: