Fractions on a number line

3/4 sits three quarters of the way from 0 to 1, and 7/4 sits the same distance past 1.

Every fraction has a place on the line

A fraction is more than a slice of pizza. It is a number, and like every number it has one exact position on the number line. Placing fractions on a line is the clearest way to see how big they are compared with each other and with whole numbers, and it is the same picture you will use later for decimals, negative numbers and measurements on a ruler.

To place 3/4, start with the stretch from 0 to 1 and cut it into 4 equal steps. Each step has length 1/4. Count three steps from 0 and you land on 3/4. The denominator decides how finely each unit is cut, and the numerator decides how many steps you take from 0.

The same procedure works past 1. On the line above, each whole unit holds four quarter steps, so 1 is at 4/4 and 2 is at 8/4. To place 7/4, take seven quarter steps: four of them bring you to 1 and three more bring you to 7/4. The point lies between 1 and 2, closer to 2.

Count the gaps, not the marks

When a number line already has tick marks, first work out what each step is worth. Count the gaps between 0 and 1, not the marks. A line with 6 gaps between 0 and 1 is marked in sixths, even though you can see 7 marks if you include the ones at 0 and 1. Counting marks instead of gaps is the most common mistake with number lines, and it gives answers that are off by one part.

Example 1. A line from 0 to 1 has 5 equal gaps and a dot at the third mark after 0. Each gap is 1/5, so the dot is at 3/5.

Example 2. A line from 0 to 2 has 8 equal gaps in total. Each whole unit holds 4 gaps, so each gap is 1/4, not 1/8. A dot six gaps from 0 is at 6/4, which is the same as 1 and a half.

Example 3. Where is 2/3 compared with 1/2? Cut the unit into sixths. 1/2 is 3/6 and 2/3 is 4/6, so 2/3 is one sixth to the right of 1/2.

Distance between two fractions

The number line also shows how far apart two fractions are. From 1/4 to 3/4 there are two quarter steps, so the distance is 2/4, which is 1/2. When two fractions have the same denominator, the distance is the difference of the numerators, counted in that size of step. When the denominators differ, you need a common step size first, which is what units 3 and 7 are about.

There is always another fraction in between

Between any two different fractions there is always another one. Between 1/2 and 1 there is 3/4. Between 1/2 and 3/4 there is 5/8, halfway between them. Between 1/2 and 5/8 there is 9/16, and you can keep halving for as long as you like. Whole numbers have gaps between them, but fractions fill the line so densely that no two of them are next-door neighbours.

A quick way to find a fraction between two others is to add their numerators and add their denominators. For 1/3 and 1/2 this gives 2/5, and indeed 1/3 is about 0.333, 2/5 is 0.4 and 1/2 is 0.5. This number, called the mediant, always lands strictly between the two starting fractions when the denominators are positive. It is a way to find an in-between value and nothing more: adding fractions this way gives the wrong sum, as unit 7 explains.

Landmarks: 0, 1/2 and 1

Three points are worth using as landmarks. A fraction is less than 1/2 when its numerator is less than half its denominator: 3/8 is less than 1/2 because 3 is less than 4. A fraction is close to 1 when its numerator is close to its denominator: 11/12 is only 1/12 below 1. It is close to 0 when the numerator is small compared with the denominator: 1/20 is a short step from 0.