Years 1–6Maths at school

Number lines: counting on, differences and fractions

Written by A Dagdelen · Reviewed

A number line is the first picture of numbers most children meet at school, and it stays with them all the way to negative numbers and decimals. Here is how schools use it at each stage, and how it helps with sums that look tricky on paper.

Illustration: Quizo School

  • Year 1numbers shown on a number line
  • Year 2estimating where a number goes
  • Year 3counting in tenths; fractions as numbers on the line
  • Years 4–6decimals and negative numbers

What it is and when it appears

  1. In Year 1 numbers get a place. The National Curriculum asks Year 1 pupils to "identify and represent numbers using objects and pictorial representations including the number line". Many classrooms have one along the wall.
  2. In Year 2 children estimate. Year 2 pupils estimate where a number goes on a line, such as 37 on a line from 0 to 100. It builds a feel for how big numbers are.
  3. Counting on finds a difference. To work out 403 − 387, a child can count on from 387 to 403 in friendly jumps: +3 to 390, +10 to 400, +3 to 403, which makes 16. Many schools teach this for subtractions where the two numbers are close, because it avoids exchanging altogether.
  4. Fractions and decimals are numbers too. From Year 3 children count up and down in tenths and begin to see fractions as numbers on the line, not only as parts of a shape. In Year 4 they place numbers with one or two decimal places on number lines.
  5. Below zero in Years 4 to 6. Year 4 counts backwards through zero to negative numbers, Year 5 uses them in context, and Year 6 works out intervals across zero, such as the rise from −4 °C to 6 °C.

Worked examples

Finding a difference by counting on: 403 − 387

A number line marked 387, 390, 400 and 403, with jumps of +3, +10 and +3 drawn as arcs. The jumps add up to 16.387390400403+3+10+33 + 10 + 3 = 16
  1. Draw a line and put 387 near the start. There is no need to mark every number.
  2. Jump to the next ten: 387 + 3 = 390.
  3. Jump to the next hundred: 390 + 10 = 400.
  4. Jump to the target: 400 + 3 = 403.
  5. Add the jumps: 3 + 10 + 3 = 16. So 403 − 387 = 16.

Tenths on a number line: decimals and fractions

A number line from 0 to 1 split into tenths and labelled 0.1 to 0.9. The point at 0.3 is marked as 3/10, and the middle point, 0.5, as 1/2.00.10.20.30.40.50.60.70.80.91310125/10 = 1/2 = 0.5
  1. Draw a line from 0 to 1 and split it into ten equal parts.
  2. Each mark is one tenth more: 0.1, 0.2, 0.3 and so on up to 1.
  3. The third mark is 0.3, which is the same as 3/10.
  4. The middle mark is 0.5, or 5/10, which is the same as 1/2.

How to help at home

“That's not how I learnt it”

If you would always use columns for 403 − 387, counting on can look like a detour. Try both: the number line takes three small jumps and needs no exchanging. Children who can choose between methods are usually the confident ones.

A 10-minute activity: the washing line

Peg cards on a string, or draw a line on paper from 0 to 100 (or 0 to 1 for older children). Call out numbers, fractions or decimals and your child places them. Then ask how far apart two of them are.

Spot real number lines

Rulers, tape measures, thermometers and kitchen scales are all number lines. Ask what each small mark is worth.

Make the jumps friendly

Encourage jumps to the next 10 or 100 first. Small, round jumps are easier to add up than one big leap.

Common mistakes

  • Counting the marks instead of the jumps. Starting the count at 1 on the first number gives an answer one too big. Count the gaps between marks, not the marks themselves.
  • Uneven spacing. When the gaps are uneven, 1/2 can end up nearer 1 than 0. On a measured line, equal steps must look equal.
  • Thinking 0.25 is bigger than 0.9. Because 25 is bigger than 9. Placing both on a line from 0 to 1 shows 0.9 is much closer to 1.
  • Stopping at zero. From −4 to 6 is 4 up to zero and 6 more, 10 in all. Stopping at zero gives 4 or 6.

Practise and explore on Quizo School

Questions parents ask

What is an empty number line?

A line with no marks printed on it. Children add only the numbers they need, so it works for numbers of any size.

Should my child still use number lines in Year 6?

Yes, for some things: differences between close numbers, negative numbers and decimals. For large numbers far apart, columns are usually quicker.

Why does my child count up to subtract?

Finding the difference between close numbers is often quicker by counting on, and it avoids exchanging.

Are number lines used in secondary school?

Yes. The Key Stage 3 curriculum uses the number line for ordering numbers, including negative numbers, fractions and decimals.

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