The bar model: what it is and why schools use it

What is a bar model?

A bar model is a picture of a maths problem drawn with rectangles. Each bar stands for a number, and the lengths show how the numbers compare. There are two main kinds: a part-whole bar model, where parts join to make a whole, and a comparison bar model, where two bars sit side by side to show a difference.

The national curriculum doesn't require schools to use bar models. They come from government guidance for teachers, which uses them alongside number lines, tens frames and part-whole models as ways to picture a calculation.

The point of a bar model isn't the drawing itself. It's that once the picture is right, the calculation is usually obvious.

Part-whole bar models

A part-whole bar model has the whole on top and the parts underneath. It's the same idea as the circles of a part-whole model, drawn as bars. Here, 20 and 35 make 55:

Government guidance asks children to find every equation one bar model shows. A bar of 37 split into 25 and 12 gives eight, including these four:

• 25 + 12 = 37 and 12 + 25 = 37
• 37 − 12 = 25 and 37 − 25 = 12

That one picture shows that addition and subtraction are inverses, which is why bar models are so useful for checking answers from column addition and column subtraction.

Missing part problems

When the whole is known but one part isn't, the missing part is found by subtracting. A question mark marks the gap:

Children often write this as 20 + ? = 50. The bar model shows that 50 − 20 gives the answer, so they don't need to count on in ones.

Comparison bar models for finding the difference

A comparison bar model puts two bars one above the other, starting at the same point. The gap at the end is the difference. Government guidance uses this for questions that say "how many more", "how many fewer" or "find the difference", from Year 2.

Sam has 35 stickers and Ali has 20. How many more does Sam have?

Children are often unsure whether "how many more" means add or take away. The picture settles it: the question asks about the gap, so subtract.

Bar model for multiplication and division

For multiplying and dividing, the bar is split into equal parts. £24 shared equally between 3 children looks like this:

The same picture works both ways. Read it as division, 24 ÷ 3 = 8. Read it as multiplication, 3 × 8 = 24, because 3 equal parts of 8 make 24. Government guidance shows bar models for both kinds of division: sharing into a number of groups (£14 shared between 2 children) and grouping into a size (14 ping-pong balls in packs of 2).

Bar models don't replace the times tables, though. Children still need to know that 3 × 8 = 24, and the 8 times table is where that comes from.

Bar modelling fractions

Finding a fraction of an amount is the same as splitting a bar into equal parts. To find three-fifths of 40, split 40 into 5 equal parts, then take 3 of them:

The guidance's own wording is "three-fifths is equal to 3 one-fifths": divide by 5 to find one-fifth, then multiply by 3. Government guidance starts with unit fractions, such as one-fifth of 15, in Year 3. This example is from its Year 5 section, where children also learn to draw their own bar models for harder problems, rather than being given one.

How a bar model helps choose the operation

The hardest part of a word problem is often deciding what to do. A bar model helps because children draw what they know first, then see what's missing.

Take two problems with the same numbers:

• Problem A. Ella read 45 pages on Monday and 27 on Tuesday. How many pages did she read?
• Problem B. Ella's book has 72 pages. She has read 45. How many are left?

In problem A, both parts are known and the whole is missing, so add:

In problem B, the whole is known and a part is missing, so subtract:

Both give the same bar. What changes is which box has the question mark. As a rough guide:

What the bar shows	What to do
The whole is missing	Add the parts
A part is missing	Subtract from the whole
Equal parts, and the whole is missing	Multiply
Equal parts, and one part is missing	Divide
Two bars and the gap is missing	Subtract to find the difference

Bar models in Year 6

By Year 6, bar models help with problems that have two unknown numbers. Government guidance gives this one: two numbers add up to 25, and their difference is 7. What are they?

Draw two bars, one 7 longer than the other. Take off the extra 7 and the bars are the same length, with 25 − 7 = 18 between them. So the smaller number is half of 18, which is 9, and the larger is 9 + 7 = 16. The finished picture:

Common mistakes to watch for

• Bars that don't match the numbers. A bar for 20 drawn longer than a bar for 35 hides the answer. The bars don't need to be measured, but bigger should look bigger.
• Putting the question mark in the wrong place. Read the question again and ask: do we know the whole, or a part?
• Unequal parts for division. If the problem shares equally, every part must be the same size.
• Comparison bars that don't line up. Both bars must start at the same point, or the difference can't be seen.

How to help at home

• Draw it together. For any word problem, start with "what do we know?" and draw a blank bar model before working anything out.
• Use your child's school picture. Ask how they draw bar models in class, and copy that.
• Ask for the other equations. Once a bar shows 25 + 12 = 37, ask what take-away sums it shows too.
• Make up real problems. Pocket money, pages of a book or sweets shared between friends all make good bar models.

Bar model questions to try

Draw a bar model for each one first.

1. 18 children are in the hall and 26 are in the playground. How many altogether?
2. A ribbon is 60 cm long. 35 cm is cut off. How much is left?
3. Mia has 42 marbles and Jack has 27. How many more does Mia have?
4. 32 sweets are shared equally between 4 friends. How many does each get?
5. What is two-thirds of 18?
6. Two numbers add up to 30, and their difference is 4. What are they?

Answers

1. 44 (the whole is missing, so add)
2. 25 cm (a part is missing, so subtract)
3. 15 more (the gap between the bars)
4. 8 each (4 equal parts of 32)
5. 12 (one-third is 6, and 2 thirds are 12)
6. 13 and 17 (30 take away 4 is 26, half of 26 is 13, and 13 + 4 = 17)

Bar models help with written methods too, such as the bus stop method for division word problems. For everything else children learn in these years, see the KS1 maths and KS2 maths guides.