How to Help With Maths Word Problems: The Bar Model

Plenty of children who are quick with their times tables freeze when the same sum arrives as a story. That isn't a gap in their arithmetic. A word problem asks for something different: turning a situation into a calculation. The good news is that there is a simple, reliable way to do it, and you can practise it together with our money word problems.

Why do children find word problems hard?

Usually the numbers are easy. The trouble is everything around them. A child has to read the story, keep it in mind, decide which numbers matter and choose an operation, all before any calculating starts. Under that load many children do what feels quickest: take the two numbers they can see and add them.

A typical mistake looks like this: "Mia has 23 stickers. That is 8 more than Leo. How many stickers does Leo have?" The word "more" pulls children towards 23 + 8 = 31. The answer is 23 − 8 = 15, because Leo has fewer. Nothing about the arithmetic was hard; the story was.

What is a bar model?

A bar model is a quick drawing of a problem using rectangles. Each amount in the story becomes a bar, the bars are drawn roughly to size, and the thing you are looking for is marked with a question mark. The method was developed in Singapore in the 1980s and is now used in many schools in England, especially those teaching maths mastery. In the US the same drawing is called a tape diagram.

The bar doesn't do the sum. It shows the shape of the problem, and once the shape is clear the calculation is usually obvious. It is the same idea as the dots and arrays in our multiplication visualiser: see it first, then work it out.

How does a part-whole bar work?

Use a part-whole bar when some pieces make up a total. Draw one long bar for the whole and split it into the parts.

Example: "Sam has £10. He buys a book for £3.70. How much does he have left?"

Whole: £10
Book: £3.70	Left: ?

The whole is known and one part is known, so the missing part is the whole take away the part: £10 − £3.70 = £6.30. If the parts were known and the whole was missing, the same drawing would say "add". For the subtraction itself, see column subtraction with borrowing.

How do you draw a comparison bar?

Use two bars, one under the other, when a problem compares two amounts ("more than", "fewer than", "how much longer"). Line them up on the left so the difference sticks out on the right.

Example: the sticker problem above. Mia's bar is 23. Leo's bar is shorter, and the gap between them is 8. Leo's bar is Mia's bar without the gap: 23 − 8 = 15. Once it is drawn, the word "more" can't trick anyone.

Comparison bars are what our measuring word problems show for questions like "How much longer is one ribbon than the other?", drawn on a ruler so the difference is a real length.

How do bar models help with multiplication and division?

A bar cut into equal parts covers both:

• Multiplication: "4 tickets cost £2.50 each. What is the total?" Four equal parts of £2.50: 4 × £2.50 = £10.
• Division by sharing: "£24 is shared between 3 friends." One bar of £24 in 3 equal parts: £24 ÷ 3 = £8 each.
• Division by grouping: "How many 250 mL glasses can you fill from a 1 litre jug?" How many 250s fit in 1,000? 4.

When the division doesn't come out exactly, the bar makes the remainder visible, which helps with methods like the bus stop method.

Can bar models be used for fractions and percentages?

Yes, and this is where they really pay off in upper primary. "3/5 of a class of 30 walk to school" becomes a bar of 30 in 5 equal parts: one part is 6, so three parts are 18. Percentages work the same way with 10 parts of 10%. Our article on finding a percentage of a number builds on exactly this, and the percentage word problems use hundred squares and bar charts for test scores and price changes.

What steps should my child follow?

1. Read it twice. Once for the story, once for the numbers.
2. Say what is known and what is asked. In their own words, out loud.
3. Draw it. A part-whole bar, a comparison bar or equal parts. A quick sketch is enough.
4. Decide the calculation from the drawing, not from keywords.
5. Calculate, then check: does the answer make sense in the story? Leo can't have more stickers than Mia.

Time problems follow the same steps with a number line instead of a bar: jump from the start time to the end time. Our time word problems draw those jumps, and the 24-hour clock guide helps with timetables.

How can I help at home without doing it for them?

• Ask, don't tell. "What do we know? What are we finding? Can you draw it?" are better than any explanation.
• Let them hold the pencil. If you draw the bar, you have solved the hard part for them.
• Use real life. Doubling a recipe, working out change, or how far is left on a car journey are all word problems. Our cooking and travel word problems are built on exactly these.
• Keep the numbers easy at first. The skill being learnt is the drawing, so use numbers your child can calculate in their head.
• Print some for later. The word problem worksheets come with a picture for each question and an answer key with the pictures filled in.