What is the bar model? Singapore maths for parents

The bar model is the drawing at the heart of Singapore maths. It is not a new kind of sum, it is a way of picturing a word problem as rectangles so a child can see what is known, what is missing and which operation to use. Instead of guessing whether to add or subtract, they draw the story, and the picture makes the maths obvious. Once a child can model a problem, the arithmetic is the easy part. Here is how the bar model works, with worked examples you can copy at the kitchen table.

What a bar model is

A bar model represents each quantity in a problem as a rectangle, drawn roughly to scale so bigger numbers get longer bars. The bars sit next to or above each other so the relationship between the numbers is visible: which is the whole, which are the parts, and which is bigger.

There are only two shapes to learn. The part-whole model, where smaller bars join to make one whole, and the comparison model, where two bars sit one above the other so you can see the difference. Almost every primary word problem is one of these two.

The part-whole model: addition

Start with the most common shape. Priya has 8 red apples and 5 green apples. How many apples does she have altogether? Draw one long bar split into two parts, one labelled 8 and one labelled 5. The whole bar underneath is what you are looking for.

The parts are known and the whole is missing, so you add: 8 plus 5 is 13. Priya has 13 apples. The picture shows at a glance that the two parts combine into one whole, which is why the operation is addition.

Two parts known, whole missing: add the parts.
8 plus 5 is 13, so the whole bar is 13.

The part-whole model: subtraction

Now flip it. Priya has 13 apples. 8 are red and the rest are green. How many are green? This time the whole (13) and one part (8) are known, and the other part is missing. Draw the whole bar of 13 with one section marked 8 and the rest unknown.

When the whole and one part are known, you subtract: 13 minus 8 is 5. There are 5 green apples. Same picture, different missing piece, and the model tells you subtraction is the move.

Whole and one part known, other part missing: subtract.
13 minus 8 is 5, so the missing part is 5.

The comparison model: how many more?

The comparison model is where the bar model really earns its keep, because 'how many more' problems confuse children who only know 'add means more'. Tom has 12 stickers and Sara has 7. How many more does Tom have? Draw Tom's bar of 12 and, directly below it, Sara's bar of 7, lined up at the left.

The extra length on Tom's bar is the difference, and you find it by subtracting: 12 minus 7 is 5. Tom has 5 more stickers. Seeing the two bars side by side makes it clear you are comparing, not combining.

Line the two bars up at the same starting edge.
The overhang is the difference: 12 minus 7 is 5.

The comparison model: working the other way

Comparison bars also handle 'more than' problems where the total is missing. Sara has 7 stickers. Tom has 5 more than Sara. How many does Tom have? Draw Sara's bar of 7, then draw Tom's bar as the same 7 plus an extra block of 5.

Tom's bar is 7 plus 5, which is 12. Tom has 12 stickers. The same picture as before, but now the difference is known and one of the totals is what you are solving for.

Multiplication and division: equal bars

For times and share problems, draw equal-sized units. One box holds 6 pencils. How many pencils are in 4 boxes? Draw 4 equal bars, each labelled 6. Four units of 6 is 4 times 6, which is 24. There are 24 pencils.

Division is the same picture read backwards. 24 pencils are shared equally into 4 boxes. Draw one bar of 24 split into 4 equal parts, and each part is 24 divided by 4, which is 6. Each box holds 6 pencils.

4 equal units of 6: 4 times 6 is 24.
24 split into 4 equal parts: 24 divided by 4 is 6.

Two-step problems: where the model shines

The bar model really pays off when a problem has more than one step, because the drawing holds the first answer while you work on the second. A shirt costs 20 dollars. The trousers cost 15 dollars more than the shirt. How much do both cost together?

First find the trousers with a comparison bar: 20 plus 15 is 35 dollars. Then find the total with a part-whole bar: 20 plus 35 is 55 dollars. Both items cost 55 dollars. The two little pictures keep the steps in order so nothing gets lost.

Step one, trousers: 20 plus 15 is 35.
Step two, total: 20 plus 35 is 55.

Fractions of an amount

Older children can use bars for fractions too. What is three quarters of 20? Draw a bar of 20 split into 4 equal parts, so each part is 20 divided by 4, which is 5. Three of those parts is 3 times 5, which is 15. So three quarters of 20 is 15.

The bar turns an abstract fraction into a picture of equal shares, which is exactly the idea a fraction represents.

Tips for using bar models at home

Read the whole problem before drawing. Decide what is known and what is missing first.
Draw bars roughly to scale. A bigger number should look bigger, so an impossible answer looks wrong.
Label every bar with its number or a question mark. The question mark is the thing you solve for.
Do not rush to the sum. The point is the picture, the arithmetic follows once the model is right.
Use the same two shapes over and over. Consistency is what makes the method automatic.

Free word-problem practice

Bar models make word problems make sense, and a little regular practice cements the habit. SproutSheets makes printable word-problem and arithmetic worksheets with answer keys computed in code, so they are never wrong. Generate a set your child can model with bars, then check their reasoning against the key.