Bus stop division, bar models and number lines, explained

This article describes the NSW Mathematics K–10 Syllabus (2022). Other states and territories have their own curriculum documents, which we haven’t checked for these methods, so ask your child’s school how it names and teaches them.

What is bus stop division?

It’s a written way to divide by a one-digit number. The number being divided sits under a bracket shaped like a bus shelter, and you divide digit by digit from the left, carrying any remainder across to the next digit. If you were taught short division at school, the layout will look familiar.

The NSW syllabus doesn’t use the name “bus stop”, or “short division”. What it does say, in Stage 3 (Years 5 and 6), is that children should select and apply strategies to divide a number with 3 or more digits by a one-digit divisor, estimate the result, use and interpret remainders, and record their strategy. That sits under outcome MA3-MR-01, which is about selecting and applying appropriate strategies to solve multiplication and division problems.

For multiplication, the same page names a formal algorithm, which is the teacher’s term for a formal written method. For division it names none. It lists informal written strategies, such as the area model. Some schools teach bus stop division as one way to record a division.

What is a bar model?

It’s a rectangle drawn to stand for a quantity and split into parts, used to show how the numbers in a word problem relate: two parts making a whole, a comparison between two amounts, or equal shares.

The Stage 2 syllabus (Years 3 and 4) names it. In its addition and subtraction content (Additive relations), the bar model is listed as one way for children to represent solutions to word problems. In Stage 3, finding a fraction of an amount, such as a quarter or a fifth of a whole number, the syllabus uses the term “tape diagram”. It doesn’t say whether that is the same thing as a bar model, so check which word your child’s teacher uses.

What is a number line?

The syllabus glossary defines it: “A number line is used to represent numbers according to their distance from a zero point.” It shows how big a number is, and where it sits among the others.

Number lines appear in the syllabus from Stage 1:

• Stage 1 (Years 1 and 2): ordering numbers and finding the nearest ten to a number.
• Stage 2 (Years 3 and 4): fractions as lengths. The outcome itself, MA2-PF-01, asks children to represent and compare halves, quarters, thirds and fifths as lengths on a number line.
• Stage 3 (Years 5 and 6): negative numbers (integers), and fractions.
• Stage 4 (Years 7 and 8): positive and negative fractions, decimals and mixed numbers, placed together to compare them.

Why does school teach these as well as the old methods?

The reasoning in the guidance is that pictures make the maths visible: a bar shows the parts and the whole, and a number line shows how big a number is. The written method then records the same idea in a shorter form.¹ That is the guidance’s reasoning, not a tested comparison of picture-first against method-first teaching.

There’s a second reason. Children use a method better when they understand how it works, and understanding helps them rebuild a step they’ve forgotten. England’s Education Endowment Foundation recommends teaching children to understand procedures, not only recall them.¹ Understanding and skill with methods also build on each other in both directions, rather than one coming strictly first.²

What does the research say about drawing a problem?

The strongest finding is about diagrams in general, not the bar model in particular. A US government expert panel rated “use concrete and semi-concrete representations” as strongly supported for primary children who need extra help with maths. It names strip diagrams, simple drawings, arrays and number lines as examples.³ The panel’s caveat is that its evidence comes mostly from small-group programs for children who need extra help, and practices were usually tested in bundles, not one at a time. The Education Endowment Foundation also reports strong evidence for using diagrams to solve problems.¹

Teaching children to spot the type of problem and draw a diagram of its structure beat general problem-solving strategies in a randomised trial with 88 eight and nine year olds.⁴ A review of 18 group studies found this kind of teaching evidence-based for children with learning difficulties.⁵ These diagrams are close to bar models but not identical, so the safe summary is that drawing a diagram of a problem’s structure helps.

Studies of the Singapore bar model itself are few and mostly descriptive. A descriptive study of 151 Singapore Year 5 children found they used it to solve problems that would otherwise need algebra, and the authors say this held for higher-ability children.⁶ That is interesting, but it is not proof that the bar model works.

What does the research say about number lines?

Number lines have the best evidence of the three. The same US panel rated “use the number line” as strongly supported,³ and England’s Education Endowment Foundation found them “a particularly effective representation” for children aged about 7 to 14.¹

• A link, not a cause. How well a child places numbers on a number line goes with how good they are at maths generally. A meta-analysis of 263 results from more than 10,000 children, with average ages from 4 to 14 across the samples, found a correlation of about 0.44, higher for fractions.⁷ That shows a link. It doesn’t show that number line practice causes maths gains.
• A trial. In a randomised trial with 259 nine and ten year olds identified as needing extra help with fractions, a tutoring program built around placing fractions on a 0 to 1 number line improved fraction understanding and calculation more than usual teaching.⁸ That is one strong trial from one research group.
• The empty number line. This is a line with no marks, where children jump in chunks, like plus 10 then plus 3. It came from Dutch classrooms, and a study of 275 Dutch seven and eight year olds found it a “very powerful model” for adding and subtracting up to 100.⁹ Both groups in that study used it, so it doesn’t show the empty number line beats other methods.

What does the research say about bus stop division?

We haven’t found a study of the bus stop layout itself, so we can’t say it’s research-based, or that the older way doesn’t work. The evidence that bears on it is general. The Education Endowment Foundation reports some evidence that, for multiplication and division, spending time on children’s own informal reasoning before formal written methods helps, and then going back to show where the informal and formal methods agree.¹ Schools want children to know why a method works, so they can check an answer and fix a slip.

Should my child keep using objects and pictures?

Not forever. The Education Endowment Foundation says manipulatives, the hands-on objects, should act as a scaffold that can be removed once the child can work without them, and that the decision should follow the child’s understanding, not their age. It says these points are likely to apply to other representations too.¹

Drawing a diagram of a problem is a different thing. It has strong evidence as a problem-solving strategy, as above, and the same guidance says that when moving away from objects, children may find it helpful to draw diagrams.

One thing to try at home is what our homework article suggests: ask your child to show you the class method and teach it to you. You could also ask them why a step works. A child who can explain why a method works is doing what the guidance asks for.

How Workings fits

Workings doesn’t replace what your child’s school teaches. It’s practice and diagnosis, and it sits alongside the teaching. The same maths is asked in ten question types, from hands-on to explain-it, so the same idea is met as a picture, a number sentence, a word problem and a riddle. Every model is drawn precisely from structured components, never by an image model.

Fractions are shown on a number line rather than as slices of pizza, because the groundwork for fractions in the syllabus is partitioning a length. And Workings never gives the answer in any mode. A child who is stuck gets a smaller question, and takes the last step themselves. Read how the curriculum is built.