Key takeaways
- A bar model is a rectangle (or set of rectangles) drawn to represent the quantities in a problem — it turns a wordy question into a picture students can reason about.
- There are two core types: the part-whole model (parts combine into a whole) and the comparison model (two or more bars set side by side).
- The same two models stretch across all four operations, plus fractions, ratio and percentages — students learn one tool, not fifteen tricks.
- It's the 'pictorial' step in the Concrete–Pictorial–Abstract sequence, bridging hands-on equipment and formal number sentences.
- Bars don't need to be perfectly to scale — but the bigger quantity must look bigger, or the picture stops helping.
Ask a class to solve "Mia has 34 stickers. Tom has 28. How many altogether?" and most will manage it. Ask them "Mia has 34 stickers. She has 6 more than Tom. How many does Tom have?" and hands go down — the numbers barely changed, but the structure did. The bar model exists for exactly this second kind of problem: it makes the relationship between the quantities visible, so students can see whether to add or subtract instead of guessing from keywords.
A bar model is simply a rectangle drawn to stand for a quantity. Longer bar, bigger number. Once students can draw the story, the calculation is usually the easy part.
Where the bar model comes from
The bar model is often called the Singapore bar method because it grew out of the maths curriculum Singapore's Ministry of Education developed in the 1980s, when the country set out to lift stubbornly low maths results. It became a signature feature of Singapore maths and later spread through mastery programmes in the UK (White Rose, NCETM's Maths Hubs) and into Australian classrooms through resources like reSolve and the national mathematics hub.
It isn't a gimmick bolted onto problem-solving. It's the pictorial stage of the Concrete–Pictorial–Abstract (CPA) approach — an idea drawn from psychologist Jerome Bruner's work on how understanding is built. Students first handle real objects (counters, cubes), then represent them as drawn bars, then move to abstract symbols like 34 + 28 = 62. The bar is the bridge in the middle, and it's the stage most curricula skip.
The two models that do all the work
Part-whole model
One long bar is the whole; the sections inside it are the parts. Use it whenever a total is split into pieces — combining amounts, or finding a missing part when you know the whole. Missing whole → add the parts. Missing part → subtract from the whole.
Comparison model
Two or more bars stacked one above the other, left ends lined up. Use it whenever a problem compares quantities — 'more than', 'fewer than', 'twice as many', 'the difference between'. The gap between the bar ends is the difference.
Almost every word problem a primary student meets is one of these two shapes. Teaching students to ask "is this a part-whole problem or a comparison problem?" is more powerful than teaching them to hunt for keywords like altogether or left, which routinely mislead ("He gave away 4 and now has more" trips up keyword-spotters every time).
Worked examples across the four operations
- 1
Addition (part-whole)
"Mia has 34 stickers, Tom has 28. How many altogether?" Draw one bar split into two parts, 34 and 28. The whole is unknown, so add the parts: 34 + 28 = 62.
- 2
Subtraction (part-whole)
"There are 62 stickers. 34 are Mia's. How many are Tom's?" Draw the whole bar (62) with one known part (34). The missing part is 62 − 34 = 28.
- 3
Comparison (finding a difference)
"Mia has 34, Tom has 28. How many more does Mia have?" Draw two bars, Mia's longer. The extra sticking out is the difference: 34 − 28 = 6.
- 4
Multiplication (equal parts)
"A packet holds 6 pencils. How many in 4 packets?" Draw one bar made of 4 equal units, each labelled 6. The whole is 4 × 6 = 24.
- 5
Division (sharing)
"24 pencils shared into 4 pots equally. How many per pot?" Draw a bar of 24 split into 4 equal units. Each unit is 24 ÷ 4 = 6.
Bar models for fractions, ratio and percentages
The reason the bar model earns its place all the way to secondary school is that it doesn't stop at whole numbers.
- Fractions: For "3/4 of 20", draw a bar of 20 split into 4 equal parts (each worth 5) and shade 3 of them: 3 × 5 = 15.
- Ratio: For "Share £40 between Ana and Ben in the ratio 3:2", draw 3 units above 2 units — 5 units in total. One unit is 40 ÷ 5 = 8, so Ana gets 3 × 8 = £24 and Ben gets 2 × 8 = £16.
- Percentages: A bar split into 10 equal parts makes 10%, 30% or 70% of a number something students can point to rather than compute blindly.
Same drawing, harder numbers. Students who built the habit in Year 2 are still using it on ratio and algebra problems years later.
When Singapore's Ministry of Education built the bar model into its national maths curriculum, the reform that later spread worldwide as 'Singapore maths'.
Source: Singapore Ministry of Education
How to teach it, step by step
- 1
Start concrete, not with a drawing
Lay out real cubes or counters in a line first. The bar is a picture of that line — introduce the rectangle only once students have physically made the quantity.
- 2
Model the drawing out loud
Think aloud as you draw: 'This whole bar is all the stickers. I'll cut it here for Mia's, and this part is Tom's.' Narrate why each bar and each cut appears.
- 3
Label everything
Every bar gets a number or a question mark. The question mark shows exactly what the problem is asking — students calculate towards it.
- 4
Keep bars roughly proportional
They needn't be measured to the millimetre, but 6 must not be drawn longer than 28. If the picture lies, it stops helping.
- 5
Solve the picture, then write the number sentence
Only after the model is drawn and labelled do students commit to 34 + 28 = 62. The abstract step comes last, never first.
- 6
Fade the support gradually
Once a student can reliably choose the right model, let them sketch quick, rough bars — the goal is a thinking tool, not a neat diagram for its own sake.
Common mistakes to head off
Reaching for the calculation first
Students who write the number sentence before drawing miss the whole point. Insist the model comes first — the picture is where the thinking happens.
Bars wildly out of scale
A tiny 'whole' and a huge 'part' means the student hasn't understood the relationship. Non-examples on the board fix this fast.
Confusing the two models
Comparison problems drawn as a single part-whole bar (or vice versa) usually signal the student hasn't yet spotted whether quantities are being combined or compared.
Never fading the scaffold
If Year 6 students still draw painstaking rulered bars for one-step sums, the tool has become a chore. Speed up and rough up the drawings as fluency grows.
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Make matching maths worksheetsFrequently asked questions
A bar model is a rectangle, or set of rectangles, drawn to represent the quantities in a problem. A longer bar means a bigger number. It turns a word problem into a picture so students can see the relationship between the numbers and decide which operation to use.
The part-whole model, where a whole bar is split into parts, and the comparison model, where two or more bars are drawn side by side. Part-whole suits problems that combine or split a total; comparison suits problems using 'more than', 'fewer than' or 'the difference'.
Simple part-whole bars can start in the first years of school (around Foundation to Year 1) once children can combine and split small quantities with concrete objects. The same models then extend through primary and into secondary for fractions, ratio and algebra.
The bar model is one of the best-known tools within Singapore maths, but it isn't the whole approach. It's the 'pictorial' stage of the Concrete–Pictorial–Abstract sequence that Singapore maths is built on, alongside a strong focus on number sense and mastery.
Not exactly, but they should be roughly proportional. A quantity of 6 should never be drawn longer than a quantity of 28. As long as bigger numbers look bigger, the picture supports the reasoning; if it doesn't, students should redraw it.