Bar Models in Maths
Bar modelling is an essential maths mastery strategy. A Singapore-style of maths model, bar modelling allows pupils to draw and visualize mathematical concepts to solve problems.
What is a bar model?
A bar model is a problem solving tool that uses boxes to represent numbers. It draws on the Concrete, Pictorial, Abstract (CPA) approach and is used extensively in the Singapore maths curriculum. Bar models are highly versatile and can be used across a wide range of concepts and topics. They often allow pupils to understand on a conceptual level what occurs when using complex formulas (for example, algebra).
How to use bar models
Bar modelling and the CPA approach
The bar model method draws on the Concrete, Pictorial, Abstract (CPA) approach — an essential maths mastery concept. The process begins with pupils exploring problems via concrete objects. Pupils then progress to drawing pictorial diagrams, and then to abstract algorithms and notations (such as the +, -, x and / symbols).
The example below explains how bar modelling moves from concrete maths models to pictorial representations.
As shown, the bar method is primarily pictorial. Pupils will naturally develop from handling concrete objects, to drawing pictorial representations, to creating abstract rectangles to illustrate a problem. With time and practice, pupils will no longer need to draw individual boxes/units. Instead, they will label one long rectangle/bar with a number. At this stage, the bars will be somewhat proportional. So, in the example above, the purple bar representing 12 cookies is longer than the orange bar representing 8 cookies.
What are the benefits of bar modelling?
On one hand, the Singapore maths model method — bar modelling — provides pupils with a powerful tool for solving word problems. However, the lasting power of bar modelling is that once pupils master the approach, they can easily use bar models year after year across many maths topics. For example, bar modelling is an excellent technique (but not the only one!) for tackling ratio problems, volume problems, fractions, and more.
Importantly, bar modelling leads students down the path towards mathematical fluency and number sense. Maths models using concrete or pictorial rectangles allow pupils to understand complex formulas (for example, algebra) on an intuitive, conceptual level. Instead of simply following the steps of any given formula, students will possess a strong understanding of what is actually happening when applying or working with formulas.
The result? A stable, transferable, and solid mathematical framework for approaching abstract concepts. Combines with other essential maths mastery strategies and concepts like number bonds, bar modelling sets students up for long-term maths success.
How can you use bar models in maths?
Bar models represent quantities visually rather than through operation-specific tricks or rules, so the same core method carries pupils through every stage of primary maths — from their first addition problems in Key Stage 1 through to fractions and ratio in Key Stage 2. Here's how the model adapts to each area.
Addition and subtraction
In the early years, bar models most often take a part-whole form: two or more known parts sit inside a single bar, and pupils either combine them to find the whole (addition) or remove one part to find what's left (subtraction). The same bar structure works in both directions. Pupils quickly see addition and subtraction as inverse operations rather than two unrelated topics, rather than memorising separate methods for each. A comparison bar model — two bars of different lengths placed side by side — is then used for "how many more/fewer" problems, where the gap between the bars represents the difference to be found.
Division
For division, a bar model usually shows a known whole split into a number of equal segments. Pupils can use the same drawing to answer two different questions: how many items are in each group (sharing), or how many groups can be made altogether (grouping) — the two structures of division that pupils often mix up when they only see the calculation. The bar is divided into equal parts, so a bar model for division also makes remainders visible as whatever's left outside a full segment, rather than treating a remainder as a rule to remember.
Multiplication
A bar model for multiplication typically shows one segment repeated an equal number of times, making the connection between multiplication and repeated addition explicit rather than abstract. As pupils move through Key Stage 2, the same structure extends naturally to scaling problems (for example, "three times as many"), giving them a consistent visual method for multiplication problems that grow more complex without needing a new strategy for each one.
Bar models for fractions
A fraction bar model divides a single bar into equal parts, with each part representing a unit fraction of the whole. Shading a given number of parts shows a specific fraction at a glance, which makes comparing fractions with the same or different denominators far more intuitive than comparing numbers alone. The same model supports adding and subtracting fractions with like denominators, and finding a fraction of an amount — for example, dividing a bar into quarters to work out three-quarters of 20. Bar modelling fractions this way keeps the part-whole relationship visible throughout, rather than something pupils have to hold in their heads.
Bar modelling in action
What's the difference between a bar model and a number line?
A bar model shows the relationship between quantities using bars, helping pupils visualise part-whole and comparison problems. A number line shows the position and order of numbers and is often used for counting, measuring intervals and understanding number sequences.
What are the different types of bar models?
There are two main types of bar model: the part-whole model, used to show how quantities combine or split apart, and the comparison model, used to show the difference between two quantities. Each type helps pupils solve different kinds of mathematical problems.
Where does the bar model come from?
The bar model method was developed in Singapore in the 1980s by Dr Kho Tek Hong's Primary Mathematics Project team, as part of a national effort to improve maths achievement, and was officially introduced into the Singapore curriculum in 1983. It was built on Jerome Bruner's Concrete-Pictorial-Abstract framework — the same broader theoretical basis, alongside the work of Zoltan Dienes and Alan J Bishop, that underpins the wider approach to bar modelling used today.
How do you draw a bar model?
To draw a bar model, identify the known and unknown values in the problem, represent each quantity as a bar and label the parts clearly. The size and structure of the bars should reflect the mathematical relationship being shown.
How does bar modelling progress through school?
Bar models can be used throughout primary education. It typically begins with simple addition and subtraction problems before progressing to multiplication, division, fractions, ratios and even algebra. As pupils move through school, bar models are used to represent increasingly complex mathematical relationships.
Can bar models be used for algebra?
Yes. Because a bar model represents unknown quantities visually rather than through symbols, pupils can access algebraic reasoning long before they learn formal notation. Maths — No Problem! has shown how a GCSE Higher-level question — solving 5x − 26 = x + 26 — can be represented and solved by Year 5 and 6 pupils using a bar model, without needing algebraic manipulation at all.
