Bar Modelling - Maths No Problem by Dr Yeap Ban Har

Bar Modelling

The bar model method is a strategy used by children to visualise mathematical concepts and solve problems. The method is a way to represent a situation in a word problem, usually using rectangles.

Key points – points – Bar model method uses the concrete pictorial and abstract (CPA) sequence when teaching certain maths topics. The process starts by using real world, tangible representations, before moving onto showing the problem using a pictorial diagrams diagrams before then introducing introducing the abstract algorithms algorithms and notations. The example below shows the concrete modelling with real objects, handling real objects and moving to the pictorial.

Taken from: https://mathsnoproblem.com/ (By Dr. Yeap Ban Har) 1


The bar model method is pictorial and it develops from children handling actual objects, to drawing pictures and then drawing boxes to represent objects. Eventually, they will no longer need to draw all the boxes, which represents individual units, instead they just draw one long bar and label it with a number. At this stage the bars do need to be somewhat proportional, so in the example above the purple bar representing 12 cookies is longer than the orange bar representing 8 cookies.

The particular power of the bar modelling pictorial approach is that it is applicable across a large number of topics. Once students have the basics of the approach secured, they can easily extend it across many topics.

A good understanding of the four operations is needed to use bar models. Children need to have strategies to add, subtract multiply and divide for them to use bar models. Bar models don’t give you an answer – it gives you an understanding of what to do do get to the answer. The what to do part is where children would normally use the four operations.

BACKGROUND TO CONCRETE, PICTORIAL AND ABSTRACT (CPA) Children and adults can find maths difficult because it is abstract. The CPA approach helps children learn new ideas and build on their existing knowledge by introducing abstract concepts in a more familiar and tangible way. The approach is so established in Singapore maths teaching, that the Ministry of Education will not approve any teaching materials which do not use the CPA approach.


CONCRETE Concrete is the “doing” stage, using concrete objects to model problems. Instead of t he traditional method of maths teaching, where a teacher demonstrates how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical objects themselves. Every new abstract concept is learned first with a “concrete” or physical experience .

For example, if a problem is about adding up four baskets of fruit , the children might first handle actual fruit before progressing to handling counters or cubes which are used to represent the fruit.


PICTORIAL Pictorial is the “seeing” stage, using representations of the objects to model problems. This stage encourages children to make a mental connection between the physical object and abstract levels of understanding by drawing or looking at pictures, circles, diagrams or models which represent the objects in the problem.

Building or drawing a model makes it easier for children to grasp concepts they traditionally find more difficult, such as fractions, as it helps them visualise the problem and make it more accessible.


Number bonds

Number bonds are a way of showing how numbers can be combined or split up. They are used to reflect the ‘part -partwhole’ relationship of numbers. Number bonds teach children how numbers join together and how they can be broken down into their component parts. From year 1, children use number bonds to build up their number sense before learning about addition and subtraction.


WHERE DO NUMBER BONDS COME FROM? Number bonds are a core element of teaching maths for mastery using Singapore Maths and have been part of Singapore’s primary curriculum since the early 1970s. However, the phrase ‘number bonds’ has been around since the 1920s and became widespread in the UK in the late 1990s as a way of developing mental arithmetic strategies.

HOW DO NUMBER BONDS LOOK? Number bonds are often represented by circles that are connected by lines as illustrated below. The ‘whole’ is written in the first circle and its ‘parts’ are written in the adjoining circles.



TAKING NUMBER BONDS FURTHER Number bonds are also used to develop strategies such as ‘making ten’ using ten frames, multilink or unifix cubes.

By mastering number bonds early on, pupils build the foundations they need for subsequent learning and are better equipped to develop mental strategies and mathematical fluency. By building a strong number sense, pupils can also decide what action to take when trying to solve problems in their head.

This example shows how a pupil would develop their number sense, or mathematical fluency, by using number bonds to perform a mental calculation.



Teaching Fractions Using the Singapore Maths Method Many children (and adults for that matter) find fractions difficult to understand. This is often because fraction notation (writing a fraction as a number, e.g. ½ ) is very confusing. Children therefore struggle to relate the symbol to the ‘thing’ and end up guessing. In Singapore, the understanding of fractions is rooted in the Concrete, Pictorial, Abstract (CPA) model, where children use paper squares and strips to learn the link between the concrete and the abstract. At the heart of understanding fractions is the ability to understand that we’re giving an equal part a name. It is simply a naming activity! Taken from the Maths — No Problem! Primary Maths Series Textbooks, here are 4 easy steps that will develop and ensure children’s understanding of fractions.

FINDING EQUAL PARTS Children need to understand what a fraction is. When we divide a whole into equal parts we create fractions. A fraction is just an equal part.

In this lesson, children are encouraged to create 4 equal parts.


Reference: Maths — No Problem! Primary Maths Series, Textbook 2B, pages 102 – 103

Tip: Ask the children to also show you 4 unequal parts (if haven’t done it already!).

2. NAMING EQUAL PARTS Once the children can make/identify equal parts (fractions), they need to give them a name.


In this lesson, children cut a pizza into a different number of equal parts. To show that the equal parts are different, the children give them different names. This is the denominator (name).

Reference: Maths — No Problem! Primary Maths Series, Textbook 2B, page 116

Tip: Call the denominator the ‘namer’ to star t with. Write out the denominator as a word as well as a number.


3. OPERATIONS INVOLVING EQUAL PARTS If children can name a fraction, they are ready to do calculations using like fractions (they have the same name).

In this lesson, children can use strips of paper to model the problem and see how it links to the written and symbolic notation.


Reference: Maths — No Problem! Primary Maths Series, Textbook 5A, page 275 – 276

Tip: Prepare 3 apples (or any other item). Show the items to the children and ask them what is 2 apples + 1 apple? Now show them 3 equal pieces of paper. Write a ‘quarter’ on each. Then ask the children what is 2 quarters + 1 quarter? Is this any harder than adding apples?

4. WHAT IF THE PARTS AREN’T EQUAL? Can we add 3 apples and 2 oranges? Is it 5 apples? Is it 5 oranges? It is neither because we cannot add things with different names. We have to give them the same name, and in this case we could rename them as ‘fruit’. They now all have the same name and so we can do the calculation (5 pieces of fruits). The same is true for fractions. We can’t add 2 quarters and 1 eighth because they have different names, however, if we can give them the same name (equivalent) it is possible.

In this lesson, children cut up a quarter to show more equal parts and name the parts.



Reference: Maths — No Problem! Primary Maths Series, Textbook 3B, pages 128 – 129

Tip: Ask the children to shade in their piece.



Bar Modelling - Maths No Problem by Dr Yeap Ban Har

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