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The Bar Model - Addition and Subtraction

Created on 24 April 2014 by ncetm_administrator
Updated on 04 August 2017 by ncetm_administrator

Addition and Subtraction

The bar model supports understanding of the relationship between addition and subtraction in that both can be seen within the one representation and viewed as different ways of looking at the same relationships.

bar model

This diagram encapsulates all of the following relationships;

a = b + c ; a = c + b ; a – b = c ; a – c = b

To prepare young children for the bar model it is a good idea to encourage them to line up objects in a linear arrangement when representing addition and subtraction problems.

Such arrangements will also help children to organise their counting. The physical objects can then be replaced, in time, with linking cubes and with a bar drawn next to it. The question can then be asked “what’s the same, what’s different?” to support the children in their reasoning and in making sense of the bar as an abstract representation of the physical objects. It is useful for children to work in pairs with one manipulating the cubes, while the other records by drawing the bars and then writing the number sentence underneath. The children can then swap roles.

Sam had 10 red marbles and 12 blue marbles. How many marbles did he have altogether?

Using the bar model to add 10 + 12

In problems involving addition and subtraction there are three possible unknowns as illustrated below and given the value of two of them the third can be found.

Diagram showing that the whole is made up of parts of the bar

The examples below illustrate a variety of ways that the bar might be used for addition and subtraction problems. A question mark is used to indicate the part that is unknown.

- two quantities combined

addition aggregation

I have 6 red pencils and 4 yellow pencils. How many pencils do I have?

(I combine two quantities to form the whole)

- a quantity is increased

addition augmentation

I have 6 red pencils and I buy 4 yellow pencils. How many pencils do I have?

(The bar I started with increases in length)

- Take Away

subtraction - take away

I had 10 pencils and I gave 6 away, how many do I have now?

(This time we know the whole but only one of the parts, so the whole is partitioned and one of the parts removed to identify the missing part)

- Comparison or Difference

subtraction - comparison or difference

Tom has 10 pencils and Sam has 6 pencils. How many more does Tom have?

(The bar is particularly valuable for seeing the difference between the two quantities)


The model can be rearranged to demonstrate equivalence in a traditional layout


Pupils need to develop fluency in using this structure to represent addition and subtraction problems in a variety of contexts using the bar model. The model will help children to see that different problems share the same mathematical structure and can be visualised in the same way. Asking children to write their own problems, using the bar as the structure will help to consolidate this understanding.


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09 October 2016 13:02
This is a very useful reminder of exploring different ways of solving problems building on pictorial representation.
05 October 2016 14:24
this looks like cuisenaire
By Walkthedog
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24 August 2016 23:49
This really is not new - I've been using this model for many years - especially when dealing with unequal sharing type problems.
By bevdex
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12 April 2016 21:24
I love Maths and I love the Singapore Bar!!!
By bradbeer
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01 January 2016 15:29
^ n.b. these hint at where the bar method (& much other maths!) may really have originated.

The use of more than one ruler is rather nice; especially so when using a millimetre scale and flipping the second ruler upside down to subtract: really emphasises how "negative" and a 180 degree turn can, usefully, be described as "the same 'thing'"...
By BW_2012
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17 August 2015 17:01
I used this for solving equations with x on both sides. My year 8 students were able to do this easily in comparison to my year 10 students whom I did not use the bar modelling.
By kemmy
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16 June 2015 16:29
I've used the Singapore bar method with a range of primary year groups (and teachers) and it's a brilliant tool for deconstructing mathematical ideas. I had a year 3 class able to solve quite advanced fraction problems in a very short space of time.
By nic9265
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04 November 2014 16:21
By smbaker1
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