# 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.

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?**

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.

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.

## Addition

Aggregation

- two quantities combined

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

(I combine two quantities to form the whole)

## Addition

Augmentation

- a quantity is increased

**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)

## 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)

## 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)

## Equivalence

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.

This is a very useful reminder of exploring different ways of solving problems building on pictorial representation.