# Addition

PDF HANDOUT OF ADDITION POLICY

The aim is that children use mental methods when appropriate but, for calculations that they cannot do in their heads, they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for addition, which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient written method for addition of whole numbers.

**Counting in ones**

- Starting from 0 and then from any number
- Counting out loud and practicing 1:1 correspondence (knowing that each object is a separate unit)
- It is also important that each number represents a group of objects (e.g. 3 = 3 teddies)

**Practical Addition**** (first ‘count all’ and then ‘count on’)**

Count all – 4 + 5 is counted 1, 2, 3, 4 and then 5, 6, 7, 8, 9 (out loud). Children are therefore not able to count on from any number yet and need to begin at 0 rather than start at 4 and count on 5 steps.

Count on – 4 + 5 is counted 4 and then 5, 6, 7, 8, 9. Children are now able to contract these steps by knowing that they have 4 and starting their counting from here to add 5.

**Simple addition using picture jottings**

**Drawing a picture**

There were 4 yellow sharks and 1 blue. How many sharks were there altogether?

**Dots or tally marks**

3 kids were on a bus and then 4 more got on. How many were on the bus in total?

**Counting in ones along a number line/track**

5 children are at school. 4 children arrive late. How many children are at school now?

Children could use a pre-drawn number line and then begin to create their own.

**Practical Addition**

**Use of numicon**

To help children to learn number facts and visualize quantities and what digits represent.

To look for patterns and relationships in number.

Use of numicon number line to add amounts.

**Practical and informal partitioning**

Use of practical apparatus such as Numicon and multibase to partition and present place value of digits.

Place value cards can be used to additional support understanding of place value.

4 7 + 1 5 = 62

**Using Addition facts to 10 to bridge the ten during addition.**

Here Numicon is used to bridge the ten. E.g. in 56 + 8 the 8 is split into two 4’s in order to form a number bond of 6 + 4. This enables the next multiple of 10 (60) to be reached before adding on the remaining 4.

Bead strings can also be used to show this method.

24 + 10 = 24 + 6 + 4

**The empty number line**

The mental methods that lead to column addition generally involve partitioning. Children need to be able to partition numbers in ways other than into tens and ones to help them make multiples of ten by adding in steps.

The empty number line helps to record the steps on the way to calculating the total.

One step in their develop when using a number line is to first be able to jump on in tens from any number and then in ones.

**Partitioning by horizontal expansion method**

The next stage is to record mental methods using partitioning into tens and ones separately. Add the tens and then the ones to form partial sums and then add these partial sums.

Partitioning both numbers into tens and ones mirrors the column method where ones are placed under ones and tens under tens.

This method builds on mental methods as each part is calculated mentally and recorded. It also makes the value of digits clear to children.

Before calculation children should be able to make a sensible estimate (e.g. 487+546= is approximately 500+500=1000) so they can check the reasonableness of their answer.

Record steps in addition using partitioning:

47 + 76 47 + 76

47 + 70 = 117 or 40 + 70 = 110

117+ 6 = 123 7 + 6 = 13

110 + 13 = 123

Partitioned numbers are then written under one another, for example :

47 = 40 + 7

76 = 70 +6

110 +13 = 123

**Vertical expansion method**

Move on to a layout showing the addition of the tens to the tens and the ones to the ones separately. To find the partial sums initially the tens, not the ones, are added first, following mental methods. The total of the partial sums can be found by adding them together.

The addition of the tens in the calculation 47 + 76 is described in the words ‘forty plus seventy equals one hundred and ten’, stressing the link to the related fact ‘four plus seven equals eleven’. As children gain confidence, ask them to start by adding the ones digits first every time.

The expanded method leads children to the more compact method so that they understand its structure and efficiency. The amount of time that should be spent teaching and practicing the expanded method will depend on how secure the children are in their recall of number facts and in their understanding of place value.

Ensure that digits are kept in the correct columns throughout and columns may be labeled with H, T and U to show place value.

Write the numbers in columns.

Adding the tens first: Adding the ones first:

4 7 4 7

7 6 7 6

1 1 0 (40+70) 1 3 (7 + 6)

1 3 (7+6) 1 1 0 (40 + 70)

1 2 3 1 2 3

Discuss how adding the ones first gives the same answer as adding the tens first. Refine over time to adding the ones digits first consistently. This will be important for the 'carrying over' in the compact method later. It is important that children practice this method with apparatus next to help them understand the 'carrying over' needed in the compact coloumn method. Please see te next stages below.

**Vertical expansion method**

Vertical expansion using multibase and place value counters.

Use of apparatus conceptually help children how that many ones can produce a ten. The ten that is produced by the ones is 'carried' to the tens column.

**Compact column method**

In this method, recording is reduced further. Carry digits are recorded below the line, using the words ‘carry ten’ or ‘carry one hundred’, **not** ‘carry one’.

Later, extend to adding three two-digit numbers, two three-digit numbers and numbers with different numbers of digits.

2 5 8 3 6 6

+ 8 7 + 5 8

3 4 5 4 2 4

1 1 1 1

Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. When children understand decimal place value it can be used to add numbers with 1 more decimal places.

0 . 2 5

1 . 8 6 3

+ 1 . 2

** **3 . 3 1 3

1 1