Progression in Mental Calculation Strategies
Children need to learn these strategies, and know that they are essentially mental methods, although some jottings or note taking may be helpful.
They also need to be able to recognise and to articulate the circumstances in which each method is appropriate to use, for example, finding the difference is a good method to use when numbers are fairly close together.
The following methods are enclosed:
Counting on and back
Finding the difference
Doubles and near doubles
Adjusting
Partitioning
Guidance on the teaching sequence for learning times tables is also included in this document.
Times tables
The teaching sequence
2 x 4= 8 and 4 x 2= 8 so 8 ÷ 4= 2 and 8 ÷ 2= 4 Also using the inverse (opposite) operation to work out an answer: 3 x _= 18 OR so 18 ÷ 3= 6


Stage 1  2, 10 and 5

Stage 2  4, 8, 3 and 6

Stage 3  11 and 9

Stage 4  7 and 12

Stage 5  Mixed tables 
Stage 6  Mixed tables including decimals 
Counting on and back  
Stage 1  Add and subtract 1 to/from any number up to 20 (using B10 and money as a model)
Eg 4+/1

Stage 2  Add and subtract 10 to/from any number up to 1 or 10 (using B10 and money as a model)
Eg. 14+/10

Stage 3  Add or subtract multiples of 1 and 10 to/from any number up to 100 (using B10 and money as a model)
Eg. 53+/20 53+/24

Stage 4  Add or subtract multiples of 100 to/from any number up to 1000
Eg 263+300
Using money as a model mentally add multiples of 10p and 1p to any amount and record as a decimal eg add 20p to £1.34

Stage 5  Mentally add a 3d number to any number by counting on 100’s tens then1s
Eg 463+221 (463…663…683…684)
Mentally add and subtract multiples of 0.1 and 0.01 to any number including in the context of money and measure

Stage 6  Add or subtract multiples of 1000 to/from any number up to 10000 and beyond
Eg. 3026+/2000, 326+4000 
Finding the difference  
Exemplification examples  
Stage 1  Find the difference between 2 amounts of objects up to 10
Find the difference between 2 numbers on a number line by counting the ‘jumps’ between them
Say how many ‘single jumps’ on the number line to get to 10 from any number less than 10

Make 2 towers of cubes with a difference of 2 how many pairs of towers can you make?
How many jumps are there from 6 to get to 10 
Stage 2  Find the difference between 2 amounts of objects up to 20 in a range of contexts
Find the difference between 2 numbers up to 20 using a number line model
Say how many ‘jumps’ to the next multiple of 10 from any 2d number

What is the difference between 2 lengths of ribbon?
What is the difference between the number of children who walk to school and the number of children who come by car?
63+7=70 It is 11:47 how many minute before 11.50?

Stage 3  Find the difference between 2 amounts up to 100 in a range of contexts
Find the difference between two 2d numbers using a number line model by finding the next multiple of 10 and counting on in tens

I need £83 to buy a pair of trainers, I have already saved £46. How much more do I need?

Stage 4  Find the difference between two 3d numbers/amounts by counting on from one number to the other using a number line as a model

I need 320ml of water for my recipe, I have already put in 180ml in the mixture how much more do I need to add? 
Stage 5  Find the difference between two numbers/amounts including larger and decimals by counting on from one number to the other using a number line as a modeland know when it is appropriate to use a written method instead

33001600
(for 33631847 it will be better to use a written method)

Doubles and near doubles  
Examples calculations and problems  
Stage 1  Know number facts for doubles up to 20, recognise and use them to derive the answers to near double calculations up to 20.  6+7
Find the total of 5p and 6p 

Stage 2  Know number facts for doubles of multiples of 10 up to 200 where no exchange is necessary, recognise and use them to derive the answers to near doubles calculations up to 200.  20+21
Cut 2 length of string, make one 20cm long, and the other 21cm long. How much string do you need.


Stage 3  Quickly calculate doubles of numbers up to 100, recognise and use them to derive the answers to near doubles calculations.  26+27
I have 26p, my friend has 1p more, how much do we have altogether? 

Stage 4  Quickly calculate doubles of numbers up to 100, recognise and use them to derive the answers to near doubles calculations and use the associated halves.  38+41
What is half of 85? (half of 84 is 42, half of 1 is ½ )


Stage 5  Quickly calculate doubles of numbers up to 1000, recognise and use them to derive the answers to near doubles calculations.  156+155
140+150
It is 283 miles from London to Newcastle and 160 miles from Newcastle to Dundee. Joe droves from Dundee to London via Newcastle. How far did he travel?


Stage 6  Know when it is appropriate to calculate mentally using doubles and near doubles, and when it is more appropriate to use a written method this is usually when no exchange is necessary, unless the numbers involved are multiples of 10.  3500+3495
4550+4500
Zoe had 2 buckets. One holds 4.5 litres, the other holds just 50ml more. How many 50 millilitre containers can she fill from both buckets? 

Stage 7  Recognise when decimal numbers are near doubles, and use number facts and place value to derive or calculate the answers when there is only one decimal place  1.5+1.4
What is the sum of 1.5 and 1.4?


Stage 8  Recognise when decimal numbers are near doubles, and use number facts and place value to derive or calculate the answers  1.82+1.9
Sarah needs 4 metres of ribbon to decorate a bag. She finds two lengths, one is 1m82cm, the other is 1m90cm. How much more does she need?


Adjusting  
Stage 1  Know that 9 and 11 are ‘nearly‘ 10  
Stage 2  Use knowledge of adding and subtracting 10 to quickly calculate addition and subtraction of 9 or 11.
Eg. 23+/9, 42+/11
Know that 19 and 21 are ‘nearly 20, and 29 and 31 are ‘nearly’ 30 etc


Stage 3  Use knowledge of adding and subtracting multiples of 10 to quickly calculate addition and subtraction of 2 digit numbers ending in 1 or 9.
Eg. 73+/10, 42+/21
Know that 101, 102, 98 and 99 are nearly 100 
Recognise when numbers are close, and use this in other calculations
Eg 16971 

Stage 4  Use knowledge of adding and subtracting 100 from a number to quickly calculate addition and subtraction of 99, 98, 101, 102
Eg 263+/98
Identify numbers which are close to multiples of 100 
Extend to large numbers
Eg 1378382 

Stage 5  Use knowledge of adding and subtracting multiples of 100 from a number to quickly calculate addition and subtraction of numbers ending in 99, 98, 97, 96, 95 101, 102, 103, 104 and 105
Eg 2263+/498,
Know that numbers ending in .1 and numbers ending in .9 are close to whole numbers and use this in mental calculations

Extend to larger numbers  
Stage 6  Know that 0.11 and 0.09 are close to 0.1 and use this in calculations  Extend to decimals
Eg 3.451.5 

Partitioning  
Stage 1  13 is the 10 and 3 
Stage 2  Partition all 2d numbers into tens and ones

Stage 3  Add two digit numbers by partitioning, adding and regrouping
Eg 37 + 48 30 + 40 = 70 7 + 8 = 15 70 + 15 = 85

Stage 4  Mentally add two digit numbers by partitioning, adding and regrouping

Stage 5  Find the difference between two digit numbers when there is no regrouping
Eg 6746 6040=20 76=1 20+1=21

Stage 6  Mentally find the difference between two digit numbers when there is no regrouping

Stage 7  Extend methods to 3d numbers

Stage 8  Extend methods to larger numbers using jottings as appropriate
