1. Progression through the teaching of addition and subtraction

2. Maths has changed!
The maths work your child is doing at school may look very different to the kind of ‘sums’ you remember. This is because children are encouraged to work mentally, where possible, using personal jottings to help support their thinking.
‘Formal’ calculations are introduced from Year 3 onwards. Children are then encouraged to use these methods for calculations they cannot solve in their heads.

3. Shall I use a pencil and paper method?
Parents Meeting on: Progression through Calculations
When faced with a problem, we want children to ask themselves….
Shall I use a pencil and paper method?
Do I need jottings ?
Can I do it in my head?
Do I need to use a calculator?
Lancashire Mathematics Team

4. How would you solve these calculations?= =
5321 – 2847 = =
27 – 5 =
81 – 35 = =

5. Laying the foundations for addition and subtraction
Partitioning
Rounding
Compensating
Counting on and back
Bridging through 10s, 100s, 1000s boundaries
Addition and subtraction facts
Lancashire Mathematics Team

6. Addition + Reception and Year 1
THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO ACHIEVE.
Reception and Year 1
Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures.

7. A number line is just a ‘picture’ of how we work out some calculations in our heads!

8. ___________________________________________ 0 1 2 3 4 5 6 7 8 9
They use number lines and practical resources to support calculation and teachers demonstrate the use of the number line.
3 + 2 = 5 +1 +1
Year 1
___________________________________________
Children then begin to use number lines to support their own calculations using a numbered line to count on in ones.
8 + 5 = 13 +1 +1 +1 +1 +1
Bead strings or bead bars can be used to illustrate addition including bridging through ten by counting on 2 then counting on 3.

9. Year 2
Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on.
First counting on in tens and ones.
= 57
+10
+10 +1 +1 +1
Then helping children to become more efficient by adding the units in one jump (by using the known fact = 7).
= 57
+10
+10 +3
Followed by adding the tens in one jump and the units in one jump.

10. Your turn! = 89
+20
+10
+10 +5 84 89 64 74

11. Year 3
Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate.
Count on from the largest number irrespective of the order of the calculation.
= 124
_______________________________________________

12. Compensation
Year 3
+50
= 122 -1
Children will begin to use informal pencil and paper methods (jottings) to support, record and explain mental methods building on existing mental strategies.
Stage 1: Adding the most significant digits first, then moving to adding least significant digits.
80 ( ) ( )
11 (7 + 4) (60 +80)
(7 + 5)
352

13. Year 3
Moving to adding the least significant digits first in preparation for ‘carrying’.
11 (7 + 4) (7+5)
80 ( ) (60+80)
(200+0)
352

14. Your turn! = 72
+ 46 8
(2 + 6)
110
( )
118

16. Year 4
Children will in Year 4 be introduced to carrying above the line.
Using similar methods, children will:
Add several numbers with different numbers of digits;
Begin to add 2 or more 3-digit sums of money, with or without adjustment from pence to pounds;
Know that the decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g £ p 1 1 1 1

17. Year 5
Following formal addition methods with carrying above the line being introduced at Year 4, children should at Year 5, extend the carrying method to numbers with at least four digits.
Children would use rounding to estimate the answer to the calculation.
So is about , which is approximately 1100.
Using similar methods, children will:
Add several numbers with different numbers of digits;
Begin to add two or more decimal fractions with up to three digits and the same number of decimal points
Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g 3.2m – 280cm

18. Year 6
Children should extend the carrying method to numbers with any number of digits.
Using similar methods, children will:
Add several numbers with different numbers of digits;
Begin to add two or more decimal fractions with up to four digits and either one or two decimal places;
Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g

19. Vocabulary. Add Plus Altogether Addition Total Count on Increase Sum
Make

20. Subtraction - Reception and Year 1
THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO ACHIEVE.
Reception and Year 1
Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures etc.

21. There are five frogs. If 2 frogs jumped into the lake how many would be left?

22. Year 1
They use number lines and practical resources to support calculation. Teachers demonstrate the use of the number line.
6 – 3 = 3 -1 -1 -1
The number line should also be used to show that means the ‘difference between 6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart.

23. Year 1
Children then begin to use numbered lines to support their own calculations - using a numbered line to count back in ones.
13 – 5 = 8 -1 -1 -1 -1 -1
Bead strings or bead bars can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back 2.
13 – 5 = 8

24. Year 2
Children will begin to use empty number lines to support calculations.
Counting back
First counting back in tens and ones.
47 – 23 = 24
- 10
- 10 -1 -1 -1

25. Subtracting the tens in one jump and the units in one jump.
Then helping children to become more efficient by subtracting the
units in one jump (by using the known fact 7 – 3 = 4).
47 – 23 = 24
- 10
- 10
- 3
Year 2
Subtracting the tens in one jump and the units in one jump.
- 20
- 3

26. Your turn! = 38
-20 -6 38 44 64

27. Counting on
If the numbers involved in the calculation are close together or near to multiples of 10, 100 etc, it can be more efficient to count on.
Year 2
73 – 68 = 5
+ 2 +3
82 – 47 = 35
+ 3
+ 10
+ 10
+ 10
+ 2

28. Year 3
Children continue to use empty number lines with increasingly large numbers.
Children are then taught this expanded method using partitioning.
89 =
= 32
Initially, the children are taught using examples that do not need the children to exchange.

29. From this the children move to exchanging; 71 = - 46 Step 1: 70 + 1
Year 3
From this the children move to exchanging; =
- 46
Step 1:
Step 2:
= 25
This would be recorded by the children as
70 + 1
= 25 60 1

30. Your turn! 73
- 26 60 1
70 + 3
20 + 6 40 + 7 = 47

31. Year 4 Partitioning and decomposition 754 = 86 Step 1 700 + 50 + 4= 86
Step
Step (adjust from T to U)
Step (adjust from H to T)
= 668

32. This would be recorded by the children as:
= 668
As decomposition this would look like this:
754
668
600
140 1
Year 4 14 1 6

33. Common calculation errors!
902

34. Year 4
Children should:
Be able to subtract numbers with different numbers of digits;
Using this method, children should also begin to find the difference between 2 3-digit sums of money, with or without adjustment from the pence to pounds;
Know that decimal points should line up under each other.
For example:
£ =
- £4.38
leading to
8.95 8 1
- 4.38
= £4.57
Alternatively, children can set the amounts to whole numbers, i.e. 895 – 438 and convert to pounds after the calculation.

35. Year 4
Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc. counting on using a number line should be used.
511 – 197 = 314
+300
+11 +3

36. Year 5
The teaching of subtraction continues from Year 4 where an expanded method will have been introduced.
This expanded method uses partitioning
Step 1: =
Step 2: (adjusting from T to U)
Step 3: (adjusting from H to T)
= 468
600
140
This would be recorded by the children as
= 468

37. Year 5
Decomposition
754
- 286
468
Children should:
Be able to subtract numbers with different numbers of digits
Begin to find the number between two decimal fractions with up to three digits and the same number of decimal places this could be in the context of money or measures
Know that decimal points should line up under each other.
Children would use rounding to estimate the answer to the calculation.
So 754 – 286 is about , which is approximately 500.

38. Year 5
Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc. counting on using a number line should be used.
1209 – 388 = 821
+ 800
+ 9
+12

39. Year 6
Decomposition
6467
- 2684
3783 13 1 5
Now using 4 digit numbers and beyond.
Children should:
be able to subtract numbers with different numbers of digits;
Be able to subtract two or more decimal fractions with up to three digits and either one or two decimal places; this could be in the context of money or measures
know that decimal points should line up under each other.

40. Year 6
Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc. counting on using a number line should be used.
= 1005
+ 1000
+ 2
+ 3

42. Subtraction Vocabulary
Take (away)
How many are left?
How many have gone?
1 less
Decrease
Difference between
How many fewer is ... than ...
How many are left over?
10 less
Count back

43. Key messages
Children need to develop skills such as counting, partitioning and recombining numbers
They need to build an awareness of the number system, value of numbers and number relationships
They need to recall facts such as halving and doubling, number bonds and multiplication facts
From all of these they learn to construct strategies that they can apply in many different areas.
The questions at the forefront of their minds: ‘Can I do it in my head? If not which method will help me?’

44. Thank you for attending our workshop on the progression through addition and subtraction.