Progression through the teaching of

addition and subtraction

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.

When faced with a problem, we want children to ask themselves….

Parents Meeting on:Parents Meeting on:Progression through CalculationsProgression through Calculations

Lancashire Mathematics Team

Can I do it in my head?

Do I need

jottings ?

Do I need to use a

calculator?

Shall I use a pencil and

paper method?

How would you solve these calculations?

76 + 75 =

47 + 19 =

5321 – 2847 =

6003 - 5997 =

27 – 5 =

52 + 30 =

81 – 35 =

Laying the foundations for addition Laying the foundations for addition and subtractionand subtraction

o PartitioningPartitioning

o RoundingRounding

o CompensatingCompensating

o Counting on and backCounting on and back

o Bridging through 10s, 100s, 1000s boundariesBridging through 10s, 100s, 1000s boundaries

o Addition and subtraction factsAddition and subtraction facts

Lancashire Mathematics Team

Addition +

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.

A number line is just a ‘picture’ of how we work out

some calculations in

our heads!

They use number lines and practical resources to support calculation and teachers demonstrate the use of the number line.

3 + 2 = 5

___________________________________________0 1 2 3 4 5 6 7 8 9

+1+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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

+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.

Year 1

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.

34 + 23 = 57

34 44 54 55 56 57

+1+1+1

Then helping children to become more efficient by adding the units in one jump (by using the known fact 4 + 3 = 7).

34 + 23 = 57

34 44 54 57

+10 +10+3

+10 +10

Followed by adding the tens in one jump and the units in one jump.

Your turn!

64 + 25 =

89

+5

8474

+10

64

+10

+20

89

Count on from the largest number irrespective of the order of the calculation.

38 + 86 = 124

+4 +30 +4

_______________________________________________86 90 120 124

Year 3

Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate.

Compensation

49 + 73 = 122

73 122 123

+50

-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. 67 267 + 24 + 85 80 (60 + 20) 200 (200 + 0) 11 (7 + 4) 140 (60 +80) 91 12 (7 + 5) 352

Year 3

Moving to adding the least significant digits first in preparation for ‘carrying’.

67 267+ 24 + 85 11 (7 + 4) 12 (7+5) 80 (60 + 20) 140 (60+80) 91 200 (200+0) 352

Year 3

Your turn!

72 + 46 =

72

+ 46

(70 + 40)110

(2 + 6)8

118

Year 4

Children will in Year 4 be introduced to carrying above the line.

625 783 367+ 48 + 42 + 85

673 825 452Using 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 £3.59+78p

11 11

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.

587 3587+ 475 + 675 11 111 1062 4262

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

Children would use rounding to estimate the answer to the calculation.So 587 + 475 is about 600 + 500, which is approximately 1100.

Year 6

Children should extend the carrying method to numbers with any number of digits.

7648 6584+ 1486 + 5848 111 111 9134 12432

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 401.2 + 26.85 + 0.71

Vocabulary.• Add

• Plus

• Altogether

• Addition

• Total

• Count on

• Increase

• Sum

• Make

Subtraction -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.

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

They use number lines and practical resources to support calculation. Teachers demonstrate the use of the number line.

-1

6 – 3 = 3

0 1 2 3 4 5 6 7 8 9 10

-1 -1

The number line should also be used to show that 6 - 3 means the ‘difference between 6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart.

0 1 2 3 4 5 6 7 8 9 10

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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

-1 -1 -1 -1 -1

Year 1

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

24 25 26 27 37 47

- 10 - 10-1-1 -1

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

24 27 37 47

- 10- 10- 3

Subtracting the tens in one jump and the units in one jump.

24 27 47

- 20- 3

Year 2

Your turn!

64 - 26 =

4438

-20-6

64

38

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.

Counting on

73 – 68 = 5

68 70 73

+ 2 +3

82 – 47 = 35

47 50 60 70 80 82

+ 3 + 10 + 10 + 10 + 2

Year 2

Year 3

Children continue to use empty number lines with increasingly large numbers.

Children are then taught this expanded method using partitioning.

89 = 80 + 9 - 57 50 + 7 30 + 2 = 32

Initially, the children are taught using examples that do not need the children to exchange.

From this the children move to exchanging;

71 = - 46

Step 1: 70 + 1 - 40 + 6

Step 2: 60 + 11 - 40 + 6 20 + 5 = 25

This would be recorded by the children as 70 + 1 - 40 + 6 20 + 5 = 25

60 1

Year 3

Your turn!

73

70 + 3

- 26

20 + 6

60 1

7+40 = 47

Year 4

Partitioning and decomposition

754 = 86

Step 1 700 + 50 + 4 - 80 + 6

Step 2 700 + 40 + 14 (adjust from T to U) - 80 + 6

Step 3 600 + 140 + 14 (adjust from H to T) - 80 + 6 600 + 60 + 8 = 668

This would be recorded by the children as:

700+ 50 + 4 - 80 + 6 600 +60 +8 = 668

As decomposition this would look like this:

754 - 86 668

600 140 1

14 16

Year 4

Common calculation errors!Common calculation errors!

945 1 1 1

- 237 2000

712 - 108

902

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:

£8.95 = - £4.38

8 + 0.9 + 0.054 + 0.3 + 0.08

8 + 0.8 + 0.154 + 0.3 + 0.08

4 + 0.5 + 0.07

= £4.57

leading to

8.95- 4.38

1

Alternatively, children can set the amounts to whole numbers, i.e. 895 – 438 and convert to pounds after the calculation.

8

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

+3

197 200 500 511

+300+11

Year 4

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: 754 = 700 + 50 + 4 - 286 200 + 80 + 6

Step 2: 700 + 40 + 14 (adjusting from T to U) - 200 + 80 + 6

Step 3: 600 + 140 + 14 (adjusting from H to T) - 200 + 80 + 6 400 + 60 + 8 = 468

This would be recorded by the children as 700 + 50 + 4200 + 80 + 6

400 + 60 + 8 = 468

140600

Decomposition

6 14 1

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 800 -300, which is approximately 500.

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

+12

388 400 1200 1209

+ 800+ 9

Year 5

Year 6

Decomposition

6467 - 2684 3783

Now using 4 digit numbers and beyond.

5 13 1

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.

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.

3002 -1997 = 1005

+ 3

1997 2000 3000 3002

+ 1000+ 2

Year 6

Subtraction Vocabulary

How many are left?

How many have gone?

How many fewer is ... than ...

1 less

10 less

Difference between

Take (away)

How many are left over?

Decrease

Count back

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: The questions at the forefront of their minds: ‘Can ‘Can I do it in my head? If not which method will help I do it in my head? If not which method will help me?’me?’

Thank you for attending our

workshop on the progression

through addition and subtraction.