÷

Written methods of calculations are based on mental strategies. Each of the four operations builds on secure mental skills which provide the

foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead

on to more formal written methods of calculation.

Strategies for calculation must be supported by familiar models and images. When approaching a new strategy it is important to start with numbers that the child can easily manipulate so that they have every

opportunity to fully grasp each concept.

The transition between stages should not be hurried as not all children will be ready to move on to the next stage at the same time, therefore the progression in this document is outlined in stages. Previous stages

may need to be revisited to consolidate understanding before progressing. Failure to secure understanding can lead to misconceptions later so it is essential learning is personalised for every child to ensure solid mathematical foundations are laid which can be built upon in the

future

. A sound understanding of the number system and the patterns within it

is essential for children to carry out calculations efficiently and accurately.

Introduction

Mathematics is NOT just a memory game

Children need a level of understanding

Children MUST be encouraged to think for themselves and to reason

Head first. Can the calculation be done mentally more

efficiently?

£5.00 – £4.99

£5.00 - £4.99

49 + 1

49 + 1

Children need to develop understanding of number

Children need to understand the position of numbers and how they relate to one another

Children need to understand the value of

numbers(Place Value)

Children need to be able to partition and recombine numbers

Learn number bonds

Learn multiplication

tables and related division

facts

Learn facts about measures e.g. 24 hours in a day, 100cm in

a metre

Learn how to tell the time on

an anologue clock

Add/subtract one to/from any number

Add/subtract ten to/from any

number

Mental Mental CalculationCalculation

Number Bonds

Year 1 – recognise and reason bonds up to 10 and then up to 20 and related subtraction facts Year 2 – practise addition and subtraction bonds up to 20 to become increasingly fluent, use knowledge of bonds to calculate and use related bonds to 100 using multiples of 10 e.g. 70 + 30Year 3 – consolidate previous learning then investigate bonds of larger numbers, bonds to 1 using tenths, fractions and decimals e.g. 0.1 + 0.9, 1/10 + 9/10Year 4 – consolidate previous learning, decimal and fraction bonds bonds using hundredths 0.99 + 0.01 – link to money and measuresYear 5 – practise fluency with bonds with one-, two- and three-decimal places, including links with money and measuresYear 6 – consolidate understanding of bonds to three-decimal places to achieve fluency

A number bond is an addition sum with two numbers.

Knowing bonds to 10 then 20 then 100 helps with addition,

both mental and written.

e.g. 18 + 7 = 18 + 2 + 5 = 20

Partition 7 into 2 and 5 so that 2 can be added to 18 to make 20 and then add 5

Number Bond

Practice

Progression in methods for addition

Compact Method

1 2 3 4 5 876 1090

Number Track

Number Line

Expanded method (partitioning and

recombining)

4 3

+ 2 8

1

7 1

4 0 + 3

2 0 + 8

6 0 + 1 1

7 0 + 1 = 7 1

Stage 1 – Understanding Addition & Number Track

1 2 3 4 5 876 1090

and

Use a puppet to practise counting on. Practise counting on/adding small numbers. If the

puppet makes a ‘mistake’ can the child spot it?

What happens if we start at 7 and add/count

on 3?

Combine two (or more) sets of objects and find out how many there all

together

Remember to use the different

words linked to ‘addition’

Stage 2 – Introducing the number line – counting on

0 1 2 3 4 5 6 7 8 9 10

Use a puppet to reinforce counting forwards. Link to number track. Start with a fully numbered number line and then progress to encouraging the children

to sketch their own to help with calculation.

+ 1+ 10

13 23 24

13 + 11

Ensure children understand place value e.g. 11 is one ten and

one unit or one

Start on the largest number

Add the tens … and then the units

Stage 3 – The Expanded Method (partitioning & recombining)

20 8

40 3

4 0 + 3

2 0 + 8

7 0 + 1 = 7 1

Use place value cards and place value apparatus

alongside written jottings. Partition the numbers into tens

and units, add, and then recombine.

1 0

4 3

+ 2 8

7 1

1

20 8

4 0 + 3

2 0 + 8

6 0 + 1 1

7 0 + 1 = 7 1

40 3

Link the expanded

method to the compact method

Stage 4 – Compact Method

Progression in methods for subtraction

Compact Method

1 2 3 4 5 876 1090

Number Track

Number Line

Expanded method (partitioning and

recombining)

4 3

- 2 7

1 6

13

40 3

- 20 7

10 and 6

10 +

30

Stage 1 – Number Track (counting back) & taking away

1 2 3 4 5 876 1090

Use a puppet to practise counting backwards. Practise taking away small numbers. If the

puppet makes a ‘mistake’ can the child spot it?

What happens if we start at 7 and take away/count

back 3?

Take away objects from a

group and count how many are

left

Remember to use the different words linked to

‘subtraction’

Stage 2 – Introducing the number line

0 1 2 3 4 5 6 7 8 9 10

Use a puppet to reinforce counting backwards. Link to number track. Start with a fully numbered

number line and then progress to encouraging the children to sketch their own to help with calculation.

- 3 - 10

14 23 33

33 - 19

Start counting back in ones and then progress to

larger jumps

Start on the largest number

Count back the tens… and then the units

- 6

20

Stage 3 – Expanded Method

40 3

- 20 7

10 and 6

10 +30

to subtract 7 units we need to exchange

a ten for ten units

43 - 27 = 16

Use place value apparatus alongside written jottings. Partition the numbers into tens and units, subtract

and then recombine

Stage 4 – Compact Method

40 3

- 20 7

10 and 6

10 +30

4 3

- 2 7

1 6

13

Is the answer

sensible?

Link the expanded

method to the compact method

Progression in methods for multiplication

Compact method

Repeated addition

Arrays

Grid method

10 2

3

10 100 20

630

100 + 30 + 20 + 6 = 156

5 6 × 2 7 1 1 2 0 (56 × 20) 3 9 2 (56 × 7) 1 5 1 2

14

1

Stage 1 – Repeated addition & …

Children need to understand that multiplication is the

same as repeated addition. Find opportunities to count in

groups e.g. socks, ‘fingers’ on 4 hand prints.

… arraysChildren need to be

able to see numbers as arrays. An array is an

arrangement of a number visually in rows and columns

4 x 13

4

10 3

40 + 12 = 524

10 3

40 12

Stage 2 – The grid method When learning the grid method use place

value equipment to help see the numbers.

Partition the numbers into tens and units. Draw a grid and place the

partitioned numbers across the top and down the side of the grid.

10 2

3

10 100 20

630

100 + 30 + 20 + 6 = 156

Multiply each of the part of the partitioned numbers and write the answers in the sections of the grid.

Lastly add together the answers to find the final total.

12 x 13

Stage 3 – Long multiplication

5 6 × 2 7 1 1 2 0 (56 × 20) 3 9 2 (56 × 7) 1 5 1 2

4

1

Because you are multiplying by ‘tens’ you must put a zero in the units column

Then multiply the two tens by the units (6) and then the tens (5)

Next multiply the seven units by the units (6) and then the tens (5). Finally add the two totals together to get a final answer

1

Progression in methods for division

Compact method

Sharing …

Chunking

… and grouping

÷96 ÷ 5 = 19 r 1

96

- 50 ( 10 lots of 5 )

46

- 25 ( 5 lots of 5 )

21

- 20

1

560 ÷ 24

2 3 r 8

2 4 5 6 0

- 4 8 0

8 0

- 7 2

8

Fact Box1 x 5 = 5

5 x 5 = 2510 x 5 = 50

Stage 1 - Sharing …

… and grouping

Share objects practically one at a time. Draw a

picture to show this. The objects do not need to be drawn these could

just be crosses.

Divide objects practically into equal groups. Draw a picture to show this. The objects do not need to be drawn these

could just be crosses.

4 shared by 2

8 divided into equal

groups of 2

Fact Box

2 x 5 = 10

5 x 5 = 25

10 x 5 = 50

Stage 2 – Using multiplication and division facts.

96 5

Using times tables knowledge to inverse division questions.

12 x 5 = 60

7 x 5 = 35

Remainder 1

96 5 = 19 r 1

What basic facts do I know about

the 5 times-table?

Children can use a number line to count up in the divided number. E.g. 30 ÷ 5.

Count up in 5s until you reach 30. How many jumps have you done?

0 5 3010 15

560 ÷ 24

2 3 r 8

2 4 5 6 0

- 4 8 0

8 0

- 7 2

8

Stage 3 – Short division and long division

10 + 3 r 5

7 70 + 26

96 7 = 13 r 5

7 9 6

1 3 r 52

Is the answer

sensible?

Progression in Calculations – by magnitudeYear 1 – U + U, U + multiple of 10, TU + multiple of 10, U – U, TU – U, TU – multiple of 10, counting groups of objects in ones, twos, fives and tens, sharing objects in equal groupsYear 2 - U + U, TU + U, TU + TU , U - U, TU - U, TU – TU, simple multiplication, simple division including with remainders Year 3 - TU + TU, HTU + TU, HTU + HTU, TU - TU, HTU - TU, HTU – HTU, TU x U, TU ÷ U including with remaindersYear 4 - TU + TU, HTU + TU, HTU + HTU, TU - TU, HTU - TU, HTU – HTU, TU x U, TU ÷ U including with remainders Year 5 – Add whole numbers and decimals to two decimal places, subtract whole numbers and decimals to two decimal places, HTU x TU, TU x TU, U x decimal, TU ÷ U, HTU ÷ U Year 6 - Add whole numbers and decimals to two decimal places, subtract whole numbers and decimals to two decimal places, TU x U, HTU x U, decimal x U, TU x TU, HTU x TU, TU ÷ U, HTU ÷ U, decimal ÷ U

Mathematical Language

Number sentence e.g. 2 + 4, 5 – 3, 6 x 3, 12 ÷ 3

Partition splitting a number up e.g. 123 … 100 + 20 + 3

Recombine putting a number back together e.g. 100 + 20 + 3 … 123

Bridging crossing over 10/100 etc

Exchanging e.g. swapping a 10 for 10 ones

Place value the value of each digit in a number e.g. hundreds, tens and ones (units)

Remember there are different words for +, -, x and ÷ to learn in order to help solve mathematical word problems