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2. 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

3. Progression in methods for addition Compact Method 123458761090 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

4. Stage 1 – Understanding Addition & Number Track 123458761090 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’

5. Stage 2 – Introducing the number line – counting on 01234 5 678 910 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 132324 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

6. Stage 3 – The Expanded Method (partitioning & recombining) 208 40 3 4 0 + 3 2 0 + 8 6 0 + 1 1 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.

7. 4 3 + 2 8 7 1 1 208 4 0 + 3 2 0 + 8 6 0 + 1 1 7 0 + 1 = 7 1 403 Link the expanded method to the compact method Stage 4 – Compact Method

8. Progression in methods for subtraction Compact Method 123458761090 Number Track Number Line Expanded method (partitioning and recombining) 4 3 - 2 7 1 6 1 3 40 3 - 20 7 10 and 6 10 + 30

9. Stage 1 – Number Track (counting back) & taking away 123458761090 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’

10. Stage 2 – Introducing the number line 01234 5 678 910 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 142333 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

11. 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

12. Stage 4 – Compact Method 40 3 - 20 7 10 and 6 10 + 30 4 3 - 2 7 1 6 1 3 Is the answer sensible? Link the expanded method to the compact method

13. Subtraction – can be viewed as ‘Comparison’ or ‘Finding the Difference’ How can you check your answer? 100 - 41 41 50 100 9 50 Counting On! ? 10 4 Emma has 10 marbles and James has 4 marbles. How many more marbles has Emma than James?

14. Progression in methods for multiplication Compact method Repeated addition Arrays Grid method 102 3 10020 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

15. 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. … arrays Children need to be able to see numbers as arrays. An array is an arrangement of a number visually in rows and columns

16. 4 x 13 4 10 3 40 + 12 = 52 4 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. 102 3 10020 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

17. 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

18. 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 Box 1 x 5 = 5 5 x 5 = 25 10 x 5 = 50

19. 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

20. Fact Box 2 x 5 = 10 5 x 5 = 25 10 x 5 = 50 Stage 2 – Chunking using a fact box 96  5 96 ÷ 5 = 19 r 1 96 - 50 ( 10 lots of 5 ) 46 - 25 ( 5 lots of 5 ) 21 - 20 1 What basic facts do I know about the 5 times-table? Take off a ‘chunk’ of the number to be divided each time until nothing is left or there is a remainder. Top tip: Make up a fact box showing basic facts then you don’t have to remember them and they can be used to work out the ‘chunks’.

21. 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 5 2 Is the answer sensible?

22. Progression in Calculations – by magnitude Year 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 groups Year 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 remainders Year 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