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Progression in Written Calculation - x ÷ +

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Introduction Written methods of calculations are based on mental strategies. Each of the four operations builds on 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 need to be supported by familiar models and images to reinforce understanding. When teaching a new strategy it is important to start with numbers that the child can easily manipulate so that they can understand the 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 when introducing a new strategy. A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.

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**Partitioning and recombining**

+ Progression in written methods for Addition + Number Track Number Line Expanded method Partitioning and recombining Formal Compact Method

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**+ + + = + = 7 3 + 2 = 3 1 1 + 2 = 3 and and Stage 1 – Number Track**

add 3 1 2 3 4 5 8 7 6 10 9 7 + 3 and + and understand addition is combining groups of objects count on using a number track use a puppet to accentuate jumps + = + = + 2 = 3 1 1 + 2 = 3 understand addition can be done in any order

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**Stage 2a – Introducing the number line**

+ 3 1 2 3 4 5 8 7 6 10 9 1 2 3 4 5 6 7 8 9 10 7 + 3 and and + link number track and number line count on using a fully numbered number line (start counting on in ones and then move on to larger jumps)

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**Stage 2b – Using a number line Partition the smallest number.**

Add the unit(s) first + 1 + 10 13 14 24 + 7 + 1 + 10 21 13 20 31 always encourage ESTIMATION first teach and encourage children to partition numbers in different ways in order to bridge to the nearest multiple of 10 start from the largest number and then count on progress from not bridging 10 to bridging through 10 progress from fully numbered line to partially numbered line then blank

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**Stage 3 – Partitioning & Recombining**

The Expanded Method 20 8 40 3 + = T U Encourage ESTIMATION reinforce place value by using place value cards to partition alongside place value apparatus (Dienes) reinforce ‘carrying’ use equipment alongside expanded written method to bridge from concrete to abstract (see appendix a for recording – transition between equipment – pictorial recording and then abstract)

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**Stage 4 – Expanded Method leading**

to Formal Compact Method 20 8 40 3 + = 7 1 Add the unit(s) column then the ten(s) column to calculate the final answer 4 3 7 1 Expanded method leading to compact method 1 9

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**Stage 4 – Expanded Method leading to Formal Compact Method (decimal)**

0.20 0.08 0.40 0.03 + = continue to encourage ESTIMATION link to money (add more than two amounts) & measurement link to using a calculator/interpreting calculator display C M √ AC % ÷ 7 4 1 8 5 2 . 9 6 3 = x - + 0.43 0.71 Remember to line up decimal points especially when number of digits differs 1 10

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**(Finding the difference Partitioning and recombining**

Progression in written methods for Subtraction - - Number Track Number Line (Finding the difference and counting back) Expanded method Partitioning and recombining Formal Compact Method

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**7 3 - Stage 1 – Number Track take away 3**

2 3 4 5 8 7 6 10 9 7 3 - seven three understand subtraction is taking away objects jump/count back along a number track use a puppet to accentuate jumps

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**Stage 2a – Introducing Number Line**

1 2 3 4 5 8 7 6 10 9 - 3 1 2 3 4 5 6 7 8 9 10 7 3 - seven three link number track and number line understand subtraction is taking away objects jump/count back along a fully numbered number line (start counting back in ones and then move on to larger jumps)

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**Stage 2bi – Using a number line (not bridging through 10)**

finding the difference 23 – 18 5 18 23 counting back 37 – 13 Partition the smallest number. Count back the units, then tens. - 3 - 10 34 37 24 ESTIMATE first understand ‘finding the difference’ AND ‘counting back’ has the same result promote finding the difference when the numbers involved are close together progress from counting back in ones to larger steps. 15

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**Stage 2bii – Using a number line**

(bridging through 10) 43 – 27 - 10 - 10 - 4 - 3 16 26 36 40 43 ESTIMATE first encourage children to partition numbers in different ways bridge through multiples of 10 ensure children have the opportunity to solve subtraction problems in a range of different contexts encourage use of vocabulary and explanation

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**take away the units and then take away the tens**

Stage 3a – Expanded Method (no exchanging) = 33 take away the units and then take away the tens 30 and 3 use place value apparatus (Dienes) to re-inforce concept of exchanging move from concrete apparatus to expanded written method (see appendix a for recording – transition between equipment – pictorial recording and then abstract) continue to encourage ESTIMATION 17

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**to subtract 7 units we need to exchange a ten for ten units**

Stage 3b – Expanded Method (with exchanging) = 16 to subtract 7 units we need to exchange a ten for ten units and 6 30 10 + use place value apparatus (Diennes) to re-inforce concept of exchanging move from concrete apparatus to expanded written method (see appendix b for recording – transition between equipment – pictorial recording and then abstract) continue to encourage ESTIMATION 18

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**- 20 7 10 and 6 - 2 7 1 6 Stage 4a – Formal Compact Method 4 3**

10 and 6 10 + 30 4 3 1 6 3 1 move from expanded written method to compact method continue to encourage ESTIMATION

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**- 2 . 7 1 . 6 Stage 4 – Formal Compact Method (decimal) 4 . 3**

1 . 6 1 3 Remember to line up decimal points especially when number of digits differs continue to encourage ESTIMATION link to money (giving change) and measurement link to using a calculator/interpreting calculator display C M √ AC % ÷ 7 4 1 8 5 2 . 9 6 3 = x - +

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**Repeated addition, arrays**

Progression in written methods for multiplication x x Repeated addition, arrays Grid method (with imagery) Long multiplication

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**2 2 2 2 + 2 4 x 4 2 x Stage 1 – Repeated addition, arrays**

+ = 8 4 x 2 = 8 2 multiplied by 4 4 lots of 2 x 4 2 2 4 x understand that multiplication is a shortened form of repeated addition understand multiplication as arrays and jumps on a number line

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**4 x 13 ‘four lots of thirteen’**

Stage 2 – Modelling grid method with place value equipment 4 x ‘four lots of thirteen’ 4 10 3 4 10 3 40 12 = 52 use place value apparatus to illustrate grid method, encourage jottings use digits of 5 and below to avoid ‘difficult’ tables and ensure method is secure

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**4 x 23 ‘four lots of twenty three’**

Stage 2a – Modelling grid method with place value equipment (multiples of 10) 4 x ‘four lots of twenty three’ 20 3 4 20 ( 2 x 10 ) 3 80 12 4 (4 x 2 x 10) (4 x 3) = 92 use place value equipment to illustrate grid method with multiples of 10 reinforce using known facts to multiply e.g. 4 x 20 = 4 x 2 x 10

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**Stage 3 – Grid method (no apparatus)**

45 x 6 40 ( 4 x 10 ) 6 240 36 6 (6 x 4 x 10) (6 x 6) = 276 47 x 52 50 2 2000 80 2000 40 350 (4 x 5 x 10 x 10) (4 x 2 x 10) 80 80 14 350 7 + 14 14 (7 x 5 x 10) (7 x 2 ) 2444 continue to reinforce using known facts to multiply e.g. 40 x 50 = 4 x 5 x 10 x 10 progress to using the grid method efficiently to multiply decimals

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**Stage 4 – Long multiplication**

× (56 × 20) (56 × 7) 4 1 1 ONLY move on to this method if understanding of grid method is secure

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**Stage 4a – Long multiplication (decimal)**

× (5.6 × 2.0) (5.6 × 0.07) 4 1 1 continue to encourage ESTIMATION (re-inforce place value) link to money and measurement link to using a calculator and interpreting display C M √ AC % ÷ 7 4 1 8 5 2 . 9 6 3 = x - +

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**÷ ÷ Division as sharing and grouping Grouping on a number line**

Progression in written methods for division ÷ Division as sharing and grouping Grouping on a number line Link division and multiplication Vertical recording Chunking (fact box) Short/long division

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**÷ ÷ 15 3 15 5 Stage 1 – Division as sharing and grouping sharing**

one at a time ÷ 15 divided into 3 equal groups ÷ 15 divided into 5 equal groups understand division as sharing, understand division as grouping understand remainders

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**Stage 2 – Grouping on a number line**

- = 6 groups r1 understand that division is repeated subtraction show division as equal groups on a number line then begin to understand remainder

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**Vertical recording (teacher model only)**

3 6 9 12 15 18 18 15 12 9 6 3 - 3 18 ÷3 = 6 18 ( 1 x 3 ) 1 5 1 2 9 ( 1 x 3 ) 6 3 ( 1 x 3 ) turn horizontal number line vertical so children can see link to vertical calculation and model recording, use to illustrate need to take ‘chunks’ for efficiency

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**Stage 3 – Linking division & multiplication leading to chunking **

– introducing fact box What facts do I know about the 5 times-table? 96 5 96 ÷ 5 = 19 r 1 96 ( 10 x 5 ) 46 ( 5 x 5 ) 21 - 20 1 Fact Box 1 x 5 = 5 2 x 5 = 10 5 x 5 = 25 10 x 5 = 50 children need to see that when numbers are larger it is more efficient to subtract larger ‘chunks’ building a fact box will help children with the size of the ‘chunks’ children need to work with and without remainders considering if answer needs rounding up or rounding down

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**Stage 4 – Chunking with a fact box**

What facts do I know about the 7 times-table? 100 ÷ 7 = 14 r 2 100 ( 10 x 7 ) 30 ( 4 x 7 ) 2 518 ÷ 7 = 74 518 ( 50 x 7 ) 168 ( 20 x 7 ) 28 ( 4 x 7 ) Fact Box 1 x 7 = 7 2 x 7 = 14 5 x 7 = 35 10 x 7 = 70 20 x 7 = 140 50 x 7 = 350 100 x 7 = 700 children need to see that when numbers are larger it is more efficient to subtract larger ‘chunks’ building a fact box will help children with the size of the ‘chunks’ children need to work with and without remainders considering if answer needs rounding up or rounding down

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**ONLY move on to this method if understanding is secure **

Stage 5 – Long division 560 ÷ 24 2 3 r 8 8 Jottings – Fact Box 20 3 4 400 80 60 12 x = 552 ONLY move on to this method if understanding is secure move on to show remainders as a fraction and decimal

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**(showing remainder as a fraction)**

Stage 5a – Long division (showing remainder as a fraction) 560 ÷ 24 2 3 r 8/24 () 8 Jottings – Fact Box 20 3 4 400 80 60 12 x = 552 ONLY move on to this method if understanding is secure move on to show remainders as a fraction and decimal

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**(showing remainder as a decimal)**

Stage 5b – Long division (showing remainder as a decimal) 560 ÷ 24 8 0 7 2 8 Jottings – Fact Box 20 3 4 400 80 60 12 x = 552 ONLY move on to this method if understanding is secure move on to show remainders as a fraction and decimal

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**Start with apparatus then show children how to record pictorially**

T - tens U - units concrete appendix a Start with apparatus then show children how to record pictorially visual

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**Start with apparatus then show children how to record pictorially**

appendix b concrete T - tens U - units Start with apparatus then show children how to record pictorially visual

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**Link division and multiplication**

12 divided into groups of 3 gives 4 groups 12 3 = 4 12 divided into groups of 4 gives 3 groups 12 4 = 3 3 x 4 = 12 or 4 x 3 = 12 12 4 = 3 or 12 3 = 4 understand that division is the inverse of multiplication reinforce division as grouping emphasise link between times table facts and division facts

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**Understanding the inverse/finding unknowns (empty boxes)**

I can work out missing numbers in a number sentence (year 1 & Year 2) Introducing the Inverse – Play Mrs/Mr Opposite. Every instruction the teacher gives the children have to do the opposite e.g. teacher says take one step forward, children take one step backwards or teacher says turn to the right, children turn to the left etc. Explain that in maths we call the opposite the inverse and that we are going to be looking at the inverse (opposite) of adding. Addition and Subtraction – Numicon Families (Using the inverse) Ten is the same as/equals nine add/plus one. Explain that the children are going to be using Numicon. Show them what it is and explain how it is structured. Model how this can be used to demonstrate the inverse Once imagery is secure replace one piece of Numicon with an empty box. Remember to move the = sign! Ten is the same as/equals nine add/plus one. + = 10 = 9 + 1

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**Understanding the inverse/finding unknowns (empty boxes)**

I can work out missing numbers in a number sentence including where = sign is used to balance an equation e.g = 3 + ? or 6 x 4 = 3 x ? (Year 3) Use Numicon and balance to model and for the children to practise in order to reinforce understanding of equality. Once imagery is secure replace one piece of Numicon with an empty box. Remember to move the = sign AND begin to explore balancing different operations. I can work out missing numbers in a number sentence including those where = balances an equation e.g. 10 – 3 = 3 + ? or 2 x 3 = 60 ? (Year 4) Again use Numicon. Emphasis on exploring balancing equations with different operations. I can work out missing numbers in more complex calculations e.g. 3?67 – 192? = 1539 or ? = 325 (Year 5) I can find the unknown in a calculation such as ? = or 5.6 ? = 0.7, drawing on knowledge of number facts and place value, including using a calculator and inverse operations to solve more complex problems such as ? = 24.7 and explain my reasoning (Year 6) + = +

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