Lowick and Holy Island C. of E. First Schools
Calculation Policy
INTRODUCTION
This policy has been written in line with the programmes of study taken from the revised National Curriculum for Mathematics (2014). It has been devised to meet the requirements for teaching and learning of mathematics and is designed to ensure pupils have a consistent and smooth progression of learning in calculations across the whole school. The content is set out in staged blocks under the following headings: addition, subtraction, multiplication and division. Statements taken directly from the programmes of study are listed at the beginning of each section.
Age and stage expectations  This policy is organised into age stage expectations as set out in the new National Curriculum 2014, however it is vital that pupils are taught according to the stage that they are currently working at, being moved onto the next level as soon as they are ready, or working at a lower stage until they are secure enough to move on. Our school’s mixed age classes supports such teaching.
Providing a context for calculation  It is important that any type of calculation is based within a real life context or problem solving approach, this enables children to understand the purpose of calculation, and to help them to recognise when to use certain operations and methods when faced with problems. In addition to this, where ever possible, links are made with our thematic/cross curricular approach (Mantle of the Expert). Children need to be taught and encouraged to use the following processes in deciding an approach they will take to a calculation to ensure they select the most appropriate method for the numbers involved:
 Can I do it in my head using a mental strategy?
 Could I use some jottings to help me?
 Should I use a written method to work this out?
AIMS OF THE POLICY
 To ensure consistency and progression in our approach to calculation
 To ensure that children develop an efficient, reliable, formal written method of calculation for all operations
 To ensure that children can use these methods accurately with confidence and understanding
HOW TO USE THIS POLICY
 Use the policy as the basis of your planning but ensure you use previous or following years’ guidance to allow for personalised learning
 Always use Assessment for Learning to identify suitable next steps in calculation for groups of children
 If, at any time, children are making significant errors, return to the previous stage in calculation
 Cross reference with mental maths for guidance on key facts, key vocabulary and mental methods
 Always use suitable resources, models and images to support children’s understanding of calculation and place value, as appropriate
 Encourage children to make sensible choices about the methods they use when solving problems
EYFS
Early Learning in number and calculation follows the ‘Development Matters’ and towards the ‘Early Years Outcomes’ EYFS documents. This calculation policy is designed to build on progressively from the content and methods established in the Early Years Foundation Stage.
Addition – Early Stages (EYFS)
Children will engage in a wide variety of songs and rhymes, games and activities. They will begin to relate addition to combining two groups of objects, first by counting all and then by counting on from the largest number.
They will find one more than a given number.
In practical activities and through discussion they will begin to use the vocabulary involved in addition.
‘You have five apples and I have three apples. How many apples altogether?’
Subtraction – Early Stages (EYFS)
Children will engage in a variety of counting songs and rhymes and practical activities. In practical activities and through discussion they will begin to use the vocabulary associated with subtraction.
They will find one less than a given number.
They will begin to relate subtraction to ‘taking away’ using objects to count ‘how many are left’ after some have been taken away.
6 – 2 = 4 ‘Take two apples away. How many are left?’
Children will begin to count back from a given number.
Multiplication – Early Stages (EYFS)
Children will engage in a wide variety of songs and rhymes, games and activities. In practical activities and through discussion they will begin to solve problems involving doubling. ‘Three apples for you and three apples for me. How many apples altogether?’
Division – Early Stages (EYFS)
Children will engage in a wide variety of songs and rhymes, games and activities. In practical activities and through discussion they will begin to solve problems involving halving and sharing. Share the apples between two people. ‘Half of the apples for you and half of the apples for me.’
Calculation Guidelines for Early Years Foundation Stage 

ADDITION 
SUBTRACTION 
MULTIPLICATION 
DIVISION 
Children begin to record in the context of play or practical activities and problems. 

Begin to relate addition to combining two groups of objects already carried out. Solve simple word problems using their fingers Can find one more to ten. Higher Ability/ Gifted and Talented children progress to using a number line. They jump forwards along the number line using finger. 
Begin to relate subtraction to ‘taking away’ activities already carried out are left. Can find one less to ten. Higher Ability/ Gifted and Talented Progression: Counting backwards along a number line using finger. 
Real life contexts and use of practical equipment to count in repeated groups Also chanting in 2s, 5s and 10s. 
Share objects into equal groups Use related vocabulary Activities might include:
Count in twos, tens How many times? How many are left/left over? Group Answer Right, wrong What could we try next? How did you work it out? Share out Half, halve 
ADDITION
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught, and to acquire, secure mental methods of calculation, and one efficient written method of calculation for addition, which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient written method for addition of whole numbers by the end of Year 4. It is however essential, that appropriate mental strategies are taught alongside the written methods in this calculation policy. Note: Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).
Addition  Stage One
Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line. Key skills for addition at Stage 1:
 Read and write numbers to 100 in numerals, incl. 1—20 in words
 Recall bonds to 10 and 20, and addition facts within 20
 Count to and across 100
 Count in multiples of 1 2, 5 and 10
 Solve simple 1step problems involving addition, using objects, number lines and pictorial representations
Children should be able to  Add with numbers up to 20. Use number lines and number tracks to add by counting on in ones, to start with largest number and count on. Children should: Have access to a wide range of counting equipment, everyday objects, number tracks and number lines and be shown numbers in different contexts. Read and write the addition (+) and equals (=) signs within number sentences. Interpret addition number sentences and solving mixing box problems using concrete objects and number line addition to solve them: 8 + 4 = ___ __ + __ = 6 This should build on prior learning of adding by combining 2 objects.
To support understanding, pupils may physically make and carry calculation with Cuisenaire Rods, Dienes Base material or arrow cards, then compare their practical version to the written form, to help the understanding of it.
Addition  Stage Two
Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, addition, column, tens boundary. Key skills for addition at Stage 2:
 Add a 2digit number and units (e.g. 27 + 6)
 Add a 2digit number and tens (e.g. 23 + 40)
 Add pairs of 2digit numbers (e.g. 35 + 47)
 Add three singledigit numbers (e.g. 5 + 9 + 7)
 Show that adding can be done in any order (the commutative law)
 Recall bonds to 20 and bonds of tens to 100 (30 + 70 etc.)
 Count in steps of 2, 3 and 5 and count in tens from any number
 Understand the place value of 2digit numbers (tens and units)
 Compare and order numbers to 100 using < > and = signs
 Read and write numbers to at least 100 in numerals and words
 Solve problems with addition, using concrete objects, pictorial representations, involving numbers, quantities and measures, and applying mental and written methods
Children should be able to  Add with 2digit numbers: Developing mental fluency with addition and place value involving 2digit numbers, then establish more formal methods. As with Stage one  to support understanding, pupils may physically make and carry calculation with Cuisenaire Rods, Dienes Base material or arrow cards, then compare their practical version to the written form, to help the understanding of it.
Addition  Stage Three
Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, vertical, =carry‘, expanded, compact. Key skills for addition at Stage 3:
 Read and write numbers to 1000 in numerals and words
 Add 2digit numbers mentally, incl. those exceeding 100
 Add a threedigit number and units mentally (175 + 8)
 Add a threedigit number and tens mentally (249 + 50)
 Add a threedigit number and hundreds mentally (381 + 400)
 Estimate answers to calculations, using inverse to check answers
 Solve problems, including missing number problems, using number facts, place value, and more complex addition
 Recognise place value of each digit in 3digit numbers (hundreds, tens, units.)
 Continue to practise a wide range of mental addition strategies, ie. Number bonds, adding the nearest multiple of 10, 100, 100 and adjusting, using near doubles, partitioning and recombining
Children should be able to  Add numbers with up to 3digits, use the formal written method with the carry going into the next column. Children need to recognise the value of the hundreds, tens and units without recording the partitioning. Pupils need to be able to add in columns. Introduce the expanded column addition method for children who are struggling with understanding of formal written method.
Stage One 
Stage Two 
Stage Three 
+ = signs and missing numbers Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’. 2 = 1+ 1 2 + 3 = 4 + 1 3 = 3 2 + 2 + 2 = 4 + 2 Missing numbers need to be placed in all possible places. 3 + 4 = = 3 + 4 3 + = 7 7 = + 4 + 4 = 7 7 = 3 + + Ñ = 7 7 = + Ñ The Number Line Children use a numbered line to count on in ones. Children use number lines and practical resources to support calculation and teachers demonstrate the use of the number line. 7+ 4 
+ = signs and missing numbers Continue using a range of equations as in Stage 1 but with appropriate, larger numbers. Extend to 14 + 5 = 10 + and 32 + + = 100 35 = 1 + + 5 Partition into tens and ones and recombine 12 + 23 = 10 + 2 + 20 + 3 = 30 + 5 = 35 Count on in tens and ones 23 + 12 = 23 + 10 + 2 = 33 + 2 = 35 The Empty Number Line: Partitioning and bridging through 10. The steps in addition often bridge through a multiple of 10 Children should be able to partition the 7 to relate adding the 2 and then the 5. 8 + 7 = 15 Add 9 or 11 by adding 10 and adjusting by 1Add 9 by adding 10 and adjusting by 135 + 9 = 44 +10
1 
+ = signs and missing numbers Continue using a range of equations as in Stage 1 and 2 but with appropriate, larger numbers. Partition into tens and ones
36 + 53 = 53 + 30 + 6 = 83 + 6 = 89 Add a near multiple of 10 to a twodigit number Secure mental methods by using a number line to model the method. Continue as in Stage 2 but with appropriate numbers
Children need to be secure adding multiples of 10 to any twodigit number including those that are not multiples of 10. 48 + 36 = 84 pencil and paper procedures 83 + 42 = 125 either or 83 80 + 3 + _42 + 40 + 2 5 120 + 5 = 125 120 125 
Addition  Stage Four
Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens’ boundary, hundreds’ boundary, increase, vertical, carry, expanded, compact, thousands, hundreds, digits, inverse. Key skills for addition at Stage 4:
 Select most appropriate method: mental, jottings or written and explain why
 Recognise the place value of each digit in a fourdigit number
 Round any number to the nearest 10, 100 or 1000
 Estimate and use inverse operations to check answers
 Solve 2step problems in context, deciding which operations and methods to use and why
 Find 1000 more or less than a given number
 Continue to practise a wide range of mental addition strategies, i.e. Number bonds, add the nearest multiple of 10, 100, 1000 and adjust, use near doubles, partitioning and recombining
 Add numbers with up to 4 digits using the formal written method of column addition
 Solve 2step problems in contexts, deciding which operations and methods to use and why
 Estimate and use inverse operations to check answers to a calculation
Children should be able to  Add numbers with up to 4 digits. Move from expanded addition to the compact column method, adding units first, and ‘carrying’ numbers onto the top line in the correct column. Also include money and measures in contexts.
Addition  Stage Five
Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens’ boundary, hundreds’ boundary, increase, carry, expanded, compact, vertical, thousands, hundreds, digits, inverse & decimal places, decimal point, tenths, hundredths, thousandths. Key skills for addition at Stage5:
 Add numbers mentally with increasingly large numbers, using and practising a range of mental strategies i.e. Add the nearest multiple of 10, 100, 100 and adjust; use near doubles, inverse, partitioning and recombining; using number bonds
 Use rounding to check answers and accuracy.
 Solve multistep problems in contexts, deciding which operations and methods to use and why
 Read, write, order and compare numbers to at least 1 million and determine the value of each digit
 Round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000
 Add numbers with more than 4 digits using formal written method of columnar addition.
 Year 5 Add numbers with more than 4 digits including money, measures and decimals with different numbers of decimal places
Children should be able to  understand the place value of tenths and hundredths and use this to align numbers with different decimal places. Add numbers with more than 4 digits including money, measures and decimals with different numbers of decimal places.
Addition  Stage Six
Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens’ boundary, hundreds’ boundary, increase, carry, expanded, compact, vertical, thousands, hundreds, digits, inverse, decimal places, decimal point, tenths, hundredths, thousandths. Key skills for addition at Stage6:
 Perform mental calculations, including with mixed operations and large numbers, using and practising a range of mental strategies
 Solve multistep problems in context, deciding which operations and methods to use and why
 Use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy
 Read, write, order and compare numbers up to 10 million and determine the value of each digit
 Round any whole number to a required degree of accuracy
 Pupils understand how to add mentally with larger numbers and calculations of increasing complexity
Children should be able to  Add several numbers of increasing complexity. Add several numbers with different numbers of decimal places (including money and measures):
 Tenths, hundredths and thousandths should be correctly aligned, with the decimal point lined up vertically including in the answer row.
 Zeros could be added into any empty decimal places, to show there is no value to add.
 Adding several numbers with more than 4 digits.
Stage Four 
Stage Five 
Stage Six 
+ = signs and missing numbers Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers. Partition into tens and ones and recombineEither partition both numbers and recombine or partition the second number only e.g. 55 + 37 = 55 + 30 + 7 = 85 + 7 = 92 Add the nearest multiple of 10, then adjust Continue as in Stage 2 and 3 but with appropriate numbers e.g. 63 + 29 is the same as 63 + 30  1 Pencil and paper procedures367 + 185 = 431 either or 367 300 + 60 + 7 +185 100 + 80 + 5 12 400 +140+12 = 552 140 400 552 leading to 367 +185 552 1 1 Extend to decimals in the context of money. 
+ = signs and missing numbers Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers. Partition into hundreds, tens and ones and recombineEither partition both numbers and recombine or partition the second number only e.g. 358 + 73 = 358 + 70 + 3 = 428 + 3 = 431 Add or subtract the nearest multiple of 10 or 100, then adjust Continue as in Stage 2, 3 and 4 but with appropriate numbers e.g. 458 + 79 = is the same as 458 + 80  1 Pencil and paper procedures Extend to numbers with at least four digits 3587 + 675 = 4262 3587 + 675 4262 1 1 1 Revert to expanded methods if the children experience any difficulty. Extend to up to two places of decimals (same number of decimals places) and adding several numbers (with different numbers of digits). +54.6 127.4 1 1 
+ = signs and missing numbers Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers. Partition into hundreds, tens, ones and decimal fractions and recombineEither partition both numbers and recombine or partition the second number only e.g. = 42.8 + 0.3 = 43.1 Add the nearest multiple of 10, 100 or 1000, then adjust Continue as in Stage 2, 3, 4 and 5 but with appropriate numbers including extending to adding 0.9, 1.9, 2.9 etc Pencil and paper procedures Extend to numbers with any number of digits and decimals with 1, 2 and/or 3 decimal places. 13.86 + 9.481 23.341 1 1 1 Revert to expanded methods if the children experience any difficulty. 
Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level 

Extend to decimals with up to 2 decimal places, including:
Use compensation by adding too much, and then compensating 
SUBTRACTION
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for subtraction which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient method for subtraction of up to 5 whole numbers by the end of Stage 5.
To subtract successfully, children need to be able to: recall all addition and subtraction facts to 20; subtract multiples of 10 (such as 160 – 70) using the related subtraction fact,16 – 7, and their knowledge of place value; partition twodigit and threedigit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14). Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for subtraction. Using and Applying  Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).
Subtraction  Stage One
Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? Key skills for subtraction at Stage 1:
 Given a number, say one more or one less
 Count to and over 100, forward and back, from any number
 Represent and use subtraction facts to 20 and within 20
 Subtract with onedigit and twodigit numbers to 20, including zero
 Solve onestep problems that involve addition and subtraction, using concrete objects (ie bead string, objects, cubes) and pictures, and missing number problems
 Read and write numbers from 0 to 20 in numerals and words
Children should be able to  Subtract from numbers up to 20. Children consolidate understanding of subtraction practically, showing subtraction on bead strings, using cubes etc. and in familiar contexts, and introduced to more formal recording using number. Subtract by taking away on number lines. Find the distance between two points. Mental subtraction  Children should start recalling subtraction facts up to and within 10 and 20, and should be able to subtract zero.
Subtraction  Stage Two
Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back , how many left, how much less is_? difference, count on, strategy, partition, tens, units Key skills for subtraction at Stage 2:
 Recognise the place value of each digit in a twodigit number
 Recall and use subtraction facts to 20 fluently, and derive and use related facts up to 100
 Subtract using concrete objects, pictorial representations, 100 squares and subtract mentally, including: a twodigit number and units, a twodigit number and tens, and two twodigit numbers
 Show that subtraction of one number from another cannot be done in any order
 Recognise and use inverse relationship between addition and subtraction, using this to check calculations and missing number problems
 Solve simple addition and subtraction problems including measures, using concrete objects, pictorial representation, and also applying their increasing knowledge of mental and written methods
 Read and write numbers to at least 100 in numerals and in words
Children should be able to  Subtract with 2digit numbers. Subtract on a number line by counting back, aiming to develop mental subtraction skills. This strategy will be used for: 2digit numbers subtracting units (by taking away / counting back) e.g. 36—7, 2digit numbers subtracting tens (by taking away / counting back) e.g. 48—30, subtracting pairs of 2digit numbers.
Subtraction  Stage Three
Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back , how many left, how much less is_? difference, count on, strategy, partition, tens, units, borrowing, decrease, hundreds, value, digit Key skills for subtraction at Stage 3:
 Subtract mentally a: 3digit number and units, 3digit number and tens, 3digit number and hundreds
 Estimate answers and use inverse operations to check
 Solve problems, including missing number problems
 Find 10 or 100 more or less than a given number
 Recognise the place value of each digit in a 3digit number
 Counting up differences as a mental strategy when numbers are close together or near multiples of 10
 Read and write numbers up to 1000 in numerals and words
 Practise mental subtraction strategies, such as subtracting near multiples of 10 and adjusting (e.g. subtracting 19 or 21), and select most appropriate methods to subtract, explaining why
Children should be able to  Subtract with 2 and 3digit numbers. Introduce partitioned column subtraction method. Approximating before calculating answer should be encouraged. Counting on as a mental strategy for subtraction: Continue to reinforce counting on as a strategy for closetogether numbers (e.g. 121—118), and also for numbers that are nearly multiples of 10, 100, 1000 or £s, which make it easier to count on (e.g. 10289, 131—79, or calculating change from £1 etc.). Start at the smaller number and count on in tens first, then count on in units to find the rest of the difference:
Stage One 
Stage Two 
Stage Three 

 = signs and missing numbers 7  3 = = 7  3 7  = 4 4 =  3  3 = 4 4 = 7   Ñ = 4 4 =  Ñ
I have saved 5p. The socks that I want to buy cost 11p. How much more do I need in order to buy the socks?
I have 11 toy cars. There are 5 cars too many to fit in the garage. How many cars fit in the garage? 5 Use the vocabulary related to addition and subtraction and symbols to describe and record addition and subtraction number sentences Recording by  drawing jumps on prepared lines  constructing own lines 
 = signs and missing numbers Continue using a range of equations as in Stage 1 but with appropriate numbers. Extend to 14 + 5 = 20  Find a small difference by counting up42 – 39 = 3 Subtract 9 or 11. Begin to add/subtract 19 or 2135 – 9 = 26 Use known number facts and place value to subtract(partition second number only) 37 – 12 = 37 – 10 – 2 = 27 – 2 = 25
Bridge through 10 where necessary 32  17 
 = signs and missing numbers Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers. Find a small difference by counting up Continue as in Stage 2 but with appropriate numbers e.g. 102 – 97 = 5
Subtract mentally a ‘near multiple of 10’ to or from a twodigit number Continue as in Stage 2 but with appropriate numbers e.g. 78 – 49 is the same as 78 – 50 + 1
Use known number facts and place value to subtract Continue as in Year 2 but with appropriate numbers e.g.97 – 15 = 72 82 87 97 5 10 With practice, children will need to record less information and decide whether to count back or forward. It is useful to ask children whether counting up or back is the more efficient for calculations such as 57 – 12, 86 – 77 or 43 – 28. Pencil and paper proceduresComplementary addition 84 – 56 = 28 +20 +4 +4 56 60 80 84 
Subtraction  Stage Four
Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units, borrowing, decrease, hundreds, value, digit, inverse. Key skills for subtraction at Stage 4:
 Subtract by counting on where numbers are close together or they are near to multiples of 10, 100 etc
 Children select the most appropriate and efficient methods for given subtraction calculations
 Estimate and use inverse operations to check answers
 Solve addition and subtraction 2step problems, choosing which operations and methods to use and why
 Solve simple measure and money problems involving fractions and decimals to two decimal places
 Find 1000 more or less than a given number
 Count backwards through zero, including negative numbers
 Recognise place value of each digit in a 4digit number Round any number to the nearest 10, 100 or 1000
 Solve number and practical problems that involve the above, with increasingly large positive numbers
Children should be able to  Subtract with up to 4digit numbers Partitioned column subtraction with borrowing. (decomposition): Mental strategies: A variety of mental strategies must be taught and practised, including counting on to find the difference where numbers are closer together, or where it is easier to count on.
Subtraction  Stage Five
Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units, borrowing, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal. Key skills for subtraction at Stage 5:
 Subtract numbers mentally with increasingly large numbers
 Use rounding and estimation to check answers to calculations and determine, in a range of contexts, levels of accuracy
 Solve addition and subtraction multistep problems in context, deciding which operations and methods to use and why
 Read, write, order and compare numbers to at least 1 million and determine the value of each digit
 Count forwards or backwards in steps of powers of 10 for any given number up to 1 million
 Interpret negative numbers in context, counting forwards and backwards with positive and negative integers through 0
 Round any number up to 1 million to the nearest 10, 100, 1000, 10 000 and 100 000
Children should be able to  Subtract with at least 4digit numbers including money, measures, decimals. Compact column subtraction (with borrowing.). Children who are still not secure with number facts and place value will need to remain on the partitioned column method until ready for the compact method. Subtracting with larger integers. Subtract with decimal values, including mixtures of integers and decimals, aligning the decimal point.
Subtraction  Stage Six
Key vocabulary: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back , how many left, how much less is_? difference, count on, strategy, partition, tens, borrowing, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal. Key skills for subtraction at Stage6:
 Solve addition and subtraction multistep problems in context, deciding which operations and methods to use and why
 Read, write, order and compare numbers up to 10 million and determine the value of each digit
 Round any whole number to a required degree of accuracy
 Use negative numbers in context, and calculate intervals across zero
 Children need to utilise and consider a range of mental subtraction strategies, jottings and written methods before choosing how to calculate
Children should be able to  Subtract with increasingly large and more complex numbers and decimal values. Use the compact column method to subtract more complex integers. Use the compact column method to subtract money and measures, including decimals with different numbers of decimal places. Pupils should be able to apply their knowledge of a range of mental strategies, mental recall skills, and informal and formal written methods when selecting the most appropriate method to work out subtraction problems.
Stage Four 
Stage Five 
Stage Six 
Find a small difference by counting upThis can be modelled on an empty number line (see complementary addition below). Children should be encouraged to use known number facts to reduce the number of steps. Subtract the nearest multiple of 10, then adjust. Continue as in Stage 2 and 3 but with appropriate numbers. Use known number facts and place value to subtract 92 – 25 = 67 Pencil and paper proceduresComplementary addition 754 – 86 = 668 For those children with a secure mental image of the number line they could record the jumps only:754 – 86 = 668
14 (100) 600 (700) 54 (754) 668 
Find a difference by counting upThis can be modelled on an empty number line (see complementary addition below). Subtract the nearest multiple of 10 or 100, then adjust. Continue as in Stage 2, 3 and 4 but with appropriate numbers. Use known number facts and place value to subtract Pencil and paper proceduresComplementary addition 754 – 286 = 468 OR754  286 = 468 14 (300) can be refined to 14 (300) 400 (700) 454 (754) 54 (754) 468 468 Reduce the number of steps to make the calculation more efficient. Extend to 2 places of decimals 
Find a difference by counting up
To make this method more efficient, the number of steps should be reduced to a minimum through children knowing:
Subtract the nearest multiple of 10, 100 or 1000, then adjust Continue as in Stage 2, 3, 4 and 5 but with appropriate numbers. Use known number facts and place value to subtract Pencil and paper proceduresComplementary addition 6467 – 2684 = 3783 OR 6467 – 2684 = 3783 16 (2700) can be refined to 316 (3000) 300 (3000) 3467 (6467) 3467 (6467) 3783 3783 Reduce the number of steps to make the calculation more efficient. Extend to 2 places of decimals 
Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level 
Mental methods Use compensation by subtracting too much, and then compensating Use jottings such as an empty number line to support or explain methods for adding mentally. Pencil and paper procedures (Written methods) Subtract more complicated fractions For Example: Extend to decimals with up to 2 decimal places, including:
digits
Complementary addition 
MULTIPLICATION
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for multiplication which they know they can rely on when mental methods are not appropriate.
These notes show the stages in building up to using an efficient method for by the end of Stage 4, twodigit by twodigit multiplication by the end of Stage 5, and threedigit by twodigit multiplication by the end of Stage 6.
To multiply successfully, children need to be able to:
 Recall all multiplication facts to 12 × 12
 Partition number into multiples of one hundred, ten and one
 Work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using the related fact 7 × 5 and their knowledge of place value
 Add two or more singledigit numbers mentally
 Add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value
 Add combinations of whole numbers using the column method (see above).
 Use short multiplication to multiply a 1 digit number by a number with up to four digits
 Use long multiplication to multiply 3 digit and four digit numbers by a number between 11 – 20 by the end of Stage 5
 Use long multiplication to multiply a two digit number with up to four digits Use short multiplication to multiply a one digit number by a number with one or two decimal places including money.
Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication.
Using and Applying  Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).
Multiplication  Stage One
Key vocabulary: groups of, lots of, times, array, altogether, multiply, count. Key skills for multiplication at Stage 1:
 Count in multiples of 2, 5 and 10
 Solve onestep problems involving multiplication, by calculating the answer using concrete objects
 Pictorial representations and arrays with the support of the teacher
 Make connections between arrays, number patterns, and counting in twos, fives and tens
 Begin to understand doubling using concrete objects and pictorial representations
Children should be able to  Multiply with concrete objects, arrays and pictorial representations. How many legs will 3 teddies have? There are 3 sweets in one bag. How many sweets are in 5 bags altogether? 3 + 3 + 3 + 3 + 3 = 15
Give children experience of counting equal group of objects in 2s, 5s and 10s. Present practical problem solving activities involving counting equal sets or groups, as above.
Multiplication  Stage Two
Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times. Key skills for multiplication at Stage2:
 Count in steps of 2, 3 and 5 from zero, and in 10s from any number
 Recall and use multiplication facts from the 2, 5 and 10 multiplication tables, including recognising odds and evens
 Write and calculate number statements using the x and = signs
 Show that multiplication can be done in any order (commutative)
 Solve a range of problems involving multiplication, using concrete objects, arrays, repeated addition, mental methods, and multiplication facts
 Pupils use a variety of language to discuss and describe multiplication
Children should be able to  Multiply using arrays and repeated addition (using at least 2s, 5s and 10s). Use repeated addition on a number line: Starting from zero, make equal jumps up on a number line to work out multiplication facts and write multiplication statements using x and = signs. Use arrays: to help teach children to understand the commutative law of multiplication, and give examples such as 3 x __ = 6. Use practical equipment, use mental recall:  Children should begin to recall multiplication facts for 2, 5 and 10 times tables through practice in counting and understanding of the operation.
Multiplication  Stage Three
Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times, ‘_ times as big as’, once, twice, three times, partition, grid method, multiple, product, tens, units, value Key skills for multiplication at Stage 3:
 Recall and use multiplication facts for the 2, 3, 4, 5, 8 and 10 multiplication tables, and multiply multiples of 10
 Write and calculate number statements using the multiplication tables they know, including 2digit x singledigit, drawing upon mental methods, and progressing to reliable written methods
 Solve multiplication problems, including missing number problems
 Develop mental strategies using commutativity (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240)
 Solve simple problems in contexts, deciding which operations and methods to use
 Develop efficient mental methods to solve a range of problems e.g. using commutativity (4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and for missing number problems x 5 = 20, 3 x = 18, x = 32
Children should be able to  Multiply 2digits by a single digit number. Introduce the grid method for multiplying 2digit by singledigits. Introduce the grid method with children physically making an array to represent the calculation (e.g. make 8 lots of 23 with 10s and 1s place value counters), then translate this to grid method format. To do this, children must be able to:
 Partition numbers into tens and units
 Multiply multiples of ten by a single digit (e.g. 20 x 4) using their knowledge of multiplication facts and place value
 Recall and work out multiplication facts in the 2, 3, 4, 5, 8 and 10 times tables.
 Work out multiplication facts not known by repeated addition or other taught mental strategies (e.g. by commutative law, working out near multiples and adjusting, using doubling etc.)
Strategies to support this are repeated addition using a number line, bead bars and arrays.
Stage One 
Stage Two 
Stage Three 
Multiplication is related to doubling and counting groups of the same size.
Looking at columns Looking at rows 2 + 2 + 2 3 + 3 3 groups of 2 2 groups of 3 Counting using a variety of practical resources Counting in 2s e.g. counting socks, shoes, animal’s legs… Counting in 5s e.g. counting fingers, fingers in gloves, toes… Counting in 10s e.g. fingers, toes… Pictures / marks There are 3 sweets in one bag. How many sweets are there in 5 bags? 
x = signs and missing numbers 7 x 2 = = 2 x 7 7 x = 14 14 = x 7 x 2 = 14 14 = 2 x x Ñ = 14 14 = x Ñ Arrays and repeated additionl l l l 4 x 2 or 4 + 4 l l l l 2 x 4 or 2 + 2 + 2 + 2 Doubling multiples of 5 up to 5015 x 2 = 30 PartitionChildren need to be secure with partitioning numbers into 10s and 1s and partitioning in different ways: 6 = 5 + 1 so
AND double 15 10 + 5
20 + 10 = 30 ORX 10 5 2 20 10 = 30 
x = signs and missing numbers Continue using a range of equations as in Stage 2 but with appropriate numbers. Arrays and repeated additionContinue to understand multiplication as repeated addition and continue to use arrays (as in Stage 2). Doubling multiples of 5 up to 5035 x 2 = 70 PartitionX 30 5 2 60 10 =70
Use the same method as above (partitioning), e.g. 32 x 3 = 96 = 96 
Multiplication  Stage Four
Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, groups of, sets of, lots of, equal groups, times, multiply, times as big as, once, twice, three times... partition, grid method, total, multiple, product, sets of, inverse Key skills for multiplication at Stage 4:
 Count in multiples of 6, 7, 9, 25 and 1000
 Recall multiplication facts for all multiplication tables up to 12 x 12
 Recognise place value of digits in up to 4digit numbers
 Use place value, known facts and derived facts to multiply mentally, e.g. multiply by 1, 10, 100, by 0, or to multiply 3 numbers
 Use commutativity and other strategies mentally 3 x 6 = 6 x 3 , 2 x 6 x 5 = 10 x 6 , 39x7 = 30 x 7 + 9 x 7
 Solve problems with increasingly complex multiplication in a range of contexts
 Count in multiples of 6, 7, 9, 25 and 1000
 Recognise the place value of each digit in a fourdigit number (thousands, hundreds, tens, and units)
Children should be able to  Multiply 2 and 3digits by a single digit, using all multiplication tables up to 12 x 12 Developing the grid method: Encourage the use of column addition when adding. Move onto short multiplication (see Stage 5) if and when children are confident and accurate multiplying 2 and 3digit numbers by a single digit this way, and are already confident in carrying for written addition. To do this, children should: Approximate before they calculate, and make this a regular part of their calculating, going back to the approximation to check the reasonableness of their answer. e.g: 346 x 9 is approximately 350 x 10 = 3500. Record an approximation to check the final answer against. Multiply multiples of ten and one hundred by a singledigit, using their multiplication table knowledge. Recall all times tables up to 12 x 12
Multiplication  Stage Five
Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, ‘_times as big as’, once, twice, three times, partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short/long multiplication, carry. Key skills for multiplication at Stage 5:
 Identify multiples and factors, using knowledge of multiplication tables to 12x12
 Solve problems where larger numbers are decomposed into their factors
 Multiply and divide integers and decimals by 10, 100 and 1000
 Recognise and use square and cube numbers and their notation
 Solve problems involving combinations of operations, choosing and using calculations and methods appropriately
Children should be able to  Multiply up to 4digits by 1 or 2 digits. Be introduced to column multiplication  Introduce by comparing a grid method calculation to a short multiplication method, to see how the steps are related, but notice how there are less steps involved in the column method. Children need to be taught to approximate first, e.g. for 72 x 38, they will use rounding: 72 x 38 is approximately 70 x 40 = 2800, and use the approximation to check the reasonableness of their answer. Use short multiplication for multiplying by a single digit. Pupils could be asked to work out a given calculation using the grid, and then compare it to the column method. What are the similarities and differences? Unpick the steps and show how it reduces the steps. Introduce long multiplication for multiplying by 2 digits. The grid could be used to introduce long multiplication, as the relationship can be seen in the answers in each row. Moving towards more complex numbers.
Multiplication  Stage Six
Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times, partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short / long multiplication, carry, tenths, hundredths, decimal. Key skills for multiplication at Stage 6:
 Recall multiplication facts for all times tables up to 12 x 12 (as Y4 and Y5)
 Multiply multidigit numbers, up to 4digit x 2digit using long multiplication
 Perform mental calculations with mixed operations and large numbers
 Solve multistep problems in a range of contexts, choosing appropriate combinations of operations and methods
 Estimate answers using round and approximation and determine levels of accuracy
 Round any integer to a required degree of accuracy. Suggested Video clips:  Moving from grid method to a compact method (YouTube)  Reinforcing rapid times table recall: (YouTube)  Demonstration of long multiplication (SLEP)
Children should be able to  use Short and long multiplication as in Stage 5, and multiply decimals with up to 2d.p by a single digit. Use rounding and place value to make approximations before calculating and use these to check answers. Use short multiplication (see Stage 5) to multiply numbers with more than 4digits by a single digit; to multiply money and measures. Use long multiplication (see Stage 5) to multiply numbers with at least 4 digits by a 2digit number.
Stage Four 
Stage Five 
Stage Six 

x = signs and missing numbers Continue using a range of equations as in Stage 2 but with appropriate numbers PartitionContinue to use arrays: 18 x 9 = 162 18 x 9 = (10 x 9) + (8 x 9) = 162 ORUse the grid method of multiplication (as below) Pencil and paper procedures Grid method 23 x 7 is approximately 20 x 10 = 200
x 20 3 7 140 21 = 161 
Partition47 x 6 = 282 47 x 6 = (40 x 6) + (7 x 6) = 282 ORUse the grid method of multiplication (as below) Pencil and paper procedures Grid method 72 x 38 is approximately 70 x 40 = 2800 2100 + 60 = 2160 560 + 16 = 576 2160 560 + 2736 Expanded Column Multiplication Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 × 7 is ‘thirty multiplied by seven’, not ‘three times seven’, although the relationship 3 × 7 should be stressed. 30 + 8 38 x 7 x 7 56 (8 x 7 = 56) 56 210 (30 x 7 = 210) 210 266 266 
Partition87 x 6 = 522 87 x 6 = (80 x 6) + (7 x 6) = 522 ORUse the grid method of multiplication (as below) Pencil and paper proceduresGrid method 372 x 24 is approximately 400 x 20 = 8000 Extend to decimals with up to two decimal places. Short Column Multiplication The recording is reduced further, with carry digits recorded below the line. 38 x 7 266 5 Children who are already secure with multiplication for TU × U and TU × TU should have little difficulty in using the same method for HTU × TU or applying decimals. 286 x 29 2574 (9 x 286 = 2574) 5720 (20 x 286 = 5720) 8294 1 

Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level 

Mental methods Use partitioningPartition either part of the product e.g. 7.3 x 11 = (7.3 x 10) + 7.3 = 80.3 ORUse the grid method of multiplication (as below). Pencil and paper procedures (Written methods)Use written methods to support, record or explain multiplication of:
Grid method = 49.92 Grid lines can become optional 

DIVISION
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for division which they know they can rely on when mental methods are not appropriate.
These notes show the stages in building up to long division through Stages 3 to 6 – first long division TU ÷ U, extending to HTU ÷ U, then HTU ÷ TU, and then short division HTU ÷ U. To divide successfully in their heads, children need to be able to:
 Understand and use the vocabulary of division – for example in 18 ÷ 3 = 6, the 18 is the dividend, the 3 is the divisor and the 6 is the quotient
 Partition twodigit and threedigit numbers into multiples of 100, 10 and 1 in different ways
 Recall multiplication and division facts to 10 × 10, recognise multiples of onedigit numbers and divide multiples of 10 or 100 by a singledigit number using their knowledge of division facts and place value
 Know how to find a remainder working mentally – for example, find the remainder when 48 is divided by 5
 Understand and use multiplication and division as inverse operations
Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division.
To carry out written methods of division successful, children also need to be able to:
 Understand division as repeated subtraction
 Estimate how many times one number divides into another – for example, how many sixes there are in 47, or how many 23s there are in 92
 Multiply a twodigit number by a singledigit number mentally
 Subtract numbers using the column method
Using and Applying  Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).
Division  Stage One
Key Vocabulary: share, share equally, one each, two each…, group, groups of, lots of, array, Key skills for division at Stage 1:
 Solve onestep problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations arrays with the support of the teacher
 Through grouping and sharing small quantities, pupils begin to understand division, and finding simple fractions of objects, numbers and quantities
 Children can make connections between arrays, number patterns, and counting in twos, fives and tens
Children should be able to  Group and share large quantities. Using objects, diagrams and pictorial representations to solve problems involving both grouping and sharing. Pupils should:  Use lots of practical apparatus, arrays and picture representations. Be taught to understand the difference between ‘grouping’ objects (How many groups of 2 can you make?) and ‘sharing’. (Share these sweets between 2 people). Be able to count in multiples of 2s, 5s and 10s. Find half of a group of objects by sharing into 2 equal groups.
Division  Stage Two
Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over Key skills for division at Stage 2:
 Count in steps of 2, 3, and 5 from 0
 Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
 Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the x, ÷ and = signs
 Show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot
 Solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts
Children should be able to  Group and share, using the ÷ and = sign Use objects, arrays, diagrams and pictorial representations, and grouping on a number line. Pose 12 ÷ 3 as ‘How many groups of 3 are in 12?’
Division  Stage Three
Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, carry, remainder, multiple . Key skills for division at Stage 3:
 Recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables (through doubling, connect the 2, 4 and 8s)
 Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for twodigit numbers times onedigit numbers, using mental and progressing to formal written methods
 Solve problems, in contexts, and including missing number problems, involving multiplication and division
 Pupils develop efficient mental methods, for example, using multiplication and division facts (e.g. using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (30 × 2 = 60, so 60 ÷ 3 = 20 and 20 = 60 ÷ 3)
 Pupils develop reliable written methods for division, starting with calculations of 2digit numbers by 1digit numbers and progressing to the formal written method of short division
Children should be able to  Divide 2digit numbers by a single digit (where there is no remainder in the final answer). Step 1: When the answer for the first column is zero (1 ÷ 5, as in example), children could initially write a zero above to acknowledge its place, and must always carry the number (1) over to the next digit as a remainder. STEP 2: Pupils move onto dividing numbers with up to 3digits by a single digit, however problems and calculations provided should not result in a final answer with remainder at this stage. Children who exceed this expectation may progress to Stage 5. Include money and measure contexts when confident. Real life contexts need to be used routinely to help pupils gain a full understanding, and the ability to recognise the place of division and how to apply it to problems .
Stage One 
Stage Two 
Stage Three 
Sharing Requires secure counting skills see counting and understanding number strand Develops importance of onetoone correspondence See appendix for additional information on x and ÷ and aspects of number
Sharing – 6 sweets are shared between 2 people. How many do they have each? lll lll Practical activities involving sharing, distributing cards when playing a game, putting objects onto plates, into cups, hoops etc. Grouping Sorting objects into 2s / 3s/ 4s etc How many pairs of socks are there? There are 12 crocus bulbs. Plant 3 in each pot. How many pots are there? Jo has 12 Lego wheels. How many cars can she make? 
÷ = signs and missing numbers 6 ÷ 2 = = 6 ÷ 2 6 ÷ = 3 3 = 6 ÷ ÷ 2 = 3 3 = ÷ 2 ÷ Ñ = 3 3 = ÷ Ñ Grouping Link to counting and understanding number strand Count up to 100 objects by grouping them and counting in tens, fives or twos;… Find one half, one quarter and three quarters of shapes and sets of objects 6 ¸ 2 can be modelled as: There are 6 strawberries. How many people can have 2 each? How many 2s make 6? 6 ¸ 2 can be modelled as:
In the context of money count forwards and backwards using 2p, 5p and 10p coins Practical grouping e.g. in PE 12 children get into teams of 4 to play a game. How many teams are there? 
÷ = signs and missing numbers Continue using a range of equations as in Stage 2 but with appropriate numbers. Understand division as sharing and grouping18 ÷ 3 can be modelled as: Sharing – 18 shared between 3 (see Year 1 diagram) ORGrouping  How many 3’s make 18? 0 3 6 9 12 15 18 Remainders16 ÷ 3 = 5 r1 Sharing  16 shared between 3, how many left over? Grouping – How many 3’s make 16, how many left over? 0 3 6 9 12 15 16 
Division  Stage Four
Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, carry, remainder, multiple, divisible by, factor. Key skills needed for division at Stage 4:
 Recall multiplication and division facts for all numbers up to 12 x 12
 Use place value, known and derived facts to multiply and divide mentally, including: multiplying and dividing by 10 and 100 and 1
 Pupils practise to become fluent in the formal written method of short division with exact answers when dividing by a onedigit number
 Pupils practise mental methods and extend this to threedigit numbers to derive facts, for example 200 × 3 = 600 so 600 ÷ 3 = 200
 Pupils solve twostep problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as three cakes shared equally between 10 children
Children should be able to  Divide up to 3digit numbers by a single digit (without remainders initially) Continue to develop short division: STEP 1: Pupils must be secure with the process of short division for dividing 2digit numbers by a single digit (those that do not result in a final remainder  see steps in Stage 3), but must understand how to calculate remainders, using this to carry remainders within the calculation process. Short division should only be taught once children have secured the skill of calculating “remainders”. Real life contexts need to be used routinely to help pupils gain a full understanding and the ability to recognise the place of division and how to apply it to problems.
Division  Stage Five
Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, carry, remainder, multiple, divisible by, factor, inverse, quotient, prime number, prime factors, composite number (nonprime). Key skills for division at Stage 5:
 Recall multiplication and division facts for all numbers up to 12 x 12 (as in Y4)
 Multiply and divide numbers mentally, drawing upon known facts
 Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
 Solve problems involving multiplication and division where larger numbers are decomposed into their factors
 Multiply and divide whole numbers and those involving decimals by 10, 100 and 1000
 Use the vocabulary of prime numbers, prime factors and composite (nonprime) numbers
 Work out whether a number up to 100 is prime, and recall prime numbers to 19
 Divide numbers up to 4 digits by a onedigit number using the formal written method of short division and interpret remainders appropriately for the context
 Use multiplication and division as inverses
 Interpret noninteger answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (e.g. 98 ÷ 4 = 24 r 2 = 24 ½ = 24.5 ˜ 25)
 Solve problems involving combinations of all four operations, including understanding of the equals sign, and including division for scaling by different fractions and problems involving simple rates.
Children should be able to  Divide up to 4 digits by a single digit, including those with remainders. Short division with remainders: Now that pupils are introduced to examples that give rise to remainder answers, division needs to have a real life problem solving context, where pupils consider the meaning of the remainder and how to express it, i.e. as a fraction, a decimal, or as a rounded number or value, depending upon the context of the problem. The answer to 5309 ÷ 8 could be expressed as 663 and five eighths, 663 r 5, as a decimal, or rounded as appropriate to the problem involved. Include money and measure contexts. See Stage 6 for how to continue the short division to give a decimal answer for children who are confident. If children are confident and accurate: Introduce long division for pupils who are ready to divide any number by a 2digit number (e.g. 2678 ÷ 19). This is a Stage 6 expectation.
Division  Stage Six
Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, carry, remainder, multiple, divisible by, factor, inverse, quotient, prime number, prime factors, composite number (nonprime), common factor. Key skill for division at Stage 6:
 Recall and use multiplication and division facts for all numbers to 12 x 12 for more complex calculations
 Divide numbers up to 4 digits by a twodigit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context
 Use short division where appropriate
 Perform mental calculations, including with mixed operations and large numbers
 Identify common factors, common multiples and prime numbers
 Solve problems involving all 4 operations
 Use estimation to check answers to calculations and determine accuracy, in the context of a problem
 Use written division methods in cases where the answer has up to two decimal places
 Solve problems which require answers to be rounded to specified degrees of accuracy
Children should be able to  Divide at least 4 digits by both singledigit and 2digit numbers (including decimal numbers and quantities). Short division with remainders: Pupils should continue to use this method, but with numbers to at least 4 digits, and understand how to express remainders as fractions, decimals, whole number remainders, or rounded numbers. Real life problem solving contexts need to be the starting point, where pupils have to consider the most appropriate way to express the remainder.
Stage Four 
Stage Five 
Stage Six 
÷ = signs and missing numbers Continue using a range of equations as in Stage 2 but with appropriate numbers. Sharing and grouping30 ÷ 6 can be modelled as: grouping – groups of 6 placed on no. line and the number of groups counted e.g. sharing – sharing among 6, the number given to each person Remainders 41 ÷ 4 = 10 r1 41 = (10 x 4) + 1Pencil and paper procedures Chunking.72 ÷ 5 lies between 50 ¸ 5 = 10 and 100 ¸ 5 = 20 * Partition the dividend into multiples of the divisor: 50 ÷ 5 = 10 22 ÷ 5 = 4r2 ® 10 + 4r2 = 14 r 2 OR 72  50 (10 groups) 22  20 (4 groups) 2 Answer : 14 remainder 2 
Sharing and groupingContinue to understand division as both sharing and grouping (repeated subtraction). RemaindersQuotients expressed as fractions or decimal fractions 61 ÷ 4 = 15 ¼ or 15.25 Pencil and paper procedures Chunking256 ÷ 7 lies between 210 ¸ 7 = 30 and 280 ¸ 7 = 40 * Partition the dividend into multiples of the divisor: 210 ÷ 7 = 30 46 ÷ 7 = 6r4 ® 30 + 6r4 = 36r4 OR 256  210 (30 groups) 46  42 (6 groups) 4 Answer: 36 remainder 4 Also, Short Division for More Able Children
Considering each column starting from the left. See Stage Six for full explanation. 
Sharing, grouping and remainders as Stage FivePencil and paper procedures Chunking977 ÷ 36 is approximately 1000 ¸ 40 = 25 * Partition the dividend into multiples of the divisor: 720 ÷ 36 = 20 180 ÷ 36 = 5 77 ÷ 36 = 2r5 ® 20 + 5 + 2r5 = 27r5 OR 977  720 (20 groups) 257  180 (5 groups) 77  72 (2 groups) 5 Answer: 27 ^{5}/_{36} Pencil and Paper procedures Short Division Method Write down how many times your divisor goes into the first number of the dividend.If there is a remainder, that's okay. Write down your remainderto the left of the next digit in the dividend.
Both methods above are necessary at this stage, to deal with the wide range of problems experienced at Stage Six. 
Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level 

Pencil and paper procedures (Written methods) Use written methods to support, record or explain division of:
Refine methods to improve efficiency while maintaining accuracy and understanding.
109.6  80 (10 groups of 8) 29.6  24 ( 3 ) 5.6  5.6 ( 0.7 ) 0.0 Answer: 13.7 
Pencil and paper procedures (Written methods) Continue to use the same method as in Year 7 and Year 8. Adjust the dividend and divisor by a common factor before the division so that no further adjustment is needed after the calculation e.g. 361.6 ÷ 0.8 is equivalent to 3616 ÷ 8 Use the inverse rule to divide fractions, first converting mixed numbers to improper fractions. Look at one half of a shape. How many sixths of the shape canyou see? (six) So, how many sixths in one half? (three) So ½ ÷ ^{1}/_{6 }= ½ x ^{6}/_{1 } _{ }= ^{6}/_{2} _{ }= 3 
Date Policy Adopted by Governing Body 
Review dates 

April 2015 
April 2017 
April 2018 
Policy developed by: C. Vanson (Headteacher) April 2015