Harcourt Primary School Calculation Policy 2016-2017 2017 (Aligned with the 2014 National Curriculum and NCETM guidance) The following calculation policy has been devised to meet the requirements of the National Curriculum 2014 for the teaching and learning of mathematics. It is also designed to provide children with a consistent progression of learning in calculations across the school. (Please note that early learning in number and calculation in Reception follows the Development Matters EYFS document, and this calculation policy is designed to build on progressively from the content and methods established in the Early Years Foundation Stage.) The policy will also seek to embed concepts of reasoning and mastery within the progression through each stage. Age related expectations The calculation policy is organised according to age related expectations as set out in the National Curriculum 2014, however it is vital that pupils are also taught according to the stage that they are currently working at, developing mastery and reasoning at greater depth where the concept is secure, or being supported through targeted teaching and support where there are gaps in their understanding, in order that they can continue to work within age related expectations. Teaching should focus on age related expectations for all children and where children are not yet meeting these expectations, interventions should be used both within the class and externally with TAs where appropriate. Children should not be held back from new concepts, or moved on to the next banding, but should be given the opportunity to work at their age related level at all times with differentiated levels of reasoning and mastery. Providing a context for calculation: It is important that any type of calculation is given a real life context and problem solving application, to help build children s understanding of the purpose of calculation, and to help them recognise when to use certain operations and methods when faced with problems. This must be a priority within all lessons. Reasoning and Mastery must be embedded in all lessons: Children must always be encouraged to explain their thinking processes, however straight forward a calculation may be. (Reasoning) Children should be given opportunities to explore concepts in unusual formats in order that they can effectively choose and apply appropriate methods. (Mastery) 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 select and use these methods appropriately, accurately with confidence and understanding HOW TO USE THIS POLICY: Use the policy as the basis of your planning but ensure you use previous 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, use targeted intervention to return to the previous stage in calculation for those children Cross reference with the Mental Maths Policy 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 and explain sensible choices about the methods they use when solving problems Choosing a calculation method: Children will use mental methods as their first port of call when appropriate, but for calculations that they cannot do in their heads, they will need to use an efficient written method accurately and with confidence. Children need to be taught and encouraged to use the following processes in deciding which approach they will take to a calculation, to ensure they select the most appropriate method for the numbers/processes involved:
General Progression and support: establish mental methods based on secure understanding of place value estimate reasonable outcomes encourage use of informal jottings to aid mental calculations develop use of empty number lines to help mental imagery and aid recording use models and images to promote understanding use partitioning and recombining to aid informal methods use inverse to check calculations introduce expanded methods develop expanded methods into compact standard written form Once a child has become secure in a particular form they should not move back to previous methods. When are children ready for formal written methods? Before formal methods can be used it is necessary for the children to have a secure understanding of the following: Addition/Subtraction: place value to 3 digits partition 3 digit numbers secure knowledge of facts to 20 add 3 single digits mentally add/subtract pairs of 2 digit numbers mentally with a strategy of their choice explain their strategy orally and record with jottings Multiplication/Division: know 2,3,4,5,10 x tables know result of x by 0 or 1 understand 0 as a place holder multiply 2/3 digit numbers by 10 and 100 double and halve 2 digits mentally use multiplication facts to mentally derive others explain mental strategies orally and record with jottings Vocabulary: A consistent approach to vocabulary must be used across the school. Terminology must be accurate and unambiguous. Use a wide range of terms for each calculation relevant to the context of the work Reinforce processes by wrote for multi step processes Decimal points do not move! In column methods exchange never borrow When multiplying an integer by 10 we do not add a 0 Effective use of questioning: Open ended questioning should be developed to allow children s reasoning skills to develop: What s the same What s different? Odd one out (multiple answers) What was the question? True or False? Which is the best method? Sentence stems (see opposite)
Rationale for KS1 Children in Years 1 and 2 will be given a really solid foundation in the basic building blocks of mental and written arithmetic. Place value Children will develop an understanding of how numbers work, so that they are confident in 2-digit numbers and beginning to read and say numbers above 100. They will be able to explain the value of digits. Number bonds First via practical hands-on experiences and subsequently using memorisation techniques, enables a good grounding in these crucial facts. All children leave Y2 knowing the pairs of numbers which make all the numbers up to 10 at least. They will also have experienced and been taught pairs to 20. Explain how they know the pairs add up. Number facts Add several single-digit numbers. Add/subtract a single digit number to/from a 2-digit number. Add/subtract 1 or 10, and to understand which digit changes and why. Extended to enable children to add and subtract multiples of ten to and from any 2-digit 2 number. Add or subtract any pair of 2- digit numbers by counting on or back in tens and ones. Children may extend this to adding by partitioning numbers into tens and ones. Children will be taught to count in 2s, 3s, 5s and 10s, and will have related this skill to repeated addition. They will have met and begun to learn the associated 2x, 3x, 5x and 10x tables. Multiplication/Division Engaging in a practical way with the concept of repeated addition and the use of arrays enables children to develop a preliminary understanding of multiplication, and asking them to consider how many groups of a given number make a total will introduce them to the idea of division. They will also be taught to double and halve numbers, and will thus experience scaling up or down as a further aspect of multiplication and division. Fractions will be introduced as numbers and as operators, specifically in relation to halves, quarters and thirds. Reasoning Children must be encouraged to discuss their thinking at every stage. They should explain how they know, find the odd one out, make connections between number facts. Children at this stage should be learning to use the correct terminology to explain the processes that they are learning. They should be given opportunities to explain errors and how they can be corrected. Mastery Children should be given a wide range of opportunities to explore problems in a range of formats (eg missing numbers/signs; a range of real life contexts and pictorial representations such as money, measure etc; unusually ordered number sentences). Cross curricular links are essential to mastery.
Key to this is for children to be taught to view an = sign as a sign of equivalence and not as a position for an answer to a question. Use visual representations to reinforce this. Rationale for Lower KS2 Strong links will be needed between the teachers of KS1 and lower KS2 to ensure a smooth progression and continued rate of progress across the key stages. Children build on the concrete and conceptual understandings they have gained in KS1 to develop a real mathematical understanding of the four operations, in particular developing Arithmetical competence in relation to larger numbers. Addition and subtraction: use place value and number facts to add and subtract numbers mentally develop a range of strategies to enable them to discard the counting in ones or fingers-based methods of KS1. add and subtract multiples and near multiples of 10, 100 and 1000 become fluent in complementary addition as an accurate means of achieving fast and accurate answers to 3-digit subtractions. Standard written methods for adding larger numbers are taught, learned and consolidated, and written column subtraction is also introduced. Multiplication and Division: This key stage is also the period during which all the multiplication and division facts are thoroughly memorised, including all facts up to the 12 x 12 table. Efficient written methods for multiplying or dividing a 2-digit or 3-digit number by as single-digit number are taught mental strategies for multiplication or division are taught, with large but friendly numbers, e.g. when dividing by 5 or multiplying by 20. Fractions/Decimals /Decimals: Children will develop their understanding of fractions, learning to reduce a fraction to its simplest form as well as finding non-unit fractions of amounts and quantities. The concept of a decimal number is introduced and children consolidate a firm understanding of one-place decimals, multiplying and dividing whole numbers by 10 and 100. Reasoning It is essential that children can explain their reasoning for their choice of calculation and the processes involved. They should explain how their work is set out to maintain place value. They should be able to use their understanding of place value and number facts to explain errors and make corrections. Mastery Children should be given a wide range of opportunities to explore problems in a range of formats (eg missing numbers/signs; a range of real life contexts such as money, measure etc; unusually ordered number sentences). Teaching of calculations should be intrinsic in all areas of Mathematics (for example, explaining calculation choices in the context of SSM or statistics. A wider range of cross curricular links should be developed.
Rationale for UKS2 Teachers in LKS2 need to try to ensure that children have secured their understanding and use of calculations at their age related level before they move into the UKS2 so that children in UKS2 can focus on mastering their skills in more advanced real life situations. Year 4 teachers should aim to ensure that all children can complete a compact written method for each operation before they progress to UKS2. Year 5 teachers should aim to fill any gaps in learning by the end of Term 2 and the SENCO should ensure interventions are in place to challenge any misconceptions particularly in place value and number facts. Children move on from dealing mainly with whole numbers to performing arithmetic operations with both decimals and fractions. They will consolidate their use of written procedures in adding and subtracting whole numbers with up to 6 digits and also decimal numbers with up to two decimal places. Mental strategies for adding and subtracting increasingly large numbers will also be taught. These will draw upon children s robust understanding of place value and knowledge of number facts. Efficient and flexible strategies for mental multiplication and division are taught and practised, so that children can perform appropriate calculations even when the numbers are large, such as 40,000 x 6 or 40,000 8. Y5 and Y6 that children extend their knowledge and confidence in using written algorithms for multiplication and division. Fractions and decimals are also added, subtracted, divided and multiplied, within the bounds of children s understanding of these more complicated numbers, and they will also calculate simple percentages and ratios. Negative numbers will be added and subtracted. The Year 6 teacher should aim to ensure curriculum coverage by the end of Term 3 in order to be able to focus on reasoning and mastery skills in Term 4. This should ensure that children are confident and well prepared to sit their external examinations and move on to the secondary curriculum. Reasoning It is essential that children can explain their reasoning for their choice of calculation and the processes involved. They should be confident enough to explain how current knowledge can be used to solve new problems (eg if I know the 12 x 7 = 84 then I can use this to find 24 x or 13 x or 18 x because.etc) They should explain why they have chosen written methods over others, why these methods are more efficient etc. They should be able to systematically find errors and correct them. Mastery Children should be exploring Mathematics in a range of situations which are cross curricular recognising opportunities where Maths skills will be useful and drawing conclusions. They should be involved in writing questions for others and deciding on the mark scheme. Teaching of calculations should be intrinsic in all areas of Mathematics (for example, explaining calculation choices in the context of SSM or statistics.
Overview of Progression Mathematics Calculation Policy: Year 1 Addition AS1.1 & AS1.2 The + and = 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. Example 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 + NPV1.4, AS1.3 & AS1.4 Use of prepared number lines and concrete objects Subtraction AS1.1 & AS1.2 The - and = signs and missing numbers The notes opposite are relevant here. 7-3 = = 7-3 7 - = 4 4 = - 3 NPV1.4, AS1.3 & AS1.4 Use of pictures, marks and concrete objects Sam spent 4p. What was his change from 10p? Number Lines NPV1.4, AS1.3 & AS1.4 Example- Counting Back/Down 11 7 0 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 Children are encouraged to record by drawing jumps on prepared lines. NPV1.4, AS1.3 & AS1.4 Example- Counting On/Up The difference between 7 and 11 0 1 2 3 4 5 6 7 8 9 10 11 12 Children are encouraged to record by drawing jumps on prepared lines and constructing their own lines. Multiplication MD1.1, F1.1 & F1.2 Use of pictures and objects There are 3 sweets in one bag. How many sweets are there in 5 bags? Division MD1.1, F1.1 & F1.2 Use of pictures and objects or marks 12 children get into teams of 4 to play a game. How many teams are there? NPV1.2 Count in multiples of one, two, five and ten Counting steps using bead string and on prepared number lines. Counting in multiples using a range of objects, e.g. pairs of legs on animals; fingers in gloves etc. NPV1.4 & MD1.1 Use of arrays Counting in rows and columns Two groups of three is six Three groups of two is six MD1.1 Sharing 6 sweets are shared between 2 people. How many do they have each? Make use of practical activities involving sharing, e.g. distributing cards when playing a game, putting objects onto plates, into cups, hoops etc. So 6 = 2 + 2 +2 or 6 = 3 + 3 Video clips: Using a range of equipment and strategies to reinforce addition statements / bonds to 10
New Curriculum Mathematics Calculation Policy: Year 2 Addition Subtraction AS2.3 & AS2.8 The + and = signs and missing numbers Continue using a range of equations (See Year 1) but with appropriate, larger numbers as specified in Year 2 gradelevel standards, i.e. extend to 14 + 5 = 10 + and 32 + + = 100 35 = 1 + + 5. AS2.6 Partition into tens and ones and recombine 12 + 23 = 10 + 2 + 20 + 3 = 30 + 5 = 35 AS2.6 Partitioning the second number only 23 + 12 = 23 + 10 + 2 = 33 + 2 +10 +2 = 35 23 33 35 AS4.2, AS2.5 & AS2.6 Example: Add 9 or 11 by adding 10 and adjusting by 1 35 + 9 = 44 AS2.3 & AS2.8 The and = signs and missing numbers Continue using a range of equations (See Year 1) but with appropriate numbers in relation to Year 2 grade-level standards, i.e. extend to 14 + 5 = 20 -. AS2.6 Find a small difference by counting up 42 39 = 3 + 1 + 2 39 40 42 AS2.4, AS2.5 & AS2.6 Example: Subtract 9 or 11 & begin to add/subtract 19 or 21 35 9 = 26 +1 25 26 35-10 AS2.6 Use known number facts and place value to subtract (Partition second number only) 27 37 12 = 37 10 2 25 37 = 27 2-2 -10 = 25 Multiplication MD2.1, MD2.2 & MD2.4 The x and = signs and missing numbers 7 x 2 = = 2 x 7 7 x = 14 14 = x 7 x 2 = 14 14 = 2 x MD2.5 Use materials, arrays, repeated addition (including solving problems in context) 4 x 2 or 4 + 4 2 x 4 Or repeated addition 2 + 2 + 2 + 2 NPV2.2 & NPV2.6 0 1 2 3 4 5 6 7 8 Partitioning Division MD2.1, MD2.2 & MD2.4 The and = signs and missing numbers 6 2 = = 6 2 6 = 3 3 = 6 2 = 3 3 = 2 MD2.5 Use materials, arrays, repeated addition (including solving problems in context) Use of sharing and grouping Sharing 6 sweets are shared between 2 people. How many do they have each? Grouping There are 6 sweets. How many people can have 2 each? (How many 2 s make 6?) 15 x 2 OR 20 + 10 = 30 x 10 5 2 20 10 0 2 4 6 F2.1 Find and name fractions of length, shape and sets of objects and quantities Use of diagrams- count all equal parts to determine denominator. Link to division into equal groups/parts. Video clips: 1. Teaching for understanding of multiplication facts 2. Practical multiplication and the commutative law
New Curriculum Mathematics Calculation Policy: Year 3* Addition Subtraction The + and = signs and missing numbers Continue using a range of equations as in Year 1 and Year 2 but with appropriate larger numbers specified in the gradelevel standards. The - and = signs and missing numbers Continue using a range of equations as in Year 1 and Year 2 but with appropriate larger numbers specified in the gradelevel standards. AS3.1, AS3.2 & AS3.3 Progression in mental calculations with larger numbers Calculate HTU + U Calculate HTU + TU Calculate HTU + HTU Progress from no crossing of boundaries to crossing of boundary. Partition into tens and ones and recombine Develop from Year 2- partitioning both numbers and recombining. Refine to partitioning the second number only: 36 + 53 = 53 + 30 + 6 +30 +6 = 83 + 6 = 89 53 83 89 Add a near multiple of 10 to a two-digit number Continue work from Year 2 but with appropriate numbers: 35 + 19 is the same as 35 + 20 1. AS3.4 Formal methods of columnar addition to add numbers with up to three digits 285 +73 8 150 200 358 AS3.4 & M3.3 Extend to decimals in the context of money 2.50 + 1.75 2.50 + 1.75 4.25 1 Find a small difference by counting up Continue from Year 2 but with appropriate numbers, e.g. 102 97 = 5 AS3.1, AS3.2 & AS3.3 Subtract mentally a near multiple of 10 to or from a two-digit number, extending to threedigit numbers Continue as in Year 2 but with appropriate numbers e.g. 78 49 is the same as 78 50 + 1 AS3.1, AS3.2 & AS3.3 Progression in mental calculations with larger numbers Calculate HTU - U Calculate HTU - T Calculate HTU - H Progress from no crossing of boundaries to crossing of boundary. + 20 Complementary addition + 4 + 4 84 56 = 28 56 60 80 84 AS3.4 Formal methods of columnar subtraction to subtract numbers with up to three digits See Appendix 1 examples in Year 5 and Year 6 section of this document. *From Year 3 onwards, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations. The expanded method should be used if children experience persisting difficulties. *From Year 3 onwards, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations.
Multiplication MD3.1 & MD3.2 The x and = signs and missing numbers Continue using a range of equations as in Year 2 but with appropriate numbers in relation to grade-level standards. MD3.2 TU x U Use known facts x3, x4, x8 (Year 3 grade-level standards) and x2, x5 and x10 (Year 2 grade-level standards). x 30 5 2 60 10 x 30 2 3 90 6 At Year 3, children progress to using more formal written methods. In this case, the grid method drawing on knowledge of place value, multiplication facts and their ability to recombine partitioned numbers to derive an answer. Division MD3.2 The and = signs and missing numbers Continue using a range of equations as in Year 2 but with appropriate numbers in relation to grade-level standards. MD3.2 TU U Grouping How many 3s make 18? 0 3 6 9 12 15 18 MD3.2 & MD3.3 Remainders 16 3 = 5 r1 Sharing There are 16 sweets shared between 3, how many left over? Grouping How many 3s make 16, how many left over? 15 16 0 3 6 9 12 Children with secure knowledge of multiplication facts and subtraction may progress to chunking where TU are divided by U. *From Year 3 onwards, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations. *From Year 3 onwards, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations. Video clips: 1. Demonstration of expanded 3-digit column addition 2, Subtraction teaching children to consider the most appropriate methods before calculating 3. Introducing partitioned column subtraction method, from practical to written
New Mathematics Calculation Policy: Year 4 Addition Subtraction The + and = signs and missing numbers The and = signs and missing numbers Continue using a range of equations as in Key Stage 1 and Continue using a range of equations as in Key Stage 1 and Year 3 but with appropriate numbers. Year 3 but with appropriate numbers. Partition into hundreds, tens and ones and recombine Either 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 Year 2, 3 and 4 but with appropriate numbers e.g. 458 + 79 = is the same as 458 + 80-1 AS4.1 Addition of numbers with at least four digits using formal method of columnar addition 358 +73 431 1 1 3587 +675 4262 1 1 1 The formal, efficient method of columnar addition will involve crossing of boundaries (at the tens, hundreds and/or thousands). Take a systematic approach to teaching this looking at crossing each boundary in turn before mixed practice. Revert to expanded method if children experience difficulties. Differences Find a difference by counting up, e.g. 8006 2993 = 5013. This can be modelled on an empty number line. DF4.6 Use known number facts and place value to subtract 6.1 0.4 = 5.7 5.7 6.0 6.1-0. 3-0.1 AS4.1 Subtraction with at least four digits using formal method of columnar subtraction For instance, 6467 2684 = 3783 Using expanded column subtraction where children experience difficulty with decomposition and need to see this. DF4.6 Extend subtraction to decimals (same number of decimals places) and adding several numbers (with different numbers of digits) As specified in Year 3, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations. DF4.6 Extend addition to decimals (same number of decimals places) and adding several numbers (with different numbers of digits). As specified in Year 3, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations. Video clips: 1. Subtraction teaching children to consider the most appropriate methods before calculating 2. Introducing partitioned column subtraction method, from practical to written 3, Moving to the compact column method of subtraction
Multiplication The x and = signs and missing numbers Continue using a range of equations but with appropriate numbers for Year 4. MD4.5 TU x U (See Year 3) and HTU x U (Introduced in Year 4 grade-level standards). Partition 23 x 4 = 92 23 x 4 = (20 x 4) + (3 x 4) = (80) + (12) = 92 Use the grid method of multiplication 23 x 7 is approximately 20 x 10 = 200 x 20 3 7 140 21 As specified in Year 3, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations. Division The and = signs and missing numbers Continue using a range of equations but with appropriate numbers for Year 4. MD4.3 Sharing and grouping 30 6 can be modelled as: Grouping groups of 6 taken away and the number of groups counted e.g. 6 + 6 + 6 + 6 + 6 + 0 6 12 18 24 30 Sharing sharing among 6, the number given to each person. Remainders Note three approaches below: 41 4 = 10 r1 +40-1 10 x 4-40 1 group MD4.5 TU U 72 5 lies between 50 5 = 10 and 100 5 = 20 72-50 (10 groups) or (10 x 5) 22-20 (4 groups) or (4 x 5) 2 Answer: 14 remainder 2 41 = (10 x 4) + 1 MD4.5 HTU U Can progress from no remainder to remainders. Where remainders are involved, care needs to be taken to ensure they are interpreted correctly in context of problems. 256 7 lies between 210 7 = 30 and 280 7 = 40 256-70 (10 groups) or (10 x 7) 186-140 (20 groups) or (20 x 7) 46-42 (6 groups) or (6 x 7) 4 (36 groups) or (36) Answer: 36 remainder 4 As specified in Year 3, teachers need to keep in mind the methods specified in grade-level standards for end of Key Stage 2 (See Year 5 and Year 6 Calculation Policy Document). Children should be developing their capacity to use formal written methods for all four number operations.
New Mathematics Calculation Policy: Year 5 and Year 6 The exemplification of formal methods here should be taken into account by all Key Stage 2 teachers so children are adequately prepared by Year 5 and into Year 6 to use the means of calculating specified in grade-level standards. Addition & Subtraction AS5.1 Columnar Addition & Subtraction MD5.5 Short Multiplication (DfE, 2013, Appendix 1) Multiplication & Division MD5.7 & ASMD6.2b Short Division (DfE, 2013, Appendix 1) MD5.5 & ASMD6.1 Long Multiplication (DfE, 2013, Appendix 1) ASMD6.2a Long Division (DfE, 2013, Appendix 1)
Year by Year with examples