Birdwell School Academy Whole School Policy For Numeracy Calculation 2014 DIVISION Written/Compiled: V.Moisey Numeracy Co-ordinator: August/September 2014 Reviewed/Agreed: Headteacher/Teaching Staff/Learning Support Staff/Governors:
About our Calculation Policy The following calculation policy has been devised to meet the requirements of the National Curriculum 2014 for the teaching and learning of mathematics, and is also designed to give children a consistent and smooth 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 (EYFS). Please see separate document for EYFS. Age Stage Expectations The calculation policy is organised according to age stage expectations as set out in the National Curriculum 2014. However, it is vital that children 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. Providing a context for calculation: It is important that any type of calculation is given a real life context or problem solving approach (not necessarily new to children) 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 calculation lessons. Problem solving, using and applying, should be threaded throughout lessons and should not be a bolt-on at the end of a series of taught lessons or at the end of a week. 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 formal written methods accurately with confidence and understanding How to use this Policy use the policy as a basis for your planning ensure that children are confident with the methods outlined in the previous year s guidance before moving on if at any time, children are making significant errors, return to the previous stage in calculation 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
Ensuring conceptual understanding to then enable children to choose an appropriate calculation method: It is vital that children have a conceptual understanding of numbers, the number system and the calculation methods they use. This will enable them to have a solid understanding in maths as well as given them tools to select appropriate calculation methods when solving mathematical problems. Choosing a calculation method: Children need to be taught and encouraged to use the following processes in deciding what 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? Steps to calculating Approximate Could I use some jottings or drawings to help me? Calculate Should I use a written method to work it out? Check it (Check reasonableness of final answer against approximation)
Written Calculation The aim for mental calculations. With mental work, the aim is to teach a child a repertoire of strategies from which they can select. With written calculations the ultimate aim is proficiency in a method for each operation with one clear progression route taught for each. Written calculation Building on the mental strategies they have used so they can understand the processes involved, children need first to be taught to record their methods in an expanded form. When ready - and this is dependent on teachers professional judgement - they are taught how to refine their recording to make it more compact. Challenges to teachers Ensuring that recall skills are established first so children can concentrate on a written method without reverting to first principles. Making sure that, once written methods are introduced, children continue to look out for and recognize the special cases that can be done mentally. Catering for children who progress at different rates; some may grasp a compact method of calculation while others may never do so without considerable help; catering for children who can carry out some standard methods successfully, eg for addition but not subtraction. Recognising that children tend to forget a standard method if they have no understanding of what they are doing and if they do not visit it regularly throughout the year. Often the compactness of a vertical method show mathematical principles are applied, e.g. children may use place value when working mentally, but may be confused in written work because they do not understand how place value relates to carrying. There can be long lasting problems for those taught compact, vertical methods before they understand what they are doing eg children can undertake decomposition for subtraction but are unable to explain the place value involved. Simply correcting children s errors may help in the short-term, but not permanently. Misunderstandings and misconceptions need to be analysed and children need to find their own errors. Children need to understand why a particular method works rather than simply following a set of rules. They can then fall back to a simple method if uncertain, or to check their answers. Estimating (make a sensible guess) and Approximating using rounding. This should be encouraged for all four operations to give children a sense of what the answer might be after calculations have been carried out.
Both express the relationship between a number of equal parts and the whole.
MULTIPLICATION AND DIVISION - YEAR 1 Multiplication and Division facts count in multiples of twos/fives/tens Mental Calculations count in multiples of twos/fives/tens Problem solving solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher MULTIPLICATION AND DIVISION - YEAR 2 Multiplication and Division facts count in steps of 2, 3, and 5 from 0, and in tens from any number, forward or backward (copied from Number and Place Value) recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers know that doubling is multiplying by 2 and halving is divided by 2 I know significant doubles (eg 10 + 10, 50+ 50=, 50p+50p= ) involving doubling multiples of 5 up to 50 Mental Calculations show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot Written Calculations calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication ( ), division ( ) and equals (=) signs Problem Solving solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts
MULTIPLICATION AND DIVISION - YEAR 3 Multiplication and Division facts count from 0 in multiples of 4, 8, 50 and 100 (copied from Number and Place Value) Please include others where you feel they are necessary. recall and use multiplication and division facts for the 3,4 and 8 multiplication tables Mental Calculations write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods (appears also in Written Methods) Through doubling, they connect the 2, 4 and 8 multiplication tables Children develop efficient mental methods, eg using commutativity and associativity (for example 4 x 12 x 5 = 4 x 5 x 12 = 20 = 240 Written Calculations write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods. Inverse Operations, Estimating and Checking estimate the answer to a calculation and use inverse operations to check answers (copied from Addition and Subtraction) Problem Solving solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects.
MULTIPLICATION AND DIVISION - YEAR 4 Multiplication and Division facts count in multiples of 6, 7, 9, 25 and 1000 recall multiplication and division facts for multiplication tables up to 12 x 12 doubles and halves of numbers up to 50 Mental Calculations use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers recognise and use factor pairs and commutativity in mental calculations understand the impact of place value when a number is multiplied or divided by 10 and 100 (Do not teach, just add a nought) halve whole numbers including odd numbers know that x 4 is doubling twice and x 8 is doubling three times Written Calculations multiply two-digit and three-digit numbers by a one-digit number using a formal written method Pupils practise to become fluent in the formal written method of short multiplication for multiplying using multi-digit numbers, and short division with exact answers when dividing by a one-digit number (see Appendix 1). Properties of Numbers - Multiples/Factors/Primes/Squares and Cube numbers recognise and use factor pairs and commutativity in mental calculations (repeated) Inverse Operations, Estimating and Checking estimate and use inverse operations to check answers to a calculation Problem Solving solve problems involving multiplication and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects. The Distributive Law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. Example: 3 (2 + 4) = 3 2 + 3 4 So the "3" can be "distributed" across the "2+4" into 3 times 2 and 3 times 4.
MULTIPLICATION AND DIVISION - YEAR 5 Multiplication and Division facts count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000 (copied from Number and Place Value) Mental Calculations multiply and divide numbers mentally drawing upon known facts including decimals 0.6 x 7 = 4.2 because 6 x 7 = 42 3.5 5 = 0.7 because 35 5 = 7 multiply and divide whole numbers and those involving decimals by 10, 100 and 1000 know that TU x 5 is TU x 10 and then divided by 2 (18 x 5 = (18 x 10) 2) know that TU x 9 is TU x 10 then subtract TU (18 x 9 = (18 x 10) 18 = 162) round and compensate for near pounds ( 4.99 x 3 = 5 x 3 3p = 14.97) use knowledge of doubles and halves of whole numbers to find doubles and halves of decimal numbers (2.3 + 2.3 = 4.6 because 23+23=46; Half of 5.8 is 2.7 because half of 58 is 27) Written Calculations multiply numbers up to 4 digits by a one- or two-digit number using a formal written method divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context Properties of Numbers - Multiples/Factors/Primes/Squares and Cube numbers identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers. know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers establish whether a number up to 100 is prime and recall prime numbers up to 19 recognise and use square numbers and cube numbers, and the notation for squared (2) and cubed (3) Inverse Operations, Estimating and Checking estimate and use inverse operations to check answers to a calculation Problem Solving solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates
MULTIPLICATION AND DIVISION - YEAR 6 Multiplication and Division facts count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000 (copied from Number and Place Value) Mental Calculations perform mental calculations, including with mixed operations and large numbers associate a fraction with division and calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction (e.g. 3/8) - (copied from Fractions) Written Calculations multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication divide numbers up to 4-digits by a one-digit whole number using the formal written method of short division where appropriate for the context divide numbers up to 4 digits by a two-digit 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 written division methods in cases where the answer has up to two decimal places (copied from Fractions (including decimals)) Properties of Numbers - Multiples/Factors/Primes/Squares and Cube numbers identify common factors, common multiples and prime numbers use common factors to simplify fractions; use common multiples to express fractions in the same denomination (copied from Fractions) calculate, estimate and compare volume of cubes and cuboids using standard units, including centimetre cubed (cm3) and cubic metres (m3), and extending to other units such as mm3 and km3 (copied from Measures) Order of Operations use their knowledge of the order of operations to carry out calculations involving the four operations Inverse Operations, Estimating and Checking use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy Problem Solving solve problems involving addition, subtraction, multiplication and division solve problems involving similar shapes where the scale factor is known or can be found (copied from Ratio and Proportion)
Children must have secure counting skills- being able to confidently count in 2s, 5s and 10s. Children should be given opportunities to reason about what they notice in number patterns. Using objects, diagrams and pictorial representations to solve problems involving both grouping and sharing. Children 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. Children are encouraged, through practical experiences, to develop physical and mental images. They make recordings of their work as they solve problems where they want to make equal groups of items or sharing objects out equally. Sharing and Grouping They solve sharing problems by using a 'one for you, one for me' strategy until all of the items have been given out. Children should find the answer by counting how many eggs 1 basket has got. They solve grouping problems by creating groups of the given number. Children should find the answer by counting out the eggs and finding out how many groups of 3 there are. They will begin to use their own jottings to record division Use arrays to support early pictorial representation for division. 15 3 = 5 There are 5 groups of 3.15 5 = 3 There are 3 groups of 5. Initially recording of calculating should be done by adults to model what children have done in pictures, symbols, numbers and words. Over time there should be an expectation that children will also become involved in the recording process. Whilst cameras are an excellent way of keeping a record of what children have done, they are not a substitute for the modelling of different ways of recording calculation procedures. SHARING 15 eggs are shared between 5 baskets. How many in each basket? First egg to the first basket, 2nd egg to the second etc. GROUPING There are 15 eggs. How many baskets can we make with 3 eggs in? This is an important stage in teaching the difference between sharing and grouping. Jumps on a number Key Vocabulary: share, share equally, one each, two each, group, groups of, lots of, array
Know and understand sharing and grouping- introducing children to the sign. Children should continue to use grouping and sharing for division using practical apparatus, arrays and pictorial representations. Continue work on arrays ask What do you see? at an array what do you see? Sharing They solve sharing problems by using a 'one for you, one for me' strategy until all of the items have been given out. Children should find the answer by counting how many eggs 1 basket has got. Grouping using a numberline They solve grouping problems by creating groups of the given number. Children should find the answer by counting out the eggs and finding out how many groups of 3 there are. They will begin to use their own jottings to record division. Group from zero in jumps of the divisor to find our how many groups of 3 are there in 15?. Jump in steps of 2 and 5. More able might jump in multiples of 10. Can they begin to estimate (have a guess) at what they think the answer might be? Can they begin to bring in inverse operations by using multiplication to check their answer? 15 -:- 3 = 5 Counting on a labelled and then blank number lines. 15 3 = 5 The use of arrays help to reinforce the link between multiplication and division. Using a bead string, children can represent division problems. They count on in equal steps based on adding multiples up to the number to be divided. When packing eggs into baskets of three they count in threes grouping If the problem requires 15 eggs to be shared between 3 baskets, the multiple of three is obtained each time all three baskets have received an egg. This represents 12 3, posed as how many groups of 3 are in 12? Pupils should also show that the same array can represent 12 4 = 3 if grouped horizontally. Grouping using a number line: Group from zero in equal jumps of the divisor to find out how many groups of in? Pupils could and using a bead string or practical apparatus to work out problems like A CD costs 3. How many CDs can I buy with 12? This is an important method to develop understanding of division as grouping. When ready, use an empty number line to count forwards in steps of the divisor. Counters can be used to support understanding. How many jumps of 3 make 12? 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
In year 3, children continue to work out unknown division facts by grouping on a number line from zero. They are also now taught the concept of remainders, as in the example. This should be introduced practically and with arrays, as well as being translated to a number line. Children should work towards calculating some basic division facts with remainders mentally for the 2s, 3s, 4s, 5s, 8s and 10s, ready for carrying remainders across within the short division method. Step 1 Children continue to work out unknown division facts by grouping on a number line from zero. They are also now taught the concept of remainders.. This should be introduced practically and with arrays, as well as being translated to a number line. Children should work towards calculating some basic division facts with remainders mentally for the 2s, 3s, 4s, 5s, 8s and 10s, ready for carrying. remainders across within the short division method. Begin to use the language of divisor (ie number we are dividing by.) Use number line chunking for 2-digit numbers divided by 1-digit numbers. Use number line chunking for 3-digit numbers divided by 1-digit Use more efficient jumps. Ensure children begin to find an approximate answer before calculating, using rounding. For all divisions. LEADING TO: Vertical chunking Appropriateness of number: these numbers do not need an exchange in the subtraction. element of the strategy. First five tables facts to build on recall and also, to promote a habit to be referred to later on in the progressive division strategies. Short' division of TO O can be introduced as a more compact recording of the mental method of partitioning. More able Year 3. See Year 4 Guidance. Do not refer to short division as the bus stop method. Chunks noted in brackets to count up (not the divisor (4) as this can lead to adding this as a chunk). First key question as a step to success is Can I take a chunk of 10x? Once children are secure with division as grouping and can demonstrate this using number lines, arrays etc., short division for larger 2-digit numbers should be introduced, initially with carefully selected examples requiring no calculating of remainders at all. Start by introducing the layout of short division by comparing it to an array. Remind children of correct place value, that 96 is equal to 90 and 6, but in short division, pose: How many 3 s in 9? = 3, and record it above the 9 tens. How many 3 s in 6? = 2, and record it above the 6 ones. 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
Ensure that children are confident with the methods outlined in the previous year s guidance before moving on. Continue to write and calculate mathematical statements for division using the multiplication tables that the children know e.g. Short division of two-digit number can be introduced to children who are confident with multiplication and division facts and with subtracting multiples of 10 mentally, and whose understanding of partitioning and place value is sound. Once children are secure with division as grouping and can demonstrate this using number lines, arrays etc., short division for larger 2- digit numbers should be introduced, initially with carefully selected examples requiring no calculating of remainders at all. Start by introducing the layout of short division by comparing it to an array. Remind children of correct place value, that 96 is equal to 90 and 6, but in short division, pose: How many 3 s in 9? = 3, and record it above the 9 tens. How many 3 s in 6? = 2, and record it above the 6 ones. Children should approximate the answer, using rounding, before calculating. Children need good understanding of place value and partitioning before using this method. For children not ready, continued use of grouping on a number line from zero is necessary. Once children demonstrate a full understanding of remainders, and also the short division method taught, they can be taught how to use the method when remainders occur within the calculation (e.g. 96-:-4), and be taught to carry. the remainder onto the next digit. If needed, children should use the number line to work out individual division facts that occur which they are not yet able to recall mentally. This step is only taught when children have a sound understanding of remainders. Ensuring an understanding of the relationship between and X, pupils build on chunking from Year 3 to use this strategy to divide 3-digit numbers by 1- and 2-digit numbers: Do not refer to short division as the bus stop method. Step 1 Step 2 Use the vocabulary of place value to ensure understanding and make the link to partitioning. Encourage use of known facts/ inverses eg 8 x 5 = 40 80 x 5 = 400 ENSURE THE DIVISOR COMES AFTER THE MULTIPLICATION SIGN 98 is partitioned into 70 ad 28 (90 = 70 + 28) Seven goes into 70 ten times and seven goes into 28 four times. Ten + four equals 14. This will lead into the formal written method of short division. Remember to still use the language of place value. Model with place value counters. First key question as a step to success is Can I take a chunk of 10x, 100x or a multiple of 10x? (This will be modelled by teacher by applying using known facts and place value. Year 4, children who are ready, will be taught that remainders can be expressed as a fraction. Here, appropriateness of number is important.
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, ie. as a fraction, a decimal, or as a rounded number or value, depending upon the context of the problem. If children are secure and confident include money and measure contexts. When ready Year 5 children should begin to express the remainder as a fraction, leading to expressing it as a decimal fraction. Pupils build on the written strategy from Year 4 and apply the noted tables facts to apply place value and subtract decimals from remainders: First key question as a step to success is Can I take a chunk of 10x, 100x or a multiple of 10x? (This will be modelled by teacher by applying using known facts and place value. Here, remainders are removed by applying place value knowledge to the noted tables facts: subtracting a chunk of 0.4x 5 in this instance. Note appropriateness of number: numbers here have remainders that can be divided and shown as a decimal remainder to one decimal place progressing to a maximum of two decimal places. Where this is not the case, year 5 children should express the quotient as a fraction or decimal. Introduce long division for Year 5 children who are ready to divide any number by a 2-digit number (e.g. 2678 19). This is a Year 6 expectation see year 6 guidance. 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 (non-prime)
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. Short division, for dividing by a single digit: e.g. 6497 8 Short division, for dividing by a single digit: This can be used to calculate the quotient (remainder). Continue with vertical chunking for children not ready for introduction of Long Division. Continue to encourage use of approximation before calculation. Line numbers up correctly for place value. ENSURE THE DIVISOR COMES AFTER THE MULTIPLICATION SIGN. Pupils use long division to calculate: 432 15 = This answer can be shown as a quotient (rather than an integer remainder): 28 12/15 = 28 4/5 Progressing to long multiplication to find a decimal remainder: Teachers might model: 2 x 15 8 x 15 0.8 x 15 Children will need to select the most effective method for each calculation/problem they meet, including whether to use the standard, formal written method of long division: Model selection of an appropriate division format dependent on size of number, efficient ability to apply larger tables facts such as 15x as shown. Here, depending on understanding of this strategy, pupils can refer this calculation to previously taught chunking. Introduce long division for pupils who are ready to divide any number by a 2-digit number (e.g. 2678 19). This is a Year 6 expectation see year 6 guidance. 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 (non-prime)