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Auckley School Mathematics Policy Calculation: multiplication and division Foundation, Key Stage 1 and 2 mathematics Updated: November 2016 This policy contains the key pencil and paper procedures that are to be taught throughout the school. It has been written to ensure consistency and progression throughout the school. This policy is a suggested guide for what method to teach in each year group and also to give each teacher an idea of the skills they can expect children to come with at the start of each school year. The policy allows teachers to use an appropriate method for each child in their class, depending on their ability and mathematical understanding, not their age. It also aids parents in understanding their child’s stage of learning. Use Numicon and other practical resources to support calculations and help to secure children’s understanding of place value, where appropriate. Ensure the children are confident with previous calculation methods before moving on. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to check their answers after calculation using an appropriate strategy. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods. Note regarding place value terminology – throughout this policy we refer to “units”; however the National Curriculum refers to “ones”. It is our opinion that both terminologies should be used with the pupils. Stages in Multiplication Method Counting songs and rhymes Pictorial, practical multiplication Example Double Rap A fisherman catches 3 fish on Monday, 3 on Tuesday and 3 on Wednesday. How many fish did he catch altogether? Group and count all (Counting to follow stages in addition) 1, 2, 3 and 1, 2, 3 make 1, 2, 3, 4, 5, 6, 7, 8, 9 nine fish and 1, 2, 3 Pictorial, practical multiplication A fisherman catches 3 fish on Monday, 3 on Tuesday and 3 on Wednesday. How many fish did he catch altogether? Group and count multiples 3 and 3 and 3 make 3 and 6 and 9 = nine fish Use of numbered lines 5 x 2 = 10 +2 +2 +2 +2 +2 Count on in multiples 0 1 2 3 4 5 6 7 8 9 10 Use of blank number line 5 x 2 = 10 +2 Count on in multiples 0 +2 2 +2 4 +2 6 +2 8 10 Introduce arrays Three lots of two make 6 two lots of three make six Leading to 3x2=6 2x3=6 Informal recording: partition the number into tens and units/ones multiply the parts then recombine 13 x 4 Possible eg. Partition 13 into 10 and 3 13 x 4 10 x 4 = 40 3 x 4 = 12 10 x 4 3x 4 Recombine 40 12 Or 13 x 4 10 x 4 = 40 + 3 x 4 = 12 52 13 x 4 = 52 40 + 12 = 52 52 13 x 4 = 52 Progress to 13 x 4 using previous step Verbally explain e.g. To calculate thirteen times four, partition thirteen into ten and three. Multiply as a mental these parts by four. Ten times four is forty, three times four is twelve. Add the answers together: method forty add twelve is fifty two. So thirteen times four equals fifty two. Formalise TU x U recording in to the expanded vertical method Vertical recording compact method TU x U 38 X 7 56 (8 x 7) 210 (30 x 7) 266 Extend to higher numbers 38 X 7 266 5 Extend to higher numbers Formalise TU x TU recording in to the expanded vertical method 38 X 17 266 5 380 646 1 Vertical recording compact method TU x TU (38 x 7) 38 X 17 266 380 646 1 (38 x 10) Extend to larger numbers Extend to larger numbers For decimals, the method used depends on the individual’s confidence with decimals. All partitioning methods will work with a decimal partition – the informal partitioning is the most straightforward to do. An alternative would be to deal with the decimal as a whole number and then replace the decimal in the final answer. 1.3 x 4 1.3 x 4 1.0 x 4 = 4 + 0.3 x 4 = 1.2 5.2 13 x 4 = 13 X 4 52 1 1.3 is ten times smaller than 13 so 1.3 x 4 is ten times smaller than 13 x 4. 1.3 x 4 = 5.2 13 x 4 = 52 1.3 x 4 = 52 ÷ 10 = 5.2 Multiplication Key Stage 1 The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the four operations, including with practical resources [for example, concrete objects and measuring tools]. By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency. Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1. Reception and Year 1 Objectives: Pupils should be taught to: Solve one-step problems involving multiplication, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher. Notes and guidance (non-statutory): Through grouping small quantities, pupils begin to understand: multiplication; doubling numbers and quantities; and finding simple fractions of objects, numbers and quantities. They make connections between arrays, number patterns, and counting in twos, fives and tens. Resources and methods: Children will experience equal groups of objects and will count in 2s, 5s and 10s and begin to count in 5s. They will work on practical problem solving activities involving equal sets or groups, e.g. Count five hops of 2 along this number track. What number will you reach? (Children will begin to move from using number tracks to number lines as appropriate through year 1 and 2). Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures, etc. They use number lines and practical resources to support calculation and teachers demonstrate the use of the number line. Children then begin to use numbered lines to support their own calculations using a numbered line to count on in equal steps. Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines Year 2 Objectives: 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 within the multiplication tables and write them using the multiplication (×) and equals (=) signs Show that multiplication of two numbers can be done in any order (commutative) Solve problems involving multiplication, using materials, arrays, repeated addition, mental methods, and multiplication facts, including problems in contexts. Notes and guidance (non-statutory): Pupils use a variety of language to describe multiplication. Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication tables and connect them to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions on the clock face. They begin to use other multiplication tables and recall multiplication facts, including using related division facts to perform written and mental calculations. Pupils work with a range of materials and contexts in which multiplication and division relate to grouping and sharing discrete and continuous quantities, to arrays and to repeated addition. They begin to relate these to fractions and measures (for example, 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4). Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Children will develop their understanding of multiplication and use jottings to support calculation. Lower Key Stage 2 The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the four operations, including number facts and the concept of place value. This should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers. At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and decimal place value. Teaching should ensure that pupils can make connections between measure and number. By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work. Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge and their knowledge of spelling. Year 3 Objectives: Pupils should be taught to: Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables 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 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. Notes and guidance (non-statutory): Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables. Pupils develop efficient mental methods, for example, using commutativity and associativity (for example, 4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and multiplication and division facts (for example, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (for example, 30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3). Pupils develop reliable written methods for multiplication, starting with calculations of two-digit numbers by one-digit numbers and progressing to the formal written methods of short multiplication. Pupils solve simple problems in contexts, deciding which of the four operations to use and why. These include measuring and scaling contexts (for example, four times as high, eight times as long etc.) and correspondence problems in which m objects are connected to n objects (for example, 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4 children; 4 cakes shared equally between 8 children). Resources and methods: Children understand the relationship between multiplication and division. For example, they state two multiplication sentences and two division sentences that relate to a particular array. Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Year 4 Objectives: Pupils should be taught to: Recall multiplication and division facts for multiplication tables up to 12 × 12 Use place value, known and derived facts to multiply mentally, including: multiplying by 0 and 1; multiplying together three numbers Recognise and use factor pairs and commutativity in mental calculations Multiply two-digit and three-digit numbers by a one-digit number using formal written layout Solve problems involving multiplying 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. Notes and guidance (non-statutory): Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency. Pupils practise mental methods and extend this to three-digit numbers to derive facts, (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6). Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers. Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60. Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu. Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Upper Key Stage 2 The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio. At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures should consolidate and extend knowledge developed in number. By the end of year 6, pupils should be fluent in written methods for all four operations, including long multiplication and division, and in working with fractions, decimals and percentages. Pupils should read, spell and pronounce mathematical vocabulary correctly. Year 5 Objectives: Pupils should be taught to: 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 Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers Multiply numbers mentally drawing upon known facts Multiply whole numbers and those involving decimals by 10, 100 and 1000 Recognise and use square numbers and cube numbers, and the notation for squared 2 and cubed 3 Solve problems involving multiplication 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. Notes and guidance (non-statutory): Pupils practise and extend their use of the formal written methods of short multiplication. They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations. Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres. Distributivity can be expressed as a(b + c) = ab + ac. They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for 2 example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9 x 10). Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13 + 24 = 12 + 25; 33 = 5 x 6 + 3). Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Year 6 Objectives: Should be taught to: Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication. Perform mental calculations, including with mixed operations and large numbers Identify common factors, common multiples and prime numbers Use their knowledge of the order of operations to carry out calculations involving the four operations Solve problems involving addition, subtraction, multiplication and division Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. Notes and guidance (non-statutory): Pupils practise multiplication for larger numbers, using the formal written methods of short and long multiplication. They undertake mental calculations with increasingly large numbers and more complex calculations. Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency. Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50 etc., but not to a specified number of significant figures. Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9. Common factors can be related to finding equivalent fractions. Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Stages in Division Method Pictorial, practical division – sharing based on stories / simple word problems. Note: there is no mathematical way of showing this form of division other than by practical and pictorial means. It is a necessary step in understanding division though and so should be taught. Example 6 fish shared between 3 bears: Giving: Each bear has two fish. Pictorial, practical division – grouping as repeated subtraction. 6 fish are put into baskets and each basket has 3 fish in it. How many baskets are needed? 6 divided by 3 Six take away one group of 3 then another group of three. 6 divided by 3 = 2 2 baskets are needed. Number lines Children will be taught to use both repeated addition and subtraction on a number line using the term ‘groupings’ – How many groups of 3 can we get from 15? How many groups of 3 can we subtract from 15? 15 ÷ 3 = 5 1x3 0 3 - 1x3 0 It would be helpful to refer to “lots of” e.g. 1 lot of 3, two lots of 3 etc. 1x3 1x3 6 - 1x3 3 1x3 9 - 1x3 6 1x3 12 - 1x3 9 15 - 1x3 12 15 Introduce problems with remainders pictorially. 8 fish shared between 3 sealions: remainder Giving: Each sealion has two fish with 2 left over. This is the remainder. Number lines and remainders Children will be taught to use both repeated addition and subtraction on a number line using the term ‘groupings’ – 27 ÷ 6 = 4 remainder 3 1x6 0 1x6 6 1x6 12 1x6 18 1 24 1 1 27 How many groups of 6 can we get from 27? How many groups of six can we subtract from 27? Informal vertical division - Chunking Children will be taught to use the ‘chunking’ method initially by single multiples of the divisor. Then progressing to larger multiples, ideally 10, 5 and 2, but also other known multiples. - 1x6 0 3 24 ÷ 5 = 4 remainder 4 24 - 5 19 - 5 14 - 5 9 - 5 4 - 1x6 9 - 1x6 15 - 1x6 21 Progressing to: 24 ÷ 5 = 4 remainder 4 (1 x 5) (1 x 5) (1 x 5) (1 x 5) Add the multiples to find the answer 1+1+1+1 = 4 24 - 10 14 - 10 4 (2 x 5) Add the multiples to (2 x 5) find the answer 2+2=4 27 Condensed Chunking Once children are confident in ‘chunking’, they may condense the number of steps by carrying out the minimum number of subtractions – subtract multiples of powers of 10 in descending size order. Eg 3000, 500, 20, 7 7427 ÷ 3 = 2475 rem 2 7427 -6000 (2000 x 3) 1427 -1200 (400 x 3) 227 - 210 (70 x 3) 17 - 15 (5 x 3) 2 The children should be discouraged from finding the answer in “one go” as this is an inefficient use of time. Informal recording: partition the number into useful multiples of the divisor, divide the parts then recombine 27÷6 = 4 remainder 3 1 + 2 + 3 + 4 remainder 3 6 )6 + 12 + 18 + 24 r 3 Progressing to: 2473 rem 2 3)7427 -6000 1427 -1200 227 - 210 17 - 15 2 Formal recording – Short division Model with base 10 how to group and exchange. 24÷6 = 4 TU 4 6 )224 Narration – can I split 2 tens into groups of 6? No: then we move the 2 tens into the next column to make 24 units. Can 24 units be split into groups of 6? Yes: 4 groups. Record 4 in the units column. Answer 4. 68 ÷ 4 = T U 1 7 4 )6 28 Narration – can I split 6 tens into groups of 4? Yes, there is 1 group of 4 tens and 2 tens remaning. Record the 1 group in the tens column and then move the 2 remaing tens into the units column to make 28 units. Can 28 units be split into groups of 4? Yes 7 groups with no remainder. Record 7 in the units column of the answer line. Answer 17. 435 ÷ 3 =145 HTU 1 4 5 3 )4 131 5 Narration – can I split 4 hundreds into groups of 3? Yes - there is 1 group of 3 hundreds and 1 hundred remining. Record the 1 group in the hundreds column and then move the the remaining 1 hundred into the tens column to make 13 tens. Can 13 tens be split into groups of 3? Yes - 4 groups with 1 ten remaining. Record the 4 groups in the tens column and then move the reminaing 1 ten into the units column to make 15 units. Can 15 units be split into groups of 3? Yes - 5 groups with no remainder. Record the 5 groups in the units column. Answer 145. Condensed Chunking Chunking can also be introduced for divisors greater than 10 124 ÷ 15 = 8 r 4 124 - 90 34 - 30 4 (6 x 15) Add the multiples to (2 x 15) find the answer 6+2=8 Formal recording – Long division Link to chunking. 494 ÷ 15 = 33 r 4 33 15 ) 4 9 9 -4 50 49 - 45 4 Help box 15x table 15,30,45,60,75,90,105,120,135,150 Narration – can I subtract a hundreds multiple of 15? (use help box to work out 100 x 15 = 1500). No. What is the largest tens multiple of 15 I can subtract? (Use the help box to work out 30 x 15 = 450). Record 3 in the Tens column then carry out the subtraction. Now ask - What is the largest units multiple of 15 I can take away from what is left? (Use the help box to work out 3 x 15 = 45). Record the 3 in the units column and then carry out the subtraction. Answer is 33 remainder 4. For decimals, the most straightforward method would be to deal with the decimal as a whole number and then replace the decimal in the final answer. e.g. 12.4 ÷ 4 = 3.1 124 ÷ 4 = 31 124 - 120 (30 x 4) 4 - 4 (1 x 4) 0 12.4 is ten times smaller than 124 so 12.4 ÷ 4 is ten times smaller than 124÷ 4 so 12.4 ÷ 4 = 31 ÷10 = 3.1 Progression in Remainders Model using number line, chunking and short/long division Initially remainders should be expressed as e.g. “remainder 3” This should progress to expressing them as a fraction of the divisor. 5÷3 1x3 0 3 4 5 6 5÷3=1 2 (5 is two steps towards the next multiple of 3) 3 Knowledge of fraction and decimal equivalences also allows these remainders to be converted to decimals e.g. 1 2/3 = 1.66 Extending the dividend beyond the decimal point: e.g. 24 ÷ 5 = 4.8 24.0 - 20.0 4.0 - 4.0 0 (4 x 5) (0.8 x 5) - from knowing 8 x 5=40 43.5 ÷ 3 =14.5 T U. te 1 4. 5 3 )4 13.1 5 Division Key Stage 1 The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the four operations, including with practical resources [for example, concrete objects and measuring tools]. By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency. Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1. Reception and Year 1 Objectives: Pupils should be taught to: Solve one-step problems involving division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher. Notes and guidance (non-statutory): Through grouping and sharing small quantities, pupils begin to understand: multiplication and division; doubling numbers and quantities; and finding simple fractions of objects, numbers and quantities. They make connections between arrays, number patterns, and counting in twos, fives and tens. Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Year 2 Objectives: Pupils should be taught to: Recall and use division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers Calculate mathematical statements for division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs Show that division of one number by another cannot be done in any order Solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts. Notes and guidance (non-statutory): Pupils use a variety of language to describe division. Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication tables and connect them to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions on the clock face. They begin to use other multiplication tables and recall multiplication facts, including using related division facts to perform written and mental calculations. Pupils work with a range of materials and contexts in which division relates to grouping and sharing discrete and continuous quantities, to arrays and to repeated addition. They begin to relate these to fractions and measures (for example, 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4). Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Lower Key Stage 2 The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the four operations, including number facts and the concept of place value. This should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers. At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and decimal place value. It should ensure that they can use measuring instruments with accuracy and make connections between measure and number. By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work. Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge and their knowledge of spelling. Year 3 Objectives: Pupils should be taught to: Recall and use division facts for the 3, 4 and 8 multiplication tables Write and calculate mathematical statements division using the multiplication tables that they know, Solve problems, including missing number problems, involving division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects. Notes and guidance (non-statutory): Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables. Pupils develop efficient mental methods, for example, using commutativity and associativity (for example, 4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and multiplication and division facts (for example, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (for example, 30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3). Pupils develop reliable written methods for division, starting with calculations of two-digit numbers by one-digit numbers and progressing to the formal written methods of short multiplication and division. Pupils solve simple problems in contexts, deciding which of the four operations to use and why. These include measuring and scaling contexts, (for example, four times as high, eight times as long etc.) and correspondence problems in which m objects are connected to n objects (for example, 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4 children; 4 cakes shared equally between 8 children). Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Year 4 Objectives: Pupils should be taught to: Recall division facts for multiplication tables up to 12 × 12; Use place value, known and derived facts to divide mentally, including: dividing by 1; Recognise and use factor pairs and commutativity in mental calculations. Notes and guidance (non-statutory): Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency. Pupils practise mental methods and extend this to three-digit numbers to derive facts (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6). Pupils practise to become fluent in the formal written method of short division with exact answers. Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60. Pupils solve two-step 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. Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Upper Key Stage 2 The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio. At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures should consolidate and extend knowledge developed in number. By the end of year 6, pupils should be fluent in written methods for all four operations, including long multiplication and division, and in working with fractions, decimals and percentages. Pupils should read, spell and pronounce mathematical vocabulary correctly. Year 5 Objectives: Pupils should be taught to: 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 Divide numbers mentally drawing upon known facts 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 Divide whole numbers and those involving decimals by 10, 100 and 1000 2 3 Recognise and use square numbers and cube numbers, and the notation for squared ( ) and cubed ( ) Solve problems involving 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 division, including scaling by simple fractions and problems involving simple rates. Notes and guidance (non-statutory): Pupils practise and extend their use of the formal written methods of short division. They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations. Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 =98/4 = 24 r 2 = 24 ½ = 24.5 ≈ 25). Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres. Distributivity can be expressed as a(b + c) = ab + ac. They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9 x 10). Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13+24 = 12 + 25; 33 = 5 x). 2 Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines. Year 6 Objectives: Pupils should be taught to: 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 Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context Perform mental calculations, including with mixed operations and large numbers Identify common factors, common multiples and prime numbers Use their knowledge of the order of operations to carry out calculations involving the four operations Solve problems involving addition, subtraction, multiplication and division Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. Notes and guidance (non-statutory) Pupils practise division for larger numbers, using the formal written methods of short and long division. They undertake mental calculations with increasingly large numbers and more complex calculations. Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency. Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50 etc., but not to a specified number of significant figures. Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9. Common factors can be related to finding equivalent fractions. Resources and methods: Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number tracks, numbered lines.