Market Drayton Town Schools Calculation Policy (Multiplication and Division)

Transcription

1 EYFS Market Drayton Town Schools Calculation Policy (Multiplication and Division) Development Matters: months: Separates a group of three or four objects in different ways, beginning to recognise that the total is still the same. ELG: They solve problems, including doubling, halving and sharing. Multiplication progression of teaching: Division progression of teaching: Children begin to recognise repetitive addition in groups of the same size. How many wheels do we need for these 3 bikes? CLIC MULTIPLICATION STEP 2 CLIC DIVISION STEP 1/2/3/4 In contexts such as role play, children may share or group objects equally e.g. 4 biscuits on 2 plates; 6 children grouped in twos. My first plant pot has 5 beans in it. I want double that in my second plant pot. How many should I put in it? Children will jump along number tracks / number lines / 100 squares in steps of 2 or 10, and will undertake practical work about grouping. Practical activities during exploration: Finding doubles of numbers to 5 and begin to recall these facts Practical activities during exploration: Finding halves of numbers to 10 and begin to recall these facts My first plant pot has 10 beans in it. I want half that in my second plant pot. How many should I put in it? My first plant pot has 5 beans in it. I want double that in my second plant pot. How many should I put in it? Year 1 Statutory Requirements - Pupils should be taught to: count in multiples of twos, fives and tens Moving onto having an amount that cannot be shared equally and refering to this as left over or remainders 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. Fractions relating to having and doubling (local curriculum) Doubling and halving They practise counting as reciting numbers and counting as enumerating objects, and counting in twos, fives and tens from different multiples to develop their recognition of patterns in the number system (for example, odd and even numbers), including varied and frequent practice through increasingly complex questions. 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. Pupils connect halves and quarters to the equal sharing and grouping of sets of objects Children can find doubles and halves of numbers to double 10 Children can find doubles and halves of numbers to double 10 Children will represent multiplications pictorially as repeated additions. They will begin to use number lines to represent repeated additions. CLIC LEARN ITS STEP 6 I have got three 5p coins. How much money do I have altogether? 3 x 5 = They will also record repeated additions as equations. CLIC DIVISION STEP 4 Children continue sharing or grouping in context. At this stage with more focus on grouping. Children count on in twos, fives and tens. Use a number stick to support this Moving onto calculations such as (linked to repeated addition) I have 15p and want to divide it by 5 people. How much will they get each? 15p 5 = 3p

2 5+5+5 = 15 CLIC MULTIPLICATION STEP 7/ (Children will need to become secure in understand the answer is worked out from counting the number of jumps they made) = 10 5 pairs 5 hops of = 30 6 groups of 5 6 hops of 5 10p + 10p + 10p + 10p + 10p = 50p 5 hops of 10 Deriving facts CLIC DIVISION STEP 9 Moving onto division with remainders There were 34 people in a class. 10 can sit at one table. How many tables will they need to make sure everyone can sit at one? = 30 (also could show on a number line as above) Which means that there are 4 remainders but they still need to have a table to sit at. Therefore the answer is 4 for this question. Grouping ITP Year 2 Statutory Requirements - Pupils should be taught to: count in steps of 2, 3, and 5 from 0, and in tens from any number, forward and backward recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication ( ), division ( ) and equals (=) signs show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts. Fractions working out fractions of numbers using grouping and sharing (local curriculum) They count in multiples of three to support their later understanding of a third. Pupils use a variety of language to describe multiplication and 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 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). They connect unit fractions to equal sharing and grouping, to numbers when they can be calculated, and to measures, finding fractions of lengths, quantities, sets of objects or shapes. Children can find doubles and halves of numbers to double 10 Children can find doubles and halves of numbers to double 10 (be secure in recalling and using these facts) Children begin to understand multiplication, as describing an array. ITP array Children begin to use the x symbol in number sentences. They continue to recognise the use of or symbols to stand for Children begin to use the x symbol in number sentences. They continue to recognise the use of or symbols to stand for an unknown number, e.g. 20 = 10. Family of four. Children are given a set of numbers and understand the relationship between them ie 20, 5 and 4 and write the number sentences relating to these 20 5 = 4 4 x 5 = = 5 5 x 4 = 20

3 an unknown number, e.g. 2 x = 20. Family of four. Children are given a set of numbers and understand the relationship between them ie 20, 5 and 4 and write the number sentences relating to these CLIC DIVISION STEP 12/13/14 Or can say the number sentences when numbers are represented in a triangle ff = 4 4 x 5 = = 5 5 x 4 = 20 CLIC MULTIPLICATION STEP 9 Or can say the number sentences when numbers are represented in a triangle ff Children understand multiplication as repeated addition, or as describing an array. By changing arrays, children will begin to understand that multiplication can be done in any order, e.g. 2 x 3 = 3 x Answering problem solving questions ie A farmer has 80 seeds and wants to plant 4 in each pot. How many seeds will he have in each pot? Children may begin to solve by repeated addition/suybtraction, move onto the number line and then use their knowledge of multiplication facts to derive the answer. Children will represent divisions pictorially and using a number line, e.g Children begin to see division as the inverse of multiplication. Children understand division as grouping. Begin to move onto deriving division facts with remainders and have the discussion about what to do with the remainder ie CLIC DIVISION STEP people are travelling to the cinema. Each car can carry 5 people. How many cars are needed? 23 5 = 4r3 You need 5 cars. Books cost 5 each. How many can you buy with 23? 23 5 = 4r3 You can only buy 4 books. Begin to relate repeated addition to multiplication facts and record this as a number sentence r = 4r3 They recognise that division is the inverse of multiplication. They begin to decide whether to round up or down after a division, depending on the context. Recall related division facts for the 2, 5 and 10 times tables. Recall facts for the 2, 5 and 10 times tables (also see you tube and 17 times tables video to support teaching this) For more able children use known facts to calculate 2 digit x1 digit number ie 23 x 5. ie 23 children brought in 5p each to school. How much money was there altogether? Begin by repeated addition 5p + 5p + 5p + 5p etc..= 115/ = 1.15

4 Move onto showing on number lines and then partitioning and using known facts Year 3 23 = = x 5p = 50p 50p + 50p = 100p 3 x 5p = 15p 100p + 15p = 115p = 1.15 Statutory Requirements - Pupils should be taught to: count from 0 in multiples of 4, 8, 50 and 100; 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. Pupils now use multiples of 2, 3, 4, 5, 8, 10, 50 and 100. 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, = = = 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 and 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). Pupils connect tenths to place value, decimal measures and to division by 10. They continue to recognise fractions in the context of parts of a whole, numbers, measurements, a shape, and unit fractions as a division of a quantity. Multiplication of 2 digit x 1 digit Children continue to use or to stand for an unknown CLIC MULTIPLICATION STEP 11 Solve 1dx2d number, and use a number line to show grouping as a more efficient way to calculate divisions with larger numbers. Children use their knowledge of multiplication as an array as an introduction to the grid layout for multiplication: Apples come in bags of 4. How many apples are there in 15 bags e.g x 4 10 x 4 = x 4 = = 60 Children are encouraged to estimate before calculating. They will use informal pencil and paper methods to support multiplication. They understand that multiplications can be done in any order. They understand that division is the inverse of multiplication. 23 people are travelling to the cinema. Each car can carry 5 people. How many cars are needed? 23 5 = 4r3 You need 5 cars. Books cost 5 each. How many can you buy with 23? 23 5 = 4r3 You can only buy 4 books. CLIC DIVISION STEP 16/17/18 Introduce children to partial x tables COIN MULTIPLICATION in order to develop children s confidence when making larger jumps of the divisor X (1, 2, 5, 10 etc are used as 5 25 these also link to coin values) Therefore when solving the problem above children would be able to see that 2 lots of 5 = 10 so 4 lots of 4 = 20 and then see that they would have 3 left over Children are introduced to the grid layout for multiplication: There are 8 classes in a school. Each class has 27 children. How many children are there altogether? Moving onto formal written grid method supported by either addition along a number line or column addition Then work on larger two digit one digit numbers Recall related facts for the 3, 4 and 8 times tables CLIC Column Method Multiplication Step 1

5 X Recall facts for the 3, 4 and 8 times tables (also see you tube and 17 times tables video to support teaching this) CLIC Column Method Multiplication Step 1 Children are encouraged to estimate before calculating. They may put an E next to it. They will continue to use informal pencil and paper methods to support multiplication. Children are developing a written method for TU x U. Year 4 Statutory Requirements Pupils should be taught to: count in multiples of 6, 7, 9, 25 and 1000 recall multiplication and division facts for multiplication tables up to 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 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. count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten (fractions) find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths (fractions) find the area of rectilinear shapes by counting squares counting in tens and hundreds, and maintaining fluency in other multiples through varied and frequent practice. 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 = 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 (see Mathematics Appendix 1). Pupils write statements about the equality of expressions (for example, use the distributive law 39 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, or three cakes shared equally between 10 children. Pupils understand the relation between non-unit fractions and multiplication and division of quantities, with particular emphasis on tenths and hundredths. Pupils understanding of the number system and decimal place value is extended at this stage to tenths and then hundredths. This includes relating the decimal notation to division of whole number by 10 and later 100. They relate area to arrays and multiplication. Children may be introduced to a vertical layout, starting with least significant number first: 27 x x x CLIC MULTIPLICATION STEP 14 Any 1dx2d Children will then move onto 3 digit numbers x 1 digit using formal methods CLIC MULTIPLICATION STEP 15 1dx3d Consolidate understanding of division as complimentary multiplication. Introduce children to a written method: Partial tables could be used to support the calculation = x x x = 14r

6 127 x x x x CLIC DIVISION STEP 19 Encourage children to estimate before calculating. Children will use informal pencil & paper methods to support division. Children begin to develop a written method for TU U. Continue to make use of number lines alongside partial tables. Consolidate grid layout, and most children introduced to the vertical layout: x x x x x x NB: Children unable to move to vertical layout will stay with the grid layout. Children can use multiplication to derive area of rectangular shapes A football pitch measures 23 metres by 9 metres. What is the area of the pitch? Children are encouraged to estimate before calculating. They may put an E next to it. They will continue to use informal pencil and paper methods to support multiplication. e.g for 78 x x 40 = 3600 (estimate) E By the end of year 4 children should be able to recall facts times tables up to 12 x 12 (also see you tube and 17 times tables video to support teaching this) Big Maths Learn It Schedule. CLIC MULTIPLICATION STEP 12 Solve any 1dx1d 89 6 = 14r r Develop complimentary multiplication encouraging larger groupings. Develop partial tables to include larger numbers x CLIC DIVISION STEP 20/1/22/ x x x x x = 32r = 32r4 Encourage children to estimate before calculating. Children will use informal pencil & paper methods to support division. Children extend the use of written methods to HTU U with integer remainders. Continue to make use of number lines alongside partial tables = 32r r = 32r r Moving onto more formal methods CLIC Column Method Division Step 3,4,5 Year 5 Statutory Requirements - 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 and 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

7 recognise and use square numbers and cube numbers, and the notation for squared ( 2 ) and cubed ( 3 ) 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. multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Mathematics Appendix 1). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations. They use and understand the terms factor, multiple and prime, square and cube numbers. 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 = = 24 r 2 = 24 = ). 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 = 92 x 10). Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, = ; 33 = 5 x ). Pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1. Pupils use their knowledge of place value and multiplication and division to convert between standard units. Children will then move onto 3 digit numbers x 1 digit using formal methods 1127 x x x x 100 8,000 8 x ,016 CLIC Division Step CLIC Column Method Division Step 7 Any 4d 2d, remainders as fractions. Moving to a more compact method (short multiplication) CLIC Multiplication Step 6 Any 4dx1d 1127 X Consolidate grid layout, and most children introduced to the vertical layout: 1127 x 28 x ,000 2, , , , x x x x 100 8,000 8 x x x 20 2, x , x ,556 CLIC Column Method Multiplication Step 7 Any 4dx2d Fractions Multiplying whole numbers by unit fractions:

8 OR OR Year 6-9 Statutory Requirements for Year 6 National Curriculum- Pupils 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 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 addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why 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. multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, = 8 1 ] divide proper fractions by whole numbers [for example, = 6 1 ] 3 associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 8 ] identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places multiply one-digit numbers with up to two decimal places by whole numbers use written division methods in cases where the answer has up to two decimal places solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison solve problems involving similar shapes where the scale factor is known or can be found solve problems involving unequal sharing and grouping using knowledge of fractions and multiples. Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar

9 addition and subtraction, short and long multiplication, and short and long division (see Mathematics Appendix 1). 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, x 3 = 5 and (2 + 1) x 3 = 9. Common factors can be related to finding equivalent fractions. Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle. Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if of a length is 36cm, then the whole length is 36 4 = 144cm). Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to two decimal places by onedigit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as = 0.8, and in practical contexts, such as measures and money. Pupils are introduced to the division of decimal numbers by one-digit whole number, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication. Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes). Pupils link percentages or 360 to calculating angles of pie charts. Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work. Pupils solve problems involving unequal quantities, for example, for every egg you need three spoonfuls of flour, of the class are boys. These problems are the foundation for later formal approaches to ratio and proportion. Children continue with vertical layout, more able may compact the working:1127 x 28 x ,000 2, , , , x x x x 100 8,000 8 x x x 20 2, x , x ,556 Encourage children to estimate before calculating. Children will use informal pencil & paper methods to support division. Children are using a written method for TO O (mixed number answers); 3d 2d (whole number answers) and numbers involving decimals e.g Continue with written method but extend to decimals. CLIC Division Step Moving to the long multiplication method 1127 X (20 x 1127) 9016 ( 8 x 1127) Children are encouraged to estimate before calculating. They will continue to use informal pencil and paper methods to support multiplication. Progression into decimal calculations. CLIC MULTIPLICATION STEP x 9 ideal context for money or measures x 9 or 4.65m x 9 Ensure children are using estimation to check if answers are reasonable. Fractions Multiplying two unit fractions:

10 Multiplying two non-unit fractions: Multiplying two proper fractions, with cross cancelling: CLIC Column Method Division Step 8-10 Multiplying mixed numbers by a whole number: Year 6 Divide a whole number by a unit fraction: 3 whole ones divided into halves Multiplying mixed numbers by a proper fraction: 2 whole ones divided into quarters Multiplying two mixed numbers: Divide a unit fraction by a whole number: 2 Year 7 to 9 Division with fractions using the multiplication of the inverse of the divisor: NB. Inverse of is (which is equal to 2) Inverse of (or ) is Trick: Change to a multiply and flip the fraction you are dividing by upside down. Division of fractions with non-unit fractions: = = = = = = = = = =

11 Division of fractions with mixed numbers: Convert mixed numbers into an improper fraction i.e. or = = = = = = Calculators will be used throughout to check calculates and pupils to understand how to interpret the display appropriately e.g. 1.2 = 1.20 when dealing with money