Calculations Policy. Staff Meeting : Autumn term 2014 Adopted at Full Governing Body Meeting :

Transcription

1 Calculations Policy Staff Meeting : Autumn term 2014 Adopted at Full Governing Body Meeting : Review Cycle : Every 3 years Review Due : Autumn term 2017 Version No. 1.0 Signed.. Chair of Governors Date:... 1

2 Introduction Stowupland Pyramid KS1 to KS2 Calculations Policy Children are introduced to the processes of calculation through practical, oral and mental activities. As children begin to understand the underlying ideas they develop ways of recording to support their thinking and calculation methods, use particular methods that apply to special cases, and learn to interpret and use the signs and symbols involved. Over time children learn how to use models and images, such as empty number lines, to support their mental and informal written methods of calculation. As children s mental methods are strengthened and refined, so too are their informal written methods. These methods become more efficient and succinct and lead to efficient written methods that can be used more generally. By the end of Year 6 children are equipped with mental, written and calculator methods that they understand and can use correctly. At whatever stage in their learning, and whatever method is being used, it must still be underpinned by a secure and appropriate knowledge of number facts, along with those mental skills that are needed to carry out the process and judge if it was successful. This calculation policy aims to support the following aims of the National Curriculum: To ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. 2

3 Written methods for addition of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for addition which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient written method for addition of whole numbers. To add successfully, children need to be able to: recall all addition pairs to and complements in 10 e.g = 10; add mentally a series of one-digit numbers, such as ; add multiples of 10 (such as ) or of 100 (such as ) using the related addition fact, 6 + 7, and their knowledge of place value; Partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways. Note: It is important that children s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for addition. Resources ITP s, number line, place value cards, hundred square, bead strings. EYFS and Year 1 (Part A) At first children will relate addition to the combining of 2 groups: Count out 3, count out 2. Put together and count out 5 For example: = 5 + = 5 Introduce the term add as appropriate. Recall number bonds to 10. Children are encouraged to make use of fingers as these are a constantly available resource for calculations at this level. Missing numbers such as 5 = + 2 Alternatively, count out 3 and then count on 2 more to make 5 EYFS and Year 1 (Part B) The next step is to be able to count one more, and then several more, on a number line: For example: = 7 +1 Number bonds to 20. Recording introduced in Year

4 Year 2 Steps in addition can be recorded on a number line. The steps often bridge through a multiple of = 15 The mental methods involve partitioning, e.g. adding the tens and ones separately, often starting with the tens. The empty number line helps to record the steps on the way to calculating the total = 84 or: Pupils should be introduced to the concept of commutativity = = 8 Number bonds can be extended to 100, in multiples of 5 and 10. Add together three one digit numbers. Introduce using inverse operations (addition and subtraction). Solve missing number problems. 4

5 Year 3 The next stage is to record mental methods using partitioning. Add the tens and then the ones to form partial sums and then add these partial sums. Record steps in addition using partitioning: = = = Move on to a layout showing the addition of the tens to the tens and the ones to the ones separately. To find the partial sums either the tens or the ones can be added first, and the total of the partial sums can be found by adding them in any order. The addition of the tens in the calculation is described in the words forty plus seventy equals one hundred and ten, stressing the link to the related fact four plus seven equals eleven Three digit numbers: 187 or Mentally, add ones, tens and hundreds to a 3 digit number. Solve words problems (could involve decimals in context of money or measurement). Carrying In this method, recording is reduced further. Carry digits are recorded below the line, using the words 'carry ten' or 'carry one hundred', not 'carry one'. Pencil and paper procedures 5

6 Year 4 Before they use a written method to add decimal numbers, children should estimate the answer. For example, they calculate , and use rounding to check that their answer is approximately 23, rounding to check that their answer is approximately Column addition should progress to adding 4 digit numbers and using inverse operations to check. Solve 2 step word problems. Year 5 Use column addition to add numbers of more than 4 digits. Mentally add increasingly large numbers. Round numbers to check answers. Solve multi step problems. Year 6 Perform mental calculations, including with mixed operations and large 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. 6

7 Written methods for subtraction of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for subtraction which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient method for subtraction of four-digit numbers and beyond. To subtract successfully, children need to be able to: recall all addition and subtraction facts to 20; subtract multiples of 10 (such as ) using the related subtraction fact,16 7, and their knowledge of place value; partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into or ). Note: It is important that children s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for subtraction. 7

8 EYFS and Y1 In the early stages, children will be taught to take away one or two objects and find the new total. For example: 5-2=3 5 take 2 away is 3 The next stage is for children to be able to work out one less or several less on a number line. For example: 6 1 = 5-1 This progresses to calculations such as 12 3 which crosses the tens boundary At an early stage children are introduced to the concept of difference and that subtraction can be worked out by counting on the difference. For example: How much longer is this row of cubes than this one? 9 3 = Finally at this stage, children should be introduced to the concept of order in subtraction. 5 2 = does not equal 3! Year 2 8

9 The counting up method Once taught, this method should be used with numbers up to 20 and multiples of 10. The mental method of counting up from the smaller to the larger number can be recorded using either number lines or vertically in columns. The number of rows (or steps) can be reduced by combining steps. With two-digit numbers, this requires children to be able to work out the answer to a calculation such as 30 + = 74 mentally. or: Some pupils may benefit from the counting back method, but the counting on method should be the primary method taught at this stage Year 3 With three-digit numbers the number of steps can again be reduced, provided that children are able to work out answers to calculations such as = 200 and = 326 mentally. After the steps have been made, pupils can use a mental or written method to calculate the solution or: The teacher should model how to use the number line and how to write working vertically in columns to the side. This will act as a link to the next efficient method of subtracting Finally at this stage, column subtraction can be introduced, initially with no borrowing then with borrowing This method could be introduced using an expanded written method supported by the use of Diene s apparatus

10 Year 4 Up to 4 digit column subtraction. Use inverse operation to check and apply estimating. Solve 2 step problems Year 5 Use column subtraction for numbers with more than 4 digits. Mentally subtract increasingly large numbers. Use rounding to check calculations. Solve multistep problems. Year 6 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. 10

11 Written methods for multiplication of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for multiplication which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient method for two-digit by one-digit multiplication, two-digit by two-digit multiplication, and three-digit by two-digit multiplication. To multiply successfully, children need to be able to: recall all multiplication facts to 12 12; partition number into multiples of one hundred, ten and one; work out products such as 70 5, 70 50, or using the related fact 7 5 and their knowledge of place value; add two or more single-digit numbers mentally; add multiples of 10 (such as ) or of 100 (such as ) using the related addition fact, 6 + 7, and their knowledge of place value; add combinations of whole numbers Note: It is important that children s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication 11

12 . EYFS and Year 1 First children are taught to count in 2 s, 10 s and 5 s using practical objects. Use term lots of in EYFS and Year 1. For example: 3 x 2 = 6 Number sentences and x symbol not to be introduced in EYFS or Year x 5 = = 15 In Year 1, pupils are introduced to arrays using objects. 5 x 2 = 10 Children should be taught how to 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. 12

13 Year 2 Number Line Jumps Introduce the x symbol in Year = 8 2 x 4 = 8 Or 4 x 2 = Number line jumps support mental methods x 6 = 78 Arrays Arrays & Repeated addition X x 2 = = 8 2 x 4 = = 8 X 5 5 x 3 = x 5 = = = 3 Children should be taught to: 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 and mental methods. 13

14 Year 3 Step 1: Progress from arrays to simple grid method. Arrays to grid 12 x 3 X x = = 36 Step 2: Partitioning units and tens, hundreds or thousands Mental methods for multiplying TU U can be based on the distributive law (partitioning) of multiplication over addition. This allows the tens and ones to be multiplied separately to form partial products. These are then added to find the total product. Either the tens or the ones can be multiplied first but it is more common to start with the tens. 43 x 4 = 40 x 4 = x 4 = Children should be taught to: (a) Jane has 3 hats and four coats. How many different outfits can she make? 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 (b) and correspondence problems (a) in which n objects are connected to m objects. (b) A recipe for one cake needs: 120g flour 140g sugar 150g butter 1 egg How much sugar do I need to make 2 cakes? 14

15 Year 4 The grid method = (30 7) + (8 7) = = 266 As a staging post, an expanded method which uses a grid can be used. This is based on the distributive law and links directly to the mental method. It is an alternative way of recording the same steps. It is better to place the number with the most digits in the left-hand column of the grid so that it is easier to add the partial products. x = = 266 Or Children should be taught to: 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 onedigit 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. 15

16 Year 5 Step 1: The grid method should be extended to calculate up to four digit numbers multiplied by one digit numbers = (3000 6) + (100 x 6) + (40 x 6) + (8 x 6) = = x x Step 2: Long multiplication for two-digit numbers 24 x Children 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 twodigit number using a formal written method, including long multiplication for two-digit numbers multiply and divide numbers mentally drawing upon known facts multiply and divide whole numbers and those involving decimals by 10, 100 and

17 Year 6 Children 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. 17

18 Written methods for division of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for division which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to long division through upper Key Stage 2 first long division TU U, extending to HTU U and then HTU TU. To divide successfully in their heads, children need to be able to: understand and use the vocabulary of division for example in 18 3 = 6, the 18 is the dividend, the 3 is the divisor and the 6 is the quotient; partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways; recall multiplication and division facts to 12 12, recognise multiples of one-digit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value; know how to find a remainder working mentally for example, find the remainder when 48 is divided by 5; understand and use multiplication and division as inverse operations. Note: It is important that children s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division. To carry out written methods of division successful, children also need to be able to: understand division as repeated addition; estimate how many times one number divides into another for example, how many sixes there are in 47, or how many 23s there are in 92; multiply a two-digit number by a single-digit number mentally; add numbers using an appropriate method. 18

19 EYFS and Year 1 Understanding division as sharing 6 cakes are shared between 2 people. How many cakes do they have each? 6 2 = 3 Children explore division through practical and problem solving activities. 10 children are grouped into teams of 2. How many teams will there be? 10 2 = = 5 In Year 1 pupils begin to use arrays to explore division. 19

20 Year 2 Understanding division as grouping I I I I I I I I I I I I r = 4 r Children should be taught to: 58 5 = 11 r 3 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. 20

21 Year 3 Using knowledge of the times tables (4x, 3x 8x) and the related inverse equations to find division answers. 6 x 5 = = = 5 Writing and calculating statements. For example: 18 3 = 6. Counting on a number line A number line can be used to record grouping or sharing. It is also possible to record counting down in this way. This could be 18 chocolate eggs that are to be shared between three children OR 18 chocolate eggs that are to be packed (grouped) in boxes of 3 Remainders 16 3 = 5 r1 Grouping How many 3 s make 16, how many left over? Sharing - 16 shared between 3, how many left over? e.g. x1 x1 x1 x1 x1 r This method supports both mental division strategies and the expanded strategy shown below. Children can extend to multiples or chunks of the divisor e.g = (40 4) + (40 4) + (12 4) + (1 4) = 23 ¼ x x4 + 3x4 = 23 r1 R1 Remainder 1 should also be seen as 1 out of a possible four and therefore 1/ Children should be taught to: 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. 21

22 Year 4 Children should be taught to: recall multiplication and division facts for multiplication tables up to Continue empty number line from Year 3 to include bigger numbers. Chunking for most able as appropriate. 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 In Year 4 there is the opportunity to consolidate mental methods as written methods for division are not introduced until Year 5. However, more able Year 4 pupils may enjoy exploring this method: Climbing/chunking up method This method, often referred to as chunking, is based on adding multiples of the divisor, or chunks. Initially children add several chunks, but with practice they should look for the biggest multiples of the divisor that they can find to add. Chunking is useful for reminding children of the link between division and repeated addition x x x x Answer: 32R1 22

23 Year 5 Step 1: Introducing short division. Short division 36 3 becomes becomes Step 2: The next step is to introduce the written form of long division to tackle HTU TU. The layout on the right, which links to chunking, is in essence the long division method. Recording the build-up to the quotient on the left (where 20 and 3 are written below the 24) of the calculation keeps the links with chunking and reduces the errors that tend to occur with the positioning of the first digit of the quotient. How many packs of 24 can we make from 560 biscuits? Start by multiplying 24 by multiples of 10 to get an estimate. As = 480 and = 720, we know the answer lies between 20 and 30 packs. We start by subtracting 480 from Answer: 23 R 8 This calculation could be completed by chunking up. 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 multiply and divide whole numbers and those involving decimals by 10, 100 and 1000 Year 6 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 use their knowledge of the order of operations to carry out calculations involving the four operations. 23