CALCULATION POLICY FIN

Clifton upon Dunsmore C of E Primary School Calculation Policy for Mathematics Mastery 2018 “The answer is only the beginning.”

Mathematics Mastery At the centre of the mastery approach to the teaching of mathematics is the belief that all children have the potential to succeed. They should have access to the same curriculum content and, rather than being extended with new learning, they should deepen their conceptual understanding by tackling challenging and varied problems. Similarly, with calculation strategies, children must not simply rote learn procedures but demonstrate their understanding of these procedures through the use of concrete materials and pictorial representations. This policy outlines the different calculation strategies that should be taught and used in Year 1 to Year 6 in line with the requirements of the 2014 Primary National Curriculum. Background The 2014 Primary National Curriculum for mathematics differs from its predecessor in many ways. Alongside the end of Key Stage year expectations, there are suggested goals for each year; there is also an emphasis on depth before breadth and a greater expectation of what children should achieve. In addition, there is a whole new assessment method, as the removal of levels gives schools greater freedom to develop and use their own systems. One of the key differences is the level of detail included, indicating what children should be learning and when. This is suggested content for each year group, but schools have been given autonomy to introduce content earlier or later, with the expectation that by the end of each key stage the required content has been covered. For example, in Year 2, it is suggested that children should be able to ‘add and subtract one-digit and two-digit numbers to 20, including zero’ and a few years later, in Year 5, they should be able to ‘add and subtract whole numbers with more than four digits, including using formal written methods (columnar addition and subtraction)’. In many ways, these specific objectives make it easier for teachers to plan a coherent approach to the development of pupils’ calculation skills. However, the expectation of using formal methods is rightly coupled with the explicit requirement for children to use concrete materials and create pictorial representations – a key component of the mastery approach. Mathematical Language The 201 4 National Curriculum is explicit in articulating the importance of children using the correct mathematical language as a central part of their learning (reasoning). Indeed, in certain year groups, the non- statutory guidance highlights the requirement for children to extend their language around certain concepts. It is therefore essential that teaching using the strategies outlined in this policy is accompanied by the use of appropriate and precise mathematical vocabulary. New vocabulary should be introduced in a suitable context (for example, with relevant real objects, apparatus, pictures or diagrams) and explained carefully to ensure good procedural fluency. High expectations of the mathematical language used are essential, with teachers only accepting what is correct and expecting answers in full sentences. How to use the policy This mathematics policy is a guide for all staff at Clifton upon Dunsmore Primary School and has been adapted from work by the NCETM and White Rose Maths Hubs. It is purposely set out as a progression of mathematical skills and not into year group phases to encourage a flexible approach to teaching and learning. It is expected that teachers will use their professional judgement as to when consolidation of existing skills is required or if to move onto the next concept. However, the focus must always remain on breadth and depth rather than accelerating through concepts. Children should not be extended with new learning before they are ready, they should deepen their conceptual understanding by tackling challenging and varied problems. The policy does not recommend one set of resources over another, rather that, a variety of resources are used. For each of the four rules of number, different strategies are laid out, together with examples of what concrete materials can be used and how, along with suggested pictorial representations.

Pupils need to be taught and encouraged to decide what approach they will take to a calcula- tion, to ensure that they select the most appropriate method for the numbers involved: s To work out a tricky calculation:

Key language: sum, total, parts and wholes, plus, add, altogether, more, ‘is equal to’ ‘is the same as’., commutative addend plus addend equals sum Concrete Pictorial Abstract Combining two parts to make a whole Children to represent the manipulatives 4 + 3 = 7 shown using part, part, whole using manipulatives (numicon/cubes/ using drawings Four is a part, 3 is a part and the whole counters) is 7 2 7 2 7 4 3 3 4 + 3 = 7 shown using Singapore bar A bar model which encourages the The abstract number line: method children to count on, rather than count What is 2 more than 4? all. What is the sum of 2 and 4? 7 What is the total of 4 and 2? 4 + 2 = 4 3

Regrouping to make 10 using Children draw the ten frame and Children develop an understanding of manipulatives (cubes/10 frame/numicon) counters/cubes equality e.g. 6 + __ = 11 6 + 5 = 5 + __ 6 + 5 = __ + 4 TO + O using base 10. Continue to devel- Children to represent the base 10 e.g. Partitioning using part, part, whole to op understanding of partitioning and lines for tens and squares for ones. move into abstract written method place value 41 + 8 TO + TO using base 10. Continue to Children to represent the base 10 in a Looking for ways to make 10 develop understanding of partitioning place value chart. and place value.

Use of place value counters to add Children to represent the counters in a HTO + TO, HTO + HTO etc. When place value chart, circling when they there are 10 ones we exchange for 1 make an exchange. ten, when there are 10 tens in the tens column we exchange for 100. 1 Record place 10 value counters like this: 100 No place value counters in KS1 As the children move on, introduce decimals with the same number of as they don’t show quantity. decimal places and then different. Money can be used here. Variation: different ways to ask children to solve 21 + 34 Calculate the sum of twenty-one and thirty-four. ? 21 + 34 = ___= 21 + 34 21 34 Missing digit problems Word problems: In year 3, there are 21 children and in year 4, there are 34 children. How many ? children in total? 21 + 34 = 55. 21 34 Prove it

Key language: take away, less than, the difference, subtract, minus, fewer, decrease minuend subtract subtrahend equals difference Concrete Pictorial Abstract Physically taking away and removing Children draw the concrete resources 4 - 3 = ____ objects from a whole (ten frames, they are using and cross out the correct Numicon, cubes and other items such as amount. The bar model can also be used. ____ = 4 - 3 bean bags could be used) 4 - 3 = 1 4 4 3 ? 3 ? Counting back (using number lines or Children to represent what they see Children to represent the calculation on number tracks) children start with 6 pictorially, e.g. a number line or number track and show and count back 2. their jumps. Encourage children to use 6 - 2 = 4 an empty number line.

Finding the difference (using cubes, Children to draw the cubes/other con- Find the difference between 8 and 5 Numicon or Cuisenaire rods, other crete objects which they have used or objects can also be used). Calculate the use the bar model to illustrate what difference between 8 and 5. they need to calculate 8 – 5, the difference is ____ Children to explore why 9 -6 =8 – 5 =7 – 4 (have the same difference). Making 10 using ten frames. Children to present the ten frame Children to show how they can make 10 14-5 pictorially and discuss what they did to by partitioning the subtrahend. make 10. Column method using base 10/dienes. Children to represent the base 10/ Column method or children could count 48-7 dienes pictorially. back 7.

Column method using base 10 or dienes Represent the base 10 or dienes Formal column method. Children must and having to exchange 41-26 pictorially, remembering to show the understand that when hey have exchange exchanged the 10, they still have 41 because 41 = 30 + 11 Column method using place value Represent the place value counters Formal column method. Children must counters. 234-88 pictorially; remembering to show what understand what has happened when has been exchanged. they have crossed out the digits. Column method including decimals (using Represent the money/place value coun- Formal column method. Children must money place value counters, and other ters pictorially; remembering to show understand what has happened when manipulatives) what has been exchanged. they have crossed out the digits. Variations; different ways to ask children to solve 391-186 Raj spent £391, Timmy spent £186. How much more did Raj spend? Calculate the difference between 391 and 186.

Year 1  read, write and interpret mathematical statements involving addition (+) , subtraction (-) and equals (=) signs  represent and use number bonds and related addition and subtraction facts within 20  add and subtract one-digit and two-digit numbers to 20, including 0  solve one-step problems that involve addition and subtraction using concrete objects and pictorial representations, and missing number problems. Year 2 solve problems with addition and subtraction:  using concrete objects and pictorial representations, including those involving numbers, quantities and measures  applying their increasing knowledge of mental and written methods  recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100  add and subtract numbers using concrete objects, pictorial representations, and mentally, including:  a two-digit number and 1s  a two-digit number and 10s  2 two-digit numbers  adding 3 one-digit numbers  show that addition of 2 numbers can be done in any order (commutative) and subtraction of 1 number from another cannot  recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Year 3 add and subtract numbers mentally, including:  a three-digit number and 1s  a three-digit number and 10s  a three-digit number and 100s  add and subtract numbers with up to 3 digits, using formal written methods of columnar addition and subtraction.  estimate the answer to a calculation and use inverse operations to check answers  solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction. Year 4  add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate  estimate and use inverse operations to check answers to a calculation  solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why show that addition of 2 numbers can be done in any order (commutative)  recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Year 5  add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)  add and subtract numbers mentally with increasingly large numbers  use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy  solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why 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 4 operations  solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why  solve problems involving addition, subtraction  use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Key language: double, times, multiplied by, the product of, groups of, lots of, equal groups, commutative factor multiplied by factor is equivalent/equal to product Concrete Pictorial Abstract Repeated grouping/ repeated addition Children to represent the practical 3 x 4 = 12 3 x 4 resources in a picture or use a bar model. 4 + 4 + 4 = 12 4 + 4 + 4 There are 3 equal groups with 4 in each group Number lines to show repeated groups Represent this pictorially alongside a Abstract number line showing three 3 x 4 number line e.g. groups of four. 3 x 4 = 12 Cuisenaire rods can be used too.

Use arrays to illustrate commutativity Children to represent the arrays Children to be able to use an array ton (counters and other objects can also be pictorially. write a range of calculations e.g. used to make these.) 10 = 2 x 5 2 x 5 = 5 x 2 5 x 2 = 10 2 + 2 + 2 + 2 + 2 = 10 10 = 5 + 5 Partition to multiply using Numicon, base Children to represent the concrete Children to be encouraged to show the 10 or Cuisenaire rods. manipulatives pictorially steps they have taken 4 x 15 4 x 15 10 5 10 x 4 = 40 5 x 4 = 20 40 + 20 = 60 Formal column method with place value Children to represent counters Children to record what it is they are counters (base 10 can also be used) pictorially doing to show understanding 3 x 23

Formal column method with place value Children represent the counters/base Formal written method counters 10 pictorially 6 x 23 Introduce alongside concrete and pictorial methods. Not in isolation. When children start to multiply 3 digit by 3 digit and 4 digit x 2 digit etc., they should be confident with the abstract: To get 744 children have solved 6 x 124 To get 2480 the children have solved 20 x 124 Variations; different ways to ask children to solve 6 x 23 Mai had to swim 25 lengths, 6 times a week. How many lengths did she swim in one week? With the counters prove that 6 x 23 = 138

Key language: share, group, divide, divided by, half Dividend divided by devisor equals quotient Concrete Pictorial Abstract Sharing using a range of objects Represent the sharing pictorially 6 ÷ 2 = 3 6 ÷ 2 Children should also be encouraged to use their times table facts Repeated subtraction using Cuisenaire Children represent subtraction Abstract number line to represent the rods above a ruler pictorially equal groups that have been subtracted 6 ÷ 2 3 groups of 2

2 digit ÷ 1 digit with remainders using Children to represent the lollipop sticks 15 ÷ 4 = 3 remainder 1 lollipop sticks. pictorially Children should be encouraged to use Squares made as dividing by 4 times table facts they could also represent addition on a number line. There are 3 whole squares with 1 left There are 3 whole squares with 1 left There are 3 whole squares with 1 left over. over. over Short division using place value counters Represent the place value counters Children record the calculation using to group 615 ÷ 5 pictorially. the short division scaffold Make 615 with place value counters N.B. Children should also consider How many groups of 5 hundreds can you make with 6 whether calculations can be done men- hundred counters? tally with jottings i.e. Exchange 1 hundred for 10 tens How many groups of 5 tens can you make with 11 615 ÷ 5 615 ÷ 10 x 2 counters?

Long division using place value counters 2544 ÷ 2 We can’t group 2 thousands into groups of 12 so we’ll exchange them. We can group 24 hundreds into groups of 12 which leaves us with 1 hundred. After exchanging the hundred, we have 14 tens. We can group 12 tens into a group of 12, which leaves 2 tens. After exchanging the 2 tens, we have 24 ones. We can group 24 ones into 2 groups of 12, which leaves no remainder. Conceptual variation; different ways to ask children to solve 615 ÷ 5 Using the part whole I have £615 and share What is the calculation? model below, how can it equally between 5 What is the answer? you divide 615 by 5 bank accounts. How much will be in each without using short account? division? 615 pupils need to be put into 5 groups. How many will be in each group?

Year 1  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.  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. Year 2  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.  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 dis- crete 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).

Year 3  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 objects.  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 de- rive 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).

Year 4  recall multiplication and division facts for multiplication tables up to 12 × 12  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 objects.  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 an- swers (see Mathematics Appendix 1).  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, or three cakes shared equally between 10 children.

Year 5  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 (nonprime) 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 re- mainders appropriately for the context  multiply and divide whole numbers and those involving decimals by 10, 100 and 1000 Mathematics – key stages 1 and 2  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 under- standing the meaning of the equals sign  solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates.

 Pupils practise and extend their use of the formal written methods of short multiplication and 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.  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, includ- ing with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 = 4 98 = 24 r 2 = 24 2 1 = 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 = 92 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 ).

Year 6  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  use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.  Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and 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 fluen- cy.  Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50 etc., but not to a speci- fied 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.