Calculation policy All Saints Benhilton C of E Primary School Jan 2014

Transcription

1 Calculation policy All Saints Benhilton C of E Primary School Jan 2014

2 Counting Year 1 - Number and Place Value (Statutory Requirements) Count to and from 100, counting backwards and forwards, beginning with 0 or 1, or from any given number. Year 2 - Number and Place Value (Statutory Requirements) Count in steps of 2, 3 and 5 from 0 and count in 10s from any number, forwards and backwards Year 3 - Number and Place Value (Statutory Requirements) Count from 0 in multiples of 4, 8, 50 and 100, finding 10 or 100 more or less than a given number Count up and down in tenths; recognise that tenths arise from a number dividing an object by ten equal parts and in dividing one-digit numbers or quantities by ten. Year 4 - Number and Place Value (Statutory Requirements) Count from 0 in multiples of 6, 7, 9, 25 and Count backwards and forwards through zero including negative numbers Count up and down in hundredths, recognising that hundredths arise from a number dividing an object by hundred in dividing tenths by ten. Year 5 - Number and Place Value (Statutory Requirements) Count forwards and backwards in steps of powers of ten for any given number up to

3 Manipulatives - used throughout the school Counting apparatus Place value apparatus Dienes Numicon Place value cards Place value counters Numbered number lines Marked but unnumbered number lines Blank number lines Hundred square Counting stick Bead strings Models and images charts ITPs number facts, ordering numbers, number grids, counting on and back in n numbers.

4 Classroom Key Words four operations Something similar will be found in every classroom

5 Addition Year 1 Subtraction Pupils should be taught to: 2A read, write and interpret mathematical statements involving addition (+), subtraction ( ) and equals (=) signs 2B represent and use number bonds and related subtraction facts within 20 add and subtract one-digit and two-digit numbers to 20, including zero 2C solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 = 9. Pupils memorise and reason with number bonds to 10 and 20 in several forms (for example, = 16; 16 7 = 9; 7 = 16 9). They should realise the effect of adding or subtracting zero. This establishes addition and subtraction as related operations. Pupils combine and increase numbers, counting forwards and backwards. They discuss and solve problems in familiar practical contexts, including using quantities. Problems should include the terms: put together, add, altogether, total, take away, distance between, difference between, more than and less than, so that pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. + = signs and missing numbers Know by heart all pairs of numbers with a total of 10 and Know that addition can be done in any order = = 5 Begin to use the and = signs to record mental calculations in a number sentence When adding mentally put biggest number first and count on Know by heart subtraction facts for numbers up to 10 and = 5 8 Use manipulatives to subtract 20 5 = = 8 Subtract single digits by counting back under the number line in ones. The answer is the number you land on 15 7 = = 8

6 Progression in difficulty when adding on a number line Begin to find the difference by counting on from the smallest number. 1 digit + 1 digit Circle both numbers then count on from the smaller to the larger number above the number - The answer is the number of steps made 15 7 = 8 2 digit + 1 digit Extend by counting on in 10s, then ones. 2 digit + 2 digit 53 = 50 and 3 Please Note: For this method to become successful and embedded, children must be able to add 10 and one from any given number using their knowledge of place value and having had plenty of experience with jumping in ones and tens on a 100 square.

7 Multiplication Year 1 Division Pupils should be taught to: 3A 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 Pictures and symbols There are 3 sweets in one bag. How many sweets are there in 5 bags? Pictures / marks 12 children get into teams of 4 to play a game. How many teams are there? (Recording on a number line modelled by the teacher when solving problems) Use of bead strings to model groups of. Know doubles and corresponding halves Using laminated sheets with circles (groups) on them, children group objects using the correct mathematical vocabulary. = signs and missing numbers 6 2 = = = 3 3 = 6 2 = 3 3 = 2 = 3 3 = Understand division as sharing and grouping Sharing 6 sweets are shared between 2 people. How many do they have each? 6 2 can be modelled as: Grouping There are 6 sweets. How many people can have 2 each? (How many 2 s make 6?) x = signs and missing numbers 7 x 2 = = 2 x 7 7 x = = x 7 x 2 = = 2 x x = = x Arrays Number Square - Timer for number facts

8 3 x 4 4 x 3 Children begin using jottings of simple multiplication with the associated vocabulary They begin by drawing the number of groups, then draw the number of dots inside the circles. They count the number of dots they have altogether to get to the answer. Children are exposed to the different ways in which multiplication can be expressed using manipulatives and linking it to real life situations. They begin to understand that repeated addition can also be expressed as multiplication using concrete materials. Expressing multiplication as repeated addition Children begin to commit multiples of 2, 5, 10 to memory and use these facts to solve problems.

9 There are 10 spiders how many legs do they have altogether? 8 X 10 = 80 When Peter behaves well in school he gets 2 sweets at the end of the day. If he behaves well for 5 days, how many sweets will he get altogether? There are 4 flower beds in a garden. Each flower bed has 3 flowers. How many flowers are in the garden altogether?

10 Addition Year 2 Subtraction Pupils should be taught to: 2A 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 2B recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100 2C add and subtract numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and ones > a two-digit number and tens > two two-digit numbers > adding three one-digit numbers 2D show that addition of two numbers can be done in any order (commutative) and subtraction of one number from another cannot 2E recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems. Pupils extend their understanding of the language of addition and subtraction to include sum and difference. Pupils practise addition and subtraction to 20 to become increasingly fluent in deriving facts such as using = 10; 10 7 = 3 and 7 = 10 3 to calculate = 100; = 30 and 70 = They check their calculations, including by adding to check subtraction and adding numbers in a different order to check addition (for example, = = ). This establishes commutatively and associativity of addition. Recording addition and subtraction in columns supports place value and prepares for formal written methods with larger numbers. + = signs and missing numbers Continue to practice the mental recall of all pairs of number bonds with a total of 10 and = 20 Know which digit changes when adding 1s or 10s to any number = = 25 The vocabulary to be used is FINDING THE DIFFERENCE. In other words, children are finding how many numbers there are in between two chosen numerals. The steps are recorded by counting up from the smaller to the larger number to find the difference. Children are taught to write the bigger number first when writing the number sentence. They then circle both numbers on the number line. Starting with the smallest number children count on. They count the number of jumps. Partitioning two and three digit numbers. Continue with partition two digit numbers into their Tens and Units and extend to partitioning three digit numbers into their H, T, U. Knowing which digit changes when adding 1s or 10s to any number = = 25 Finding the difference between two 2 digit numbers by counting on in tens first, then counting on in 1s using a marked number line.

11 In Year 2 children learn to add using a blank number line. Progression in difficulty when adding on a blank number line Finding the difference between two 2 digit numbers by counting on using an empty number line. Children are taught to jump in tens and ones from the smaller number to get to the larger number. The jumps are added to get the answer. 1 digit by 1 digit They begin to combine their jumps. 2 digit by 1 digit Finding the difference using their knowledge of number bonds to the next multiple of digit by 2 digit Essential knowledge: Being able to decompose and recompose any number under10.

12 2 digit by 2 digit and combine their jumps Begin bridging through 10 then teach the formal method of column addition. For children to be confident with this method they must be able to quickly recompose numbers. Suggested mental maths activity is the use of cluster cards.

13 Multiplication Year 2 Pupils should be taught to: 3A recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers 3B calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication ( ), division ( ) and equals (=) signs 3C show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot 3D 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 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 commutatively and inverse relations to develop multiplicative reasoning (for example, 4 5 = 20 and 20 5 = 4). x = signs and missing numbers Count in steps of 2s, 3s, 5s, 10s and 20s forward and back from 0 and from any of its multiples using the 100 / 200 square and taking the opportunity to discuss patterns that are recognised. Division = signs and missing numbers Count in steps of 2s, 3s, 5s, 10s and 20s forward and back from 0 and from any of its multiples using the 100 / 200 square and taking the opportunity to discuss patterns that are recognised. Continue using a range of equations as in Year 1 but with appropriate numbers. Understand division as sharing and grouping 18 3 can be modelled as: Sharing 18 shared between 3 OR Know doubles and corresponding halves and extend to partitioning numbers then double / partitioning numbers then halve. Or Grouping - How many 3 s make 18? Use fingers to work out doubles up to double 5.

14 3 Double 3 = 6 Children continue using jottings of simple multiplication with the associated vocabulary and those who still find this difficult will use the laminated sheets with circles to group concrete objects. They begin by drawing the number of groups, then draw the number of dots inside the circles. They count the number of dots they have altogether to get to the answer. Know doubles and corresponding halves and extend to partitioning numbers then double / partitioning numbers then halve Teach using jottings when multiplying multiples of ten by writing T in each of the groups. They begin by drawing the number of groups, then write the number of T s inside the circles. They count the number of T s using their knowledge of counting in tens to obtain an answer. corresponding division facts Use known multiplication facts to work out Teach jumping on a marked number line in multiples of 2, 3, 5, 10. This method requires children to keep the jumps equal in size as they count the number of jumps.this is a challenging process, however it further embeds the understanding of repeated addition. The constant re-enforcement of vocabulary groups of is very important Children continue to use concrete materials and physical resources to share objects equally with or without the help of the laminated grouping sheets. 12 muffins shared between 3 people = 4

15 I have 18 strawberries. Put them into groups of 3. How many groups have we got altogether? As children become confident with counting in multiples of 2, 3, 5, 10 they begin to use the empty number line to solve multiplication problems. In this method there are strong links with the activity of counting choir using 100 / 200 squares and the recognition of patters with each of the multiples. Children write their own number after each time they make a jump. They further develop their skills of problem solving using multiplication and begin to relate it to the area of a rectangle / square. Children investigate the number of multilink cubes needed to create a block with a given number of length and width. Grouping with the use of jottings. Children first draw the total number of items using dots, then put circles around the given number of dots. They count the number of groups to obtain an answer. In year 2 children are exposed to grouping in all multiples between 2-9. I have 18 multilink cubes. If I put them into groups of 3, how many groups have I got? Children begin to use a marked number line to solve division problems. Please note: Although division is associated with repeated subtraction, we teach children to COUNT ON in jumps of the given number because when it comes to teaching division with remainders on a number line, this is the only way it will work. This eliminates confusion. This method requires children to find out how many jumps of 3 can they make between 0 and 18. They circle 0 and 18 on the number line before they commence their equal jumps of 3. The constant re-enforcement of vocabulary into groups of is very important.

16 More confident children, who are able to reliably count in multiples of 2, 3, 5, 10 use an empty number line to make their jumps. They write their own numbers underneath the number-line each time they complete a jump to keep track of where they are. The challenge in this process is to remember to stop once they got to the required number. In this case,18. Reinforce division as grouping through arrays and jottings and introduce remainders.

17 Pupils should be taught to: 2A add and subtract numbers mentally, including: Addition Year 3 Subtraction a three-digit number and ones - a three-digit number and tens - a three-digit number and hundreds 2B add and subtract numbers with up to three 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. Pupils practise solving varied addition and subtraction questions. For mental calculations with two-digit numbers, the answers could exceed 100. Pupils use their understanding of place value and partitioning, and practise using column addition and subtraction with increasingly large numbers up to three digits to become fluent. (see Mathematics Appendix 1) + = signs and missing numbers Continue using a range of equations as in Year 2 but with appropriate numbers. Partition into tens and ones and recombine Either partition both numbers and recombine or partition the second number only e.g = = = Begin with teaching this method without carrying. Carried digits are recorded below the line, using the words carry ten or carry one hundred, not carry one. Later, extend to adding three-digit and two-digit numbers, two three-digit numbers and numbers with varied number of digits. - = signs and missing numbers Continue using a range of equations as in Year 2 but with appropriate numbers. Find a small difference by counting up e.g = 7 This can be modelled on an empty number line (see complementary addition below). Subtract the nearest multiple of 10, then adjust. Continue as in Year 2 but with appropriate numbers. Use known number facts and place value to subtract = Pencil and paper procedures Complementary addition = 668 Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable

18 Written methods = = Children should be encouraged to use inverse operations to check if their answer is correct. This gives them opportunity to practise both operations (addition and subtraction) at the same time. Explicit teaching needs to point out that if they add the bottom number to the answer they should end up with the top number. Calculation: Checking using the inverse Please note: This is purely the written method of subtraction. When calculating mentally using smaller numbers, teachers should model counting on using an empty number line. When calculating with time and finding time-differences, the number line method should be used EVERY TIME

19 Multiplication Year 3 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. x = signs and missing numbers Continue using a range of equations as in Year 2 but with appropriate numbers Because children have to get used to a new layout which does not necessarily provide understanding, it is important that the multiplication method is taught on split screen which shows the conceptual understanding alongside the procedural. Children must have secure times tables knowledge to 10 x 10 in order for them to see the benefits of this quick efficient method. Division = signs and missing numbers Children must have secure division facts knowledge to 10 x 10 in order for them to see the benefits of this quick efficient method. Teaching must however follow the order of difficulty to overcome possible misconceptions. Those who are not yet ready for this method should carry on with grouping through the use of arrays as the models in previous pages show. Because children have to get used to a new layout which does not necessarily provide understanding, it is important that the multiplication method is taught on split screen which shows the conceptual understanding alongside the procedural. The carrying of digits further complicate the learning of this method, therefore the following progression in the teaching is recommended. Partitioning and grid methods are also used 14 x 6 = 10 x 6 = = 84 4 x 6 = x 16 = X = 224 Continue using a range of equations as in Year 2 but with appropriate numbers. Sharing and grouping 30 6 can be modelled as: grouping groups of 6 taken away and the number of groups counted e.g sharing sharing among 6, the number given to each person Remainders 41 4 = 10 r

20 The carrying of digits further complicate the learning of this method, therefore the following progression in the teaching is recommended. Begin with numbers where carrying is not involved. Example: 32 x x 3 Always start multiplying by the unit number. So 3 is multiplied by 2 first, then 3 is multiplied by 3. Again, begin by multiplying the units groups 10 x OR 41 = (10 x 4) + 1 Pencil and paper procedures Suggested mental maths starter before teaching the division method with remainders is to find remainders when dividing numbers mentally. Example: 27 5 = 5r2 or 38 6 = 6r2 or 82 9 = 9r1 OR All children should be able to calculate using this method by the end of year 3.

21 Addition Year 4 Subtraction Pupils should be taught to: 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 Pupils continue to practise both mental methods and columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics Appendix 1). Notes and guidance (non-statutory requirements) + = signs and missing numbers Continue using a range of equations as in Year 3 but with appropriate numbers. Partition into hundreds, tens and ones and recombine Either partition both numbers and recombine or partition the second number only e.g = = = = signs and missing numbers Continue using a range of equations as in Year 3 but with appropriate numbers. Extend to = 20 - Find a small difference by counting up = Begin with teaching this method without carrying. Carried digits are recorded below the line, using the words carry ten or carry one hundred, not carry one. Later, extend to adding three-digit and two-digit numbers, two three-digit numbers and numbers with varied number of digits. Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. Children however need to be careful how they set out the numbers when calculating with decimals Subtract 9 or 11. Begin to add/subtract 19 or = 26 Use known number facts and place value to subtract (partition second number only) = = 27 2 = Use number square to begin with

22 Written methods = = = Children should be encouraged to use inverse operations to check if their answer is correct. This gives them opportunity to practise both operations (addition and subtraction) at the same time. Explicit teaching needs to point out that if they add the bottom number to the answer they should end up with the top number. Calculation: Checking using the inverse Please note: This is purely the written method of subtraction. When calculating mentally using smaller numbers, teachers should model counting on using an empty number line. When calculating with time and finding time-differences, the number line method should be used EVERY TIME

23 Multiplication Year 4 Pupils 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 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. 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. x = signs and missing numbers Continue using a range of equations as in Year 3 but with appropriate numbers Partition 47 x 6 = x 6 = (40 x 6) + (7 x 6) = ( 240 ) + ( 42 ) = 282 OR Use the grid method of multiplication (as below) Pencil and paper procedures Grid method 72 x 38 is approximately 70 x 40 = 2800 x The carrying of digits further complicate the learning of this method, therefore the following progression in the teaching is recommended. Begin with numbers where carrying is not involved. Example: 32 x x 3 Division = signs and missing numbers Continue using a range of equations as in Year 3 but with appropriate numbers. Sharing and grouping Continue to understand division as both sharing and grouping (repeated subtraction). Remainders Quotients expressed as fractions or decimal fractions 61 4 = 15 ¼ or groups 5 groups groups groups Pencil and paper procedures The next level of difficulty is to write 0 above the digit into which the divisor doesn t go into. OR

24 Always start multiplying by the unit number. So 3 is multiplied by 2 first, then 3 is multiplied by 3. Again, begin by multiplying the units. When knowledge is secure, higher numbers are used to introduce carrying. Examples: 643 x x 8 Carrying must be recorded as shown. All children should be able to do this by the end of year 4. Year 4 should move onto 2D x 2D or 3D x 2D in the summer term but only those children who are secure with their multiplication facts up to 10 x 10.

25 Addition Year 5 Subtraction Pupils should be taught to: 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. Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics Appendix 1). They practise mental calculations with increasingly large numbers to aid fluency (for example, = ). Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics Appendix 1). They practise mental calculations with increasingly large numbers to aid fluency (for example, = ). Model using negative number line + = signs and missing numbers Continue using a range of equations as in Year 4 but with appropriate numbers. Partition into hundreds, tens, ones and decimal fractions and recombine Either partition both numbers and recombine or partition the second number only e.g = = = = signs and missing numbers Continue using a range of equations as in Year 4 but with appropriate numbers. Find a small difference by counting up Continue as in Year 4but with appropriate numbers e.g = 5 Subtract mentally a near multiple of 10 to or from a two-digit number Continue as in Year 4 but with appropriate numbers e.g is the same as Use known number facts and place value to subtract Continue as in Year 4 but with appropriate numbers e.g = Begin with teaching this method without carrying. Carried digits are recorded below the line, using the words carry ten or carry one hundred, not carry one. Later, extend to adding three-digit and two-digit numbers, two three-digit numbers and numbers with varied number of digits Written methods = = = = Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. Children however need to be careful how they set out the numbers when calculating with decimals. Children should be encouraged to use inverse operations to check if their answer is correct. This gives them opportunity to practise both operations (addition and subtraction) at the same time.

26 Explicit teaching needs to point out that if they add the bottom number to the answer they should end up with the top number. Calculation: Checking using the inverse Use a zero as a place holder. In these examples children need to understand that the decimal points are always written underneath each other when using column addition. Please note: This is purely the written method of subtraction. When calculating mentally using smaller numbers, teachers should model counting on using an empty number line. When calculating with time and finding time-differences, the number line method should be used EVERY TIME

27 Multiplication Year 5 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. Multiply and divide whole numbers and those involving decimals by 10, 100 and 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. 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 = 4 2 = = ). 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 ). x = signs and missing numbers Continue using a range of equations as in Year 4 but with appropriate numbers Partition 87 x 6 = x 6 = (80 x 6) + (7 x 6) = ( 480 ) + ( 42 ) = 522 OR 87 X6 42 ( 6 x 7) 480 ( 6 x 80) 522 (units, then tens, hundreds etc) OR Use the grid method of multiplication (as below) Pencil and paper procedures Grid method 372 x 24 is approximately 400 x 20 = 8000 x When knowledge is secure, higher numbers are used to introduce carrying. Division = signs and missing numbers Continue using a range of equations as in Year 4 but with appropriate numbers. Sharing and grouping Continue to understand division as both sharing and grouping (repeated subtraction). Remainders Quotients expressed as fractions or decimal fractions = groups groups 4 groups groups Pencil and paper procedures Make a point of teaching the following: when the divisor doesn t go into the last digit of the dividend we write the 0 but we will also write that number as a remainder. OR Carrying must be recorded as shown.

28 Teach estimating the approximate answer to the multiplication using mental methods. In the below example children are encouraged to multiply the whole numbers of 6 and 5 to get the answer of 30. This will help them gauge whether the magnitude of the number they get as a result is right x X = Decimal points are taken out the calculation is completed just like whole numbers. Once an answer is obtained, the number of digits after the decimal point in both numbers are counted to indicate the number of digits after the decimal point in the answer.

29 Addition Multiplication Year 6 Subtraction Division Please note that addition, subtraction, multiplication and division all merge into one for yr 6. 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. 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 (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. x = signs and missing numbers Begin with teaching this method without carrying. Carried digits are recorded below the line, using the words carry ten or carry one hundred, = signs and missing numbers Written methods for subtraction not carry one. Later, extend to adding three-digit and two-digit numbers, two three-digit numbers and numbers with varied number of digits = = = = Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. Children however need to be careful how they set out the numbers when calculating with decimals Children should be encouraged to use inverse operations to check if their answer is correct. This gives them opportunity to practise both operations (addition and subtraction) at the same time. Explicit teaching needs to point out that if they add the bottom number to the answer they should end up with the top number. Use a zero as a place holder.

30 In these examples children need to understand that the decimal points are always written underneath each other when using column addition. Calculation: Checking using the inverse Begin by multiplying the unit with each of the digits. Children need to be taught that the 0 in the second row is written as a placeholder because we are now multiplying the tens with each digit. Teach estimating the approximate answer to the multiplication using mental methods. In the below example children are encouraged to multiply the whole numbers of 6 and 5 to get the answer of 30. This will help them gauge whether the magnitude of the number they get as a result is right x 5.4 = Please note: This is purely the written method of subtraction. When calculating mentally using smaller numbers, teachers should model counting on using an empty number line. When calculating with time and finding timedifferences, the number line method should be used EVERY TIME Children should move on to decimal subtraction e.g money remembering to change the unit of measurement from to p. Continue using a range of equations as in Year 5 but with appropriate numbers. Make a point of teaching the following: when the divisor doesn t go into the last digit of the dividend we write the 0 but we will also write that number as a remainder. Decimal points are taken out of both numbers and calculate multiplication just like whole numbers. Once an answer is obtained, the number of digits after the decimal point in both numbers are counted to indicate the number of digits after the decimal point in the answer. This method is followed on from the short division however uses a different format to make finding the remainder easier to calculate. Children must be taught to express long division as decimals as well as a

31 mixed number fraction. To express remainders as a decimal number, we must carry on with the division by bringing down a zero until we have remainders. Children should use their knowledge of place value and conversions between fractions and decimals to express the answer as a decimal as well as a mixed number fraction. In both of the above methods children should check if their answer is correct using inverse operations by multiplying their answer by the divisor and adding the remainder to their answer. Using and applying: Once confident with this method, provide children with plenty of opportunity to be able to use and apply their newly gained skills to solve problems that involves getting answers with remainders and decimals.