Birdwell School Academy

Whole School Policy For Numeracy Calculation 2014

MULTIPLICATION Written/Compiled: V.Moisey – Numeracy Co-ordinator: August/September 2014 Reviewed/Agreed: Headteacher/Teaching Staff/Learning Support Staff/Governors:

About our Calculation Policy The following calculation policy has been devised to meet the requirements of the National Curriculum 2014 for the teaching and learning of mathematics, and is also designed to give children a consistent and smooth progression of learning in calculations across the school. Please note that early learning in number and calculation in Reception follows the ‘Development Matters EYFS document, and this calculation policy is designed to build on progressively from the content and methods established in the Early Years Foundation Stage (EYFS). Please see separate document for EYFS.

Age Stage Expectations The calculation policy is organised according to age stage expectations as set out in the National Curriculum 2014. However, it is vital that children are taught according to the stage that they are currently working at, being moved onto the next level as soon as they are ready, or working at a lower stage until they are secure enough to move on.

Providing a context for calculation: It is important that any type of calculation is given a real life context or problem solving approach (not necessarily new to children) to help build children’s understanding of the purpose of calculation, and to help them recognise when to use certain operations and methods when faced with problems. This must be a priority within calculation lessons. Problem solving, using and applying, should be threaded throughout lessons and should not be a ‘bolt-on’ at the end of a series of taught lessons or at the end of a week.

Aims of the Policy   

To ensure consistency and progression in our approach to calculation To ensure that children develop an efficient, reliable, formal written method of calculation for all operations To ensure that children can use formal written methods accurately with confidence and understanding

How to use this Policy   

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use the policy as a basis for your planning ensure that children are confident with the methods outlined in the previous year’s guidance before moving on if at any time, children are making significant errors, return to the previous stage in calculation always use suitable resources, models and images to support children’s understanding of calculation and place value, as appropriate encourage children to make sensible choices about the methods they use when solving problems

Ensuring conceptual understanding to then enable children to choose an appropriate calculation method: It is vital that children have a conceptual understanding of numbers, the number system and the calculation methods they use. This will enable them to have a solid understanding in maths as well as given them tools to select appropriate calculation methods when solving mathematical problems.

Choosing a calculation method: Children need to be taught and encouraged to use the following processes in deciding what approach they will take to a calculation, to ensure they select the most appropriate method for the numbers involved:

Can I do it in my head using a mental strategy?

Steps to calculating

Approximate

Could I use some jottings or drawings to help me?

Calculate

Check it

Should I use a written method to work it out?

(Check reasonableness of final answer against approximation)

Written Calculation The aim for mental calculations. With mental work, the aim is to teach a child a repertoire of strategies from which they can select. With written calculations the ultimate aim is proficiency in a method for each operation with one clear progression route taught for each. Written calculation Building on the mental strategies they have used so they can understand the processes involved, children need first to be taught to record their methods in an expanded form. When ready - and this is dependent on teachers’ professional judgement - they are taught how to refine their recording to make it more compact. Challenges to teachers  Ensuring that recall skills are established first so children can concentrate on a written method without reverting to first principles. 

Making sure that, once written methods are introduced, children continue to look out for and recognize the special cases that can be done mentally.

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Catering for children who progress at different rates; some may grasp a compact method of calculation while others may never do so without considerable help; catering for children who can carry out some standard methods successfully, eg for addition but not subtraction.

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Recognising that children tend to forget a standard method if they have no understanding of what they are doing and if they do not visit it regularly throughout the year.

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Often the compactness of a vertical method show mathematical principles are applied, e.g. children may use place value when working mentally, but may be confused in written work because they do not understand how place value relates to ‘carrying’. There can be long lasting problems for those taught compact, vertical methods before they understand what they are doing eg children can undertake decomposition for subtraction but are unable to explain the place value involved.

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Simply correcting children’s errors may help in the short-term, but not permanently. Misunderstandings and misconceptions need to be analysed and children need to find their own errors. Children need to understand why a particular method works rather than simply following a set of rules. They can then fall back to a simple method if uncertain, or to check their answers.

Estimating (make a sensible guess) and Approximating using rounding. This should be encouraged for all four operations to give children a sense of what the answer might be after calculations have been carried out.

Both express the relationship between a number of equal parts and the whole.

MULTIPLICATION AND DIVISION - YEAR 1 Multiplication and Division facts  count in multiples of twos/fives/tens Mental Calculations  count in multiples of twos/fives/tens Problem solving  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

MULTIPLICATION AND DIVISION - YEAR 2 Multiplication and Division facts  count in steps of 2, 3, and 5 from 0, and in tens from any number, forward or backward (copied from Number and Place Value)  recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers  know that doubling is multiplying by 2 and halving is divided by 2  I know significant doubles (eg 10 + 10, 50+ 50=, 50p+50p= ) involving doubling multiples of 5 up to 50 Mental Calculations  show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot Written Calculations  calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs Problem Solving  solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts

In order to prepare the children for written methods they should be exposed to a range of multiplication strategies using practical equipment such as counting groups of objects ( I have 3 groups of 5, how many do I have altogether?). Following this links should be made to repeated addition (3 groups of 5 is the same as 5 + 5 + 5) and the use of arrays. Children should be taught times tables as age appropriate according to the curriculum and framework guidance. Children will count repeated groups of the same size in practical contexts. They will use the vocabulary associated with multiplication in practical contexts. They will solve practical problems that involve combining groups of 2, 5 or 10 using visual aids to support understanding (combining groups of the same number. Understand multiplication is related to doubling and combing groups of the same size (repeated addition) Washing line, and other practical resources for counting. Concrete objects. Numicon; bundles of straws, bead strings Use repeated addition to carry out multiplication supported by the use of counters/cubes etc. /cubes.

Use cuisenaire rods to develop the vocabulary relating to ‘times’ – “Pick up five, 4 times”. Use arrays to understand multiplication can be done in any order (commutative ). They will see everyday versions of arrays, e.g. egg boxes, baking trays, ice cube trays, wrapping paper etc. and use this in their learning answering questions such as; 'How many eggs would we need to fill the egg box? How do you know? See Year 2 guidance for arrays.

Link to repeated addition.

Children should use pictorial representations/jottings and may use rings to show eg 3 groups of 2 and 2 groups of 3 when introducing the commutative law of multiplication.

Initially recording of calculating should be done by adults to model what children have done in pictures, symbols, numbers and words. Over time there should be an expectation that children will also become involved in the recording process. Whilst cameras are an excellent way of keeping a record of what children have done, they are not a substitute for the modelling of different ways of recording calculation procedures. Key vocabulary: groups of, lots of, times, array, rows, columns, altogether, multiply, count

Children continue to use repeated addition to carry out multiplication tasks and represent their counting on a bead string or a number line. On a bead string, children count out three lots of 5 then count the beads altogether. On a number line. Children count on in groups of 5. These models illustrate how multiplication relates to repeated addition.

Expressing multiplication as a number sentence using x. Using understanding of the inverse and practical resources to solve missing number problems. 7 X 2 =  ;  = 2 X 7; 14 = 2 X  Further develop understanding of multiplication using array and number lines (see Year 1). Include multiplications not in the 2, 5 or 10 times tables. It is important to be able to visualise multiplication as a rectangular array. This helps children develop their understanding of the commutative law i.e. 3 x 4 = 4 x 3. The rectangular array allows the total to be found by repeated addition and the link can be made to the ‘x’ sign and associated vocabulary of ‘lots of’ ‘groups of’ etc. The relationship between the array and the number line showing both repeated additions should be demonstrated alongside each other. For more direct comparison, this could then be demonstrated on a single number line as appropriate. Begin to develop understanding of multiplication as scaling (3 times bigger/taller)

Doubling numbers up to 10 + 10 Link with understanding scaling Using known doubles to work out double 2d numbers (double 15 = double 10 + double 5)

Children begin pattern work on a 100 square to help them begin to recognise multiples and rules of divisibility.

Towards written methods Use jottings to develop an understanding of doubling two digit numbers.

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times..., tens, ones

MULTIPLICATION AND DIVISION - YEAR 3 Multiplication and Division facts  count from 0 in multiples of 4, 8, 50 and 100 (copied from Number and Place Value) Please include others where you feel they are necessary.  recall and use multiplication and division facts for the 3,4 and 8 multiplication tables Mental Calculations  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 (appears also in Written Methods)  Through doubling, they connect the 2, 4 and 8 multiplication tables  Children develop efficient mental methods, eg using commutativity and associativity (for example 4 x 12 x 5 = 4 x 5 x 12 = 20 = 240 Written Calculations  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. Inverse Operations, Estimating and Checking  estimate the answer to a calculation and use inverse operations to check answers (copied from Addition and Subtraction) Problem Solving  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.

Missing number problems - Continue with a range of equations as in Year 2 but with appropriate numbers. Mental methods Doubling 2 digit numbers using partitioning Demonstrating multiplication on a number line – jumping in larger groups of amounts - 13 x 4 = 10 groups 4 = 3 groups of 4 Written methods (progressing to 2d x 1d) Developing written methods using understanding of visual images. Introduce the grid method initially by linking it with arrays (with counters or Deines/Base 10), with children physically making an array to represent the calculation, then translate this to grid method format. To do this, children must be able to:  Partition numbers into tens and ones  Multiply multiples of ten by a single digit (e.g. 20 x 4) using their knowledge of multiplication facts and place value  Recall and work out multiplication facts in the 2, 3, 4, 5, 8 and 10 times tables.  Work out multiplication facts not known by repeated addition or other taught mental strategies (e.g. by commutative law, working out near multiples and adjusting, using doubling etc.) Strategies to support this are repeated addition using a number line, bead bars and arrays: Children should be shown how this model shows 13 x 4 but the calculation steps are ‘made easier’ by partitioning the 13 into 10 and 3. The use of Dienes.

This is the first exposure to the distributive law of multiplication and children should be given plenty of opportunity to explore this. The link between arrays and the grid method should be made clear to children by the use of place value apparatus such as place value counters and Dienes.

emphasises the distributive law.

OR

Leading to Please ensure that children line numbers up correctly to give them their correct place value.

Children who are secure and accurate could begin year 4 work – Expanded short multiplication. (See Year 4 guidance). Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated ad-dition, column, row, commutative, sets of, equal groups, times, _times as big as, once, twice, three times..., partition, grid method, multiple, product, tens, ones, value

MULTIPLICATION AND DIVISION - YEAR 4 Multiplication and Division facts  count in multiples of 6, 7, 9, 25 and 1000  recall multiplication and division facts for multiplication tables up to 12 x 12  doubles and halves of numbers up to 50 Mental Calculations  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  understand the impact of place value when a number is multiplied or divided by 10 and 100 (Do not teach, just add a nought)  halve whole numbers including odd numbers  know that x 4 is doubling twice and x 8 is doubling three times Written Calculations  multiply two-digit and three-digit numbers by a one-digit number using a formal written method Pupils practise to become fluent in the formal written method of short multiplication for multiplying using multi-digit numbers, and short division with exact answers when dividing by a one-digit number (see Appendix 1).

Properties of Numbers - Multiples/Factors/Primes/Squares and Cube numbers 

recognise and use factor pairs and commutativity in mental calculations (repeated)

Inverse Operations, Estimating and Checking  estimate and use inverse operations to check answers to a calculation Problem Solving  solve problems involving multiplication 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. The Distributive Law says that multiplying a number by a group of numbers 

added together is the same as doing each multiplication separately. Example: 3 × (2 + 4) = 3×2 + 3×4 So the "3" can be "distributed" across the "2+4" into 3 times 2 and 3 times 4.

Continue with a range of missing number using appropriate numbers. Teach alongside inverse operation. Mental methods Counting in multiples of 6, 7, 9, 25 and 1000, and steps of 1/100. Solving practical problems where children need to scale up. Relate to known number facts. (e.g. how tall would a 25cm sunflower be if it grew 6 times taller?) See next page for advice on adding rows or columns when using the grid method.

Written methods (progressing to 3d x 2d) Children to embed and deepen their understanding of the grid method to multiply up 2d x 2d. Ensure this is still linked back to their understanding of arrays and place value counters. Use Dienes/Base 10/place value counters where needed. Use column addition to add the partial products.

Children should begin to estimate the answer by rounding (Approximation). eg 412 x 3 = 400 x 3 = 1200 (Before calculating the actual answer. The should check against estimate to see if their answer is sensible. 4 x 3 = 12 40 x 3 = 120 412 x 3 400 x 3 = 1200 Model and encourage use of known facts: 400 x 3 = 4 x 3 = 12 40 x 3 = 120 400 x 3 = 1200

Line numbers up accurately to ensure numbers are given correct place value and to ensure accurate addition.

See next page for guidance on adding rows or columns. When confident and secure with the grid method children should begin to work towards using an Expanded Short method.

Step 1

Model alongside the grid method. Children should make comparisons/ find relationships. Ensure numbers are lined up to give them their correct place value. LEADING TO: Refine the method in preparation for Formal short multiplication.

After calculating check answer against approximation. Children should check to see if their actual answer is reasonable.

Use the language of place value to ensure understanding. Multiply least significant digit first. Use inverse operations to check answers. It might be more appropriate to start with multiplying the most significant digit initially so that clear links can be made between the methods. Children should be moved towards starting with the column of smallest value as soon as their understanding of the relationship between the methods allows, to move towards longmultiplication.

Step 2

Firstly represent the method of recording in a column format, but showing the working. Draw attention to the links with the grid method.

98 x 8 784

Step 1 Year 5

6 Short multiplication. ONLY FOR YEAR 4 CHILDREN SECURE IN PREVIOUS METHOD, modelled alongside use of base 10. See Year 5 guidance.

move towards long multiplication. Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, groups of, sets of, lots of, equal groups, times, multiply, times as big as, once, twice, three times... partition, grid method, total, multiple, product, sets of, inverse

Using the grid method for two- by two-digit numbers. Adding the rows or adding the columns Having calculated the sections of the grid, children will decide whether to add the rows or columns first as they become more confident with recognising efficient calculations and depending on the numbers that are produced through the calculation. Adding the columns: eg 53 x 12 (Children should find an approximate answer first by using rounding and predict if their answer will be bigger or smaller than their estimate.) 53 x 12 is approximately 50 x 10 = 500 Ensure children line numbers up correctly to give them their correct place value. This also helps them to avoid making errors when adding the partial calculations. Remember to use known facts. Children should check their actual answer against their approximation to see if it is reasonable.

partial calculations

Adding the rows - is most efficient in this case. eg 53 x 12 (Children should find an approximate answer first by using rounding and predict if their answer will be bigger or smaller than their estimate.)

MULTIPLICATION AND DIVISION - YEAR 5 Multiplication and Division facts  count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000  (copied from Number and Place Value) Mental Calculations  multiply and divide numbers mentally drawing upon known facts including decimals  0.6 x 7 = 4.2 because 6 x 7 = 42  3.5 ÷ 5 = 0.7 because 35 ÷ 5 = 7  multiply and divide whole numbers and those involving decimals by 10, 100 and 1000  know that TU x 5 is TU x 10 and then divided by 2 (18 x 5 = (18 x 10) ÷ 2)  know that TU x 9 is TU x 10 then subtract TU (18 x 9 = (18 x 10) – 18 = 162)  round and compensate for near pounds (£4.99 x 3 = £5 x 3 – 3p = £14.97)  use knowledge of doubles and halves of whole numbers to find doubles and halves of decimal numbers (2.3 + 2.3 = 4.6 because 23+23=46; Half of 5.8 is 2.7 because half of 58 is 27) Written Calculations  multiply numbers up to 4 digits by a one- or two-digit number using a formal written method  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 Properties of Numbers - Multiples/Factors/Primes/Squares and Cube numbers    

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 recognise and use square numbers and cube numbers, and the notation for squared (2) and cubed (3)

Inverse Operations, Estimating and Checking  estimate and use inverse operations to check answers to a calculation Problem Solving  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

Continue with a range of equations with appropriate numbers. Also include equations with missing digits. Use practical resources and jottings. Written methods - move towards use of more complex numbers Start with recapping/revising the expanded short multiplication using more complex numbers. (Year 4 guidance). Short multiplication/Expanded Long multiplication/Compact Long multiplication: Each digit continues to be multiplied by each digit, but the totals are recorded in a more compact form, using ‘carrying’ Children’s understanding of place value is vital so they recognise when they are multiplying tens, hundreds etc they record their answer in the correct columns. Children should be able to explain each step of the process, initially relating it back to previous methods and experiences. They should be able to articulate the different stages of this calculation with the true values of the digits they are dealing with.

Short multiplication

Do not use term: ‘under the doorstep’

The array using base 10 becomes the basis for understanding short multiplication. (Initially this could be done without exchanging.) Continue to approximate first. Remember to give numbers their correct place value. Use language of place value. eg 6 x 4 = 24. Put 2 in the ‘Ones’ column, carry 2 tens under the ‘Tens’ colum. 2 x 6 (Relate to 20 lots of 6) = 12 and the ‘2’ carried = 144. Place Value Language is vital. 4x 6 = 24 so record the 4 in the Expanded Long multiplication. ones and carry the 20 (2) into the tens 6 x 20 = 120 + (the carried) 20 = LEADING TO: 140 so record the 40 in the tens OR and carry the 100 (1) into the Check against hundreds column . 6 x 100 = 600 estimate. + (the carried) 100 = 700. Record as 7 in the hundreds. Children 20 x 4 = 80 so record this on a should decide new if their answer answer row in the correct is reasonable. columns. 20 x 20 = 400. Record the 4 in the hundreds column. 20 x 100 = 2000 so record this Note modelling of noting steps to help with selfappropriately. checking and ensuring knowledge of place value. Use column addition to add the two Key vocabulary groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, totals total, together, resulting in 3224. commutative, sets of, equal groups, _times as big as, once, twice, three times..., partition, grid method, multiple, product,

Compact Long multiplication.

inverse, square, factor, integer, decimal, short/long multi-plication, ‘carry‘

MULTIPLICATION AND DIVISION - YEAR 6 Multiplication and Division facts  count forwards or backwards in steps of powers of 10 for any given number up to 1,000,000  (copied from Number and Place Value) Mental Calculations  perform mental calculations, including with mixed operations and large numbers  associate a fraction with division and calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction (e.g. 3/8) - (copied from Fractions) Written Calculations  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 one-digit whole number using the formal written method of short division where appropriate for the context  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  use written division methods in cases where the answer has up to two decimal places (copied from Fractions (including decimals)) Properties of Numbers - Multiples/Factors/Primes/Squares and Cube numbers  

identify common factors, common multiples and prime numbers use common factors to simplify fractions; use common multiples to express fractions in the same denomination (copied from Fractions)  calculate, estimate and compare volume of cubes and cuboids using standard units, including centimetre cubed (cm3) and cubic metres (m3), and extending to other units such as mm3 and km3 (copied from Measures) Order of Operations 

use their knowledge of the order of operations to carry out calculations involving the four operations

Inverse Operations, Estimating and Checking  use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy Problem Solving  solve problems involving addition, subtraction, multiplication and division  solve problems involving similar shapes where the scale factor is known or can be found  (copied from Ratio and Proportion)

Continue with a range of equations with appropriate numbers. Also include equations with missing digits. Mental methods Identifying common factors and multiples of given numbers Solving practical problems where children need to scale up. Relate to known number facts. Written methods Continue to practise and develop the formal short multiplication method and formal long multiplication method with larger numbers and decimals throughout Y6. Return to an expanded form of calculation initially, if necessary (see Y5 guidance). As in year 5, children should be able to explain each step of the process, initially relating it back to previous methods and experiences. They should be able to articulate the different stages of this calculation with the true values of the digits they are dealing with. Short and long multiplication as in Y5, and multiply decimals with up to 2d.p by a single digit. Pupils progress towards multiplying Th H T O x T O and H T O . t h x T using formal written method of long multiplication. Progress to multiplication of decimals, in the context of money is recommended to ensure a concrete understanding of the concept and value of the digits.

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times... partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short / long multiplication, „carry‟, tenths, hundredths, decimal