AIM HIGH ACADEMY TRUST NEWBOTTLE PRIMARY ACADEMY Mathematics

AIM HIGH ACADEMY TRUST NEWBOTTLE PRIMARY ACADEMY Mathematics Calculation Policy 2014 This calculation policy is intended to bring consistency, continuity and progression as methods build upon each other from Reception to Year 6. Rapid recall strategies and mental calculation methods will serve to reinforce and supplement these written methods.

Teaching Sequence The teaching sequence will be dependent upon the pupils maturity and the efficiency of the method to solve a number sentence or problem. The methods are, therefore, not essentially in any order. A number line can be as effective as an algorithmic method when solving a complex calculation – needs of the calculation must be taken into account if the pupil is to become fluent in performing written and mental calculations and mathematical techniques.

Challenges • Ensure that recall skills are established first so pupils can concentrate on the written method without reverting to first principles. • Ensure that, once written methods are introduced, pupils continue to look for and recognise situations where calculations can be completed using a mental strategy. • Understand that pupils progress at different rates and methods need to be adapted to meet those needs. • Pupils need to understand the method as well as applying it to various situations.

Written Method – Compact Algorithm Method • High understanding alongside mastering the techniques to empower pupils to become numerate. • There will be a common and consistent approach to mathematics across the school. • To provide pupils with a personalised, focused and progressive approach to calculation. • Pupils need to look at the numbers involved and select the most efficient method to use.

Outer skills of Numeracy 1. Maths Breadth: this needs to include wider subject areas within the mathematics curriculum, such as measures, probability etc. 2. Problem solving: these are purely mathematical problems that place number acquisition in a different context. 3. Word Problems: real life scenarios. 4. Multi-methods: looking at different ways of solving the same mathematical question.

Key Areas to be kept consistent throughout the school: 1. When solving a word problem - use RUCSAC mnemonic to aid children's approach: R = Read the question U = Understand what the question is asking you C = Choose which operation you will need to use (+ - x ÷ ) S = Solve it! A = Answer the question C = Check your answer! 1. n 2. When solving calculations, encourage children to approximate first. 3. From Yr 2 onwards, ensure children understand the INVERSE methods of calculation e. g. addition / subtraction and multiplication / division 4. When using NUMBER LINES: Adding = jump forwards above the line Subtracting = jump backwards below the line 5. Introduce and vary the language used for the four basic calculation operations: ADDITION: add, sum of, total, count on, increase by, plus, altogether SUBTRACTION: take away, subtract, less than, minus, find the difference MULTIPLICATION: multiply, times, lots of / groups of, product DIVISION: divide by, share, groups of, quotien

EYFS - Counting, Addition & Subtraction • Children will be taught to say numbers in familiar contexts such as number rhymes or in role play. This will develop into the • counting of everyday objects. • The children will be taught to say the number names in order and recognise the numerals from 1 - 20 • Children will be taught to recognise count and order numbers up to 20. • Wherever possible the children will be given the opportunity to solve simple problems involving the use of the skills listed above. • The children will be taught to use the vocabulary involved in addition and subtraction. • They will be taught to recognise differences in quantity of everyday objects and to find one more or one less. • The children will learn to relate addition to the combining of 2 groups and subtraction as taking away from a group. • The children will use the vocabulary of addition and subtraction in practical activities and in solving practical problems. • Children will be introduced to number bonds to 10

Key Stage 1 • The principal focus of mathematics teaching in is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This will involve working with numerals, words and the four operations, including with practical resources (e. g. concrete objects and measuring tools). • Pupils will develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. Teaching will involve using a range of measures to describe and compare different quantities such as length, mass, capacity/volume, time and money. • By the end of Year 2, pupils will know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency.

Lower Key stage 2 – Year 3 and 4 • The principal focus of mathematics teaching will ensure that pupils become increasingly fluent with whole numbers and the four operations, including number facts and the concept of place value. This will ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers. • Pupils will develop their ability to solve a range of problems, including with simple fractions and decimal place value. Teaching will ensure that pupils draw with increasing accuracy and develop mathematical reasoning so they can analyse shapes and their properties, and confidently describe the relationships between them. It will ensure that pupils can use measuring instruments with accuracy and make connections between measure and number. • By the end of Year 4, pupils will have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work.

Upper Key Stage 2 – Year 5 and 6 • The principal focus of mathematics teaching will ensure that pupils extend their understanding of the number system and place value to include larger integers. This will develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio. • Pupils will develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures will consolidate and extend knowledge developed in number. Teaching will also ensure that pupils classify shapes with increasingly complex geometric properties and that they learn the vocabulary they need to describe them. • By the end of Year 6, pupils will be fluent in written methods for all four operations, including long multiplication and division, and in working with fractions, decimals and percentages.

Addition - Objects 3 + 5 = 7 + 2 8 9 Remember to: • Find out how many there are in the larger group • Count on from the larger group • Count on each one carefully =

Addition 3+5= +5 3 Remember to: Find the start number Count on the right amount… One jump for each number See where you have landed This is the answer 8 +3 16 + 3 = 10 16 19 20 +7 16 + 7 = 16 20 23

Addition - Partitioning 47 + 68 = 40 7 60 0. 4 + 0. 5 3. 4 + 2. 5 = 8 + 40+60=100 7+ 8= 15 115 Remember to: • Partition the numbers • Write out the two new questions • Add the tens • Add the units answer to the tens answer 3+2 5 + 5. 9 0. 9 Remember to: Partition the numbers Write out the two new questions Add the units Add the tenths Add the units answer to the tenths

Addition - Partitioning £ 3. 85 + £ 8. 67 = £ 3. 00 + £ 8. 00 = £ 11. 00 £ 0. 80 + £ 0. 60 = £ 1. 40 £ 0. 05 + £ 0. 07 = £ 0. 12 £ 12. 52 Remember to: • Partition the numbers • Write out the new questions • Add the pounds • Add the tens • Add the pence • Add the totals together

Addition - Algorithm 7 6 6 8 6 + 4 8 + 5 4 9 1 8 · 7 + 5 6 · 4 1 2 3 5 7 5 · 1 1 1 Remember to: • Line up the numbers in their columns • Add them together one column at a time – right to left • Carry the tens, hundreds etc to be added in next column • Add the totals

Subtraction - Objects 5 - 3 = 7 2 Remember to: • Set out the number of objects to start with • Tell me how many we will be taking away • Take away the objects • Count how many are left - 2 5 =

Subtraction – Going Backwards 5 -3= -5 3 Remember to: Find the start number Count back the right amount… One jump for each number See where you have landed This is the answer 8 -3 16 - 3 = 10 16 19 20 -7 16 - 7 = 16 20 23

Subtraction – Going Forwards 84 - 73 = 11 7 73 4 80 84 37 63 - 26 = 33 4 26 30 Remember to: Find the gap to the multiple of 10 Find the gap from the multiple of 10 Add the gaps together 19 5. 69 5. 57 0. 12 8. 57 – 2. 88 = 2. 84 8. 57 3. 00 63

Subtraction - Algorithm 6 1 7 6 8 17 1 9 8 5 7 1 1 1 8 • 6 2 5 - 4 8 - 5 9 6 - 4 • 9 0 8 2 8 3 8 9 3 • 7 1 7 Remember to: • Line up the numbers in their columns • Subtract one column at a time – right to left • Reduce the tens/hundreds/tenths column etc by 1 (crossing through original digit and writing a new digit reduced by 1 and then, where necessary, writing a mini 1 next to the digit on the right with the amended digit) • Subtract as before.

Multiplication - Objects 4 lots of 3 blocks 12 Remember to: • Set out one lot of … • Set out another lot of… • Continue until the correct amount of groups have been arranged • Add together 5 X 3 = 5 + 5 = 15 Remember to: • X = ‘lots of’ • Find out which number to keep adding • Find out how many times to add it • Write it out as an addition number sentence

Multiplication - Tables Know your tables to 12 X 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 70 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144

Multiplication - Partitioning 4 X 23 = 6 X 725 = 4 X 20 = 80 6 X 700 = 4200 6 X 20 = 120 + 6 X 5 = 30 4350 4 X 3 = 12 92 Remember to: • Partition the 2 d number • Write out the 2 questions • Times the tens • Times the units • Add the totals together Remember to: • Partition the 3 d number • Write out the 3 questions • Times the hundreds • Times the tens • Times the units

Multiplication – Partitioning/Grid 38 X 69 = 4 X 7. 6 = 4 X 7 = 28 4 X 0. 6 = 2. 4 32. 4 Remember to: • Partition the number • Write out the 2 questions • Times the units • Times the tenths • Add the totals together X 30 8 60 9 1800 270 480 72 Remember to: • Partition the numbers • Set out in grid • Times the tens • Times the units • Times the tens and units • Add the answers together NB: Grid may be necessary for the pupil as a transitional method

Multiplication - Algorithm 3 8 5 X 6 5 1 0 5 X 2 4 7 3 4 8 1 9 1 8 5 7 6 1 5 6 0 0 0 X 2 3 1 0 1 3 1 2 5 6 1 4 6 • 2 4 • 4 4 • 8 0 • 2 4 1 Remember to: • Do the tables bit! • Carry the tens, hundreds etc to be added in next column • Add the totals together

Division - Objects Sharing 12 ÷ 2 = 6 Remember to: • Give each pile one object at a time • Check at the end that each pile has the same amount Grouping 12 ÷ 2 = 6 • Remember to: • Set out the correct number in the starting pile • Know what size of smaller pile you are making • Pull out smaller piles until the starting pile has all gone • Count how many ‘lots of…’ There are

Division - Objects Grouping 13 ÷ 2 = 6 r 1 • Remember to: • Set out the correct number in the starting pile • Know what size of smaller pile you are making • Pull out smaller piles until the starting pile has all gone • Count how many ‘lots of…’ There are • Record how many were left over Tables 15 ÷ 5 = 3 17 ÷ 3 = 5 r 2 Remember to: • Learn table facts to solve division calculations. X 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 70 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144

Division – Through Multiplication 280 ÷ 14 = 20 286 ÷ 14 = 20 r 6 X 14 1 14 2 28 5 70 10 140 20 280 50 700 Remember to: Fill out the table to locate the answer by: • Find ‘lots of…’ • Multiply by 10 • Double answers • Halve answers • Find the remainder where necessary 1 14 2 28 5 70 10 140 20 280 50 700 NB: A fundamental part of being numerate is the ability to find multiples of numbers. r 6

Division 669 ÷ 8 = 83 r 5 (⅝) 6721 ÷ 22 = 305 • 5 81 ÷ 3 = 27 8 2 7 3 2 8 1 Remember to: 8 6 6 3 2 9 r 5 2 3 0 5 • 5 2 1 • 0 1 2 1 1 1 0 1 1 2 6 7 - 6 6 - 0 • Use your tables knowledge to find ‘lots of…’ NB: Can be a remainder • Use the remainder as a mini tens etc digit or a fraction • Calculate as before • Record amounts • Use remainder, fraction of decimal where necessary

Division - chunking 7 3 3 5 3 7 1 7 0 (7 x 10) Remember to: 0 1 7 2 3 7 1 371 ÷ 7 = 53 6 7 • Multiply divisor by 10 and subtract from amount 0 (7 x 10) • Continue until within tables amount 1 • Multiply divisor by a tables number and 0 (7 x 10) subtract from amount • Add together number of chunks 1 (multiples) to find answer 0 (7 x 10) 9 1 7 0 (7 x 10) 2 1 0 0 (7 x 3) NB: Chunking may be required depending upon the needs of the pupil – transition from one method to another or as the efficient method for a particular calculation. Caution needs to be taken when the answer has a remainder.

Fact Families Link calculations together to ensure progression and continuity and consistency. 7 + - 7 7 + 8 =15 8 + 7 = 15 15 - 8 = 7 15 -7 = 8 8 7 apples + 8 apples =15 apples 8 grams + 7 grams = 15 grams 15 tens - 8 tens = 7 tens 15 tenths - 7 tenths= 8 tenths 8 7 x 8 =56 8 x 7 = 56 56 ÷ 8 = 7 56 ÷ 7= 8 x ÷ 7 tens x 8 tens =56 tens 8 tens x 7 tens= 56 tens ÷ 8 tens = 7 tens 56 tens ÷ 7 tens= 8 tens

Percentages • Teach pupils how to recognise percentages as fractions and calculate. For example • 50% = ½ • 25% = ¼ • 75% = ¾ • 33% = ⅓ • 66% = ⅔ • All fifths and tenths • For example: 25% 0 f 60 = ¼ of 60

Calculating Percentages of Amounts • Turn percentage into a fraction and calculate. • 25% of 80 ¼ of 80 = 20 • Finding 10% and adjust accordingly. • 35% of 80 10% = 8 • 30% = (3 x 8 = 24) 5% = (half of 8 =4) • 24 + 4 = 28. Therefore 35% of 80 = 28

Calculating Percentages of Amounts • Find 1% and multiply by original percentage amount. • 36% 0 f 76 • 1% = 0. 76 (calculation would be 36 x 0. 76) • Divide percentage by 100 and multiply the answer by the amount. • 45% of 86 • 0. 45 x 86

Calculating Fractions of Amounts • Divide the amount by the denominator and multiply the answer by the numerator. • ¾ of 60 Numerator Denominator • 60 ÷ 4 = 15 Denominator Numerator 15 x 3 = 45 ¾ of 60 = 45

Fractions When teaching all four operations for fractions, make sure that working out is detailed and accurate (one equals sign per line and annotate the lines of working with the appropriate methods as necessary. )

Fractions – Addition and Subtraction Start with the same denominator (pictorial representation may help to reinforce the idea): Simply add/ subtract the numerators when the denominators are the same:

Fractions – Addition and Subtraction When the denominators are not the same, make the fractions equivalent so that the denominators are the same and then add/subtract numerators as before, simplifying if possible. (by dividing the numerator and denominator by the highest common factor).

Fractions – Addition and Subtraction Change the improper fractions to a mixed number by working out how many times the denominator goes into the numerator and what the remainder is. Mixed numbers and/or improper fractions are worked out in the same way. First change the mixed numbers into improper (top heavy) fractions:

Fractions – Addition and Subtraction Answers are returned in the form they are asked (usually the simplest form), unless otherwise stated. So in this instance either the improper fraction or the mixed number would be just as valid. Alternatively add/subtract the whole numbers and add/subtract the fraction parts separately.

Fractions - Multiplication When multiplying, simply multiply numerator by numerator and denominator by denominator: Simplify where possible by dividing the numerator and denominator by the highest common factor (hcf). (using the hcf of 4)

Fractions - Multiplication Alternatively, simplify first: Cancelling down first is an important tool for when the numerators and/or denominators are very large.

Fractions - Division When dividing, invert (flip the numerator and denominator) the second fraction only and then multiply as before. It is always worth including whole numbers and fractions from the start so pupils know that the method works in the same way for a mixture of whole numbers and fractions too. Again simplify if possible.