## ACADEMIC

## Calculation

Policy

## Calculation Policy

In Lower Key Stage 2, children build on the concrete and conceptual understandings they have gained in Key Stage 1 to develop a real mathematical understanding of the four operations, in particular developing arithmetical competence in relation to larger numbers.

## Addition and Subtraction:

Children are taught to use place value and number facts to add and subtract numbers mentally and they will develop a range of strategies to enable them to discard the ‘counting in 1s’ or fingers-based methods of Key Stage 1. In particular, children will learn to add and subtract multiples and near multiples of 10, 100 and 1000, and will become fluent in complementary addition as an accurate means of achieving fast and accurate answers to 3-digit subtractions. Standard written methods for adding larger numbers are taught, learned and consolidated, and written column subtraction is also introduced.

## Multiplication and Division:

This key stage is also the period during which all the multiplication and division facts are thoroughly memorised, including all facts up to 12 × 12. Efficient written methods for multiplying or dividing a 2-digit or 3-digit number by a 1-digit number are taught, as are mental strategies for multiplication or division with large but ‘friendly’ numbers, e.g. when dividing by 5 or multiplying by 20.

## Fractions:

Children will develop their understanding of fractions, learning to reduce a fraction to its simplest form, as well as finding non-unit fractions of amounts and quantities. The concept of a decimal number is introduced and children consolidate a firm understanding of 1-place decimals, multiplying and dividing whole numbers by 10 and 100.

## Mental calculation:

- Know pairs with each total to 20; e.g. 2 + 6 = 8, 12 + 6 = 18, 7 + 8 = 15
- Know pairs of multiples of 10 with a total of 100
- Add any two 2-digit numbers by counting on in 10s and 1s or by using partitioning
- Add multiples and near multiples of 10 and 100
- Perform place-value additions without a struggle; e.g. 300 + 8 + 50 = 358
- Use place value and number facts to add a 1-digit or 2-digit number to a 3-digit number ; e.g. 104 + 56 is 160 since 104 + 50 = 154 and 6 + 4 = 10; 676 + 8 is 684 since 8 = 4 + 4 and ; 76 + 4 + 4 = 84
- Add pairs of ‘friendly’ 3-digit numbers; e.g. 320 + 450
- Begin to add amounts of money using partitioning

## Written calculation

- Use expanded column addition to add two or three 3-digit numbers or three 2-digit numbers
- Begin to use compact column addition to add numbers with 3 digits
- Begin to add like fractions; e.g. 3⁄8 + 1⁄8 + 1⁄8
- Recognise fractions that add to 1; e.g. 1⁄4 + 3⁄4 e.g. 3⁄5 + 2⁄5

## All:

- Know pairs of numbers which make each total up to 10, and which total 20
- Add two 2-digit numbers by counting on in 10s and 1s; e.g. 56 + 35 is 56 + 30 and then add the 5
- Understand simple place-value additions; e.g. 200 + 40 + 5 = 245
- Use place value to add multiples of 10 or 100

## Mental calculation:

- Know pairs with each total to 20; e.g. 8 – 2 = 6; e.g. 18 – 6 = 12; e.g. 15 – 8 = 7
- Subtract any two 2-digit numbers
- Perform place-value subtractions without a struggle; e.g. 536 – 30 = 506
- Subtract 2-digit numbers from numbers > 100 by counting up; e.g. 143 – 76 is done by starting at 76. Then add 4 (80), then add 20 (100), then add 43, making the difference a total of 67
- Subtract multiples and near multiples of 10 and 100
- Subtract, when appropriate, by counting back or taking away, using place value and number facts
- Find change from £1, £5 and £10

## Written calculation

- Use counting up as an informal written strategy for subtracting pairs of 3-digit numbers; e.g. 423 – 357
- Begin to subtract like fractions; e.g. 7⁄8 – 3⁄8

## All:

- Know pairs of numbers which make each total up to 10, and which total 20
- Count up to subtract 2-digit numbers; e.g. 72 – 47
- Subtract multiples of 5 from 100 by counting up; e.g. 100 – 35
- Subtract multiples of 10 and 100

## Mental calculation:

- Know by heart all the multiplication facts in the ×2, ×3, ×4, ×5, ×8 and ×10 tables
- Multiply whole numbers by 10 and 100
- Use place value and number facts in mental multiplication; e.g. 30 × 5 is 15 × 10
- Partition teen numbers to multiply by a 1-digit number; e.g. 3 × 14 as 3 × 10 and 3 × 4
- Double numbers up to 50

## Written calculation

- Use partitioning (grid multiplication) to multiply 2-digit and 3-digit numbers by ‘friendly’ 1-digit numbers

## All:

- Know by heart the ×2, ×3, ×5 and ×10 tables
- Double given tables facts to get others
- Double numbers up to 25 and multiples of 5 to 50

## Mental calculation:

- Know by heart all the division facts derived from the ×2, ×3, ×4, ×5, ×8 and ×10 tables
- Divide whole numbers by 10 or 100 to give whole number answers
- Recognise that division is not commutative
- Use place value and number facts in mental division; e.g. 84 ÷ 4 is half of 42
- Divide larger numbers mentally by subtracting the 10th multiple as appropriate, including those with remainders; e.g. 57 ÷ 3 is 10 + 9 as 10 × 3 = 30 and 9 × 3 = 27
- Halve even numbers to 100, halve odd numbers to 20

## Written calculation

- Perform divisions just above the 10th multiple using horizontal or vertical jottings and understanding how to give a remainder as a whole number
- Find unit fractions of quantities and begin to find non-unit fractions of quantities

## All:

- Know by heart the division facts derived from the ×2, ×3, ×5 and ×10 tables
- Halve even numbers up to 50 and multiples of 10 to 100
- Perform divisions within the tables including those with remainders; e.g. 38 ÷ 5

## Mental calculation:

- Add any two 2-digit numbers by partitioning or counting on
- Know by heart/quickly derive number bonds to 100 and to £1
- Add to the next 100, £1 and whole number; e.g. 234 + 66 = 300; e.g. 3·4 + 0·6 = 4
- Perform place-value additions without a struggle; e.g. 300 + 8 + 50 + 4000 = 4358
- Add multiples and near multiples of 10, 100 and 1000
- Add £1, 10p, 1p to amounts of money
- Use place value and number facts to add 1-, 2-, 3- and 4-digit numbers where a mental calculation is appropriate; e.g. 4004 + 156 by knowing that 6 + 4 = 10 and that 4004 + 150 = 4154 so the total is 4160

## Written calculation

- Column addition for 3-digit and 4-digit numbers
- Add like fractions; e.g. 3⁄5 + 2⁄5 = 7⁄5 = 1 2⁄5
- Be confident with fractions that add to 1 and fraction complements to 1; e.g. 2⁄3 + _ = 1

## All:

- Add any 2-digit numbers by partitioning or counting on
- Number bonds to 20
- Know pairs of multiples of 10 with a total of 100
- Add ‘friendly’ larger numbers using knowledge of place value and number facts
- Use expanded column addition to add 3-digit numbers

## Mental calculation:

- Subtract any two 2-digit numbers
- Know by heart/quickly derive number bonds to 100
- Perform place-value subtractions without a struggle; e.g. 4736 – 706 = 4030
- Subtract multiples and near multiples of 10, 100, 1000, £1 and 10p
- Subtract multiples of 0·1
- Subtract by counting up; e.g. 503 – 368 is done by adding ; 368 + 2 + 30 + 100 + 3 (so we added 135)
- Subtract, when appropriate, by counting back or taking away, using place value and number facts
- Subtract £1, 10p, 1p from amounts of money
- Find change from £10, £20 and £50

## Written calculation

- Use expanded column subtraction for 3- and 4-digit numbers
- Use complementary addition to subtract amounts of money, and for subtractions where the larger number is a near multiple of 1000 or 100; e.g. 2002 – 1865
- Subtract like fractions; e.g. 4⁄5 – 3⁄5= 1⁄5
- Use fractions that add to 1 to find fraction complements to 1; e.g. 1 – 2⁄3 = 1⁄3

## All:

- Use counting up with confidence to solve most subtractions, including finding complements to multiples of 100; e.g. 512 – 287; e.g. 67 + _ = 100

## Mental calculation:

- Know by heart all the multiplication facts up to 12 × 12
- Recognise factors up to 12 of 2-digit numbers
- Multiply whole numbers and 1-place decimals by 10, 100, 1000
- Multiply multiples of 10, 100 and 1000 by 1-digit numbers ; e.g. 300 × 6; e.g. 4000 × 8
- Use understanding of place value and number facts in mental multiplication; e.g. 36 × 5 is half of 36 × 10 ; e.g. 50 × 60 = 3000
- Partition 2-digit numbers to multiply by a 1-digit number mentally; e.g. 4 × 24 as 4 × 20 and 4 × 4
- Multiply near multiples by rounding; e.g. 33 × 19 as (33 × 20) – 33
- Find doubles to double 100 and beyond using partitioning
- Begin to double amounts of money; e.g. £35·60 doubled is £71·20

## Written calculation

- Use a vertical written method to multiply a 1-digit number by a 3-digit number (ladder method)
- Use an efficient written method to multiply a 2-digit number by a number between 10 and 20 by partitioning (grid method)

## All:

- Know by heart multiplication tables up to 10 × 10
- Multiply whole numbers by 10 and 100
- Use the grid method to multiply a 2-digit or a 3-digit number by a number ≤ 6

## Mental calculation:

- Know by heart all the division facts up to 144 ÷ 12
- Divide whole numbers by 10, 100, to give whole number answers or answers with 1 decimal place
- Divide multiples of 100 by 1-digit numbers using division facts; e.g. 3200 ÷ 8 = 400
- Use place value and number facts in mental division ; e.g. 245 ÷ 20 is half of 245 ÷ 10
- Divide larger numbers mentally by subtracting the 10th or 20th multiple as appropriate; e.g. 156 ÷ 6 is 20 + 6 as 20 × 6 = 120 and 6 × 6 = 36
- Find halves of even numbers to 200 and beyond using partitioning
- Begin to halve amounts of money; e.g. half of £52·40 is £26·20

## Written calculation

- Use a written method to divide a 2-digit or a 3-digit number by a 1-digit number
- Give remainders as whole numbers
- Begin to reduce fractions to their simplest forms
- Find unit and non-unit fractions of larger amounts

## All:

- Know by heart all the division facts up to 100 ÷ 10
- Divide whole numbers by 10 and 100 to give whole number answers or answers with 1 decimal place
- Perform divisions just above the 10th multiple using the written layout and understanding how to give a remainder as a whole number
- Find unit fractions of amounts

## Mental calculation:

- Know number bonds to 1 and to the next whole number
- Add to the next 10 from a decimal number; e.g. 13.6 + 6.4 = 20
- Add numbers with 2 significant digits only, using mental strategies; e.g. 3.4 + 4.8; e.g. 23 000 + 47 000
- Add 1- or 2-digit multiples of 10, 100, 1000, 10 000 and 100 000; e.g. 8000 + 7000; e.g. 600 000 + 700 000
- Add near multiples of 10, 100, 1000, 10 000 and 100 000 to other numbers; e.g. 82 472 + 30 004
- Add decimal numbers which are near multiples of 1 or 10, including money; e.g. 6.34 + 1.99; e.g. £34.59 + £19.95
- Use place value and number facts to add two or more ‘friendly’ numbers, including money and decimals; e.g. 3 + 8 + 6 + 4 + 7; e.g. 0.6 + 0.7 + 0.4; e.g. 2056 + 44

## Written calculation

- Use column addition to add two or three whole numbers with up to 5 digits
- Use column addition to add any pair of 2-place decimal numbers, including amounts of money
- Begin to add related fractions using equivalences; e.g. 1⁄2 + 1⁄6 = 3⁄6 + 1⁄6
- Choose the most efficient method in any given situation

## All:

- Add numbers with only 2 digits which are not zeros; e.g. 3.4 + 5.8
- Derive swiftly and without any difficulty number bonds to 100
- Add ‘friendly’ large numbers using knowledge of place value and number facts
- Use expanded column addition to add pairs of 4- and 5-digit numbers

## Mental calculation:

- Subtract numbers with 2 significant digits only, using mental strategies; e.g. 6.2 – 4.5; e.g. 72 000 – 47 000
- Subtract 1- or 2-digit multiples of 10, 100, 1000, 10 000 and 100 000: e.g. 8000 – 3000; e.g. 60 000 – 200 000
- Subtract 1- or 2-digit near multiples of 10, 100, 1000, 10 000 and 100 000 from other numbers: e.g. 82 472 – 30 004
- Subtract decimal numbers which are near multiples of 1 or 10, including money; e.g. 6.34 – 1.99; e.g. £34.59 – £19.95
- Use counting up subtraction, with knowledge of number bonds to 10, 100 or £1, as a strategy to perform mental subtraction; e.g. £10 – £3.45; e.g. 1000 – 782
- Recognise fraction complements to 1 and to the next whole number; e.g. 1 2⁄5 + 3⁄5 = 2

## Written calculation

- Use compact or expanded column subtraction to subtract numbers with up to 5 digits
- Use complementary addition for subtractions where the larger number is a multiple or near multiple of 1000
- Use complementary addition for subtractions of decimal numbers with up to 2 places, including amounts of money
- Begin to subtract related fractions using equivalences; e.g. 1⁄2 – 1⁄6 = 2⁄6
- Choose the most efficient method in any given situation

## All:

- Derive swiftly and without difficulty number bonds to 100
- Use counting up with confidence to solve most subtractions, including finding complements to multiples of 1000; e.g. 3000 – 2387

## Mental calculation:

- Know by heart all the multiplication facts up to 12 × 12
- Multiply whole numbers and 1- and 2-place decimals by 10, 100, 1000, 10 000
- Use knowledge of factors and multiples in multiplication; e.g. 43 × 6 is double 43 × 3; e.g. 28 × 50 is ½ of 28 × 100 = 1400
- Use knowledge of place value and rounding in mental multiplication; e.g. 67 × 199 as 67 × 200 – 67
- Use doubling and halving as a strategy in mental multiplication ; e.g. 58 × 5 is half of 58 × 10; e.g. 34 × 4 is 34 doubled twice
- Partition 2-digit numbers, including decimals, to multiply by a 1-digit number mentally; e.g. 6 × 27 as 6 × 20 (120) plus 6 × 7 (42) ; e.g. 6.3 × 7 as 6 × 7 (42) plus 0.3 × 7 (2.1)
- Double amounts of money by partitioning; e.g. £37.45 doubled is £37 doubled (£74) plus 45p doubled (90p) giving a total of £74.90

## Written calculation

- Use short multiplication to multiply a 1-digit number by a number with up to 4 digits
- Use long multiplication to multiply 3-digit and 4-digit numbers by a number between 11 and 20
- Choose the most efficient method in any given situation
- Find simple percentages of amounts; e.g. 10%, 5%, 20%, 15% and 50%
- Begin to multiply fractions and mixed numbers by whole numbers ≤ 10; e.g. 4 × 2⁄3 = 8⁄3 = 2 2⁄3

## All:

- Know multiplication tables to 11 × 11
- Multiply whole numbers and 1-place decimals by 10, 100 and 1000
- Use knowledge of factors as aids to mental multiplication; e.g. 13 × 6 is double 13 × 3; e.g. 23 × 5 is 1⁄2 of 23 × 10
- Use the grid method to multiply numbers with up to 4 digits by 1-digit numbers
- Use the grid method to multiply 2-digit numbers by 2-digit numbers

## Mental calculation:

- Know by heart all the division facts up to 144 ÷ 12
- Divide whole numbers by 10, 100, 1000, 10 000 to give whole number answers or answers with 1, 2 or 3 decimal places
- Use doubling and halving as mental division strategies; e.g. 34 ÷ 5 is (34 ÷ 10) × 2
- Use knowledge of multiples and factors, as well as tests for divisibility, in mental division; e.g. 246 ÷ 6 is 123 ÷ 3; e.g. We know that 525 divides by 25 and by 3
- Halve amounts of money by partitioning; e.g. 1⁄2 of £75.40 = 1⁄2 of £75 (£37.50) plus half of 40p (20p) which is £37.70
- Divide larger numbers mentally by subtracting the 10th or 100th multiple as appropriate; e.g. 96 ÷ 6 is 10 + 6, as 10 × 6 = 60 and 6 × 6 = 36; e.g. 312 ÷ 3 is 100 + 4 as 100 × 3 = 300 and 4 × 3 = 12
- Know tests for divisibility by 2, 3, 4, 5, 6, 9 and 25
- Know square numbers and cube numbers
- Reduce fractions to their simplest form

## Written calculation

- Use short division to divide a number with up to 4 digits by a number ≤ 12
- Give remainders as whole numbers or as fractions
- Find non-unit fractions of large amounts
- Turn improper fractions into mixed numbers and vice versa
- Choose the most efficient method in any given situation

## All:

- Know by heart division facts up to 121 ÷ 11
- Divide whole numbers by 10, 100 or 1000 to give answers with up to 1 decimal place
- Use doubling and halving as mental division strategies
- Use an efficient written method to divide numbers ≤ 1000 by 1-digit numbers
- Find unit fractions of 2- and 3-digit numbers

## Mental calculation:

- Know by heart number bonds to 100 and use these to derive related facts e.g. 3.46 + 0.54
- Derive, quickly and without difficulty, number bonds to 1000
- Add small and large whole numbers where the use of place value or number facts makes the calculation do-able mentally; e.g. 34 000 + 8000
- Add multiples of powers of 10 and near multiples of the same ; e.g. 6345 + 199
- Add negative numbers in a context such as temperature where the numbers make sense
- Add two 1-place decimal numbers or two
- 2-place decimal numbers less than 1; e.g. 4.5 + 6.3; e.g. 0.74 + 0.33
- Add positive numbers to negative numbers; e.g. Calculate a rise in temperature or continue a sequence beginning with a negative number

## Written calculation

- Use column addition to add numbers with up to 5 digits
- Use column addition to add decimal numbers with up to 3 decimal places
- Add mixed numbers and fractions with different denominators

## All:

- Derive, swiftly and without difficulty, number bonds to 100
- Use place value and number facts to add ‘friendly’ large or decimal numbers; e.g. 3.4 + 6.6 ; e.g. 26 000 + 54 000
- Use column addition to add numbers with up to 4-digits
- Use column addition to add pairs of 2-place decimal numbers

## Mental calculation:

- Use number bonds to 100 to perform mental subtraction of any pair of integers by complementary addition; e.g. 1000 – 654 as 46 + 300 in our heads
- Use number bonds to 1 and 10 to perform mental subtraction of any pair of 1-place or 2-place decimal numbers using complementary addition and including money; e.g. 10 – 3.65 as 0.35 + 6; e.g. £50 – £34.29 as 71p + £15
- Use number facts and place value to perform mental subtraction of large numbers or decimal numbers with up to 2 places; e.g. 467 900 – 3005; e.g. 4.63 – 1.02
- Subtract multiples of powers of 10 and near multiples of the same
- Subtract negative numbers in a context such as temperature where the numbers make sense

## Written calculation

- Use column subtraction to subtract numbers with up to 6 digits
- Use complementary addition for subtractions where the larger number is a multiple or near multiple of 1000 or 10 000
- Use complementary addition for subtractions of decimal numbers with up to 3 places, including money
- Subtract mixed numbers and fractions with different denominators

## All:

- Use number bonds to 100 to perform mental subtraction of numbers up to 1000 by complementary addition; e.g. 1000 – 654 as 46 + 300 in our heads
- Use complementary addition for subtraction of integers up to 10 000; e.g. 2504 – 1878
- Use complementary addition for subtractions of 1-place decimal numbers and amounts of money; e.g. £7.30 – £3.55

## Mental calculation:

- Know by heart all the multiplication facts up to 12 × 12
- Multiply whole numbers and decimals with up to 3 places by 10, 100 or 1000; e.g. 234 × 1000 = 234 000 ; e.g. 0.23 × 1000 = 230
- Identify common factors, common multiples and prime numbers and use factors in mental multiplication; e.g. 326 × 6 is 652 × 3 which is 1956
- Use place value and number facts in mental multiplication; e.g. 4000 × 6 = 24 000 ; e.g. 0.03 × 6 = 0.18
- Use doubling and halving as mental multiplication strategies, including to multiply by 2, 4, 8, 5, 20, 50 and 25; e.g. 28 × 25 is a quarter of 28 × 100 = 700
- Use rounding in mental multiplication; e.g. 34 ×? 19 as (34 × 20) – 34
- Multiply 1- and 2-place decimals by numbers up to and including 10 using place value and partitioning; e.g. 3.6 × 4 is 12 + 2.4; e.g. 2.53 × 3 is 6 + 1.5 + 0.09
- Double decimal numbers with up to 2 places using partitioning; e.g. 36.73 doubled is double 36 (72) plus double 0.73 (1.46)

## Written calculation

- Know by heart all the multiplication facts up to 12 × 12
- Multiply whole numbers and 1- and 2-place decimals by 10, 100 and 1000
- Use an efficient written method to multiply a 1-digit or a teen number by a number with up to 4 digits by partitioning (grid method)
- Multiply a 1-place decimal number up to 10 by a number ≤ 100 using the grid method

## Mental calculation:

- Know by heart all the division facts up to 144 ÷ 12
- Divide whole numbers by powers of 10 to give whole number answers or answers with up to 3 decimal places
- Identify common factors, common multiples and primes numbers and use factors in mental division; e.g. 438 ÷ 6 is 219 ÷ 3 which is 73
- Use tests for divisibility to aid mental calculation
- Use doubling and halving as mental division strategies, for example to divide by 2, 4, 8, 5, 20 and 25; e.g. 628 ÷ 8 is halved three times: 314, 157, 78.5
- Divide 1- and 2-place decimals by numbers up to and including 10 using place value; e.g. 2.4 ÷ 6 = 0.4; e.g. 0.65 ÷ 5 = 0.13 ; e.g. £6.33 ÷ 3 = £2.11
- Halve decimal numbers with up to 2 places using partitioning
- e.g. Half of 36.86 is half of 36 (18) plus half of 0.86 (0.43)
- Know and use equivalence between simple fractions, decimals and percentages, including in different contexts
- Recognise a given ratio and reduce a given ratio to its lowest terms

## Written calculation

- Use short division to divide a number with up to 4 digits by a 1-digit or a 2-digit number
- Use long division to divide 3-digit and 4-digit numbers by ‘friendly’ 2-digit numbers
- Give remainders as whole numbers or as fractions or as decimals
- Divide a 1-place or a 2-place decimal number by a number ≤ 12 using multiples of the divisors
- Divide proper fractions by whole numbers

## All:

- Know by heart all the division facts up to 144 ÷ 12
- Divide whole numbers by 10, 100, 1000 to give whole number answers or answers with up to 2 decimal places
- Use an efficient written method, involving subtracting powers of 10 times the divisor, to divide any number of up to 1000 by a number ≤ 12; e.g. 836 ÷ 11 as 836 – 770 (70 × 11) leaving 66 which is 6 × 11, giving the answer 76
- Divide a 1-place decimal by a number ≤ 10 using place value and knowledge of division facts