Peatmoor Community Primary School Calculation Policy
Introduction
Children are introduced to the processes of calculation through practical, oral and mental activities. As children begin to understand the underlying ideas they develop ways of recording to support their thinking and calculation methods, use particular methods that apply to special cases, and learn to interpret and use the signs and symbols involved. Over time children learn how to use models and images, such as empty number lines, to support their mental and informal written methods of calculation. As children’s mental methods are strengthened and refined, so too are their informal written methods. These methods become more efficient and succinct and lead to efficient written methods that can be used more generally. By the end of Year 6 children are equipped with mental and written methods that they understand and can use correctly. When faced with a calculation, children are able to decide which method is most appropriate and have strategies to check its accuracy.
A child who is confident with number and calculation will have:
- A robust understanding of place value
- A really good bank of memorised number facts
- A set of models/images on which they draw when doing calculations
- A real competence in doubling and halving
Aims
The national curriculum for mathematics aims to ensure that all pupils:
- becomefluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
- can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
Information and communication technology (ICT)
Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils’ conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure.
An overview of KS1
Children in Years 1 and 2 will be given a really solid foundation in the basic building blocks of mental and written arithmetic. Through being taught place value, they will develop an understanding of how numbers work, so that they are confident in 2-digit numbers and beginning to read and say numbers above 100. A focus on number bonds, first via practical hands-on most important application of this knowledge is their ability to add or subtract any pair of 2-digit numbers by counting on or back in tens and ones. Children may extend this to adding by partitioning numbers into tens and ones. Children will be taught to count in 2s, 3s, 5s and 10s, and will have related this skill to repeated addition. They will have met and begun to learn the associated 2x, 3x, 5x and 10x tables. Engaging in a practical way with the concept of repeated addition and the use of arrays enables children to develop a preliminary understanding of multiplication, and asking them to consider how many groups of a given number make a total will introduce them to the idea of division. They will also be taught to double and halve numbers, and will thus experience scaling up or down as a further aspect of multiplication and division. Fractions will be introduced as numbers and as operators, specifically in relation to halves, quarters and thirds.
Year 1 / Year 2Mental Addition / Using place value
* Count in 1s e.g. 45 + 1
* Count in 10s e.g. 45 + 10 without counting on is
* Add 10 to any given 2-digit number
Counting on
* Count on in 1s e.g. 8 + 3 as 8, 9, 10, 11
* Count on in 10s e.g. 45 + 20 as 45, 55, 65
* Add by putting the larger number first
Using number facts
* ‘Story’ of 4, 5, 6, 7, 8, and 9
e.g. 7 = 7+0, 6+1, 5+2, 4+3
* Number bonds to 10
e.g. 5 + 5, 6 + 2, 7 + 3, 9 + 1, 10 + 0
* Use number facts to add single-digit numbers to two-digit numbers,
e.g. use 4 + 3 = 7 to work out 24 + 3, 34 + 3…
* Add two single-digit numbers
* Add three single-digit numbers spotting doubles or pairs to 10 / Using place value
* Know 1 more or 10 more than any number
e.g. 1 more than 67, 10 more than 85
Partitioning
e.g. 55 + 37 as 50 + 30 and 5 + 7, then finally combine the two totals: 80 + 12
Counting on
* Add 10 and small multiples of 10 to any given 2-digit number
e.g. 76 + 20 as 76, 86, 96 or in one hop: 76 + 20 = 96
* Add any pair of 2-digit numbers by counting on in 10s and then in 1s
e.g. 55 + 37 as 55 + 30 (85) + 7 = 92
* Add near multiples of 10 e.g. 46 + 19, 63 + 21
Using number facts
* Know pairs of numbers which make the numbers up to and including 12 e.g. 8 = 4 + 4, 3 + 5, 2 + 6, 1 + 7, 0 + 8
* Know number bonds to 20 e.g. 18 + 2, 15 + 5, 14 + 6, 11 + 9
* Add a single-digit number to any 2-digit number using number facts e.g. use 6 + 3 = 9 to work out 36 + 3 = 39, 56 + 3 = 59
* Bridging 10
e.g. 57 + 5 = 57 + 3 (60) + 2 =62
* Add two or three single-digit numbers, spotting bonds to 10 or doubles e.g. 3 + 5 + 3 = 6 + 5 = 11
e.g. 8 + 2 + 4 = 10 + 4 = 14
Year 1 / Year 2
Mental Subtraction / Using place value
* Count back 1s e.g. 53 - 1
* Count back 10s e.g. 53 - 10 without counting back is
* Subtract 10 from any given 2-digit number
Taking away
* Count back in 1s
e.g. 11 – 3 as 11, 10, 9, 8
e.g. 14 – 3 as 14, 13, 12, 11
* Count back in 10s
e.g. 53 – 20 as 53, 43, 33
Using number facts
* ‘Story’ of 4, 5, 6, 7, 8 and 9 e.g. ‘Story’ of 7 is 7 – 1 = 6, 7 – 2 = 5
* Number bonds to 10
e.g. 10 – 1 = 9, 10 – 2 = 8, 10 – 3 = 7
* Use number facts to subtract single-digit numbers from two-digit numbers, e.g. use 7 – 2 to work out 27 – 2, 37 – 2… / Using place value
* Know 1 less of 10 less than any number
e.g. 1 less than 74, 10 less than 82
* Partitioning
e.g. 55 – 32 as 50 – 30 and 5 – 2 and combine the answers: 20 + 3
Taking away
* Subtract 10 and small multiples of 10 from any given 2-digit number
e.g. 76 – 20 as 76, 66, 56 or in one hop: 76 – 20 = 56
* Subtract any pair of 2-digit numbers by counting back in 10s, then in 1s
e.g. 67 – 34 as 67 – 30 (37) then count back 4 (33)
* Subtract near multiples of 10 e.g. 74 – 21, 57 - 19
Using number facts
* Number bonds – knowing all the pairs of numbers which make all the numbers up to and including 12 and derive related subtraction facts
e.g. 10 – 6 = 4, 8 – 3 = 5, 5 – 2 = 3
* Subtract using patterns of known facts
e.g. 9 – 3 = 6 so we know that 39 – 3 = 36, 69 – 3 = 66
* Bridging 10
e.g. 52 – 6 as 52 – 2 (50) – 4 = 46
Counting up
* Find the difference between two numbers on a line where the numbers are close together
e.g. 51 – 47
Year 1 / Year 2
Mental Multiplication / Counting in steps (‘clever’ counting)
* Count in 2s
* Count in 10s
* Begin to count in 5s
* Begin to say what three 5s are by counting in 5s or what four 2s are by counting in 2s, etc.
Doubling and halving
* Find doubles up to 5 using fingers e.g. double 3
* Double numbers to 10
Grouping
* Begin to use visual and concrete arrays and sets of objects to find the answers to ‘three lots of four’ or ‘two lots of five’
e.g. three lots of four
/ Counting in steps (‘clever’ counting)
* Count in 2s, 5s and 10s
* Begin to count in 3s
Doubling and halving
* Begin to know doubles multiples of 5 to 100
e.g. double 35 is 70
* Begin to double two-digit numbers less than 50 with 1s digits of 1, 2, 3 4 or 5
Grouping
* Use arrays to find answers to multiplication and relate to ‘clever’ counting
e.g. 3 x 4 as three lots of four things
e.g. 6 x 5 as six steps in the 5s count as well as six lots of five
* Understand that 5 x 3 can be worked out as three 5s of five 3s
Year 1 / Year 2
Mental Division / Counting in steps (‘clever’ counting)
* Count in 2s
* Count in 10s
* Begin to count in 5s
Doubling and halving
* Find half of even numbers to 12 and know it is hard to halve odd numbers
e.g. half of 6 = 3
Grouping
* Begin to use visual and concrete arrays and ‘sets of’ objects to find the answer to questions such as ‘How many towers of four can I make with twelve cubes?’
Sharing
* Find half of even numbers by sharing
e.g. find half of 16 cubes by giving one each repeatedly to two children. / Counting in steps (‘clever’ counting)
* Count in 2s, 5s and 10s
* Begin to count in 3s
Doubling and halving
* Find half of numbers up to 40, including realising that half of an odd number gives a remainder of 1 or an answer containing a ½
e.g. ½ of 11 = 5½
* Begin to know half of multiples of 10 to 100
e.g. half of 70 = 35
Grouping
* Relate division to multiplication by using arrays or towers of cubes to find answers to division
e.g. ‘How many towers of five cubes can I make from twenty cubes?’
as _ x 5 = 20 and also as 20 ÷ 5 = _
*Relate division to ‘clever’ counting and hence to multiplication
e.g.’ How many fives do I count to get to twenty?’
Sharing
* Begin to find half or a quarter of a quantity using sharing
e.g. find a quarter of 16 cubes by sorting the cubes into four piles
* Find ½, 1/3, ¼ and ¾ of a quantity of objects and of amounts (whole number answers)
Using number facts
* Know half of even numbers up to 24
* Know x2, x5 and x10 division facts
* Begin to know x3 division facts
Overview of LKS2
In the lower juniors, children build on the concrete and conceptual understandings they have gained in the Infants to develop a real mathematical understanding of the four operations, in particular developing arithmetical competence in relation to larger numbers. In addition and subtraction, they are taught to use place value and number facts to add and subtract numbers mentally and will develop a range of strategies to enable them to discard the ‘counting in ones’ or fingers-based methods of the infants. In particular, they 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. This key stage is also the period during which all the multiplication and division facts are thoroughly memorised, including all facts up to the 12 x 12 table. Efficient written methods for multiplying or dividing a 2-digit or 3-digit number by as single-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. 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 one-place decimals, multiplying and dividing whole numbers by 10 and 100.
Year 3 - ADDITION / Using place value
* Count in 100s
e.g. Know 475 +200 as 475, 575, 675
* Add multiples of 10, 100 and £1
e.g. 476 + 200
e.g. 476 + 40
e.g. £6.34 + £5 as £6 + £5 and 34p
* Partitioning
e.g. £8.50 + £3.70 as £8 + £3 and 50p + 70p and then combine the totals: £11 + £1.20
e.g. 347 + 36 as 300 and 40 + 30 and 7 + 6 and combine the totals: 300 + 70 + 13 = 383
e.g. 68 + 74 as 60 + 70 and 8 + 4 and combine the totals: 130 + 12 = 142
Counting on
* Add two 2-digit numbers by adding the multiple of 10
e.g. 67 + 55 as 67 + 50 (117) + 5 = 122
* Add near multiples of 10 and 100
e.g. 67 + 39
e.g. 364 + 199
* Add pairs of ‘friendly’ 3-digit numbers
e.g. 548 + 120
* Count on from 3-digit numbers
e.g. 247 + 34 as 247 + 30 (277) + 4 = 281
Using Number Facts
* Know pairs which total each number to 20
e.g. 7 + 8 = 15
e.g. 12 + 6 = 18
* Know number bonds to 100
e.g. 35 + 65 = 100
e.g. 46 + 54 = 100
e.g. 73 + 27 = 100
* Know pairs of multiples of 10 with a total of 100
e.g. 60 + 40 = 100
e.g. 20 + 80 = 100
* Add to the next 10 and the next 100
e.g. 176 + 4 = 180
e.g. 435 + 65 = 500 / * Use expanded column addition to add two or three 3-digit numbers or three 2-digit numbers
e.g. 476 + 358
* Begin to use compact column addition to add numbers with three digits
e.g. 347 + 286 + 495
* Begin to add like fractions
e.g. 3/8 + 1/8 + 1/8 = 5/8
* Recognise fractions that add to 1
e.g. 1/4 + 3/4
e.g. 2/5 + 3/5
Mental Calculation / Written Calculation
Year 3 - SUBTRACTION / Taking away
* Use place value to subtract
e.g. 348 – 300
e.g. 348 – 40
e.g. 348 - 8
* Take away multiples of 10, 100 and £1
e.g. 476 – 40 = 436
e.g. 476 – 300 = 176
e.g. £4.76 - £2 = £2.76
* Partitioning
e.g. 68 – 42 as 60 – 40 and 8 – 2
e.g. £6.84 - £2.40 as £6 - £2 and 80p – 40p
* Count back in 100s, 10s then 1s
e.g. 763 – 121 as 763 – 100 (663)- 20 (643) – 1 = 642
* Subtract near multiples of 10 and 100
e.g. 648 – 199
e.g. 86 – 39
Counting up
* Find a difference between two numbers by counting up from the smaller to the larger
e.g. 121 – 87 = 34
Using number facts
* Know pairs which total each number to 20
e.g. 20 – 14 – 6
* Know number bonds to 100
e.g. 100 – 48 = 52
e.g. 100 – 65 = 35
* Subtract using number facts to bridge back through a 10
e.g. 42 – 5 = 42 – 2 (40) – 3 = 37 / * Use counting up as an informal written strategy for subtracting pairs of three-digit numbers
e.g. 423 – 357 is
+3 +40 +23 = 66
357 360 400 423
* Use counting up subtraction to find change from £1, £5 and £10
e.g. £10.00 - £6.84 - £3.16
*Subtract like fractions
e.g. 7/8 – 3/8
* Recognise complements of any fraction to 1
e.g. 1 – 1/4 = 3/4
e.g. 1 – 3/5 = 2/5
Mental Calculation / Written Calculation
Year 3 - MULTIPLICATION / Counting in steps (‘clever’ counting)
* Count in 2s, 3s, 4s, 5s, 8s and 10s
Doubling and Halving
* Double numbers to 50 using partitioning
e.g. double 48
* Use doubling as a strategy for multiplying by 2
e.g. 18 x 2 is double 18 = 36
Grouping
* Recognise that multiplication is commutative
e.g. 4 x 8 = 8 x 4
* Multiply multiples of 10 by 1-digit numbers
e.g. 30 x 8 = 240
* Partition teen numbers to multiply by a single-digit number
e.g. 3 x 14 as 3 x 10 and 3 x 4
Using number facts
* Know doubles to 20
e.g. double 15 is 30
* Know doubles of multiples of 5 to 100
e.g. double 85 is 170
* Know x2, x3, x4, x5, x8, x10 tables facts / Build on partitioning to develop grid multiplication
e.g. 23 x 4
Mental Calculation / Written Calculation
Year 3 - DIVISION / Counting in steps (‘clever’ counting)
* Count in 2s, 3s, 4s, 5s, 8s and 10s
Doubling and Halving
* Find half of even numbers to 100 using partitioning
e.g. half of 48
* Use halving as a strategy for dividing by 2
e.g. 36 ÷ 2 is half of 36 = 18
* Find half of odd numbers up to 20
Grouping
* Recognise that division is not commutative
e.g. 16 ÷ 8 does not equal 8 ÷ 16
* Relate division to multiplications ‘with holes in’
e.g. _ x 5 = 30 is the same calculation as 30 ÷ 5 = _ thus we can count in 5s to find the answer
* Divide multiples of 10 by 1-digit numbers
e.g. 240 ÷ 8 = 30
* Begin to use subtraction of multiples of 10 of the divisor to divide numbers above the tenth multiple
e.g. 52 ÷ 4 is 10 x 4 (40) and 3 x 4 (12) = 13
Using number facts
* Know half of even numbers to 40
* Know half of multiples of 10 to 200
e.g. half of 170 is 85
* Know x2, x3, x4, x5, x8 and x10 division facts / * Perform divisions just above the 10th multiple using written jottings, understanding how to give a remainder as a whole number.
* Use division facts to find a unit and simple non-unit fractions of amounts within times tables
e.g. 3/4 of 48 is 3 x (48 ÷ 4) = 36
Mental Calculation / Written Calculation
Year4 - ADDITION / Using Place Value
* Count in 1000s
e.g. Know 3475 + 2000 as 3475, 4475, 5475
* Partitioning
e.g. 746 + 40
e.g. 746 + 203 as 700 + 200 and 6 + 3
e.g. 134 + 707 as 100 + 700 and 4 + 7
Counting on
* Add 2-digit numbers to 2-, 3- and 4-digit numbers by adding the multiple of 10 and then the 1s
e.g. 167 + 55 as 167 + 50 (217) + 5 = 222
* Add near multiples of 10, 100 and 1000
e.g. 467 + 199
e.g. 3462 + 2999
* Count on to add 3-digit numbers and money
e.g. 463 + 124 as 463 + 100 (563) + 20 (583) + 4 = 587
e.g. £4.67 + £5.30 as £9.67 + 30p
Using number facts
* Know by heart/quickly derive number bonds to 100 and to the next multiple of 100
e.g. 288 + 12 = 300
e.g. 1353 + 47 = 1400
e.g. 463 + 37 = 500
* Know by heart/quickly derive number bonds to £1 and to the next whole pound
e.g. 63p + 37p = £1
e.g. £3.45 + 55p = £4
* Add to the next whole number
e.g. 4.6 + 0.4
e.g. 7.2 + 0.8 / * Build on expanded column addition to develop compact column addition with larger numbers
e.g. 1466 + 4868
* Compact column addition with larger numbers
e.g. 5437 + 2286 + 1495
* Use expanded and compact column addition to add amounts of money
* Add like fractions
e.g. 3/5 + 4/5 = 7/5 = 1 2/5.
* Be confident with fractions that add to 1 and fraction complements to 1
e.g. 2/3 + ? = 1
Mental Calculation / Written Calculation
Year 4 - SUBTRACTION / Taking away
* Use place value to subtract
e.g. 4748 – 4000
e.g. 4748 - 8
* Take away multiples of 10, 100, 1000, £1, 10p or 0.1
e.g. 8392 – 50
e.g. 6723 – 3000
e.g. £3.74 – 30p
e.g. 5.6 – 0.2
* Partitioning
e.g. £5.87 - £3.04 as £5 - £3 and 7p – 4p
e.g. 7493 – 2020 as 7000 – 2000 and 90 – 20
* Count back
e.g. 6482 – 1301 as 6482 – 1000 (5482) – 300 (5182) – 1 = 5181
* Subtract near multiples of 10, 100, 1000 or £1
e.g. 3522 – 1999
e.g. £34.86 - £19.99
Counting up
* Find a difference between two numbers by counting up from the smaller to the larger
e.g. 506 – 387
e.g. 4000 - 2693
Using number facts
* Know by heart/quickly derive number bonds to 10 and 100
e.g. 100 – 76 = 24
e.g. 1 – 0.6 = 0.4
* Know by heart/quickly derive number bonds to £1 and £10
e.g. £1.00 – 86p = 14p
e.g. £10.00 - £3.40 = £6.60 / * Use expanded column subtraction for 3-digit and 4-digit numbers
* Begin to develop compact column subtraction
e.g. 726 – 358
* Use complementary addition (counting up subtraction) to subtract amounts of money, and for subtractions where the larger number is a near multiple of 1000 or 100
E.g. 2002 – 1865
+5 +30 +102 = 137
1865 1870 1900 2002
*Subtract like fractions
e.g. 3/8 – 2/8 = 1/8
* Use fractions that add to 1 to find fraction complements to 1
e.g. 1 – 2/3 = 1/3
Mental Calculation / Written Calculation
Year 4 - MULTIPLICATION / Counting in steps (sequences)
* Count in 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, 11s, 12s, 25s, 50s, 100s and 1000s
Doubling and halving
* Find doubles to double 100 and beyond using partitioning
e.g. double 126
* Begin to double amounts of money
e.g. £3.50 doubled is £7
* Use doubling as a strategy in multiplying by 2, 4 and 8
e.g. 34 x 4 is double 34 (68) doubled again = 136
Grouping
* Use partitioning to multiply 2-digit numbers by 1-digit numbers
e.g. 24 x 5
* Multiply multiples of 100 and 1000 by 1-digit numbers using tables facts
e.g. 400 x 8 = 3200
* Multiply near multiples by rounding
e.g. 24 x 19 as (24 x 20) – 24 = 456
Using number facts
* Know by heart all the multiplication facts up to 12 x 12
* Recognise factors up to 12 of two-digit numbers
* Multiply whole numbers and one-place decimals by 10, 100, 1000 / * Use the grid method to multiply 3-digit numbers by 1-digit numbers
e.g. 253 x 6
* Use a vertical written method (ladder) to multiply 3-digit numbers by 1-digit numbers
e.g. 253 x 6
* Use grid method to multiply 2-digit numbers by 2-digit numbers
e.g. 48 x 16
Mental Calculation / Written Calculation
Year 4 - DIVISION / Counting in steps (sequences)
* Count in 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, 11s, 12s, 25s, 50s, 100s and 1000s
Doubling and halving
* Find half of even numbers to 200 and beyond using partitioning
e.g. half of 258
* Begin to half amounts of money
e.g. £9 halved is £4.50
* Use halving as a strategy in dividing by 2,4 and 8
e.g. 164 ÷ 4 is half of 164 (82) halved again = 41
Grouping
* Use multiples of 10 times the divisor to divide by 1-digit numbers above the tables facts
e.g. 45 ÷ 3 as 10 x 3 (30) and 5 x 3 (15)
* Divide multiples of 100 by 1-digit numbers using division facts
e.g. 3200 ÷ 8 = 400
Using number facts
* 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 one decimal place / * Use a written method to divide a 2-digit or a 3-digit number by a single-digit number
e.g. 86 ÷ 3 as 20 x 3 (60) and 8 x 3 (24), remainder 2
* Give remainders as whole numbers
*Use division facts to find unit and non-unit fractions of amounts within the times tables
e.g. 7/8 of 56 is 7 x (56 ÷ 8) = 49
OVERVIEW OF UKS2
Children move on from dealing mainly with whole numbers to performing arithmetic operations with both decimals and fractions. They will consolidate their use of written procedures in adding and subtracting whole numbers with up to 6 digits and also decimal numbers with up to two decimal places. Mental strategies for adding and subtracting increasingly large numbers will also be taught. These will draw upon children’s robust understanding of place value and knowledge of number facts. Efficient and flexible strategies for mental multiplication and division are taught and practised, so that children can perform appropriate calculations even when the numbers are large, such as 40,000 x 6 or 40,000 ÷ 8. In addition, it is in Y5 and Y6 that children extend their knowledge and confidence in using written algorithms for multiplication and division. Fractions and decimals are also added, subtracted, divided and multiplied, within the bounds of children’s understanding of these more complicated numbers, and they will also calculate simple percentages and ratios. Negative numbers will be added and subtracted.
Mental Calculation / Written CalculationYear 5 - ADDITION / Using place value
* Count in 0.1s, 0.01s
e.g. Know what 0.1 more than 0.51 is
* Add one or two-digit multiples of 10, 100, 1000, 10,000 and 100,000
e.g. 8000 + 7000
e.g. 600,000 + 700,000