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CARTERHATCH JUNIOR SCHOOL CALCULATION POLICY September 2013 Introduction: The aim is for all children to have one written method for each of the four operations which is reliable and efficient Each page refers to a different operation, i.e +, -, x and ÷. Stages refer to the guidance paper on written calculation strategies in the renewed framework. There is progression in calculation from N to Y6. In order for progression to run smoothly vocabulary, mental calculation strategies and rapid recall facts should be introduced at the appropriate stage and reinforced. Throughout KS1 and KS2 it is assumed that mathematical vocabulary will be introduced and used in context. Refer to the electronic Renewed framework Numerical examples have been included to illustrate progression from year to year as well as development within the year. The different approaches to mental and written calculation should be made explicit. It is important to notice consistency between methods of written calculations (+ and – starting with units; when dividing in Y5 and Y6, be consistent with subtraction method needed for children to divide by "chunking", i.e. repeated subtraction), therefore when adding or subtracting in earlier years, begin with the least significant digits. Use the Framework for Teaching Mathematics to ensure the appropriate numbers are used for appropriate year groups or levels of attainment. Some methods rely heavily on: A firm understanding of place value Interpretation of written signs Familiarity with a range of vocabulary / Efficient mental calculation Addition Progression examples: To add successfully, children need to be able to: 23 +42 315 +624 recall all addition pairs to 9 + 9 and complements in 10; 1. No 'carrying' add mentally a series of one-digit numbers, such as 5 + 8 + 4; 2. Extra digit in answer 94 +73 561 +718 add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value; 3. Carrying U (units/ones) to T (tens) 47 +25 237 +516 partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways. 4. Carrying T to H 371 +485 293 +541 5. Carrying U to T and T to H 376 +485 295 +547 6. More than two numbers to be added 463 921 +759 7. Different numbers of digits 24 375 + 48 4756 20375 + 752 £4.21 +£3.87 24.90kg + 7.25kg Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for addition. 8. Real life problems involving money or measures 9. Numbers with decimals 2.13 +5.62 Addition Pre-stage Stage 2 Partitioning (by the end of year 3) Counting on fingers Steps in addition can be recorded on a Record steps in addition using Count on one digit from a number line. The steps often bridge partitioning: two digit number through a multiple of 10. 47 + 76 = 47 + 70 + 6 = 117 + 6 Counting forwards and 8 + 7 = 15 = 123 backwards on a number 47 + 76 = 40 + 70 + 7 + 6 = 110 line + 13 = 123 Partition into multiples of 10 Partitioned numbers are then and 1 and recombine 48 + 36 = 84 written under one another: Derive fact families using 47 40 7 addition and subtraction 76 70 6 20 – 7 = 13 20 – 13 = 7 110 13 123 or: 13 + 7 = 20 7 + 13 = 20 Say how a set of objects can be separated into 2 groups 8 is 5 and 3 Stage 1 The Empty Number Line Children need to be able to partition numbers in ways other than into tens and ones to help them make multiples of ten by adding in steps. Stage 3 Expanded method in columns Practical apparatus to be used when learning carrying and exchanging Write the numbers in columns. Adding the tens first: 47 76 110 13 123 Adding the ones first: 47 76 13 110 123 Discuss how adding the ones first gives the same answer as adding the tens first. Refine over time to adding the ones digits first consistently. Stage 4 Column Method (by the end of Year 4) 47 258 366 76 87 458 824 123 345 11 11 11 Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. Carry digits are recorded below the line, using the words ‘carry ten’ or ‘carry one hundred’, not ‘carry one’. Addition N R Through practical activities in meaningful contexts. Counting on fingers in a consistent way Partitioning say how a set of objects can be separated into 2 groups 8 is 5 and 3 counting on using fingers Y1 Adding multiples of 10. Y4 Stage 4 by the end of year 4 67 + 24 10 + 5 = 15 Stage 3 Deliberately count on the wrong number. Ask the children how to put it right. Y3 Stage 2 70 + 30 = 100 42 + 30 = 72 45 + 13 30 + 20 = 50 Adding multiples of 10 = 45 + 3 + 10 = 58 and units. Illustrate number stories with number sentences Model and encourage use of mathematical language count on altogether one more, etc. Model interpretation of and sign using appropriate language Combine numbers of objects – count all the objects Y2 +3 +10 45 48 58 combining sets to make a total (units first) steps along a number line partition into tens (counting on) and ones (units record using + first) and = signs Add a one digit number to a one 45 + 13 of two digit 40 + 5 + 10 + 3 number Stage 1 – 8 + 50 = 58 number line Derive fact families Using add / subtract eg 20-7=13 20-13=7 13 + 7 = 20 7 + 13 = 20 367 + 85 12 140 300 452 = 60 + 7 + 20 + 4 = 11 + 80 = 91 + 67 24 11 80 91 Y5 7587 675 12 150 1100 7000 8262 (units first) + add mentally from top to bottom Y6 7648 1486 14 120 1000 8000 9134 (units first) + (units first) (7 + 4) (60 +20) (units first) "carrying" Extend to 3 digit numbers Relies heavily on firm understanding of place value + 3587 675 4262 1 1 1 + 7648 1486 9134 1 1 1 Real life money problems decimals £4.21 72.5 km 124.90 kg £3.87 + 54.6 km + 7.25 kg 0.08 127.1 km 132.15 kg 1 1 1 1 1.00 7.00 Numbers with any amount of digits, e.g. £8.08 72 3 numbers less than 567 1000 26 + + Addition 367 569 826 + 22 140 1600 Leading to ‘carrying’ 367 85 + 452 11 Units first. Call each digit by its value, e.g. “60 plus 80 is 150” * links to Y2 partitioning Subtraction Progression examples: To subtract successfully, children need to be able to: recall all addition and subtraction facts to 20; subtract multiples of 10 (such as 160 – 70) using the related subtraction fact,16 – 7, and their knowledge of place value; partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14). Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for subtraction. 1. No adjustment 47 -23 864 -621 2. Adjustment T to U 51 -36 432 -217 3. Adjustment H to T 437 -182 618 -217 4. Adjustment H to T and T to U 432 -187 5. Noughts 470 -142 700 -485 604 -347 Subtraction Pre-stage Counting back from a larger number One less, two less Counting on from a smaller number Find the difference by counting up – important to use visual images Stage 1 The Empty Number Line Steps in subtraction can be recorded on a number line. The steps often bridge through a multiple of 10. 15 – 7 = 8 74 – 27 = 47 worked by counting back: The Counting Up Method 74 27 3 40 4 47 30 70 74 or: ______ _______________ The steps may be recorded in a different order: or combined: Children need to be able to partition numbers in ways other than into tens and ones to help them make multiples of ten by adding in steps. 74 27 3 44 47 30 74 Stage 2 Partitioning Subtraction can be recorded using partitioning: 74 – 27 = 74 – 20 – 7 = 54 – 7 = 47 74 – 27 = 70 + 4 – 20 – 7 = 60 + 14 – 20 – 7 = 40 + 7 This requires children to subtract a single-digit number or a multiple of 10 from a two-digit number mentally. The method of recording links to counting back on the number line. Stage 3 Expanded method in columns Practical apparatus to be used when learning carrying and exchanging Partitioned numbers are then written under one another: Example: 74 − 27 70 4 20 7 60 6 14 14 70 4 20 7 40 7 7 4 27 4 7 Example: 741 − 367 700 40 1 300 60 7 600 130 11 700 40 1 300 60 7 300 70 4 6 13 11 7 41 3 67 374 The expanded method leads children to the more compact method so that they understand its structure and efficiency. The amount of time that should be spent teaching and practising the expanded method will depend on how secure the children are in their recall of number facts and in their understanding of place value. Subtraction N R Y1 Y2 Y3 Y4 counting back by partitioning counting back Through practical activities in meaningful contexts "How many are left?" "Take away" "The difference between Taking away and count how many are left. 1 2 3 4 5 counting back from the larger number one less, etc. -10 5 -8 15 -50 23 5 15 +10 Illustrate number stories with number sentences Model interpretation of – sign using appropriate language Comparing numbers of objects How many are left – count back How many have been removed – count on How many more are needed – count on 84 23 Extend to 3 digit numbers -30 40 78 +8 70 – 30 = 40 multiples of 10 50 – 20 = 30 -6 70 - 50 20 11 7 4 754 – 86 600 140 80 600 - 200 400 14 6 Record using – and = signs 5 4 6 140 14 80 6 60 8 = 468 leading to 6 70 (units first) 754 – 286 700 50 - 200 80 81 – 57 = 24 28 14 – 12 = 2 15 – 8 = 7 21 – 10 = 11 Y6 (units first) 84 – 56 = 28 Count back from the larger number. Stage 3 (units first) 23 – 18 23 – 8 – 10 Counting on from smaller number. 0 Stage 2 (units first) (units first) 5–2=3 Y5 Stage 3 by the end of stage 4 decomposition 754 – 86 600 140 80 600 60 14 6 8 - 7 2 4 14 5 8 6 1 4 6 8 = 668 extend to decimals using chosen method Using an empty number line Find the difference by counting up Understand inverse is addition 20-13 = 7 13 = 7 = 20 Fact families Call each digit by its value. £6.28 - £3.15 - 6 2 3 13 4 6 7 1 6 8 8 7 4 3 Multiplication Progression examples: To multiply successfully, children need to be able to: recall all multiplication facts to 10 × 10; partition number into multiples of one hundred, ten and one; work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using the related fact 7 × 5 and their knowledge of place value; add two or more single-digit numbers mentally; add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value; add combinations of whole numbers using the column method (see above). Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication. 1. No 'carrying' 2. Extra digit 3. 'Carrying' but keeping in same decade 4. 'Carrying' and going into next decade 5. Noughts 6. Multiplying by multiples of 10 7. 'Long' multiplication 32 44 x3 x2 32 51 x4 x4 83 34 x4 x7 78 68 x7 x8 202 430 x4 x6 87 416 x10 x 60 47 832 x23 x74 Multiplication Stage 1 Mental multiplication using partitioning Informal recording in Year 4 might be: Also record mental multiplication using partitioning: Stage 2 The Grid Method (introduced in year 4) 38 × 7 = (30 × 7) + (8 × 7) = 210 + 56 = 266 x 30 8 7 210 56 Stage 3 Expanded short multiplication (by end of year 4) 30 8 7 210 56 266 30 7 210 8 7 56 38 7 210 56 266 266 14 3 (10 4) 3 (10 3) (4 3) 30 12 42 43 6 (40 3) 6 (40 6) (3 6) 240 18 258 Note: These methods are based on the distributive law. Children should be introduced to the principle of this law (not its name) in Years 2 and 3, for example when they use their knowledge of the 2, 5 and 10 Derive and recall ALL times-tables to work out multiples of 7: multiplication facts up to 10 x 10, the corresponding division facts and multiples 7 3 (5 2) 3 (5 3) (2 3) 15 6 21 of numbers to 10 up to the tenth multiple. Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 × 7 is ‘thirty multiplied by seven’, not ‘three times seven’, although the relationship 3 × 7 should be stressed. Stage 4 Short Multiplication 38 7 266 Stage 5 Two-digit by Two-digit products (by end of year 5) 56 × 27 is approximately 60 × 30 = 1800. 5 The step here involves adding 210 and 50 mentally with only the 5 in the 50 recorded. This highlights the need for children to be able to add a multiple of 10 to a two-digit or three-digit number mentally before they reach this stage. If, after practice, children cannot use the compact method without making errors, they should return to the expanded format of stage 3. 56 × 27 is approximately 60 × 30 = 1800. 56 27 1000 50 20 1000 120 6 20 120 350 50 7 350 42 6 7 42 1512 1 56 × 27 is approximately 60 × 30 = 1800. 56 27 1120 56 20 392 56 7 1512 1 Multiplication N R Through practical activities in meaningful contexts Grouping objects in twos or threes, then adding groups of the same number counting in 2s, 10s Doubles up to 5 + 5 Y1 Y2 Y3 Doubles up to 10 + 10 Counting in 2s,5’s 10s Y4 Y5 Y6 Stage 4 by the end of stage 4 double by partitioning and recombining Doubles to 20+20 Multiples of 5 to 50 Doubles of multiples of 10 to 100 Counting in 2s, 10s + 5s. Reading 4x2=8 ∆∆∆∆ ∆∆∆∆ or 2x4=8 ∆∆ ∆∆ ∆∆ ∆∆ Record using x and = signs Understand 3x2 as 2+2+2 Repeated addition Understand 15x3 is (10x3) + (5x3) understand multiplication as repeated addition 15 x 3 15 added together 3 times 15 + 15 + 15 32 x 3 30 x 3 = 90 2x3=6 90 + 6 = 96 Record + as x and vice versa. Multiply 1 or 2 digit nos by 10 or 100 Family facts for x / 143 100 40 3 x2 x2 x2 x2 487 400 80 7 Stage 5 x2 x2 x2 x2 Stage 1 200 + 80 + 6 = 286 800 + 160 + 14 = 974 Stage 2 grid method x 8 Stage 5 grid method 2 digit by 2 digit 23 x 8 20 3 160 24 3620 3000 600 20 x2 x2 x2 x2 6000+1200+40 = 7240 372 x 24 X 300 70 2 20 6000 1400 40 4 1200 160 + 24 = 184 72 x 38 x 70 2 30 2100 60 8 560 16 280 8 6000+1400+1200+2 80+40+8 = 8928 2100+560+60+16= 2736 Decimals x 9 346 x 9 300 40 6 4.9 x 3 x 4 3 12 0.9 2.7 12 + 2.7 = 14.7 Multiply by 10 and 100 numbers to a 1000 Stage 3 expanded Short multiplication vertically 23 20 + 3 7 x 7 346 9 54 360 2700 x (6x9) (40x9) (300x9) 4.92 x 3 x 4 0.9 0.02 3 12 2.7 0.06 12 + 2.7 + 0.06 = 14.76 4346 8 48 320 2400 x (6x8) (40x8) (300x8) Multiplication 21 140 161 (3x7) (20x7) 3114 32000 34768 (4000x8) leading to: leading to: leading to: 23 7 161 2 x 72 38 576 2160 2736 x (72x8) (72x30) 352 27 2464 7040 9504 Units first (relies on mental calculation strategies x Division To divide successfully in their heads, children need to be able to: understand and use the vocabulary of division – for example in 18 ÷ 3 = 6, the 18 is the dividend, the 3 is the divisor and the 6 is the quotient; partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways; recall multiplication and division facts to 10 × 10, recognise multiples of one-digit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value; know how to find a remainder working mentally – for example, find the remainder when 48 is divided by 5; understand and use multiplication and division as inverse operations. Single-digit division Examples: To carry out written methods of division successful, children also need to be able to: understand division as repeated subtraction; estimate how many times one number divides into another – for example, how many sixes there are in 47, or how many 23s there are in 92; multiply a two-digit number by a single-digit number mentally; subtract numbers using the column method. Two digit division Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division. 1. No remainder, no carrying 3 69 2. Remainder, no carrying 3 68 3. No remainder, carrying 3 45 4. Remainder, carrying 3 47 5. Placing of the quotient 7 287 6. Noughts in quotient 4 816 8 5608 7. No remainder 32 64 31 93 8. Similar but remainder 13 29 31 97 9. Quotient not so apparent 22 56 41 92 10. Placing the quotient 21 126 32 224 11. No remainder 21 483 32 224 12. Remainder 33 718 13. Noughts in quotient 17 6834 14. Divisors like 29, 39, 48 15. Divisors like 45, 37, 24, 56 3 264 Division Stage 1 Grouping and sharing Stage 2 Repeated subtraction Stage 3 Stage 4 Expanded short division leading to compact short division (by end Long Division of year 4) (by end of year 5) Counting in multiples Division as Grouping 97 ÷ 9 How many packs of 24 can we make Use hands: from 560 biscuits? Start by multiplying A bag of 6 sweets, how many 9 97 How many groups of 5 in 15? 24 by multiples of 10 to get an estimate. children can have 2 sweets each. 90 9 10 How many 5s have been counted? As 24 × 20 = 480 and 24 × 30 = 720, we The division sign (÷) ‘divided into How many more 5s do we need to 7 know the answer lies between 20 and 30 groups of’ Answer: 10 R 7 reach 25? packs. We start by subtracting 480 from Use number square to 560. demonstrate counting in multiples. Division as Sharing 6 196 24 560 Using a number line: Share equally 60 6 10 12 ÷ 3 = 4 20 480 24 20 136 Share a bag of 6 sweets between 12 - 3 - 3 - 3 – 3 = 4 80 60 6 10 2 children – one for you, one for 3 72 24 3 76 me… 8 0 3 6 9 12 60 6 10 Answer: 23 R 8 16 In effect, the recording above is the long 72 ÷ 5 12 6 2 Can we subtract 10 lots of 5? division method, though conventionally 4 32 How many other lots of 5 can we the digits of the answer are recorded Answer: 32 R 4 subtract? above the line as shown below. To find 196 ÷ 6, we start by multiplying 6 by 10, 20, 30, … to find 10 x 5 2 4x5 23 that 6 × 30 = 180 and 6 × 40 = 240. The multiples of 180 and 240 24 560 trap the number 196. This tells us that the answer to 196 ÷ 6 is 0 2 22 72 480 between 30 and 40. Turn number line around 80 Start the division by first subtracting 180, leaving 16, and then 72 20 subtracting the largest possible multiple of 6, which is 12, leaving 4. 8 -4 1x4 Answer: 23 R 8 6 196 16 -4 1x4 12 -4 1x4 -4 1x4 8 4 -4 0 1x4 180 6 30 16 12 6 2 4 32 Answer: 32 R 4 The quotient 32 (with a remainder of 4) lies between 30 and 40, as predicted. Division N R Y1 Y2 Y3 Halving up to 10 understanding 8÷2 as half of 8 Through practical activities in meaningful contexts Grouping objects equally 10 grouped into 2s *How many groups?* Y5 Y6 Halving by partitioning and recombining Understand division as grouping (repeated subtraction) Interpret 8 ÷ 2 as how many 2s make 8? ∆∆/∆∆/∆∆/∆∆ and ‘8 put into groups of 2’ Halving of numbers Use multiplication up to 20 facts and count up/back Halves of multiples 20 ÷ 4 = 5 count up/back in 4s of 10 to 100 20 Grouping 2,5,10 Represent number stories using sign. Y4 Record using ÷ and = signs Know related division facts for 2 x 5 x 10 x tables Recognise relationship between x and ÷ Include calculations with remainders -4 1x4 -4 1x4 -4 1x4 -4 1x4 -4 1x4 16 12 8 4 0 remainders i.e. 21 ÷ 4 = 5 r 1 21 -4 17 -4 13 72 ÷ 2 70 ÷ 2 = 35 2 ÷2 = 1 35 + 1 = 36 3476 3000 400 70 6 ÷2 ÷2 ÷2 ÷2 ÷2 150 + 25 + 4 = 179 1500+200+35+3 = 1738 72 ÷ 5 256 ÷ 7 977 ÷ 36 14r2 5 ) 72 - 20 52 - 20 32 - 20 12 - 10 2 36r4 7 ) 256 - 210 (30x7) 46 - 42 (6x7) 4 27r5 36 ) 977 - 720 (20x36) 257 - 180 (5x36) 77 - 72 (2x36) 5 (4x5) (4x5) (4x5) (2x5) bracketed numbers represent how many groups -4 5 ÷2 ÷ 2 = 150 ÷ 2 = 25 ÷2 =4 Concrete ‘acting out’ of repeated subtraction repeated subtraction (chunking) -4 9 358 300 50 8 -4 1 0 Inverse of x Remainders rounded depending on context Decimals Express remainder as fraction. 7 ) 87.5 Vocabulary Addition and subtraction Reception add, more, and make, sum, total altogether score double one more, two more, ten more… how many more to make… ? how many more is… than…? take (away), leave how many are left/left over? how many have gone? one less, two less… ten less… how many fewer is… than…? difference between is the same as Year 1 double, how much more is…? - , subtract, minus how much less is…? half, halve Year 3 Year 4 Year 5 addition, one hundred more, subtraction, one hundred less, tens boundary ones, units, tens hundreds boundary Increase, decrease, inverse units boundary, tenths boundary partition 1 digit 2 digit 3 digit inverse Thousands 4 digits decimal point decimal place tenths/hundredths negative ten thousands million lots of, groups of x, times, multiply, multiplied by multiple of once, twice, three times… ten times… times as (big, long, wide… and so on) repeated addition array row, column share equally one each, two each, three each… group in pairs, threes… tens equal groups of ÷, divide, divided by, divided into multiplication, product factor, quotient, divisible by inverse left left over remainder round up/down grid row column divisor Year 6 integer Tens, ones Count up Number sentence digit double, halve share, left, left over half Multiplication and division Year 2 near double count in 2s,5s,10s groups of multiplication dividend