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Christ Church C.E. Primary School, Moreton MATHEMATICS Calculation Policy A school community, nurturing achievers, who in Jesus’ sight are: Respectful Resilient Resourceful Reliable Reviewed policy agreed by Governing Body on: Reviewed policy shared with staff on: Policy to be reviewed again on: Progression in Addition EYFS to Year 1 Add and subtract onedigit and two-digit numbers to 20, including zero. Solve one step problems that involve addition and subtraction, using concrete objects and pictorial representations and missing number problems. 2+5= Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. 2+5= Add and subtract numbers using concrete objects, pictorial representations and mentally, including: a two-digit number and ones, Count on from the first number (cover first number or display as numeral). + + adding three one-digit numbers + 6 + 18 6 + 58 By counting on from the largest number By partitioning the smaller number through the multiple of 10. 6 + 8 becomes 8+2+4 5+ Recognise the largest number in the calculation and count on from it mentally or using number line. 5+2 (without counters) Recognise the largest number in the calculation and count on from it (using objects or fingers for smaller number if necessary). + TU + TU within 100 37 + 44 Show addition jumps above the numberline – ‘rainbow jumps’. Addition of three single digits – look for bonds you know and doubles Partition the smaller number and use the tens number to bridge calculation. 5 + 17 becomes 17 + 3 + 2 Special cases: Adding 9 9 + 33 58 + 2 + 4 58 60 30 + 46 44 64 By counting on in tens a two-digit number and tens, two two-digit numbers 2+5 5+8 4 + 13 11 + 7 Leading to… Count out each set then find the total (using ‘concrete’ objects or pictorial representations). 2 Year 2 2+5= 22 + 50 46 56 66 76 By counting in groups of ten and one from largest number. 50 70 72 74 80 81 or 40 + 30 = 70 7 + 4 = 11 70 + 11 = 81 or 44 + 40 – 3 = 81 Recall of facts to 20 and recall of adding multiples of 10 will support this thinking. 6+9+3 6+3=9 Double 9 = 18 +10 33 42 43 -1 Using Doubles 29 + 30 is the same as 30 + 30 – 1 Progression in Addition Year 3 Add and subtract numbers mentally, including: a three digit number and ones, a three digit number and tens, a three digit number and hundreds, two two-digit numbers across 100 (nonstatutory guidance). Partitioning the numbers for TU + TU across 100 55 + 78 70 + 50 = 120 8 + 5 = 13 120 + 13 = 133 Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. Special cases 66 + 79 80 + 66 – 1 = 145 Partitioning Adding ones and tens to a 3 digit number Using doubles 356 + 8 356 + 4 + 4 = 364 76 + 78 55 + 78 78 + 50 = 128 128 + 2 + 3 = 133 Double 70 + double 6 + 2 Double 70 + double 8 – 2 Add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction. Recall of number bonds for all numbers to 20 and by adding multiples of 10 will support this thinking. Recall of number bonds for all numbers to 20 and by adding multiples of 10 will support this thinking. Year 4 Use mental strategy where appropriate Addition of 3 digit + 3 digit; 4 digit + 4 digit Add and subtract numbers with up to four digits using the formal written methods of columnar addition and subtraction where appropriate. 1460 + 499 1460 + 500 – 1 = 1959 Use mental strategies to approximate 576 +369 7268 +5179 945 12447 356 + 70 350 + 70 + 6 = 420 356 + 600 300 + 600 + 56 = 956 Addition of numbers with decimal places 1.5 + 1.5 Double 1 + Double 0.5 1.6 + 1.7 1.7 + 0.3 + 1.3 = 3.3 Addition of 3 digit + 2 digit numbers and 3 digit + 3 digit numbers 268 +179 1 7 (8 + 9) 1 3 0 (60 + 70) 3 0 0 (200 + 100) 447 Then as above but without brackets. Then… 6 5 + 7 3 6 8 + 7 3 5 7 6 + 3 6 9 1 3 8 141 9 4 5 1 Addition of numbers to 2 decimal places 4.45 +3.55 57.89 +46.67 8.00 104.56 Progression in Addition Year 5 Add and subtract numbers mentally, with increasingly large numbers e.g. 5 digit – 4 digit multiple of 10. Using mental calculation by counting on 45678 + 3500 = 49178 45678 + 3000 = 48678 42678 = 500 = 49178 Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction). 5.78 + 2.45 = 8.23 5.78 + 2 = 7.78 5.73 + 0.4 = 8.18 5.33 + 0.05 = 8.23 Year 6 Partitioning Perform mental calculations, including with mixed operations and large numbers. 4.578 + 0.008 = 4.586 Solve addition and subtraction multi-step problems in context, deciding which operations and methods to use and why. Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. Column addition Mixed decimals 58765 +29648 57.89 + 46.6 + 23.785 88413 23.785 57.890 +46.600 2 6.568 + 0.079 = 6.647 6.568 + 0.07 = 6.638 6.638 + 0.009 = 6.647 128.275 Column addition with 5 or 6 digits 58765 +29648 88413 include 0 as placeholder where necessary Using all 4 operations 6 + 7 x 8 = 62 because multiplication first then addition where there are no brackets. 2780 – 910 + 1220 can be reordered to 2780 + 1220 – 910 = 3090 as long as the symbol moves with the number. Misconceptions • Models & Images Linked Vocabulary Confusion with ‘…teen’ and ‘…ty’ numbers. When using the number line, children may count the start number so answer is out by 1. Setting out when working in columns – confusion over the place value. Reliance on rules and procedures without conceptual understanding • • • Bar Models / Thinking Boxes When solving word problems, children will be encouraged to draw bar models / thinking boxes. This will help them to visualise the problem and decide which operation(s) they will need to use to solve the problem. Remember… • • • • Addition is commutative. Children should estimate first to see if their answer makes sense. Subtraction is the inverse of addition – the sooner children understand this, the better. Children need to solve missing number problems: 3+4=? 3+?=7 ?+4=7 ?+?=7 ?=3+4 7=3+? 7=3+? 7=?+4 + Add More Sum Total Make Altogether Greater Plus Addition Increase Exchanging / Regrouping Inverse Commutativity Tens boundary Hundred boundary Units boundary Tenths boundary + + ? Part Part Progression in Subtraction EYFS to Year 1 Add and subtract onedigit and two-digit numbers to 20, including zero. Solve one step problems that involve addition and subtraction, using concrete objects and pictorial representations and missing number problems. Year 2 Add and subtract numbers using concrete objects, pictorial representations and mentally, including: a two-digit number and ones, a two-digit number and tens, two two-digit numbers adding three one-digit numbers Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. 5–2 7–2 Count out 5 and remove 2 to find the answer Count back on the numbertrack by saying ‘start on 7, count back 1,2. What number are you on?’ Draw pictures and cross out 8–2 15 – 5 Difference Use tens and ones when the calculation doesn’t bridge 10 7 – 6 or find the difference between 7 and 6 14 – 3 13 – 5 becomes 13 – 3 – 2 7–3 Using a 10 frame – children may subitise how many are remaining without having to count them all Show subtraction jumps below the number line – ‘smiley faces’. Subtracting by counting backwards in tens and ones Subtracting in groups of ten (rather than counting in tens) or groups of ones (by partitioning number being subtracted through multiple of 10) 28 – 4 Count backwards mentally or using a numberline Numberline – secure partitioning of the second number, then counting back in tens then ones. Special cases 27 – 14 = 13 28 – 9 28 – 10 + 1 32 – 7 32 – 2 – 5 45 – 20 Use tens and ones when the calculation doesn’t bridge 10 Partitioning 28 – 8 = 20 76 – 70 = 6 Difference When subtracting 9 or 19 23 – 19 +4 -4 27 13 14 15 16 17 65 – 40 Use a numberline or models and images Partitioning the number being subtracted through the multiple of 10 mentally or using a numberline. - 10 -4 +1 18 19 -10 28 When numbers are close together, count on from the smallest number through the multiple of 10 or count back from the largest to the smallest through the multiple of 10. Progression in Subtraction Year 3 Add and subtract numbers mentally, including: a three digit number and ones, a three digit number and tens, a three digit number and hundreds, two two-digit numbers across 100 (nonstatutory guidance). Use mental strategies where appropriate e.g. Partitioning Subtracting ones and tens from a 3 digit number 567 – 60 = 507 745 – 700 = 45 832 – 2 = 830 Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. TU – TU By counting back in tens and ones 56 60 61 91 356 – 70 356 – 50 – 20 = 286 93 – 39 as 93 – 40 + 1 £10.00 - £3.59 +1p +40p +£6 £3.59 £3.60 £4.00 £10.00 Year 4 Partitioning 3 1 429 - 132 429 - 132 297 +1 53 54 93 -40 1678 – 600 = 1078 2689 – 80 = 2609 6839 – 9 = 6830 7484 – 1100 = 6384 46 – 28 Difference (see also subtraction up to three digits) Count on using numberline when working with money £6 + 40p + 1p = £6.41 Add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction. Use mental strategies where appropriate 46 - 28 18 Special cases Add and subtract numbers with up to four digits using the formal written methods of columnar addition and subtraction where appropriate. Use models and images to support understanding of column subtraction e.g. Dienes 3 1 91 – 35 91 – 30 – 1 – 4 364 – 8 364 – 4 – 4 = 356 956 – 600 956 – 600 = 356 Subtraction up to three digits Using mental calculation when appropriate by counting back 5678 – 2342 = 5678 – 2000 = 3678 3678 – 300 = 3378 3378 – 40 = 3338 3338 – 2 = 3336 See difference too ‘Exchanging’ ‘Regrouping’ Subtraction up to four digits Subtraction up to four digits Difference 5003 – 3897 = 1106 £50 - £28.25 = £21.75 +75p £28.25 £29 Column subtraction +£20 +£1 £30 £50 N.B. Children tend to make fewer errors if they count on using a numberline (rather than using standard algorithm) when dealing with money. 2 12 11 1 3 3 2 6 + 2 6 7 8 0 6 4 8 ‘Exchanging’ ‘Regrouping’ +103 3897 4000 +1003 5003 Progression in Subtraction Year 5 Partitioning Add and subtract numbers mentally, with increasingly large numbers e.g. 5 digit – 4 digit multiple of 10. Mentally 6.76 – 0.06 = 6.7 7.47 – 0.4 = 7.07 Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction). Using mental calculation by counting back Difference Use number bonds to 100 to support 45678 – 3500 = 42178 45678 – 3000 = 42678 42678 – 500 = 42178 £10 - £7.71 = £2.29 Use numberline or jottings Partitioning Perform mental calculations, including with mixed operations and large numbers. 4.578 – 0.008 = 4.57 6.378 – 0.07 = 6.308 £7.71 £8.00 = 29p £8 £10 = £2 7 – 2.45 = 4.55 2.45 3 = 0.55 37=4 5.78 – 2.45 = 3.33 5.78 – 0.05 = 5.73 5.73 – 0.4 = 5.33 5.33 – 2 = 3.33 Year 6 Solve addition and subtraction multi-step problems in context, deciding which operations and methods to use and why. Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. Difference using larger numbers and number facts Difference (used mixed decimals) 6.45 – 1.7 = 4.75 £100 – 67.23 = £32.77 77p £67.23 £68 1.7 2 = 0.3 2 6.45 = 4.45 £32 £100 Column subtraction 2 1 5 3 8 7 6 15 - 1 9 2 4 8 1 9 5 1 7 ‘Exchanging’ ‘Regrouping’ As above with 5 or 6 digits Misconceptions • • • • • • • Confusion with ‘…teen’ and ‘…ty’ numbers. When using number line – children may count the start number, so calculation is out by 1. Setting out when working in columns confusion over the place value Misunderstanding regarding place value and concept of exchanging tens for ones, hundreds for tens etc. Lack of understanding that when subtracting from a number that the answer will be smaller than start number. Children switch the digits around to be able to ‘do’ the calculation (believe it is commutative as with +/x). Reliance on rules and procedures without conceptual understanding Models & Images Whole Part When solving word problems, children will be encouraged to draw bar models / thinking boxes. This will help them to visualise the problem and decide which operation(s) they will need to use to solve the problem. Remember… • • • Children should estimate first to see if their answer makes sense. Subtraction is the inverse of addition – the sooner children understand this, the better. Children need to solve missing number problems: 12 - 4 = ? 12 - ? = 3 ? -4=3 ? - ?=3 ? = 12 - 4 3 = 12 - ? 3=? - 4 Linked Vocabulary ? Take Take away Leaves Left Fewer Less than Minus Subtract Subtraction Remove Decrease Difference between Reduced Inverse Progression in Multiplication Year 1 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. Count in multiples of twos, fives and tens. Year 2 Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs. Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers. Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and development of mathematical language. There are two apples on one plate. How many apples on 3 plates? + Recall and derive doubles + Recall and derive doubles 25 × 2 7 + 7 = 14 7 × 2 = 14 Arrays 20 × 2 5 × 4 = 20 4 × 5 = 20 40 5×2 + 25 Repeated addition 40 6 x 2 = 12 1 10 50 2 3 4 5 6 10 = 50 Progression in Multiplication Year 3 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. Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables. Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and development of mathematical language. Use arrays and numberlines to count in multiples Using partitioning to multiply Partitioning using the grid method 4 x 13 = 52 Multiply single digits by 20, 30, 40, 50 and 80. x 3 20 60 4 12 57 × 2 50 × 2 100 7×2 + 14 = 114 57 × 2 72 0 1 2 3 4 5 6 7 8 9 10 11 100 14 114 Year 4 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. Multiply and divide twodigit and three-digit numbers by a one-digit number using formal written layout. Recall multiplication and division facts for multiplication tables up to 12x12 (facts for 6, 7, 9, 11 and 12 are new). Multiply single digits by 60, 70 and 90. Mental Using place value, multiply by 10 and 100 e.g. 24 x 100 Th 2 H 4 T U 2 4 0 0 Partitioning 267 × 2 200 × 2 = 400 60 × 2 = 120 7 × 2 = 14 400 + 120 + 14 = 534 Partitioning using the grid method x Expanded method 3 20 60 x 400 6 2400 30 7 180 42 2622 (6x3) (10x3) (20x3) (100x3) 4 12 72 437 x 6 = 2622 (5x3) Then move from the expanded method above to standard algorithm. 1 Progression in Multiplication Year 5 Multiply numbers up to 4 digits by a one or two-digit number using a formal written method, including long multiplication for two-digit numbers. Multiply and divide numbers mentally drawing upon known facts. Multiply and divide whole numbers and those involving decimals by 10, 100 and 1000. Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and development of mathematical language. Mental calculation Partitioning 407 × 4 400 × 4 = 1600 0×4=0 7 × 4 = 28 1600 + 28 = 1628 Rounding and adjusting £3.99 × 6 £4 × 6 = £24 £24.00 - £0.06 = £23.94 TU × TU by partitioning 24 × 32 = 2 8 × 3 9 × 20 4 30 600 120 2 40 8 640 128 28 × 19 28 × 10 × 2 = 560 560 – 28 = 532 Year 6 Multiply multidigit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication. Perform mental calculations, including with mixed operations and large numbers. Mental calculation Partitioning 5.7 × 6 5 × 6 = 30 0.7 × 7 = 4.2 30 + 4.2 = 34.2 5.3 × 19 5.3 × 10 × 2 = 106 106 – 5.3 = 100.7 Standard algorithm 124 x 26 2 744 2480 1 Expanded then moving to standard algorithm 1 3224 7 1 8 2 4 6 0 2 (8×9) 0 ( 20 × 9 ) 0 ( 8 × 30 ) 0 ( 20 × 30 ) 10 9 2 768 567 × 86 × 5 6 7 8 6 4 4 3 4 0 2 5 5 4 5 3 6 0 4 8 7 6 2 Misconceptions Models & Images • Understanding of multiplying by 10/100 and what happens to place value of the number • Rapid recall of multiplication tables is not secure and is impacting on fluency and accuracy of calculation • Interpretation of digits in the T/H columns as single digits e.g. 4x3 instead of 4x30 •Multiplication does not always increase the number i.e. when multiplying a positive number by a fraction which is less than 1. Repeated addition Groups of Lots of Multiply Times Multiplication Product Array Row Column Inverse Double Multiplied by Once, twice, three times… as (big, wide, long and so on) Scaling Rate When solving word problems, children will be encouraged to draw bar models / thinking boxes. This will help them to visualise the problem and decide which operation(s) they will need to use to solve the problem. • • • • Multiplication is commutative. Children should estimate first to see if their answer makes sense Since multiplication and division are inverse operations (i.e. one is the mathematical ‘opposite’ of the other) they should be taught alongside each other rather than as two separate entities. Children need to solve missing number problems: 3x4=? ?=3x4 3 x ? = 12 12 = 3 x ? ? x 4 = 12 12 = ? X 4 ? X ? = 12 Linked Vocabulary ? 6 6 6 6 6 Progression in Division Year 1 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. Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. 12 cakes shared equally between 3 people There are 8 oranges. Can you share them equally? Each cake box holds 3 cakes. If I have 12 cakes, how many boxes will I need? How many groups of 3 are there in 12? How many times can I subtract 3 from 12? Year 2 Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs. Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers. Counting Recall and derive halves Relate division to counting and multiplication facts. Count in 5s to see that there are four 5s in 20. How many 10s in 40? Repeated subtraction 1 2 9÷3=3 3 Look at halves of even numbers and see halves of odd numbers as one left over or ½ . Division by sharing 10 ÷ 5 = 2 10 footballs shared between 5 children. Half of 5 Division by grouping 10 ÷ 5 = 2 10 footballs. 5 in each bag. How many bags? Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. Progression in Division Year 3 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. Year 4 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. Multiply and divide twodigit and three-digit numbers by a one-digit number using formal written layout. Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables. Use facts for numbers up to 10 times the divisor e.g. 28 ÷ 3 This is between 27 ÷ 3 = 9 and 30 ÷ 3 = 10 So 9 remainder 1 Division facts for multiplication tables up to 12x12. Use facts for numbers up to 10 times the divisor e.g. 75 ÷ 9 This is between 72 ÷ 9 = 8 and 81 ÷ 9 = 9 So 8 remainder 3 Counting Relate division to step counting and multiplication facts. e.g. count in 4s to see that there are 6 fours in 24. 1 0 2 4 3 8 12 4 16 5 Division as grouping 13 ÷ 3 = 4 r1 10 ×3 +30 6 20 0 1 0 Division as grouping Combine multiples of the divisor to support you Children may first create a ‘bank’ or key if necessary e.g. 10 × 6 = 60 5 × 6 = 30 2 × 6 = 12 87 ÷ 6 = 14 r3 (How many 6s in 87?) 4×6 +60 0 60 4×3 +12 30 r.1 42 43 24 Array shows 6 groups of 4 so 24÷4=6 10×6 43 ÷ 3 = 14 r1 +24 r.3 84 87 2 3 3 6 4 9 r1 12 13 Children may first create a ‘bank’ or key e.g. 10 × 3 = 30 5 × 3 = 15 2×3=6 Division by grouping, leading to formal division 87 ÷ 6 = 14 r3 (How many 6s in 87?) 6 1 8 6 2 2 4 r3 7 0 7 4 r3 Provide context based experiences at each level of development - real life, money, measures… Use a variety of models and images to support understanding and the development of mathematical language. Progression in Division Year 5 Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately by the context. Multiply and divide numbers mentally drawing upon known facts. Divide numbers by 10 and 100. H T U 2 7 1/10 · ·2 Short division 6725 ÷ 7 0 9 6 0 r5 6 4 7 6 7 2 5 1/100 7 Year 6 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 known facts • Know 378 is a multiple of 3 because 300, 60 and 18 are all multiples of 3. • Know 385 is a multiple of 7 because 350 and 35 are multiples of 7. Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context. Use place value and division facts Short division 432 ÷ 15 Long division 560 ÷ 24 = 23r8 432 ÷ 15 = 28 r12 1.32÷3 = 1/100 of 132÷3 132÷3 = 44 44÷100=0.44 So 1.32÷3=0.4 - remainder as a decimal remainder as a fraction Misconceptions • Understanding of dividing by 10/100 and what happens to place value of the number • Rapid recall of multiplication tables is not secure and impacting on fluency and accuracy of calculation •Assuming that division is commutative. •Writing a remainder which is larger than the divisor. •Discarding the remainder and not understanding its significance. When solving word problems, children will be encouraged to draw bar models / thinking boxes. This will help them to visualise the problem and decide which operation(s) they will need to use to solve the problem. 6 • • • ? Children should estimate first to see if their answer makes sense Since multiplication and division are inverse operations (i.e. one is the mathematical ‘opposite’ of the other) they should be taught alongside each other rather than as two separate entities. Children need to solve missing number problems: 12 ÷ 3 = ? ? = 12 ÷ 3 12 ÷ ? = 4 4 = 12 ÷ ? ? ÷3=4 4= ? ÷3 ?÷?=4 Models & Images 6 24 ? Linked Vocabulary Groups Sharing Fair Equal Left over Repeated subtraction Divide Division Inverse Quotient Dividend Divisor Remainder Half Array Factor Divided by Divisible by Divided into Supporting Materials • NCETM Progression Maps • Wirral’s ‘Counting into Calculating – A Guide to Progression Number: Addition and Subtraction NUMBER BONDS Year 1 represent and use number bonds and related subtraction facts within 20 Year 2 Year 3 Year 4 Year 5 Year 6 recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100 MENTAL CALCULATION add and subtract onedigit and two-digit numbers to 20, including zero add and subtract numbers using concrete objects, pictorial representations, and mentally, including: * a two-digit number and ones * a two-digit number and tens * two two-digit numbers * adding three onedigit numbers read, write and interpret mathematical statements involving addition (+), subtraction (-) and equals (=) signs show that addition of two numbers can be done in any order (commutative) and subtraction of one number from another cannot (appears also in Written Methods) add and subtract numbers mentally, including: * a three-digit number and ones * a three-digit number and tens * a three-digit number and hundreds add and subtract numbers mentally with increasingly large numbers perform mental calculations, including with mixed operations and large numbers use their knowledge of the order of operations to carry out calculations involving the four operations Number: Addition and Subtraction Year 1 read, write and interpret mathematical statements involving addition (+), subtraction (-) and equals (=) signs Year 2 (appears also in Mental Calculation) Year 1 solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7=-9 WRITTEN METHODS Year 3 Year 4 add and subtract add and subtract numbers numbers with up to with up to 4 digits using three digits, using the formal written formal written methods methods of columnar of columnar addition addition and subtraction and subtraction where appropriate Year 5 add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction) Year 6 INVERSE OPERATIONS, ESTIMATING AND CHECKING ANSWERS estimate the answer to a estimate and use inverse use rounding to check recognise and use the calculation and use operations to check answers to calculations and inverse relationship inverse operations to determine, in the context of answers to a calculation between addition and a problem, levels of check answers subtraction and use this to accuracy check calculations and solve missing number problems. use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy. PROBLEM SOLVING Year 3 Year 4 solve problems, solve addition and including missing subtraction two-step number problems, using problems in contexts, number facts, place deciding which value, and more operations and methods complex addition and to use and why subtraction Year 6 solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Year 2 solve problems with addition and subtraction: * using concrete objects and pictorial representations, including those involving numbers, quantities and measures * applying their increasing knowledge of mental and written methods solve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change (copied from Measurement) Year 5 solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Solve problems involving addition, subtraction, multiplication and division Number: Multiplication and Division MULTIPLICATION & DIVISION FACTS Year 3 Year 4 Year 1 Year 2 count in multiples of twos, fives and tens (copied from Number and Place Value) 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) count from 0 in multiples of 4, 8, 50 and 100 (copied from Number and Place Value) count in multiples of 6, 7, 9, 25 and 1 000 (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 recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables recall multiplication and division facts for multiplication tables up to 12 × 12 show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot Year 1 Year 2 calculate mathematical statements for multiplication and division Year 5 MENTAL CALCULATION write and calculate mathematical use place value, statements for multiplication and known and derived division using the multiplication facts to multiply and tables that they know, including divide mentally, for two-digit numbers times one- including: multiplying digit numbers, using mental and by 0 and 1; dividing progressing to formal written by 1; multiplying methods (appears also in Written together three Methods) numbers recognise and use factor pairs and commutativity in mental calculations multiply and divide numbers mentally drawing upon known facts perform mental calculations, including with mixed operations and large numbers multiply and divide whole numbers and those involving decimals by 10, 100 and 1000 associate a fraction with division and calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction 3 (e.g. /8) (copied from Fractions) (appears also in Properties of Numbers) WRITTEN CALCULATION Year 3 Year 4 write and calculate multiply two-digit mathematical and three-digit statements for numbers by a one- Year 6 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) Year 5 multiply numbers up to 4 digits by a one- or two-digit number Year 6 multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of Number: Multiplication and Division Year 1 Year 2 calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs WRITTEN CALCULATION Year 3 Year 4 multiply two-digit write and calculate and three-digit mathematical numbers by a onestatements for digit number using multiplication and formal written layout 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 Year 5 multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for twodigit numbers Year 6 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 number using the formal written method of short division and interpret remainders appropriately for the context divide numbers up to 4-digits by a two-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 (appears also in Mental Methods) use written division methods in cases where the answer has up to two decimal places (copied from Fractions (including decimals)) Number: Multiplication and Division Year 1 PROPERTIES OF NUMBERS: MULTIPLES, FACTORS, PRIMES, SQUARE AND CUBE NUMBERS Year 2 Year 3 Year 4 Year 5 recognise and use factor identify multiples and pairs and commutativity factors, including finding in mental calculations all factor pairs of a (repeated) number, and common factors of two numbers. know and use the vocabulary of prime numbers, prime factors and composite (nonprime) 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 2 notation for squared ( ) 3 and cubed ( ) Year 6 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 3 cubed (cm ) and cubic 3 metres (m ), and extending to other units such as mm 3 3 and km (copied from Measures) Year 1 Year 2 ORDER OF OPERATIONS Year 3 Year 4 Year 5 Year 6 use their knowledge of the order of operations to carry out calculations involving the four operations INVERSE OPERATIONS, ESTIMATING AND CHECKING ANSWERS estimate the answer to a calculation and use inverse operations to check answers (copied from Addition and Subtraction) estimate and use inverse operations to check answers to a calculation (copied from Addition and Subtraction) use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy Number: Multiplication and Division PROBLEM SOLVING Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 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 solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts 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 solve problems involving multiplying 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 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 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 solve problems involving similar shapes where the scale factor is known or can be found (copied from Ratio and Proportion)