Progression through the teaching of addition and subtraction

Maths has changed! The maths work your child is doing at school may look very different to the kind of ‘sums’ you remember. This is because children are encouraged to work mentally, where possible, using personal jottings to help support their thinking. ‘Formal’ calculations are introduced from Year 3 onwards. Children are then encouraged to use these methods for calculations they cannot solve in their heads.

Shall I use a pencil and paper method? Parents Meeting on: Progression through Calculations When faced with a problem, we want children to ask themselves…. Shall I use a pencil and paper method? Do I need jottings ? Can I do it in my head? Do I need to use a calculator? Lancashire Mathematics Team

How would you solve these calculations? 76 + 75 = 47 + 19 = 5321 – 2847 = 6003 - 5997 = 27 – 5 = 81 – 35 = 52 + 30 =

Laying the foundations for addition and subtraction Partitioning Rounding Compensating Counting on and back Bridging through 10s, 100s, 1000s boundaries Addition and subtraction facts Lancashire Mathematics Team

Addition + Reception and Year 1 THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO ACHIEVE. Reception and Year 1 Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures.

A number line is just a ‘picture’ of how we work out some calculations in our heads!

___________________________________________ 0 1 2 3 4 5 6 7 8 9 They use number lines and practical resources to support calculation and teachers demonstrate the use of the number line. 3 + 2 = 5 +1 +1 Year 1 ___________________________________________ 0 1 2 3 4 5 6 7 8 9 Children then begin to use number lines to support their own calculations using a numbered line to count on in ones. 8 + 5 = 13 +1 +1 +1 +1 +1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Bead strings or bead bars can be used to illustrate addition including bridging through ten by counting on 2 then counting on 3.

Year 2 Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on. First counting on in tens and ones. 34 + 23 = 57 +10 +10 +1 +1 +1 34 44 54 55 56 57 Then helping children to become more efficient by adding the units in one jump (by using the known fact 4 + 3 = 7). 34 + 23 = 57 +10 +10 +3 34 44 54 57 Followed by adding the tens in one jump and the units in one jump.

Your turn! 64 + 25 = 89 +20 +10 +10 +5 84 89 64 74

Year 3 Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate. Count on from the largest number irrespective of the order of the calculation. 38 + 86 = 124 +4 +30 +4 _______________________________________________ 86 90 120 124

Compensation Year 3 +50 49 + 73 = 122 -1 73 122 123 Children will begin to use informal pencil and paper methods (jottings) to support, record and explain mental methods building on existing mental strategies. Stage 1: Adding the most significant digits first, then moving to adding least significant digits. 67 267 + 24 + 85 80 (60 + 20) 200 (200 + 0) 11 (7 + 4) 140 (60 +80) 91 12 (7 + 5) 352

Year 3 Moving to adding the least significant digits first in preparation for ‘carrying’. 67 267 + 24 + 85 11 (7 + 4) 12 (7+5) 80 (60 + 20) 140 (60+80) 91 200 (200+0) 352

Your turn! 72 + 46 = 72 + 46 8 (2 + 6) 110 (70 + 40) 118

Year 4 Children will in Year 4 be introduced to carrying above the line. 625 783 367 + 48 + 42 + 85 673 825 452 Using similar methods, children will: Add several numbers with different numbers of digits; Begin to add 2 or more 3-digit sums of money, with or without adjustment from pence to pounds; Know that the decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g £3.59+78p 1 1 1 1

Year 5 Following formal addition methods with carrying above the line being introduced at Year 4, children should at Year 5, extend the carrying method to numbers with at least four digits. Children would use rounding to estimate the answer to the calculation. So 587 + 475 is about 600 + 500, which is approximately 1100. 587 3587 + 475 + 675 11 111 1062 4262 Using similar methods, children will: Add several numbers with different numbers of digits; Begin to add two or more decimal fractions with up to three digits and the same number of decimal points Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g 3.2m – 280cm

Year 6 Children should extend the carrying method to numbers with any number of digits. 7648 6584 + 1486 + 5848 111 111 9134 12432 Using similar methods, children will: Add several numbers with different numbers of digits; Begin to add two or more decimal fractions with up to four digits and either one or two decimal places; Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g 401.2 + 26.85 + 0.71

Vocabulary. Add Plus Altogether Addition Total Count on Increase Sum Make

Subtraction - Reception and Year 1 THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO ACHIEVE. Reception and Year 1 Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures etc.

There are five frogs. If 2 frogs jumped into the lake how many would be left?

Year 1 They use number lines and practical resources to support calculation. Teachers demonstrate the use of the number line. 6 – 3 = 3 -1 -1 -1 0 1 2 3 4 5 6 7 8 9 10 The number line should also be used to show that 6 - 3 means the ‘difference between 6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart. 0 1 2 3 4 5 6 7 8 9 10

Year 1 Children then begin to use numbered lines to support their own calculations - using a numbered line to count back in ones. 13 – 5 = 8 -1 -1 -1 -1 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Bead strings or bead bars can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back 2. 13 – 5 = 8

Year 2 Children will begin to use empty number lines to support calculations. Counting back First counting back in tens and ones. 47 – 23 = 24 - 10 - 10 -1 -1 -1 24 25 26 27 37 47

Subtracting the tens in one jump and the units in one jump. Then helping children to become more efficient by subtracting the units in one jump (by using the known fact 7 – 3 = 4). 47 – 23 = 24 - 10 - 10 - 3 Year 2 24 27 37 47 Subtracting the tens in one jump and the units in one jump. - 20 - 3 24 27 47

Your turn! 64 - 26 = 38 -20 -6 38 44 64

Counting on If the numbers involved in the calculation are close together or near to multiples of 10, 100 etc, it can be more efficient to count on. Year 2 73 – 68 = 5 + 2 +3 68 70 73 82 – 47 = 35 + 3 + 10 + 10 + 10 + 2 47 50 60 70 80 82

Year 3 Children continue to use empty number lines with increasingly large numbers. Children are then taught this expanded method using partitioning. 89 = 80 + 9 - 57 50 + 7 30 + 2 = 32 Initially, the children are taught using examples that do not need the children to exchange.

From this the children move to exchanging; 71 = - 46 Step 1: 70 + 1 Year 3 From this the children move to exchanging; 71 = - 46 Step 1: 70 + 1 - 40 + 6 Step 2: 60 + 11 - 40 + 6 20 + 5 = 25 This would be recorded by the children as 70 + 1 - 40 + 6 20 + 5 = 25 60 1

Your turn! 73 - 26 60 1 70 + 3 20 + 6 40 + 7 = 47

Year 4 Partitioning and decomposition 754 = 86 Step 1 700 + 50 + 4 754 = 86 Step 1 700 + 50 + 4 - 80 + 6 Step 2 700 + 40 + 14 (adjust from T to U) - 80 + 6 Step 3 600 + 140 + 14 (adjust from H to T) - 80 + 6 600 + 60 + 8 = 668

This would be recorded by the children as: 700+ 50 + 4 - 80 + 6 600 +60 +8 = 668 As decomposition this would look like this: 754 - 86 668 600 140 1 Year 4 14 1 6

Common calculation errors! 945 1 1 1 - 237 2000 712 - 108 902

Year 4 Children should: Be able to subtract numbers with different numbers of digits; Using this method, children should also begin to find the difference between 2 3-digit sums of money, with or without adjustment from the pence to pounds; Know that decimal points should line up under each other. For example: £8.95 = 8 + 0.9 + 0.05 - £4.38 4 + 0.3 + 0.08 leading to 8.95 8 + 0.8 + 0.15 8 1 4 + 0.3 + 0.08 - 4.38 4 + 0.5 + 0.07 = £4.57 Alternatively, children can set the amounts to whole numbers, i.e. 895 – 438 and convert to pounds after the calculation.

Year 4 Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc. counting on using a number line should be used. 511 – 197 = 314 +300 +11 +3 197 200 500 511

Year 5 The teaching of subtraction continues from Year 4 where an expanded method will have been introduced. This expanded method uses partitioning Step 1: 754 = 700 + 50 + 4 - 286 200 + 80 + 6 Step 2: 700 + 40 + 14 (adjusting from T to U) - 200 + 80 + 6 Step 3: 600 + 140 + 14 (adjusting from H to T) - 200 + 80 + 6 400 + 60 + 8 = 468 600 140 700 + 50 + 4 200 + 80 + 6 This would be recorded by the children as 400 + 60 + 8 = 468

Year 5 Decomposition 6 14 1 754 - 286 468 Children should: Be able to subtract numbers with different numbers of digits Begin to find the number between two decimal fractions with up to three digits and the same number of decimal places this could be in the context of money or measures Know that decimal points should line up under each other. Children would use rounding to estimate the answer to the calculation. So 754 – 286 is about 800 -300, which is approximately 500.

Year 5 Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc. counting on using a number line should be used. 1209 – 388 = 821 + 800 + 9 +12 388 400 1200 1209

Year 6 Decomposition 6467 - 2684 3783 13 1 5 Now using 4 digit numbers and beyond. Children should: be able to subtract numbers with different numbers of digits; Be able to subtract two or more decimal fractions with up to three digits and either one or two decimal places; this could be in the context of money or measures know that decimal points should line up under each other.

Year 6 Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc. counting on using a number line should be used. 3002 -1997 = 1005 + 1000 + 2 + 3 1997 2000 3000 3002

Subtraction Vocabulary Take (away) How many are left? How many have gone? 1 less Decrease Difference between How many fewer is ... than ... How many are left over? 10 less Count back

Key messages Children need to develop skills such as counting, partitioning and recombining numbers They need to build an awareness of the number system, value of numbers and number relationships They need to recall facts such as halving and doubling, number bonds and multiplication facts From all of these they learn to construct strategies that they can apply in many different areas. The questions at the forefront of their minds: ‘Can I do it in my head? If not which method will help me?’

Thank you for attending our workshop on the progression through addition and subtraction.