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St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation March 2013 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation One of the key features of written calculations is that children are only expected to move on when they have fully grasped the concepts behind what it is they are doing. We set a rough guide of year groups for different methods as often this is where they are introduced, but it is the progression of stages which today attempts to make clear. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Practical and Mental Methods Bead strings Numicon Number lines Number bonds Adding multiples of 10 Partitioning St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Partitioning 276 200 + 70 + 6 Repartitioning 100 + 170 + 6 Addition St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Prerequisite Skills Partitioning (Splitting up a number by place value e.g. 126 = 100 + 20 + 6) adding multiples of 10 addition facts to 20 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 1 Mental method, using partitioning: Initially carried out with jottings and then mentally. 47 + 76 47 + 76 = (40 + 70) = 110 + 13 = 123 + (7 + 6) St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 2 Vertical layout, expanded working, moving from adding the most significant digit first to adding the least significant digit first: The next step is to show children the vertical format (units under units, tens under tens, etc.) and link it to the mental method. They first practise this method with calculations they can do mentally, and then extend to three-digit numbers, which provides justification for developing a written method. An important point is that children should be able to describe what they are doing by referring to the actual values of the digits in the columns i.e. ‘20 + 50’ or ‘2 tens + 5 tens’, never ‘2 + 5’. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 2 47 + 76 47 + 76 Adding the least significant digit first Adding the most significant digit first Leading to: 47 76 + 110 (40 + 70) 13 123 (7 + 6) 47 76 13 (7 + 6) 110 123 (40 + 70) + St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 3 Vertical layout, contracting the working to a compact, efficient form: A conventional compact, efficient method, with carrying, is introduced. If, with a little practise, they cannot use the compact method without making errors, they should return to the expanded format. Again, when describing each step, children should refer to the actual values of the digits. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 3 H TU 13 110 123 Becomes + 47 76 47 76 1 2 3 1 1 Place value becomes really important, as does presentation + St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 4 Extend to bigger numbers and decimals: When returning to written calculations at a later stage, e.g. to revise or to extend to decimals or numbers with more digits, it is a good idea to start again with informal, expanded methods. This helps children to retain their understanding of the link between different methods, and makes it easier for them to resort to an expanded method if they need to do so. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Addition Stage 4 13.2 + 12.6 TU. t 13.2 12.6 2 5. 8 + Subtraction 1 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Complementary Addition Counting Up Prerequisite Skills Rapid recall of all pairs of numbers with a total of 10. Complements to 100. Counting on in steps of 10/100 etc Rounding to the next nearest multiple of 10. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Complementary Addition Counting Up Stage 1 Show the children the vertical layout for a calculation they can do mentally. Link the steps to those on an empty number line. 74 - 27 74 27 3 ( 30) 4 0 ( 70) 4 ( 74) 47 +3 27 30 +40 +4 70 74 - St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Complementary Addition Counting Up Stage 2 Show how this form of recording can help organise the steps involve in subtracting a three-digit number from another three-digit number. 326 - 178 326 178 2 ( 180) 2 0 ( 200) 1 0 0 ( 300) 2 6 ( 326) 148 - St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Complementary Addition Counting Up Stage 2 You can reduce the number of stages further, by using knowledge of pairs of numbers that total 100. 326 - 178 326 178 2 2 ( 200) 1 2 6 ( 326) 148 - St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Complementary Addition Counting Up Stage 3 Extend to bigger numbers and decimals. 22.4 - 17.8 22.4 17.8 0 . 2 ( 18) 4 . 0 ( 22) 0 . 4 ( 22.4) 4.6 - Subtraction 2 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Decomposition Prerequisite Skills Children need to have a good understanding of how to partition and repartition numbers. e.g. 63 = 60 + 3 =50 + 13 (by making the tens 10 less and the units 10 more) St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Decomposition Stage 1 Expanded vertical method The expanded vertical method of decomposition involves partitioning and repartitioning. The subtraction calculation is carried out in columns, starting with the least significant digits (the units). St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Decomposition Stage 1 263 – 141 342 – 126 30 200 + 60 + 3 100 + 40 + 1 - 100 + 20 + 2 = 122 Leading To Repartitioning 300 + 40 + 2 100 + 20 + 6 12 200 + 10 + 6 = 216 - St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Decomposition Stage 2 Standard compact method of decomposition A conventional compact, efficient method, with carrying below the line, is introduced by linking it directly to the expanded method, starting with the units. Children should be able to explain the link and appreciate that the compact method saves time recording their working. If, with a little practice, they cannot use the compact method without making errors, they should return to the expanded format or to counting up on a number line. *Again, when describing each step, children should refer to the actual values of the digits. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Decomposition Stage 2 725 – 367 6 11 1 725 367 3 58 - St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Subtraction – Decomposition Stage 3 Extend to bigger numbers and decimals When returning to written calculations at a later stage, e.g. to revise or to extend to decimals or numbers with more digits, it is a good idea to start again with informal, expanded methods. This helps children to retain their understanding of the link between different methods, and makes it easier for them to resort to an expanded method if they need to do so. Multiplication St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Prerequisite Skills Partitioning Doubling and halving Multiplying by 10 Multiplication facts/calculation strategies St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 1 Mental method, using partitioning: 38 x 7 (30 x 7) = 210 = 266 + + (8 x 7) 56 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 2 Grid method, expanded working The mental method from which written methods are developed involves partitioning, and then multiplying the tens and ones separately. It is common to start with the tens when working mentally. A useful way of recording intermediate steps is the ‘grid’ method. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 2 38 x 7 x 30 7 210 (30 x 7) 8 + 56 (8 x 7) 266 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 3: Extended to bigger numbers: 56 x 27 x 20 7 50 1000 6 + (50 x 20) 350 (50 x 7) 120 1120 (6 x 20) + 42 (6 x 7) 392 1512 1 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 4: Extended to decimals 23.5 x 12 x 10 2 20 200 3 + (20 x 10) 40 (20 x 2) 30 0.5 + (3 x 10) + 6 (3 x 2) 5 235 (0.5 x 10) + 1 (0.5 x 3) 47 282 1 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 5 Vertical format, expanded working Eventually, children may be introduced to a vertical format. They should first practise this with calculations they can do mentally. They can also investigate starting with the ones first rather than the tens. Children should describe what they do by referring to the actual values of the digits in the columns. The method is then extended to multiplying by two-digit numbers. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 5 56 x 27 38 7 x 2 1 0 (30 x 7 = 210) 5 6 (8 x 7 = 56) 266 56 27 x 38 x 7 1 0 0 0 (50 x 20 = 1000) 1 2 0 (6 x 20 = 120) 3 5 0 (50 x 7 = 350) 4 2 (6 x 7 = 42) 1512 1 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 6 Vertical format, compact method Eventually, children may be introduced to a vertical format. They should first practise this with calculations they can do mentally. They can also investigate starting with the ones first rather than the tens. Children should describe what they do by referring to the actual values of the digits in the columns. The method is then extended to multiplying by two-digit numbers. St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Multiplication Stage 6 56 x 27 38 7 x 26 6 5 56 27 x 38 x 7 1 1 2 0 (56 x 20) 3 9 2 (56 x 7) 1512 1 Division St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Stage 1 Understanding division as: Sharing equally occurs when a quantity is shared out equally into a given number of portions, and we can work out how many are in each portion. e.g. 6 ÷ 2 (share 6 sweets between 2 children) St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Stage 1 Understanding division as: Grouping (or repeated subtraction) occurs when we are asked to find how many groups of a given size are equivalent to the original quantity. For example how many groups of 2 marbles are in a set of 6 marbles, the calculation 6 ÷ 2 (how many 2s in 6?) -2 Shown/calculated on a number line: 0 -2 2 4 6 -2 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Prerequisite Skills Counting back in steps of equal size Halving Multiplication and division facts/strategies St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Stage 2 Informal written methods – subtracting multiples of the divisor (‘Chunking’) Division by subtracting multiples of the divisor… St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Key facts: 10 7 = 70 5 7 = 35 2 7 = 14 17=7 Stage 2 100 ÷ 7 2 lots 2 lots -14 0 2 10 lots -14 16 -70 30 100 ÷ 7 = 14 r2 100 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Stage 3 Progressing to an expanded vertical method: 234 ÷ 13 234 130 104 65 39 26 13 13 0 (10 x 13) (5 x 13) (2 x 13) (1 x 13) Key facts: 10 13 = 130 5 13 = 65 2 13 = 26 1 13 = 13 234 ÷ 13 = 18 St Augustine’s Catholic Primary School Parent Numeracy Workshop Progression in Calculation Division Stage 4 Standard written methods: 196 ÷ 6 6 196 180 16 12 (30) (2) 4 196 ÷ 6 = 32 r4 or 196 ÷ 6 = 32 4/6 Useful Sites National curriculum http://www.education.gov.uk/schools/teachingandlearning Information and services for parents https://www.education.gov.uk/publications/standard/publicationDetail/Page1/DCSF-00267-2010 www.parentchannel.tv www.parentlineplus.org.uk www.direct.gov.uk/en/EducationAndLearning Homework and enrichment www.bbc.co.uk/schools www.bbc.co.uk/learning www.channel4learning.com/apps/homeworkhigh www.gridclub.com www.schoolzone.co.uk www.nrich.maths.org www.sumdog.com (Children have been signed up to this site) http://www.educationcity.com