Pencil and Paper Methods – Part II LEA Recommendations for Key Stages 1 – 3 & 4..? Leicestershire Numeracy Team 2004

The approach to calculation When faced with a calculation children should ask themselves; Can I do this in my head? Can I do this in my head using drawings or jottings? Do I need to use a pencil or paper procedure? Do I need a calculator? This policy looks at the progression in developing consistent pencil and paper procedures across the school.

Key messages The policy shows the route MOST children should be able to follow successfully and with understanding. You might want to introduce other methods to more able pupils so that they can have the opportunity to explore and use them too. The methods build on the mental strategies children have/ are developing.

Addition objectives Understand addition, as it applies to whole numbers and decimals; know how to use the laws of arithmetic and inverse operations Consolidate and extend mental methods of calculation, to include decimals, fractions and percentages accompanied where appropriate by suitable jottings. Addition objectives 82–5 Understand addition of fractions and integers; use the laws of arithmetic and inverse operations. 86–7 Use the order of operations, including brackets. Addition Year 7 Year 8 Year 9 Structure of policy…

Addition – key questions Which methods for addition have the feeder high schools taught? Do we have enough emphasis on the use of number lines throughout the school? Are they used to model all four operations? Are calculations presented with the equals sign and empty numbers in a variety of places? ‘Partitioning and recombining’, ‘empty number lines’, ‘vertical methods’ – which method do we teach?.

Mental methods Use compensation by adding too much, and then compensating Use jottings such as an empty number line to support or explain methods for adding mentally e.g. Partition and deal with the most significant digits first Pencil and paper procedures (Written methods) Extend to decimals with up to 2 decimal places, including: sums with different numbers of digits; totals of more than two numbers. e.g = Addition Year 7 Year 8 Year 9 Consolidate methods learned and used in Year 6 and extend to harder examples of sums with different numbers of digits and differing numbers of decimals. e.g = = Add and subtract fractions – use diagrams to illustrate.

Subtraction – key questions Do we use a range of vocabulary for subtraction or do students only associate it with take away? Do we place enough emphasis on difference (how many more/how many less) throughout the department? Do we agree to teach complementary addition instead of decomposition? Do we want to show how complementary addition can be written vertically or do we just want to teach it using an empty number line?

Mental methods Use jottings such as an empty number line to support or explain methods for adding mentally. Use compensation by subtracting too much, and then compensating Subtract more complex fractions For example: Subtraction Year 7 Year 8 Year 9 Pencil and paper procedures (Written methods) Extend to decimals with up to 2 decimal places, including: differences with different numbers of digits totals of more than two numbers. Complementary addition Consolidate methods learned and used in previous years and extend to harder examples of differences with different numbers of digits. Counting back

Multiplication – key questions Which method for multiplication is taught in the department/across the school? Is the grid method of multiplication taught in the school? Do students need another method for multiplication (e.g. long multiplication)?

Multiplication Year 7 Year 8 Year 9 Pencil and paper procedures (Written methods) The grid method can be used to develop the understanding that algebra is a way of generalising from arithmetic, for example, when expanding the product of two linear expressions. Multiply a fraction by a fraction Mental methods Partition either part of the product e.g.7.3 x 11 = (7.3 x 10) = x 1.4 = (10 x 1.4) + (3 x 1.4) = 18.2 OR Use the grid method of multiplication (as below). Pencil and paper procedures (Written methods) Use written methods to support, record or explain multiplication of:  3 digit by a 2 digit number  a decimal with one or two decimal places by a single digit  decimals with up to two decimal places.

Key questions - division Is chunking used for division in the High Schools? Have empty number lines been used to demonstrate the idea of ‘repeated subtraction’? Has enough emphasis been placed on division by grouping (or do pupils only know how to share)?

Pencil and paper procedures (Written methods) Continue to use the same method as in Year 7 and Year 8. Adjust the dividend and divisor by a common factor before the division so that no further adjustment is needed after the calculation e.g ÷ 0.8 is equivalent to 3616 ÷ 8 Use the inverse rule to divide fractions, first converting mixed numbers to improper fractions. Look at one half of a shape. How many sixths of the shape can You see? (six) So, how many sixths in one half? (three) So ½ ÷ 1 / 6 = ½ x 6 / 1 = 6 / 2 = 3 Pencil and paper procedures (Written methods) Use written methods to support, record or explain division of:  a three-digit number by a two-digit number  a decimal with one or two decimal places by a single digit. Refine methods to improve efficiency while maintaining accuracy and understanding ÷ 8 is approximately 110 ÷ 10 = (10 groups of 8) ( 3 ) ( 0.7 ) 0.0 Answer: 13.7 Mental methods Use mental or informal written methods to calculate e.g.6785 ÷ 25 = 6785 ÷ 5 ÷ 5 Use inverses to check results e.g.703 ÷ 19 = 37 appears to be about right, because 36 x 20 = 720 Division Year 7 Year 8 Year 9

Additional Questions for Heads of Maths to consider… Are pupils encouraged to estimate the size of an answer and use it to check that their answer to a written calculation is reasonable and sensible? Are pupils encouraged to estimate the size of an answer and use it to check that their answer to a written calculation is reasonable and sensible? Are pupils given regular opportunities to use and apply written calculation methods efficiently to solve a range of problems, including word problems? Are pupils given regular opportunities to use and apply written calculation methods efficiently to solve a range of problems, including word problems? Do you know the approaches to written calculation used by your feeder schools? If not, how can you find out? Do you know the approaches to written calculation used by your feeder schools? If not, how can you find out? Do you emphasise the same approaches to calculation within your school, to those used in your feeder schools, particularly for those working below expected levels of attainment? Do you emphasise the same approaches to calculation within your school, to those used in your feeder schools, particularly for those working below expected levels of attainment?

Additional Questions for Heads of Maths to consider continued… Do you have an agreed statement of practice for written calculation within your department and across your school? Do you have an agreed statement of practice for written calculation within your department and across your school? Does emphasis continue to be placed on the development of mental strategies which help pupils in their recording of written methods? Does emphasis continue to be placed on the development of mental strategies which help pupils in their recording of written methods? Do you use pupil’s errors and misconceptions in written methods as part of a ‘cognitive conflict’ teaching approach to help improve understanding and accuracy? Do you use pupil’s errors and misconceptions in written methods as part of a ‘cognitive conflict’ teaching approach to help improve understanding and accuracy? And finally … And finally … Is there whole school agreement on the methods in the policy? If not are there alternatives, which all staff agree on? How will consistency and progression be maintained? Is there whole school agreement on the methods in the policy? If not are there alternatives, which all staff agree on? How will consistency and progression be maintained?