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Written methods of calculations are based on mental strategies. Each of the four operations builds on secure mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead on to more formal written methods of calculation. Strategies for calculation must be supported by familiar models and images. When approaching a new strategy it is important to start with numbers that the child can easily manipulate so that they have every opportunity to fully grasp each concept. The transition between stages should not be hurried as not all children will be ready to move on to the next stage at the same time, therefore the progression in this document is outlined in stages. Previous stages may need to be revisited to consolidate understanding before progressing. Failure to secure understanding can lead to misconceptions later so it is essential learning is personalised for every child to ensure solid mathematical foundations are laid which can be built upon in the future. A sound understanding of the number system and the patterns within it is essential for children to carry out calculations efficiently and accurately. Introduction

Mathematics is NOT just a memory game Children need a level of understanding Children MUST be encouraged to think for themselves and to reason Head first. Can the calculation be done mentally more efficiently? £5.00 – £4.99 £ £

Children need to develop understanding of number Children need to understand the position of numbers and how they relate to one another Children need to understand the value of numbers (Place Value) Children need to be able to partition and recombine numbers

Learn number bonds Learn multiplication tables and related division facts Learn facts about measures e.g. 24 hours in a day, 100cm in a metre Learn how to tell the time on an anologue clock Add/subtract one to/from any number Add/subtract ten to/from any number Mental Calculation

Number Bonds Year 1 – recognise and reason bonds up to 10 and then up to 20 and related subtraction facts Year 2 – practise addition and subtraction bonds up to 20 to become increasingly fluent, use knowledge of bonds to calculate and use related bonds to 100 using multiples of 10 e.g Year 3 – consolidate previous learning then investigate bonds of larger numbers, bonds to 1 using tenths, fractions and decimals e.g , 1/10 + 9/10 Year 4 – consolidate previous learning, decimal and fraction bonds bonds using hundredths – link to money and measures Year 5 – practise fluency with bonds with one-, two- and three-decimal places, including links with money and measures Year 6 – consolidate understanding of bonds to three-decimal places to achieve fluency A number bond is an addition sum with two numbers. Knowing bonds to 10 then 20 then 100 helps with addition, both mental and written. e.g = = 20 Partition 7 into 2 and 5 so that 2 can be added to 18 to make 20 and then add 5

Number Bond Practice

Progression in methods for addition Compact Method Number Track Number Line Expanded method (partitioning and recombining) = 7 1

Stage 1 – Understanding Addition & Number Track and Use a puppet to practise counting on. Practise counting on/adding small numbers. If the puppet makes a ‘mistake’ can the child spot it? What happens if we start at 7 and add/count on 3? Combine two (or more) sets of objects and find out how many there all together Remember to use the different words linked to ‘addition’

Stage 2 – Introducing the number line – counting on Use a puppet to reinforce counting forwards. Link to number track. Start with a fully numbered number line and then progress to encouraging the children to sketch their own to help with calculation Ensure children understand place value e.g. 11 is one ten and one unit or one Start on the largest number Add the tens … and then the units

Stage 3 – The Expanded Method (partitioning & recombining) = 7 1 Use place value cards and place value apparatus alongside written jottings. Partition the numbers into tens and units, add, and then recombine. 1 0

= Link the expanded method to the compact method Stage 4 – Compact Method

Progression in methods for subtraction Compact Method Number Track Number Line Expanded method (partitioning and recombining) and

Stage 1 – Number Track (counting back) & taking away Use a puppet to practise counting backwards. Practise taking away small numbers. If the puppet makes a ‘mistake’ can the child spot it? What happens if we start at 7 and take away/count back 3? Take away objects from a group and count how many are left Remember to use the different words linked to ‘subtraction’

Stage 2 – Introducing the number line Use a puppet to reinforce counting backwards. Link to number track. Start with a fully numbered number line and then progress to encouraging the children to sketch their own to help with calculation Start counting back in ones and then progress to larger jumps Start on the largest number Count back the tens … and then the units

Stage 3 – Expanded Method and to subtract 7 units we need to exchange a ten for ten units = 16 Use place value apparatus alongside written jottings. Partition the numbers into tens and units, subtract and then recombine

Stage 4 – Compact Method and Is the answer sensible? Link the expanded method to the compact method

Progression in methods for multiplication Compact method Repeated addition Arrays Grid method = × (56 × 20) (56 × 7)

Stage 1 – Repeated addition & … Children need to understand that multiplication is the same as repeated addition. Find opportunities to count in groups e.g. socks, ‘fingers’ on 4 hand prints. … arrays Children need to be able to see numbers as arrays. An array is an arrangement of a number visually in rows and columns

4 x = Stage 2 – The grid method When learning the grid method use place value equipment to help see the numbers. Partition the numbers into tens and units. Draw a grid and place the partitioned numbers across the top and down the side of the grid = 156 Multiply each of the part of the partitioned numbers and write the answers in the sections of the grid. Lastly add together the answers to find the final total. 12 x 13

Stage 3 – Long multiplication 5 6 × (56 × 20) (56 × 7) Because you are multiplying by ‘tens’ you must put a zero in the units column Then multiply the two tens by the units (6) and then the tens (5) Next multiply the seven units by the units (6) and then the tens (5). Finally add the two totals together to get a final answer 1

Progression in methods for division Compact method Sharing … Chunking … and grouping ÷ 96 ÷ 5 = 19 r ( 10 lots of 5 ) ( 5 lots of 5 ) ÷ r Fact Box 1 x 5 = 5 5 x 5 = x 5 = 50

Stage 1 - Sharing … … and grouping Share objects practically one at a time. Draw a picture to show this. The objects do not need to be drawn these could just be crosses. Divide objects practically into equal groups. Draw a picture to show this. The objects do not need to be drawn these could just be crosses. 4 shared by 2 8 divided into equal groups of 2

Fact Box 2 x 5 = 10 5 x 5 = x 5 = 50 Stage 2 – Using multiplication and division facts. 96  5 Using times tables knowledge to inverse division questions. 12 x 5 = 60 7 x 5 = 35 Remainder 1 96  5 = 19 r 1 What basic facts do I know about the 5 times-table? Children can use a number line to count up in the divided number. E.g. 30 ÷ 5. Count up in 5s until you reach 30. How many jumps have you done?

560 ÷ r Stage 3 – Short division and long division r  7 = 13 r r 5 2 Is the answer sensible?

Progression in Calculations – by magnitude Year 1 – U + U, U + multiple of 10, TU + multiple of 10, U – U, TU – U, TU – multiple of 10, counting groups of objects in ones, twos, fives and tens, sharing objects in equal groups Year 2 - U + U, TU + U, TU + TU, U - U, TU - U, TU – TU, simple multiplication, simple division including with remainders Year 3 - TU + TU, HTU + TU, HTU + HTU, TU - TU, HTU - TU, HTU – HTU, TU x U, TU ÷ U including with remainders Year 4 - TU + TU, HTU + TU, HTU + HTU, TU - TU, HTU - TU, HTU – HTU, TU x U, TU ÷ U including with remainders Year 5 – Add whole numbers and decimals to two decimal places, subtract whole numbers and decimals to two decimal places, HTU x TU, TU x TU, U x decimal, TU ÷ U, HTU ÷ U Year 6 - Add whole numbers and decimals to two decimal places, subtract whole numbers and decimals to two decimal places, TU x U, HTU x U, decimal x U, TU x TU, HTU x TU, TU ÷ U, HTU ÷ U, decimal ÷ U Mathematical Language Number sentence e.g , 5 – 3, 6 x 3, 12 ÷ 3 Partition splitting a number up e.g. 123 … Recombine putting a number back together e.g … 123 Bridging crossing over 10/100 etc Exchanging e.g. swapping a 10 for 10 ones Place value the value of each digit in a number e.g. hundreds, tens and ones (units) Remember there are different words for +, -, x and ÷ to learn in order to help solve mathematical word problems