Teaching Calculations in our School October 2017

Welcome This presentation is intended to help you understand how we teach Mathematics in our school. * It sets out to explain the concept of numeracy. *It goes through our written calculation strategies, from Foundation Stage to Year 6.

Rationale Throughout the school children learn how to use practical apparatus, as well as models and images (such as empty number lines) to support mental and informal written methods of calculation. Children’s mental methods are strengthened and refined to allow them to develop more efficient written methods. To begin with, children need a secure foundation of number facts, whatever the stage in their learning. Children are introduced to the processes of calculation through practical, oral and mental activities. As they develop their understanding of the concepts, they develop ways of recording to support their thinking, using appropriate signs and symbols. By the end of year 6, children should be equipped with a range of mental and written methods that they understand and can use correctly. They should be able to make appropriate decisions about which method to use when faced with a calculation.

What are the key aims? Become fluent in the fundamentals of mathematics: developing conceptual understanding, recall and application of key facts Reason mathematically – following lines of enquiry, spotting relationships, making generalisations Solve problems: logic, word, visual, all possibilities, sequences, algebra

Introduction to Addition In the Foundation Stage, children will be expected to learn to count reliably through lots of practical activities. They will be taught about the concept of zero (none) They will begin to learn the vocabulary associated with addition – one more, altogether, and, how many more . . . They will combine sets of amounts/numbers and begin to learn number bonds to 10.

Addition – Year 1 In Year One: The children need plenty of experience of subitising (recognising amounts without having to count) which underpins all the concepts of number. This should be started with patterns of numbers (e.g: as on a dice, in arrays) before moving to random selections. The children will use number lines to count on and find number bonds to 10. This will all be done with practical work and apparatus rather than written recordings. As children progress to adding larger numbers, they are encouraged to use a more efficient method through the use of dienes materials.

Addition – Year 2 As they develop towards using larger numbers children will use the expanded column method for addition, which relies on being able to partition numbers. The children partition both numbers into tens and units. They then add together the tens and units to find their answer.

Addition – Year 3 At the beginning of Key Stage Two the earlier methods are further developed and refined, leading to a reliable and efficient procedure for adding together numbers which children can not complete mentally. In Year 3 the children will continue to use the expanded column method, using their knowledge of partitioning: Initially this should be done without crossing the 10/100/1000 boundary. When the children are secure they will cross boundaries.

Addition – Year 4 In Year 4 the children will be expected to make their workings for addition more efficient to involve the process of carrying digits into the next column. : It is essential for the children to total the units first when using this method.

Addition – Year 5 By Year 5, the children should have a good grasp of the column method of addition using the process of “carrying” digits in to the next column. This method can be used for decimal numbers as well as whole numbers. This children should be able to make appropriate choices about which is the most efficient method to use for addition.  

Addition – Year 6 By Year 6, the children will have a good grasp of the column method of addition, working with whole numbers and decimal numbers. They will be able to use numbers with different numbers of digits, lining up columns correctly, as well as for adding a series of numbers.  

Introduction to Subtraction In the Foundation Stage the children are introduced to the concept of comparing quantities using practical apparatus for counting on and back. They will use the terms: more than, less than, one more than, one less than, in their counting. The vocabulary of subtraction is also introduced – difference, one less, fewer, take away. They will relate subtraction to “taking away” and counting “how many are left”.

Subtraction – Year 1 Throughout Key Stage One the children will gain further experience of using practical apparatus and mental strategies to solve subtraction problems, involving finding the difference and taking away. In Year 1 the children use number lines and practical apparatus to solve subtraction problems. Once again, the focus is on practical work rather than written recording. This will continue to support their work on number bonds to 10 and 20 and within 20, with addition and subtraction work.

Subtraction – Year 2 As the children gain greater confidence, and begin to use larger numbers, they will use a number line to support the process of “finding the difference” by counting on from the smaller number to the larger number. 51 – 23 = + 10 + 10 + 7 + 1 23 33 43 50 51 10 + 10 + 7 + 1 = 28 51 – 23 = 28

Subtraction – Year 3 In Year 3 the children will use a more formal method of subtraction for written calculations. This will only involve use of the expanded method. This should begin without crossing any boundaries.

Subtraction – Year 4 Children will use the expanded column method for subtraction. To do this, they need to be able to partition numbers in different ways. This is to support exchanging. By the end of the year, they will have refined the expanded column method to the compact method.  

Subtraction – Year 5  Children will use formal methods of subtraction with large numbers and decimal values, as well as extending to using numbers including zeros.   Children should be able to make appropriate choices about which is the most efficient method to use: mental, jottings, written.

Subtraction – Year 6 Children should be able to use numbers with different numbers of digits, lining up columns correctly, as well as working with numbers including zero digits.  

Introduction to Multiplication In the Foundation Stage children will begin to experience the concept of multiplication through learning to double numbers, Initially this will be supported with models and images and in time, would be instant recall of number facts. Double 3 is 3+3=6 +

Multiplication – Year 1 In Key Stage One the basic skills of multiplication will start to be developed. Children will to use the language associated with multiplication – double, times, multiply, multiplied, lots of. They will begin to recognise the symbols and signs associated with multiplication. In Year one multiplication is introduced as repeated addition: 5 added together 3 times is :- 5+5+5 = 15 or 3 lots of 5 = 15 or 3 times 5 = 15 This will be extended to looking at arrays as a visual model of multiplication.

Multiplication – Year 2 In Year 2 the concept of multiplication is reinforced through arrays: 3 x 4 This helps them to understand that multiplication can be performed in either order. They also use a number line to model counting in ‘lots of’ or ‘groups of’:

Multiplication In Key Stage Two children are introduced to a variety of mental methods to support their multiplication. Using doubling and halving 13 x 4 = 13 x 2 x 2 = 52 36 x 50 = (36 x 100) / 2 = 1800 Using known facts If 7 x 5 = 35, then 7 x 50 = 350 and 7 x 500 = 3500 If 8 x 9 = 72, then 0.8 x 9 = 7.2 and 0.08 x 9 = 0.72

Multiplication – Year 3 Written methods are taught to be used when the children are unable to work calculations out mentally. Children use the grid method to work with larger numbers, using their understanding of partitioning (TU x U, TU x TU) 24 x 4 80 + 16 = 96 24 x 4 = 96 X 20 4 80 16

Multiplication – Year 4 In Year 4 the grid method is used for increasingly larger numbers (HTU x U) The children will then adopt the expanded column method

By the end of the year, the children will be using the compact method By the end of the year, the children will be using the compact method. Initially, this will be supported through concrete materials.

Multiplication – Year 5 In Year 5 the children will build upon the compact method to include multiplying by 2-digit numbers.

Multiplication – Year 6 In Year 6 the children continue to use the column method to make their workings more efficient.  They need to be able to multiply one digit numbers with up to 2d.p. by whole numbers.

Introduction to Division The aim in the Foundation Stage is to introduce the children to the concept of division through halving. They begin to use the language associated with division – one each, two each . . ., share, halve, divide, left over, divided by . . ., equal groups of.

In year 1 children continue with practical activities to develop an understanding of division in two ways: Sharing (one for you, one for you, one for you) 6 ÷ 3 Grouping (how many groups of . . . can you get?) Division – Year 1

Division – Year 2 In Year 2, the children are taught division in two ways: Through the use of an array... 12÷3=4 Through the use of a number line: When the children are confident with this method, they will be given number sentences which will result in remainders.

Division – Year 3 In Year 3, the children will be re-arranging the dividend as an introduction to written division.

In Year 4, the children will use the short division method. Division – Year 4 In Year 4, the children will use the short division method.

In Year 5 children will continue to use the short division method Division – Year 5 In Year 5 children will continue to use the short division method

Division – Year 6 In Year 6, the children will use the compact method to divide numbers up to 4-digits by a 2-digit divisor. Children should record the multiples of the divisor alongside the written method for efficiency.

Division – Year 6 In Year 6 children will make use of short division for numbers including decimals. 34.2 / 6 6 34.2   05.7 4