# Bonneygrove Primary School Calculations Policy

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2 understand how the principles of commutative, associative and distributive laws apply or do not apply to multiplication and division understand the effects of scaling by whole numbers and decimal numbers or fractions understand correspondence where n objects are related to m objects investigate and learn rules for divisibility Progression in addition and subtraction Addition and subtraction are connected. Part Part Whole Addition names the whole in terms of the parts and subtraction names a missing part of the whole.

4 Start with 7, then count on 8, 9, 10, 11, 12 Bead strings: Bead strings: Make a set of 7 and a set of 5. Then count on from 7. Make a set of 8 and a set of 2. Then count the gap. Using Numicon: = Multilink Towers: Multilink Towers: Cuisenaire Rods: 8 Cuisenaire Rods: Number tracks: 5 3 Number tracks: Start on 5 then count on 3 more Start with the smaller number and count the gap to the larger number. 1 set within another (part-whole model) The quantity in the whole set and one part are known, and may be used to find out how many are in the unknown part. 8 2 = 6 Counters:

6 Bridging through 10s This stage encourages children to become more efficient and begin to employ known facts.

7 Bead string: Bead string: is decomposed / partitioned into The bead string illustrates how many more to the next multiple of 10? (children should identify how their number bonds are being applied) and then if we have used 3 of the 5 to get to 10, how many more do we need to add on? (ability to decompose/partition all numbers applied) Number track: 12 7 is decomposed / partitioned in The bead string illustrates from 12 how many to the last/previous multiple of 10? and then if we have used 2 of the 7 we need to subtract, how many more do we need to count back? (ability to decompose/partition all numbers applied) Number Track: Steps can be recorded on a number track alongside the bead string, prior to transition to number line. Number line Steps can be recorded on a number track alongside the bead string, prior to transition to number line. Number Line: Counting up or Shop keepers method Bead string: 12 7 becomes Starting from 7 on the bead string how many more to the next multiple of 10? (children should recognise how their number bonds are being applied), how many more to get to 12?. Number Track: Number Line:

8 Compensation model (adding 9 and 11) (optional) This model of calculation encourages efficiency and application of known facts (how to add ten) Bead string: Bead string: Children find 7, then add on 10 and then adjust by removing 1. Number line: Children find 18, then subtract 10 and then adjust by adding 1. Number line:

9 Working with larger numbers Tens and ones + tens and ones Ensure that children have been transitioned onto Base 10 equipment and understand the abstract nature of the single tens sticks and hundreds blocks Ensure that the children have been transitioned onto Numicon equipment and understand the abstract nature of trading one plate for another. Partitioning (Aggregation model) Take away (Separation model) = 57 Base 10 equipment: = 34 Base 10 equipment: Children create the two sets with Base 10 equipment and then combine; ones with ones, tens with tens. Numicon equipment: Children remove the lower quantity from the larger set, starting with the ones and then the tens. In preparation for formal decomposition. Number Line: Children create the two sets with Numicon equipment and replace with ten plates for efficiency. Partitioning (Augmentation model) At this stage, children can began to use an informal method to support, record and explain their method (optional) (50 + 7) - (20 + 3)

10 Number line: At this stage, children can begin to use an informal method to support, record and explain their method. (optional) Bridging with larger numbers Once secure in partitioning for addition, children begin to explore exchanging. What happens if the ones are greater than 10? Introduce the term exchange. Using the Base 10 equipment, children exchange ten ones for a single tens rod, which is equivalent to crossing the tens boundary on the bead string or number line. Base 10 equipment: = 52 Base 10 equipment: = 15 Discuss counting on from the larger number irrespective of the order of the calculation.

11 Expanded Vertical Method (optional) Children are then introduced to the expanded vertical method to ensure that they make the link between using Base 10 equipment, partitioning and recording using this expanded vertical method. Base 10 equipment: = 91 Base 10 equipment: = Compact method Compact decomposition Leading to X X X X

12

13 Vertical acceleration By returning to earlier manipulative experiences children are supported to make links across mathematics, encouraging If I know this then I also know thinking. Decimals Ensure that children are confident in counting forwards and backwards in decimals using bead strings to support. Bead strings: Each bead represents 0.1, each different block of colour equal to 1.0 Base 10 equipment: Addition of decimals Aggregation model of addition Counting both sets starting at zero = 0.9 Subtraction of decimals Take away model = 0.7 X X Augmentation model of addition Starting from the first set total, count on to the end of the second set = 0.9 Finding the difference (or comparison model): = Bridging through 1.0 Encouraging connections with number bonds = 1.2 Partitioning = 5.2 Bridging through 1.0 Encourage efficient partitioning = = 0.7 Partitioning = 3.4 Leading to

14 Gradation of difficulty- addition: 1. No exchange 2. Extra digit in the answer 3. Exchanging ones to tens 4. Exchanging tens to hundreds 5. Exchanging ones to tens and tens to hundreds 6. More than two numbers in calculation 7. As 6 but with different number of digits 8. Decimals up to 2 decimal places (same number of decimal places) 9. Add two or more decimals with a range of decimal places Gradation of difficulty- subtraction: 1. No exchange 2. Fewer digits in the answer 3. Exchanging tens for ones 4. Exchanging hundreds for tens 5. Exchanging hundreds to tens and tens to ones 6. As 5 but with different number of digits 7. Decimals up to 2 decimal places (same number of decimal places) 8. Subtract two or more decimals with a range of decimal places

15 Progression in Multiplication and Division Multiplication and division are connected. Both express the relationship between a number of equal parts and the whole. Part Part Part Part Whole The following array, consisting of four columns and three rows, could be used to represent the number sentences: - 3 x 4 = 12, 4 x 3 =12, = 12, =12. And it is also a model for division 12 4 = = = = 0

16 Multiplication Early experiences Children will have real, practical experiences of handling equal groups of objects and counting in 2s, 10s and 5s. Children work on practical problem solving activities involving equal sets or groups. Division Children will understand equal groups and share objects out in play and problem solving. They will count in 2s, 10s and 5s. Repeated addition (repeated aggregation) 3 times 5 is = 15 or 5 lots of 3 or 5 x 3 Children learn that repeated addition can be shown on a number line. Sharing equally 6 sweets get shared between 2 people. How many sweets do they each get? A bottle of fizzy drink shared equally between 4 glasses. Children learn that repeated addition can be shown on a bead string. Children also learn to partition totals into equal trains using Cuisenaire Rods Grouping or repeated subtraction There are 6 sweets. How many people can have 2 sweets each?

17 Scaling This is an extension of augmentation in addition, except, with multiplication, we increase the quantity by a scale factor not by a fixed amount. For example, where you have 3 giant marbles and you swap each one for 5 of your friend s small marbles, you will end up with 15 marbles. This can be written as: = 3 scaled up by = 15 For example, find a ribbon that is 4 times as long as the blue ribbon. Repeated subtraction using a bead string or number line 12 3 = 4 The bead string helps children with interpreting division calculations, recognising that 12 3 can be seen as how many 3s make 12? Cuisenaire Rods also help children to interpret division calculations. We should also be aware that if we multiply by a number less than 1, this would correspond to a scaling that reduces the size of the quantity. For example, scaling 3 by a factor of 0.5 would reduce it to 1.5, corresponding to 3 x 0.5 = 1.5. Grouping involving remainders Children move onto calculations involving remainders = 3 r1 Commutativity Children learn that 3 x 5 has the same total as 5 x 3. This can also be shown on the number line. 3 x 5 = 15 5 x 3 = 15 Or using a bead string see above. Children learn that division is not commutative and link this to subtraction. Arrays Children learn to model a multiplication calculation using an array. This model supports their understanding of commutativity and the development of the grid in a written method. It also supports the finding of factors of a number. quantities. Children learn to model a division calculation using an array. This model supports their understanding of the development of partitioning and the bus stop method in a written method. This model also connects division to finding fractions of discrete

18 4 lots of 5 - using Numicon Inverse operations Trios can be used to model the 4 related multiplication and division facts. Children learn to state the 4 related facts. 3 x 4 = 12 4 x 3 = = = 3 Children use symbols to represent unknown numbers and complete equations using inverse operations. They use this strategy to calculate the missing numbers in calculations. This can also be supported using arrays: e.g. 3 X? = 12

19 Partitioning for multiplication Arrays are also useful to help children visualise how to partition larger numbers into more useful representation. 9 x 4 = 36 Partitioning for division The array is also a flexible model for division of larger numbers 56 8 = 7 Children should be encouraged to be flexible with how they use number and can be encouraged to break the array into more manageable chunks. 9 x 4 = Children could break this down into more manageable arrays, as well as using their understanding of the inverse relationship between division and multiplication = (40 8) + (16 8) = = 7 Which could also be seen as 9 x 4 = (3 x 4) + (3 x 4) + (3 x 4) = = 36 Or 3 x (3 x 4) = 36 And so 6 x 14 = (2 x 10) + (4 x 10) + (4 x 6) = = 84 To be successful in calculation learners must have plenty of experiences of being flexible with partitioning, as this is the basis of distributive and associative law. Associative law (multiplication only) :- E.g. 3 x (3 x 4) = 36 The principle that if there are three numbers to multiply these can be multiplied in any order. Distributive law (multiplication):- E.g. 6 x 14 = (2 x 10) + (4 x 10) + (4 x 6) = = 84 This law allows you to distribute a multiplication across an addition or subtraction. Distributive law (division):- E.g = (40 8) + (16 8) = = 7 This law allows you to distribute a division across an addition or subtraction.

20 Arrays leading into the grid method Children continue to use arrays and partitioning, where appropriate, to prepare them for the grid method of multiplication. Arrays can be represented as grids in a shorthand version and by using place value counters to show multiples of ten, hundred etc. Arrays leading into chunking and then long and short division Children continue to use arrays and partitioning where appropriate, to prepare them for the chunking and short method of division. Arrays are represented as grids as a shorthand version. e.g = Grid method This written strategy is introduced for the multiplication of TO x O to begin with. It may require column addition methods to calculate the total = (30 3) + (30 3) + (18 3) = = 26 The vertical method- chunking leading to long division See above for example of how this can be modelled as an array using place value counters =

21 Short multiplication multiplying by a single digit The array using place value counters becomes the basis for understanding short multiplication first without exchange before moving onto exchanging 24 x 6 Short division dividing by a single digit Whereas we can begin to group counters into an array to show short division working 136 4

22 Gradation of difficulty (short multiplication) 1. TO x O no exchange 2. TO x O extra digit in the answer 3. TO x O with exchange of ones into tens 4. HTO x O no exchange 5. HTO x O with exchange of ones into tens 6. HTO x O with exchange of tens into hundreds 7. HTO x O with exchange of ones into tens and tens into hundreds 8. As 4-7 but with greater number digits x O 9. O.t x O no exchange 10. O.t with exchange of tenths to ones 11. As 9-10 but with greater number of digits which may include a range of decimal places x O Gradation of difficulty (short division) 1. TO O no exchange no remainder 2. TO O no exchange with remainder 3. TO O with exchange no remainder 4. TO O with exchange, with remainder 5. Zero in the quotient e.g = As 1-5 HTO O 7. As 1-5 greater number of digits O 8. As 1-5 with a decimal dividend e.g or Where the divisor is a two digit number See below for gradation of difficulty with remainders Dealing with remainders Remainders should be given as integers, but children need to be able to decide what to do after division, such as rounding up or down accordingly. e.g.: I have 62p. How many 8p sweets can I buy? Apples are packed in boxes of 8. There are 86 apples. How many boxes are needed? Gradation of difficulty for expressing remainders 1. Whole number remainder 2. Remainder expressed as a fraction of the divisor 3. Remainder expressed as a simplified fraction 4. Remainder expressed as a decimal Long multiplication multiplying by more than one digit Children will refer back to grid method by using place value counters or Base 10 equipment with no exchange and using synchronised modelling of written recording as a long multiplication model before moving to TO x TO etc. Long division dividing by more than one digit Children should be reminded about partitioning numbers into multiples of 10, 100 etc. before recording as either:- 1. Chunking model of long division using Base 10 equipment 2. Sharing model of long division using place value counters See the following pages for exemplification of these methods.

23 Chunking model of long division using Base 10 equipment This model links strongly to the array representation; so for the calculation 72 6 =? - one side of the array is unknown and by arranging the Base 10 equipment to make the array we can discover this unknown. The written method should be written alongside the equipment so that children make links. Begin with divisors that are between 5 and 9 1. Make a rectangle where one side is 6 (the number dividing by) grouping 6 tens After grouping 6 lots of 10 (60) we have 12 left over 2. Exchange the remaining ten for ten ones Complete the rectangle by grouping the remaining ones into groups of 6

24 Move onto working with divisors between 11 and 19 Children may benefit from practise to make multiples of tens using the hundreds and tens and tens and ones. 1. Make a rectangle where one side is 12 (the number dividing by) using hundreds and tens With 49 remaining 2. Make groups of 12 using tens and ones No more groups of 12 can be made and 1 remains

25 More long division (for dividing by 2 digit numbers) Remainders Quotients expressed as fractions or decimal fractions 61 4 = 15 ¼ or Appendix Children will learn the Gelosian method of multiplication and the Russian method and subtraction using negative numbers. Gelosian method

26 Russian method Subtraction in negative numbers

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