# Swavesey Primary School Calculation Policy. Addition and Subtraction

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4 Development of Subtraction KS1 The default method for carrying out and recording subtraction problems will be the use of number lines to find the difference by counting on from the smaller to the larger number. As children move through the school they will be encouraged to use the number lines in a more efficient way. In the foundation stage children will be taught to relate subtraction to taking-away by using practical counting equipment alongside the following models and images: Counting back in familiar contexts such as number rhymes or stories. Count back in 1 s from any given number. Find lots less on a number line/number track. Find lots less by using bead strings to takeaway. Teachers will begin to link subtraction to finding the difference, using this vocabulary to help children understand the concept. This will be reinforced throughout the school as children begin to use informal and formal jottings to record their calculations. As children begin to move through KS1 they will be encouraged to use number lines and number tracks more frequently as a means of carrying out their subtraction calculations by counting on to find the difference between the smaller and larger number. 5-3 = As children increase in confidence they will be asked to use the - and = symbols to record mental calculations. Children will continue to use practical equipment in their maths lessons to continue to support them in understanding subtraction as taking lots away, such as: Practical counting equipment; Bead strings; Coins; Board games. Once children have displayed clear understanding they will be taught how to record their calculations by: Using a blank number line to jump on in 1 s to find the difference. 5 3 = Using a blank number line to jump in 10 s and 1 s = As understanding of place value and partitioning begins to play a greater importance in children s ability to calculate efficiently they will be taught to partition numbers starting with the highest place value.

5 Key Objectives KS2 Year 3 Counting and understanding number Read, write and order whole numbers to at least 1000 and position them on a number line; count on from and back to zero in single-digit steps or multiples of 10 Partition three-digit numbers into multiples of 100, 10 and 1 in different ways Round two-digit or three-digit numbers to the nearest 10 or 100 and give estimates for their sums and differences Derive and recall all addition and subtraction facts for each number to 20, sums and differences of multiples of 10 and number pairs that total 100 Use knowledge of number operations and corresponding inverses to estimate and check calculations Add or subtract mentally combinations of one-digit and two-digit numbers Develop and use written methods to record, support or explain addition and subtraction of two-digit and three-digit numbers Year 4 Year 5 Counting and understanding number Recognise and continue number sequences formed by counting on or back in steps of constant size Partition, round and order four-digit whole numbers; use positive and negative numbers in context and position them on a number line; state inequalities using the symbols and (e.g. -3-5, -1 1) Use decimal notation for tenths and hundredths and partition decimals; relate the notation to money and measurement; position one-place and two-place decimals on a number line Use knowledge of addition and subtraction facts and place value to derive sums and differences of pairs of multiples of 10, 100 or 1000 Use knowledge of rounding, number operations and inverses to estimate and check calculations Add or subtract mentally pairs of twodigit whole numbers (e.g , 91 35) Refine and use efficient written methods to add and subtract two-and three-digit whole numbers and.p Use a calculator to carry out one and two step calculations involving all 4 operations, recognise negative numbers on the display, correct mistaken entries and interpret display correctly in the context of money Counting and understanding number Count from any given number in whole-number and decimal steps, extending beyond zero when counting backwards; relate the numbers to their position on a number line Explain what each digit represents in whole numbers and decimals with up to two places, and partition, round and order these numbers Use knowledge of place value and addition and subtraction of two-digit numbers to derive sums and differences of decimals (e.g ) Use knowledge of rounding, place value, number facts and inverse operations to estimate and check calculations Extend mental methods for whole number calculations, for example to subtract one near multiple of 1000 from another (eg ) Use efficient written methods to add and subtract whole numbers and decimals with up to two places Use a calculator to solve problems including those involving decimals and interpret the display correctly in the context of measurement Year 6 Counting and understanding number Find the difference between a positive and a negative integer, or two negative integers, in context Use decimal notation for tenths, hundredths and thousandths; partition, round and order decimals with up to three places, and position them on the number line Calculate mentally with integers and decimals: U.t U.t, Use efficient written methods to add and subtract integers and decimals. Use a calculator to solve problems involving multi step calculations

6 The default method for carrying out and recording addition problems will be the use of number lines to count on. As children move through the school they will be encouraged to use the number lines with increasing efficiency, however practical resources such as bead strings may still be used to support children s understanding. If children show the required level of understanding through their own application of these methods they will be taught how to use column methods, however they will still be encouraged to use number lines when faced with different sized numbers which make calculations difficult for them and to use invisible number lines for their mental calculations. To develop their understanding and reinforce correct application of methods children will be encouraged to record their calculations by: Using a blank number line to jump on in 1 s. As children s understanding develops and their confidence improves children will be taught how to use number lines more efficiently by: Adding the units on in one jump Once children have displayed a full level of understanding they will be taught how to use column methods to increase the efficiency of their calculations. This will be taught in steps, which begins by using their understanding of partitioning: Development of Addition KS = Using a blank number line to jump in 10 s and 1 s = = Adding the tens in one jump = Bridging through ten = Horizontal expansion: = = 552 Vertical expansion Units 12 (7+5) Tens 140 ( ) Hundreds 400 (30000) Before moving onto the next stage it is important that children revisit their understanding of partitioning and recognise that they are now required to partition by starting with the digit with the lowest place value. Vertical

7 Development of Subtraction KS2 The default method for carrying out and recording subtraction problems will be the use of number lines to find the difference by counting on. As children move through the school they will be encouraged to use the number lines with increasing efficiency. If children show the required level of understanding through their own application of these methods they will be taught how to use column methods, however they will still be encouraged to use number lines when faced with different sized numbers which make calculations difficult for them. To develop their understanding and reinforce correct application of methods children will be encouraged to record their calculations by: Using a blank number line to jump on in 1 s to find the difference. 5 3 = Using a blank number line to jump in 10 s and 1 s = As children s understanding develops and their confidence improves children will be taught how to use number lines more efficiently by: Adding the units on in one jump = Adding the tens in one jump = Bridging through ten = Before moving onto the next stage it is important that children revisit their understanding of partitioning and recognise that they are now required to partition by starting with the digit with the lowest place value. Once children have displayed a full level of understanding they will be taught how to use column methods to increase the efficiency of their calculations. This will be taught in steps, which begins by using their understanding of partitioning: Horizontal expansion: Horizontal expansion with adjustment: Vertical * * Please note that children should be encouraged to explain their understanding of place value by using the language of exchanging a ten for ten units.

8 Multiplication and Division

9 Key Objectives KS1 Foundation Stage Say and use number names in order in familiar contexts Know that a number identifies how many objects in a set Count reliably up to 10 everyday objects Estimate how many objects they can see and check by counting Count aloud in ones, twos, fives and tens Use language such as more or less to compare two numbers Use ordinal numbers in different contexts Recognise numerals 1 to 9 Observe number relationships and patterns in the environment and use these to derive facts. Select two groups of objects to make a given total of objects Count repeated groups of the same size Share objects into equal groups and count how many in each group Year 1 Count reliably at least 20 objects, recognising that when rearranged the number of objects stays the same; estimate a number of objects that can be checked by counting Compare and order numbers, using the related vocabulary; use the equals ( ) sign Read and write numerals from 0 to 20 then beyond; use knowledge of place value to position these numbers on a number track and number line Use the vocabulary of halves and quarters in context Count on and back in ones, twos, fives and tens and use this knowledge to derive the multiplies of 2, 5 and 10 to the tenth multiple Recall the doubles of all numbers to at least 10 Solve practical problems that involve combining groups of 2, 5 and 10, or sharing into equal groups Counting in equal groups using real life objects and pictures 3 lots of 2 or 2 lots of 3 Year 2 Read and write two-digit and three-digit numbers in figures and words; describe and extend number sequences and recognise odd and even numbers Count up to 100 objects by grouping them and counting in tens, fives or twos; explain what each digit in a two-digit number represents, including numbers where 0 is a place holder; partition twodigit numbers in different ways, including into multiples of 10 and 1 Order two-digit numbers and position them on a number line; use the greater than ( ) and less than ( ) signs Estimate a number of objects; round two-digit numbers to the nearest 10 Find one half and three quarters of shapes and sets of objects Understand that halving is the inverse of doubling and derive and recall doubles of all numbers to 20, and the corresponding halves Derive and recall multiplication facts for the 2,5 and 10 times tables and the related division facts, recognise multiplies of 2,5 and 10 Use knowledge of number facts and operations to estimate and check answers to calculations Represent repeated addition and arrays as multiplication and sharing and repeated subtraction (grouping) as division, use practical and informal methods and related vocabulary to support multiplication and division, including calculations with remainders. Use the symbols = - x and to record and interpret number sentences involving all four operations; calculate The value of an unknown in i a number sentence e.g. 2 = 6

10 The default method for carrying out and recording multiplication problems in KS1 is the use of arrays. This method not only develops understanding of multiplication as grouping but also reinforces the commutative nature. Development of Multiplication KS1 In the foundation stage children will be introduced to the idea of multiplication by setting up groups in role play using a wide range of practical models and images. These may include: Setting the table with knives, forks and plates for 4 people; Putting 5 flowers in each vase; Giving 5 children 2 apples each. This practical approach will also be use to introduce the children to the concept of doubling as finding 2 lots of. This will lead the children into using arrays to represent multiplication problems. Peg boards, counters or other practical equipment may be used to show this practically before children begin to record this themselves. Children will also be encouraged to develop their understanding of multiplication as lots of by counting on in 2 s, 10 s and then 5 s. As children s understanding develops they will be taught how to record arrays by drawing dots to carry out calculations for themselves: x 3 = x 5=15 At this stage children will begin to record their answer in a number sentence. As their ability of counting on improves children will be introduced to the idea of using number lines to carry out repeated addition: Use of number lines to carry out repeated addition will naturally progress to using a blank number line to count on in familiar steps: 5 x 3 = = = Throughout KS1 children will continue to use practical equipment to create arrays and link their multiplication calculations to grouping. 5 x 3 = = =

11 The default method for carrying out and recording division problems is using repeated subtraction. This will be recorded on a number line once the children have been taught how to use them and will provide a link to their understanding of division as grouping. In the foundation stage children will be introduced to the concept of division as sharing or grouping. Firstly, children will be taught to practically share objects out between 2: Through KS1 children will continually reinforce their of sharing and grouping by using practical examples and will begin to record their calculations in number sentences using the symbol. Development of Division KS1 6 sweets are shared between 2 people, how many do they get each? As their understanding of the concept of sharing deepens children will begin to use larger numbers as their divisor. The second way children will be taught to divide is by grouping: There are 6 sweets, how many children can have 2 sweets each? By grouping children are beginning to think about division as taking away lots of the divisor, which will be useful as they begin to use written methods. The concept of division as grouping will be used to introduce children to using written methods for their calculations through repeated subtraction. This will be done initially using vertical number lines and as the children s confidence increases, using blank number lines: 12 3 = To help develop their understanding of division by counting back, firstly in 10 s, then 2 s and then 5 s. Towards the end of KS1 the children may be introduced to the idea that sometimes objects cannot be shared or grouped equally. They will be taught that the word for anything which has been left over is remainder.

12 Key Objectives KS2 Year 3 Year 4 Year 5 Year 6 Read, write and order whole numbers to at least 1000 and position them on a number line; count on from and back to zero in single-digit steps or multiples of 10 Partition three-digit numbers into multiples of 100, 10 and 1 in different ways Round two-digit or three-digit numbers to the nearest 10 or 100 and give estimates for their sums and differences Read and write proper fractions (e.g. ¾ ) interpreting the denominator as the parts of a whole and the numerator as the number of parts: identify and estimate fractions of shapes, use diagrams to compare fractions and establish equivalence. Derive and recall multiplication facts for the 2,3,4,5,6 and 10 times tables and the corresponding division facts; recognise multiplies of 2,5 or 10 up to 1000 Multiply one digit and two digit numbers (e.g. 13 x 3, 50 4) round remainders up or down depending on the context Understand that division is the inverse of multiplication and vice versa, use this to derive and record related multiplication and division number sentences Find unit fractions of numbers and quantities e.g. 1/2 1/3 1/4 and 1/6 of 12 litres Recognise and continue number sequences formed by counting on or back in steps of constant size Use decimal notation for tenths and hundredths and partition decimals; relate the notation to money and measurement; position one-place and two-place decimals on a number line Recognise the equivalence between decimal and fraction forms of one half, quarters, tenths and hundredths Use diagrams to identify equivalent fractions e.g. ¾ and 6/8 interpret mixed numbers and position them on a number line e.g. 3 ½ Use the vocabulary of ratio and proportion to describe the relationship between two quantities e.g. There are two red beads to every 3 blue beads or 2 in every 5 beads are red. Estimate a proportion about one quarter of the apples in the box are green Identify the doubles of two-digit numbers; use these to calculate doubles of multiples of 10 and 100 and derive the corresponding halves Derive and recall multiplication facts up to 10 x 10, the corresponding division facts and multiples of numbers up to 10 up to the tenth multiple Use knowledge of rounding, number operations and inverses to estimate and check calculations Identify pairs of fractions that total one Multiply and divide numbers to 1000 by 10 and then 100 (whole number answers) understanding the effect, relate to scaling up or down Develop and use written methods to record, support and explain multiplication and division of two digit numbers by a one digit number including division and remainders (e.g. 15x9, 98 6 ) Find fractions of numbers, quantities or shapes e.g. 1/3 of plums Use a calculator to carry out one and two step calculations involving all four operations recognising negative numbers in the display, correct mistaken entries and interpret the display correctly in the context of money Count from any given number in wholenumber and decimal steps, extending beyond zero when counting backwards; relate the numbers to their position on a number line Explain what each digit represents in whole numbers and decimals with up to two places, and partition, round and order these numbers Express a smaller number as a fraction of a larger one ( recognise 5 out of 8 is 5/8 ) find equivalent fractions 7/10 = 14/20 and relate fractions to their decimal representations Understand percentages as the number of parts in every hundred and express tenths and hundredths as percentages Use sequences to scale numbers up or down, solve problems involving proportions of quantities e.g. decrease quantities in a recipe designed to feed 6 people Recall quickly multiplication facts up to 10 x10 and use them to multiply pairs of multiples of 10 and 100; derive quickly corresponding division facts Identify pairs of factors of two digit whole numbers and find common multiples (e.g. for 6 and 9) Use knowledge of rounding, place value, number facts and inverse operations to estimate and check calculations Use understanding of place value to multiply and divide whole numbers and decimals by 10,100 or 1000 Refine and use efficient written methods to multiply and divide htu x u tu x tu u.t x u and htu u Find fractions using division e.g. 1/100 of 5kg and percentages of numbers and quantities 10% of 80 Use a calculator to solve problems involving decimals and fractions and interpret the display in the context of measures Express a larger whole number as a fraction of a smaller one e.g. recognise that 8 slices of a 5 slice pizza represents 8/5 or 1 3/5 pizza. Simplify fractions by cancelling common factors; order a set of fractions by converting them to fractions with a common denominator Express one quantity as a percentage of another e.g. express 400 as a percentage of 1000; find equivalent percentages, decimals and fractions Solve simple problems involving direct proportion by scaling quantities up or down Use knowledge of place value and multiplication facts to 10 x 10 to drive related multiplication and division facts involving decimals e.g. 0.8 x Use knowledge of multiplication facts to derive quickly squares of numbers to 12 x 12 and the corresponding squares of multiples to 10 Recognise that prime numbers have only two factors and identify prime numbers less than 100, find the prime factors of two digit numbers Use approximations, inverse operations and tests of divisibility to estimate and check results Calculate mentally with integers and decimals TU x U, TU U, U.t x U, U.t U Use efficient methods to multiply and divide integers and decimals by a one digit integer and to multiply two digit and three digit integers by a two digit integer Relate fractions to multiplication and division e.g. 6 2 = ½ of 6 = 6 x ½ express a quotient as a fraction or decimal e.g = 13.4 or 13 and 2/5. Find fractions and percentages of whole number quantities e.g 5/8 of 96 65% of 260 Use a calculator to solve problems involving multi step calculations

13 Development of Multiplication KS2 The default method for carrying out and recording multiplication problems is using the grid method. This method builds on children s prior knowledge of the use of arrays and ensures the understanding of the calculation they are carrying out. As children s mental recall of multiplication facts Once full understanding of the concept of improves they will be introduced to using the grid multiplication has been met the children may be method. This may be linked to their use of arrays in introduced to expanded vertical methods of their teaching to draw on prior learning: multiplication: x 20 4 In KS2 children will continue to use arrays to represent their calculations and some children may still use practical representations of this: x 3 = x 5=15 Understanding of multiplication of finding lots of will be reinforced by using number lines to represent repeated addition, either with or without the numbers shown depending on how familiar the children are with the steps required: 5 x 3 = = = x 3 = = Children will first be taught to use this method for multiplying UxTU: 4 x 24 = 76 x This will then naturally progress to multiplying UxHTU: 3 x 124 = 372 x And continuing to multiplying TUxTU: 72 x 38 = 2736 x =80 dots + 16 dots 38 x 7 56 x x 7 56 (8x7) (30x7) x (6x7) 350 (50x7) 120 (6x20) 000 (30x7) 1512 Some children may be introduced to using the full vertical method: 136 x x The children must have a secure knowledge of times table facts and a deep understanding of place value to be able to carry out the parts of the calculation.

14 Development of Division KS2 The default method for carrying out and recording division problems is the vertical number line method. This links closely to children s understanding of division as grouping through repeated subtraction and provides a visual representation of the calculation they are carrying out. As children move into KS2 practical equipment will be used to carry out division problems to reinforce their understanding of division as sharing and grouping. The use of number lines to carry out division by repeated subtraction will be reinforced and children will continue to use this method until their understanding is secure. Some children will need to continue to use numbered vertical number lines where as some children s understanding may be clear enough to use blank vertical number lines: 12 3 = The idea that sometimes division problems may have a remainder will be re-introduced, firstly using practical equipment and then by recording on a number line. It is here that we see the importance of counting back, rather than counting on as we would do to find the difference: 7 3 = 2r As children s recall of multiplication facts improves they will be able to use the vertical number line method to carry out division problems with larger numbers and with greater efficiency: 52 4 = = x 4 3 x 4 Finally they will use this method using their times table facts to jump efficiently for division problems involving remainders: = r2 = 31r x4 10x4 10x4 1x4 While the vertical number line method will still be used as a default method some children may be introduced to using chunking as an introduction to short division: 72 3 = = (10x3) (10x3) (4x3) 0 The children will be taught to take away the chunks that they feel confident with. This method may also be used to carry out calculations with remainders: r (30x12) (2x12) 3 Finally the children may be shown how to use the short division bus stop method. Please note this should only be taught to children whose understanding is very secure

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