Stradbroke Primary School Calculation Policy Introduction Children are introduced to the processes of calculation through practical, oral and mental activities. As children begin to understand the underlying ideas they develop ways of recording to support their thinking and calculation methods, use particular methods that apply to special cases, and learn to interpret and use the signs and symbols involved. Over time children learn how to use models and images, such as empty number lines, to support their mental and informal written methods of calculation. As children's mental methods are strengthened and refined, so too are their informal written methods. These methods become more efficient and succinct and lead to efficient written methods that can be used more generally. By the end of Year 6 children are equipped with mental, written and calculator methods that they understand and can use correctly. When faced with a calculation, children are able to decide which method is most appropriate and have strategies to check its accuracy. At whatever stage in their learning, and whatever method is being used, it must still be underpinned by a secure and appropriate knowledge of number facts, along with those mental skills that are needed to carry out the process and judge if it was successful. The overall aim is that when children leave primary school they: have a secure knowledge of number facts and a good understanding of the four operations; are able to use this knowledge and understanding to carry out calculations mentally and to apply general strategies when using one- digit and two-digit numbers and particular strategies to special cases involving bigger numbers; make use of diagrams and informal notes to help record steps and part answers when using mental methods that generate more information than can be kept in their heads; have an efficient, reliable, compact written method of calculation for each operation that children can apply with confidence when undertaking calculations that they cannot carry out mentally; Mental strategies need to be established. These will be based on a solid understanding of place value and a sense of number. It is important that children's mental methods of calculation are practised and secured alongside their learning and use of efficient written methods for calculation. Mental strategies involve: Remembering number facts and recalling them without hesitation Doubling and halving Using known facts to calculate unknown facts Re-ordering numbers Understanding and using relationships between addition & subtraction, multiplication and division Checking answers Estimating answers Seeing/breaking numbers up in different ways eg 64 = 50 + 14 Children will need a repertoire of mental strategies in order to solve calculations in their heads: Count on /back in 1s, 10s, 100s from any start point doubles / near doubles bridging 10 / bridging 20 adding 9 by +10 & -1 (compensation) Stages in Addition To add successfully, children need to be able to: recall all addition pairs to 9 + 9 and complements in 10; add mentally a series of one-digit numbers, such as 5 + 8 + 4; add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value; partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways Informal jottings will help children with their calculations. Stage one They use numberlines and practical resources to support calculation and teachers demonstrate the use of the numberline. 3+2=5 +1 +1 ___________________________________________ 0 1 2 3 4 5 6 7 8 9 Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones. 8 + 5 = 13 +1 +1 +1 +1 +1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on. Stage two The empty number line The mental methods that lead to column addition generally involve partitioning, e.g. adding the tens and ones separately, often starting with the tens. Children need to be able to partition numbers in ways other than into tens and ones to help them make multiples of ten by adding in steps. The empty number line helps to record the steps on the way to calculating the total. Steps in addition can be recorded on a number line. The steps often bridge through a multiple of 10. 8 + 7 = 15 48 + 36 = 84 or: WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS? Addition and subtraction Do they know addition and subtraction facts to 20? Do they understand place value and can they partition numbers? Can they add three single digit numbers mentally? Can they add and subtract multiples of 10? Can they add and subtract any pair of two digit numbers mentally? Can they explain their mental strategies orally and record them using informal jottings? Stage 3: Partitioning The next stage is to record mental methods using partitioning. Add the tens and then the ones to form partial sums and then add these partial sums. Partitioning both numbers into tens and ones mirrors the column method where ones are placed under ones and tens under tens. This also links to mental methods. Record steps in addition using partitioning: 47 + 76 = 40 + 70 = 7+6= 47 + 76 = 47 + 70 + 6 = 117 + 6 = 123 47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123 47 + 76 Stage 4: Expanded method in columns Move on to a layout showing the addition of the tens to the tens and the ones to the ones separately. To find the partial sums either the tens or the ones can be added first, and the total of the partial sums can be found by adding them in any order. As children gain confidence, ask them to start by adding the ones digits first always. The addition of the tens in the calculation 47 + 76 is described in the words 'forty plus seventy equals one hundred and ten', stressing the link to the related fact 'four plus seven equals eleven'. The expanded method leads children to the more compact method so that they understand its structure and efficiency. The amount of time that should be spent teaching and practising the expanded method will depend on how secure the children are in their recall of number facts and in their understanding of place value. Write the numbers in columns. Adding the tens first: Adding the ones first: Discuss how adding the ones first gives the same answer as adding the tens first. Refine over time to adding the ones digits first consistently. Stage 4: Column method In this method, recording is reduced further. Carry digits are recorded below the line, using the words 'carry ten' or 'carry one hundred', not 'carry one'. Later, extend to adding three two-digit numbers, two three-digit numbers and numbers with different numbers of digits. Stage 4 Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. Stages in Subtraction To subtract successfully, children need to be able to: recall all addition and subtraction facts to 20; subtract multiples of 10 (such as 160 - 70) using the related subtraction fact, f16 - 7, and their knowledge of place value; partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14). Stage 1 Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures etc. Informal jottings will help children with their calculations They use numberlines and practical resources to support calculation. Teachers demonstrate the use of the numberline. Stage 2 6–3=3 -1 -1 -1 ___________________________________ 0 1 2 3 4 5 6 7 8 9 10 The numberline should also be used to show that 6 - 3 means the ‘difference between 6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart. 0 1 2 3 4 5 6 7 8 9 10 Children then begin to use numbered lines to support their own calculations - using a numbered line to count back in ones. 13 – 5 = 8 -1 -1 -1 -1 -1 . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Counting back First counting back in tens and ones. 47 – 23 = 24 -1 -1 -1 - 10 - 10 24 25 26 27 37 47 Then helping children to become more efficient by subtracting the units in one jump (by using the known fact 7 – 3 = 4). 47 – 23 = 24 -10 -10 -3 24 27 37 47 Subtracting the tens in one jump and the units in one jump. 47 – 23 = 24 -20 -3 24 27 47 Bridging through ten can help children become more efficient. 42 – 25 = 17 -20 -3 -2 17 20 22 42 Counting on If the numbers involved in the calculation are close together or near to multiples of 10, 100 etc, it can be more efficient to count on. Count up from 47 to 82 in jumps of 10 and jumps of 1. 82 - 47 +10 +10 +10 +1 +1 +1 +1 +1 47 48 49 50 60 70 80 81 82 Help children to become more efficient with counting on by: Subtracting the units in one jump; Subtracting the tens in one jump and the units in one jump; Bridging through ten. The children must begin to learn subtraction through their use of number lines. They will learn to calculate the difference between numbers. This method is essential and can be used with all ages. WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS? Addition and subtraction Do they know addition and subtraction facts to 20? Do they understand place value and can they partition numbers? Can they add three single digit numbers mentally? Can they add and subtract multiples of 10? Can they add and subtract any pair of two digit numbers mentally? Can they explain their mental strategies orally and record them using informal jottings? Stage 3 DECOMPOSITION – EXPANDED METHOD This method makes the move to traditional decomposition easier. The children must understand the value of the digits they are using and develop an appreciation of the need to take from the next column and the effect this has on the sum. At this stage the children must be taught to set out their work accurately. Stage 4 TRADITIONAL DECOMPOSITION This is only taught once the children have mastered the above methods. By this stage the children should be able choose the most efficient method for the calculation they have to complete. 4 15 1 5 6 3 - 2 7 8 2 8 5 Stages in Multiplication To multiply successfully, children need to be able to: recall all multiplication facts to 10 × 10; partition number into multiples of one hundred, ten and one; work out products such as 70 × 5, 70 × 50, 700 × 5 or700 × 50 using the related fact 7 × 5 and their knowledge of place value; add two or more single-digit numbers mentally; add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value; add combinations of whole numbers using the column method (see above). Multiplication is linked to division Count numbers in groups Recognise that multiplication can be done in any order – commutative law Stage 1 Children will experience equal groups of objects and will count in 2s and 10s and begin to count in 5s. They will work on practical problem solving activities involving equal sets or groups. Stage 2 Children will develop their understanding of multiplication and use jottings to support calculation: Repeated addition 3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or 5 x 3 Repeated addition can be shown easily on a number line: 5x3=5+5+5 5 5 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 x3=5+5+5 Commutativity Children should know that 3 x 5 has the same answer as 5 x 3. This can also be shown on the number line. 5 5 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3 3 3 3 3 Arrays Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method. 5 x 3 = 15 3 x 5 = 15 Before moving pupils to stage 3: Do they know the 2, 3, 4, 5 and 10 times table Do they know the result of multiplying by 0 and 1? Do they understand 0 as a placeholder? Can they multiply two and three digit numbers by 10 and 100? Can they double and halve two digit numbers mentally? Can they use multiplication facts they know to derive mentally other multiplication facts that they do not know? Can they explain their mental strategies orally and record them using informal jottings? The above lists are not exhaustive but are a guide for the teacher to judge when a child is ready to move from informal to formal methods of calculation. Stage 3 The children should be taught the basic mental method using partitioning. It is essential that at this stage they are taught to estimate their answer. For example Estimate: 30 x 5 = 150 Actual: 27 x 5= (20 x 5) + (7 x 5) 100 + 35 = 135 Stage 4 Once the children have grasped this method they should be taught the grid method. The children should practice 2digit x 1 digit (no carrying) before moving forward to gradually introduce carrying and finally moving to 3 digit x 1 digit. Example: 23 x 8 Estimate: 20 x 8 = 160 Actual: x 20 3 8 160 24 184 The children should be taught how to work with larger numbers. The principles of estimate and check (inverse operation) should also be modelled. Example: 157 x 23 Estimate: 160 x 20 = 320 Actual: x 100 50 7 20 2000 1000 140 3140 3 300 150 21 471 3611 The children should be taught how to work with larger numbers and decimals. The principles of estimate and check (inverse operation) should also be modelled. Example: 27.5 x 12 = 330 Estimate: 30 x 12 = 360 Actual: x 20 7 0.5 10 200 70 5 275 2 40 14 1 55 330 Stage 5 Once the children are confident users of the grid method they should be taught how to use the traditional vertical format. The children should be shown the two methods alongside each other to enable the children to identify the similarities between the formats. Again the children should be taught the expanded methods before moving to the compact method in step six. Example: 124 x 6 = Estimate: 120 x 6 = 720 Actual: 124 X6 24 (6 x 4) 120 (6 x 20) 600 (6 x 100) 744 Stage 6 This final step includes the use of both compact and traditional ‘long multiplication’. It is essential that the children recognise the similarities with the grid method and so we should continue to make teaching comparisons. Example: 124 x 6 = Estimate: 120 x 6 = 720 Actual: 124 X6 744 12 Example: 124 x 25 = Estimate: 120 x 30 = 3600 Actual: 124 X25 620 (5 x 124) 2480 (20 x 124) 3100 11 Stages in Division To divide successfully in their heads, children need to be able to: understand and use the vocabulary of division - for example in 18 ÷ 3 = 6,the 18 is the dividend, the 3 is the divisor and the 6 is the quotient; partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways; recall multiplication and division facts to 10 × 10, recognise multiples of one-digit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value; know how to find a remainder working mentally - for example, find the remainder when 48 is divided by 5; Understand and use multiplication and division as inverse operations. Dividing by 1 leaves the number unchanged Stage 1 Children will understand equal groups and share items out in play and problem solving. They will count in 2s and 10s and later in 5s. Children will develop their understanding of division and use jottings to support calculation Sharing equally 6 sweets shared between 2 people, how many do they each get? Grouping or repeated subtraction There are 6 sweets, how many people can have 2 sweets each? Stage 2 Repeated subtraction using a number line or bead bar 12 ÷ 3 = 4 0 1 2 3 4 5 6 7 8 9 10 11 12 3 3 3 3 The bead bar will help children with interpreting division calculations such as 10 ÷ 5 as ‘how many 5s make 10?’ This builds upon a mental method. The children should be taught the repeated subtraction method using a number line. Stage 3 The children should understand the links with multiplication and would be able to use their table knowledge and numbers lines to make larger jumps. When moving onto the vertical method the children must be given the opportunity to begin with sums they can do mentally. They should also be taught to estimate and check using the inverse operation. Example: 62 / 6 = 10 r 2 Estimate: 60 / 6 = 10 Actual: 6 / 62 - 60 (10 x 6) 2 For larger numbers, when it becomes inefficient to count in single multiples, bigger jumps can be recorded using known facts. Example without remainder: 81 ÷ 3 30 30 21 10 10 7 0 30 60 81 This could either be done by working out the numbers of threes in each jump as you go along (10 threes are 30, another 10 threes makes 60, and another 7 threes makes 81. That’s 27 threes altogether) or by counting in jumps of known multiples of 3 to reach 81 (30 + 30 + 21) then working out the number of threes in each jump. Example with remainder: 158 ÷ 7 70 70 14 2 10 10 2 0 70 140 154 158 10 sevens are 70, add another 10 sevens is 140, add 2 more sevens is 154 add 2 makes 158. So there are 22 sevens with a remainder of 2. The remainder is indicated above the jump rather than inside it, so that children do not mistakenly add 10, 10, 2 and 2 and get an answer of 24. The same method should be used with larger numbers. Example: 162 / 6 = 27 Estimate: 160 / 10 = 16 Actual: 6 / 162 - 60 (10 x 6) 102 - 60 (10 x 6) 42 -42 (7 x 6) 0 Stage 4 Once pupils are more confident with mental strategies they should be taught the contracted method. Example: 163 / 6 = 27 r1 Estimate: 160 / 10 = 16 Actual: 6 / 163 -120 (20 x 6) 43 - 42 (7 x 6) 1 Stage 5 This method should be extended further to include larger numbers and decimals. Children will continue to use written methods to solve short division TU ÷ U. Children can start to subtract larger multiples of the divisor, e.g. 30x Short division HTU ÷ U 196 ÷ 6 32 r 4 6 ) 196 - 180 30x6 16 - 12 2x6 4 Answer : 32 remainder 4 or 32 r 4 Children will continue to use written methods to solve short division TU ÷ U and HTU ÷ U. Long division HTU ÷ TU 972 ÷ 36 27 36 ) 972 - 720 20x36 252 - 252 7x36 0 Answer : 27 Any remainders should be shown as fractions, i.e. if the children were dividing 32 by 10, the answer should be shown as 3 2/10 which could then be written as 3 1/5 in it’s lowest terms. Summary children should always estimate first always check the answer, preferably using a different method eg. the inverse operation always decide first whether a mental method is appropriate pay attention to language - refer to the actual value of digits children who make persistent mistakes should return to the method that they can use accurately until ready to move on children need to know number and multiplication facts by heart discuss errors and diagnose problem and then work through problem - do not simply re-teach the method when revising or extending to harder numbers, refer back to expanded methods. This helps reinforce understanding and reminds children that they have an alternative to fall back on if they are having difficulties. Can I do it in my head? Can I change it to make it easier? Can I use jottings to help me work it out? Can I use a calculator?

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