Calculations Policy. Introduction


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1 Thousands Hundreds Tens Units Tenth Hundredth thousandth Calculations Policy Introduction This Calculations Policy has been designed to support teachers, teaching assistants and parents in the progression of all four number operations (addition, subtraction, multiplication and division). Year groups have been shown for each calculation technique to show the expected teaching level for the majority of children. More able children may embed the skills more quickly and therefore may be taught skills which are above their current year group; conversely, children who are less able may be working at a level below their year group. It should be noted that children should not move beyond the teaching for their year group until they have not only mastered the skill but can also apply the skills in a variety of contexts and reallife problems. Index Page 2 Page 3 Page 6 Page 9 Page 12 Vocabulary Addition Subtraction Multiplication Division Notes on the recording of work 1. When using squared paper, one digit should be placed in each square. 2. The decimal place should be put on the vertical line between 2 squares. 3. Commas should be used to separate polydigit numbers into groups of three starting from the right hand side and working to the left e.g. 23,400, 234,567 etc. to help in pronunciation of numbers. 4. Children should always aim to estimate the answer first before carrying out the calculation. At the end, the answer should be checked against the estimate so that unnecessary mistakes are not made. 5. When using number lines, addition jumps go above the line and subtraction jumps go below the line. 6. When writing vertical sums, the symbol should be placed on the left hand side. 7. Jottings should be shown in mathematical thinking books. 8. When labelling columns, the following notation should be used: Th H T U t h th Barrs Court Calculation Policy 1 July 2012
2 Vocabulary Associated with Number Operation Addition Vocabulary (+) Subtraction Vocabulary () add subtract addition take (away) plus leave more minus and how many have gone? sum one less, two less, ten less... total how many fewer is... than...? altogether difference between score double Note addition and subtraction are inverse how many more to make...? operations of each other. For example: how many more is... than...? = 10 and 10 6 = = 10 and 10 4 = 6 Multiplication Vocabulary (x) Division Vocabulary ( ) lots of halve groups of share, share equally times one each, two each, three each... multiplication group in pairs, threes... tens multiply equal groups of multiplied by divide, division, multiple of divided by, divided into product remainder times as (big, long, wide, and so on) factor repeated addition divisible by double Note multiplication and division are inverse operations of each other. For example: 6 x 4 = 24 and 24 6 = 4 4 x 6 = 24 and 24 4 = 6 Other Key Vocabulary Related to Number Operation Decimal  A fraction written as a number with a decimal point. Decimal Point  Separates the integer part of a number from its decimal fraction. Denominator  Bottom number of a fraction showing the parts the whole is divided into Digit  symbol used to show a number, so 18 is a 2digit number. Estimate  An approximate or rough calculation, often based on rounding. Even number  A number that is divisible by 2. Numbers that end in 2, 4, 6, 8 or 0. Factor  A number is a factor of another if it divides into it with no remainders. (To make 12 you can do 1x12, 2x6, 3x4, so factors of 12 = 1, 2, 3, 4, 6) Integer  A whole number and not a decimal number. Numerator  A number above the line of a fraction showing how many parts of the whole. Odd number  Number that is not divisible by 2. Numbers that end in 1, 3, 5, 7 or 9. Partition  Where you split the whole number into parts, so 123 = 100 and 20 and 3 Place Value  The value of a digit in a number (the 6 in 2,645 is 600) Prime Number  A number that is only divisible by the number 1 and itself i.e. a number that has only two factors. The prime numbers up to 30 are 2, 3, 7, 11, 13, 17, 19, 23, 29. N.B. 1 is not prime. Round  Change a number to a more convenient value such as nearest whole number Remember rule of 5 where 5 and above rounds up. Square Number  The answer when a number is multiplied by itself. 1x1=1, 2x2=4, 3x3=9, 4x4=16 etc. Barrs Court Calculation Policy 2 July 2012
3 YEAR ADDITION FOUNDATION Children start Foundation by learning number names and the values of each digit with lots of pictorial representation and handson maths. They then learn to count forwards and backwards using a number track/line. They learn to recognise one more or one less (initially up to 10 and extend up to 20) and move on to slightly bigger numbers. They find the total using counting on when one group of objects is hidden and select two groups of objects to make a given total. They record their calculations as a number sentence. Children are introduced to the concept of number bonds/pairs to 10 and doubles up to = YEAR 1 In Year 1, children learn how to count forwards and backwards, using number lines, 100 squares and dienes cubes. They learn how to add a single digit to 10 and then 20. Number bonds to 10 (and moving on to 20) also feature heavily in this year group as does learning number doubles to = = Using Dienes Cubes: = 25 + =25 YEAR 2 Year 2 extend their knowledge of addition using 1 and 2digit numbers up to 100 (and beyond for the more able). Children continue to use partitioning of numbers on number lines as well as 100 squares, pictorial representations and dienes cubes. Also, children are expected to know their number bonds to 20. Partitioning: = = = 37 Add the 10s first Barrs Court Calculation Policy 3 July 2012
4 = = YEAR Year 3 move on to short addition in the vertical format. Teachers may use the expanded form to help children understand the new skill. Next Steps to Success Column Addition: 1. Estimate the answer first. 2. Write the 2 numbers vertically where the biggest number is on the top. 3. Keep the digits in their HTU column. 4. Start with the units and add the 2 together, putting the answer below the line. 5. Add the tens digits together and put the answer below the line. 6. Underline the answer and then check against the estimate. Short Column Addition Expanded Column Addition (5 + 2) 7 0 ( ) 7 7 Next Steps Column Addition (carrying): 1. Estimate the answer first. 2. Write the 2 numbers vertically where the biggest number is on the top. 3. Keep the digits in their HTU column. 4. Start with the units and add the 2 together, putting the answer below the line. 5. If the total is greater than 9, then carry the tens into the next column. 6. Add the tens digits together and put the answer below the line. 7. If the tens total is greater than 10 then carry the 100s into the next column. 8. Underline the answer and then check against the estimate. Short Column Addition Expanded Column Addition (6 + 7) ( ) Barrs Court Calculation Policy 4 July 2012
5 Children move on to 3 and 4 digit numbers when they are ready YEAR 4 YEAR 5 Year 4 consolidate column addition and extend learning into the skills of adding decimals, particularly with measures and money. Next Steps Column Addition with Decimals 1. Ensure all of the appropriate place value headings are labelled. 2. Fix the decimal place between the units and tenths digits and ensure they line up. 3. If writing numbers in squares, put the decimal place on the vertical line between 2 squares. 4. Complete the sum as you would for whole numbers. T U. t cm cm cm 1 1 Year 5 learn to deal with mixed decimals as well as multiple addition sums. Next Steps Column Addition with Mixed Decimals = 1. Ensure all digits are in their place value columns. 2. Add a zero to ensure both numbers have the same number of digits after the decimal place (to hold the place value) 3. Complete the sum as you would for normal decimals starting with the numbers on the right hand side YEAR 6 In Year 6, children consolidate their knowledge of addition. Barrs Court Calculation Policy 5 July 2012
6 YEAR FOUNDATION SUBTRACTION Children start Foundation by learning number names and the values of each digit with lots of pictorial representation and handson maths. They then learn to count backwards and forwards using a number track/line. They learn to remove a smaller number from a larger number and find how many are left by counting back from the larger number (using money and objects). They also recognise how many more are needed to make a larger number by counting on. Children learn to record their calculations as number sentences. 9 4 = YEAR 1 In Year 1, children learn how to count forwards and backwards, using number lines, 100 squares and dienes cubes. They learn how to subtract a single digit to 10 and then 20 and understands the operation of subtraction as takeaway or difference = = 17 Using cubes: 25 4 = 21  = 21 YEAR 2 Children in Year 2 move on to counting forwards and backwards on an empty number line. They learn to add and subtract multiples of 10, without crossing 100 but the more able will go beyond = Barrs Court Calculation Policy 6 July 2012
7 74 27 = 47 worked out by counting back: The steps may be recorded in a different order but it is normal to subtract the tens first. Subtraction can also be recorded using partitioning: = = 54 7 = 47 Or: = = = 47 YEAR 3 In Year 3, children start to learn a more formal written method which is the precursor to column subtraction. In this, the numbers are partitioned so that children can recognise the technique. Next Steps Column subtraction with Partitioning Estimate the answer first. 1. Write the 2 numbers vertically where the biggest number is on the top. 2. Partition the numbers but keep them in their place value columns. 3. Start with the units and subtract the bottom number from the top number, putting the answer below the line. Continue if there are hundreds. 4. Subtract the tens digits from the one above and put the answer below the line. 5. Underline the answer and then check against the estimate. It is good practise to check by doing the inverse operation (so in the eg below, = 75) = = Children move on to numbers that require decomposition (otherwise known as exchanging or borrowing. In this, the golden rule is that if a number is bigger on the bottom then you cannot take it away from a smaller number. In the first example below, the 8 cannot be taken away from 5, so one ten is exchanged for ten units making it 158=7. Next Steps Subtraction with Exchange 1. Set the calculation out as shown before and estimate the answer. 2. Where the units on the bottom are bigger than the top, exchange a ten for ten units so that the number on the top is now bigger. 3. Finish the calculation as normal and check against the estimate using the inverse operation = = YEAR 4 In Year 4, children move on to the standard formal written method known as column subtraction. Initially, teachers will show the expanded method alongside the short method. Barrs Court Calculation Policy 7 July 2012
8 Process Steps to Success Vertical Subtraction with Exchange 1. Estimate the answer first. 2. Write the 2 numbers vertically where the biggest number is on the top. 3. Keep the digits in their HTU column. 4. Start with the units and take the bottom one away from the top, putting the answer below the line. (If the top number is smaller, then one ten needs to be exchanged for ten units.) 5. Take the bottom tens digit away from the top one and put the answer below the line. (Again, if the top number is smaller, then one hundred needs to be exchanged for ten tens.) Underline the answer and then check against the estimate using the inverse operation Remember, if you can do the calculation mentally then this is better to do so. In the example below, this would be quite awkward to do in the formal column method. In this case, it would be much easier to count on or back using a number line: 6,003 1,997 = 4, YEAR 5 Year 5 learn to deal with subtraction of decimals and mixed decimals. Next Steps Column Subtraction with Decimals 1. Ensure tenths and hundredths are in the appropriate column. 2. Fix the decimal place between the units and tenths digits and ensure they line up. 3. If writing numbers in squares, put the decimal place on the line between 2 squares. 4. Complete the calculation as you would for whole numbers cm 8. 5 cm cm Next Steps Column Subtraction with Mixed Decimals (As with decimals except it is essential that there is a place holder added if the numbers do not have the same number of decimal digits.) 1. Ensure all digits are in their place value columns. 2. Add a zero to ensure both numbers have the same number of digits after the decimal place (to hold the place value) 3. Complete the calculation as you would for normal decimals starting with the numbers on the right hand side = Barrs Court Calculation Policy 8 July 2012
9 YEAR 6 YEAR In Year 6, children consolidate their knowledge of subtraction. MULTIPLICATION FOUNDATION Children start by counting on in 2 s and 10 s. They also practise doubling and grouping numbers which is then linked to problem solving e.g. how many pieces of bread to make 5 sandwiches. To support children, teachers will use lots of physical equipment such as cubes, counters and other objects that the children can practise grouping and doubling. This is particularly important in Key Stage 1 as pencil and paper procedures only develop once the links are made in children s understanding. YEAR 1 Children continue counting on in 2 s, 5 s and 10 s and more able count on and back using other numbers. Children also learn to double up to Children will use pictures/diagrams to help them make groups/sets. Making sets: 3 x 5 ( 3 sets of 5 or 3 lots of 5 ) Leading to: + + = 15 Times Tables Children learn to count in twos, fives and tens. YEAR 2 Children in Year 2 continue with hands on maths based around grouping, doubling but start to use informal pencil and paper procedures, particularly a number line. Children learn to recognise that multiplication is repeated addition. They also use number squares and bigger numbers. They may start using arrays to organise their multiplication and learn common doubles of multiples of five and ten up to one hundred. E.g. Three groups of six, or six groups of three to show eighteen. Number Line: Use number lines with divisions before empty number lines. Repeated Addition: = 3 lots of 5 = 3 x 5 = 15 Times Tables Children learn the 2x, 10x and 5x tables and also all the related division facts. Children start to count in threes and fours. Barrs Court Calculation Policy 9 July 2012
10 YEAR 3 In Year 3, children start to work with bigger numbers on an empty number line and are introduced to partitioning 2digit numbers. YEAR 4 Partitioning: = 13 x 5 = (10 x 5) + (3 x 5) = = 65 Times Tables  Children learn the 2x, 10x, 5x, 3x, 4x and 6x tables as well as the corresponding division facts. Times Tables  children learn all their times tables up to 12x and the corresponding division facts. Written methods of multiplication become more formal and introduce both the grid method and the expanded standard method. Once introduced, these should be taught side by side. Next Steps Grid Multiplication (2digit x 1digit) 1. Estimate the answer first. 2. Create the grid as shown below. 3. Put the single digit on the top and partition the 2digit number down the left hand side. 4. Multiply the numbers down the side with the number on top and fill in the grid. 5. Add together the numbers in the grid and check answer against the estimate. Next Steps Grid Multiplication (2digit x 2digit) 1. Start as for 1digit but partition both numbers. 2. Multiply the numbers down the side with the number on the top and fill in the grid. 3. Add together each row and create a vertical addition sum to the right. 4. Complete the addition sum. 2digit x 1digit: X digit x 2digit: X = = Next Steps Expanded Standard written Multiplication 1. Estimate the answer first. 2. Line the numbers up vertically as shown below. 3. Take the bottom right digit and multiply it by both the top digits. 4. Write the calculation you have done and put it in brackets to the side. 5. (2digit x 2digit) repeat with the number on the bottom left. 6. Add together the numbers to get the answer and check against the estimate. 2digit x 1digit: x 1 5 (5 x 3) 5 0 (5 x 10) 6 5 Note, once children have mastered the concept of this method, they can stop putting the numbers in brackets. 2digit x 2digit: x 5 6 (7 x 8) (7 x 30) 8 0 (10 x 8) (10 x 30) Barrs Court Calculation Policy 10 July 2012
11 YEAR 5 Times Tables  children learn to mentally apply their knowledge of multiplication and division sums up to 12x to bigger numbers, eg, 17 x 5 and 23 x 4. In Year 5, children move on to short Standard written multiplication, starting with TUxU, then moving through progressive levels of difficulty of HTUxU to TUxTU. The key skills to remember are: 1 You are using the digits and not their value, so the 8 in 84 is an 8 and not When completing TU x TU sums, remember to put in the 0. If children ask, this is because you are multiplying all numbers by a number in the Ten s column so place value must be taken into account. Next Steps Short Multiplication (TU x U and HTU x U) 1. Estimate the answer first using mental methods. 2. Set the calculation out the same as the vertical expanded method. 3. Start with the bottom right hand number and multiply that with the top right hand number. 4. Put the answer below the line. If it is a 2digit number then put a small tens digit in the next column. 5. Continue with the bottom right hand number and multiply that with the other digits on the top. 6. Underline the answer and check against the estimate. Next Steps Short Multiplication (TU x TU) 1. When you have completed using the bottom right hand digit, you need to move on to the left hand bottom digit. 2. Before you do any calculations on the second line, put in the zero. 3. Then complete the calculation in the same way (2 x 6 and then 2 x 3) 4. Add together the lines to arrive at the answer. TU x U = HTU x U = TU x TU = x x x Remember zero YEAR 6 Times Tables  Children continue to practise mentally applying their knowledge of multiplication and division sums up to 12x to bigger numbers as well as decimals. In Year 6, children develop their skills of multiplication with the introduction of decimals. The key thing to remember is that unlike for all other vertical methods shown before, this is the only one where the numbers do not need to be kept in HTU columns. Next Steps Short Multiplication with Decimals (U.t x U) 1. Estimate the answer. 2. Put the decimal number on the top and the other number below. 3. Complete the calculation as you would without the decimals but ensure that a decimal point is placed in the answer in line with the decimal point in the question x Barrs Court Calculation Policy 11 July 2012
12 Next Steps Short Multiplication with Decimals (U.t x U.t) 1. Estimate the answer 2. Change the two decimal numbers into whole numbers by moving the digits up the required number of places until they are all to the left of the decimal point (keep track of the total number of times the digits have been moved to the left in both numbers of the calculation.) e.g. 2.3 x 4.5 > 23 x 45 (2.3 has been moved one place to the left, 4.5 has been moved one place to the left so a total of two moves to the left have been made. 3. Calculate the altered problem using the standard written method of multiplication or grid multiplication. (23 x 45= 1035) 4. Move the current answer s digits the total number of places to the right as originally made to the left (in this case two) to get the correct answer. (1035 > 10.35) YEAR DIVISION FOUNDATION Children learn to group / share numbers using physical apparatus. Sharing: e.g. ( 18 sweets shared between 3 children ) Their plates 6 each YEAR 1 Children continue grouping/sharing numbers and also learn to halve numbers up to 20. Grouping: 18 grouped in 3 s = 6 groups Children start to use number lines to divide single digit numbers by single digit numbers and begin to recognise the idea of remainders. E.g. 9 2= 4 r Barrs Court Calculation Policy 12 July 2012
13 YEAR 2 In Year 2, children learn that multiplication and division are inverse operations of each other. They use number lines, 100 squares and physical apparatus to practise division as repeated subtraction. They also learn there may be numbers left over (remainders) Repeated Subtraction: = = 5 r 1 Children to start arranging these sorts of problems into arrays to help with multiplication facts. E.g. YEAR 3 Three groups of six, or six groups of three to show eighteen. In Year 3, children continue with informal methods of division using such apparatus as number lines and number squares. Each of these jumps are lots of three which need to be recorded for the final answer = 53 r YEAR 4 Note it may help to get children to ring the remainder to make it clear. In Year 4, children develop their knowledge of chunking using a number line into a formal written method called the chunking method. Next Steps Division by Chunking for TU U and HTU U 1. Set out the numerator (larger number) inside a bus stop with the denominator to the left. 2. Calculate how many lots of 10 you can get from the number and record in brackets to the right. This might be lots of 10 (examples 1 and 2) or multiples of 10 (example 3). 3. Take this away from the original number to leave the remainder. 4. If you can take some more lots of 10 away then repeat. Otherwise take away some singles and record in brackets to the right. 5. The answer is the numbers in brackets added together (example 1). The answer might not work out exactly, in which case there will be also be a remainder (examples 2 and 3). 6. Display the final answer below or to the side of the calculation. Barrs Court Calculation Policy 13 July 2012
14 Example = (10x) (2x) Example = 32 r (10x) (10x) (10x) (2x) 4 Example = 32 r (30x) (2x) 4 YEAR 5 In Year 5, children move on to the short method of division, sometimes know as the bus stop method. Process Steps to Success Short Division 1. Estimate an approximate answer. 2. Set out the bus stop with the number being divided by, by the bus stop and the number being divided into, into the bus stop. 3. Calculate how many of the outer number go into the first digit of the inner number. If the first digit is smaller than the outer number then use the first 2 digits. 4. Put the answer above the bus stop and carry the remainder to the next digit. 5. Repeat until you reach the decimal place. There may or may not be a remainder. 6. Put the answer back into the number sentence and check against the estimate = 32 r r Note, the remainder can be shown as a fraction with the denominator (4/6ths) Barrs Court Calculation Policy 14 July 2012
15 YEAR 6 In Year 6, children learn to deal with remainders as decimals as well as numbers that have decimals in them. They also learn long division which is any number divided by a 2digit number. Next Steps Short Division with Decimals 1. The first part of the sum is the same as for remainders. 2. When you reach the end of the number and have a remainder, instead of stopping there, add a decimal place and as many zeros as you may need (this does not change the number) 3. Add the remainder to the first zero and carry on. 4. You probably will not want to go further than 2 or 3 decimal places = If the decimal part of the answer keeps repeating (recurring) then show 2 digits the same and put a dot above the second digit. So one third would be shown as 0.33 Next Steps Division by Chunking for HTU TU 1. Set out the numerator (larger number) inside a bus stop with the denominator to the left. 2. Calculate how many lots of 10 you can get from the number and record in brackets to the right. This might be lots of 10 (examples 1 and 2) or multiples of 10 (example 3). 3. Take this away from the original number to leave the remainder. 4. If you can take some more lots of 10 away from the remainder then repeat. Otherwise take away lots of fives or some singles and record in brackets to the right. 5. The answer is the numbers in brackets added together, with any remainder being displayed at the bottom of the calculation (example 1). 6. Display the final answer below or to the side of the calculation. Example (10x) (5 x) =15r (10x) (1 x) =11 Barrs Court Calculation Policy 15 July 2012
16 Next Steps Long Division The method for long division will only be taught to those children deemed necessary, as the Chunking method for Long division is currently a sufficient technique for use in calculation at Year 6. Barrs Court Calculation Policy 16 July 2012
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