Calculation Policy Overview of Calculation Methods and Strategies used at Hove Junior School

Calculation Policy Overview of Calculation Methods and Strategies used at Hove Junior School

Overview of Calculation Methods and Strategies used at Hove Junior School: Guide for Parents/Carers 2014/15 This policy contains the key methods of calculation that are taught at Hove Junior School. It has been written to ensure consistency and progression throughout the school. The methods we are advocating are in line with the new National Curriculum. We hope this will be helpful to you and that you will be able to support your child if they need help. The methods that we use in school may or may not be familiar to you. Children can often become confused when they ask parents for help at home and they try to teach the methods that they themselves were taught. Knowing how the methods in this booklet work will help you to help your children. All staff in school work from this document so that we can ensure the consistency of our approach and can make sure that the children move onto the next step when they are ready. The National Curriculum for mathematics aims to ensure that young people become fluent in the fundamentals of mathematics, reason mathematically and are able to problem-solve by applying mathematical thinking. By the time they leave junior school they should: have a secure knowledge of number facts and a good understanding of the four operations. estimate / mental methods. make effective use of diagrams and informal notes to help record steps/part answers when using mental methods. have an efficient, reliable, formal, written method of calculation for each operation that they can apply with confidence when undertaking calculations that they cannot carry out mentally. How calculation is taught is key. At Hove Junior School our children are taught a variety of methods, both mental and written, so that they develop the skills required to select an efficient method to carry out a calculation with confidence. The four operations that are covered by this booklet are addition, subtraction, multiplication and division. Whichever operation is being taught the child needs to experience all of the following steps to completely master it. When they first start school, the first step children will begin by using objects/manipulatives before any recording occurs. They are then introduced to resources such as number lines. Number lines provide a mental strategy for addition and subtraction and enable the child to clearly see the calculation that they are working on. 2

As they progress, they will begin using more formal written methods (in an expanded form) before using a compact written method. Concrete apparatus are available for the child to access at any point throughout a lesson. Children routinely use whiteboards and have resources (such as number lines) at hand, should they wish to use them. We hope this booklet demonstrates how we move towards using formal, compact and written methods. All terminology is outlined in our Glossary section and prefaces the step-by-step pictorial and written description of methods and strategies. In order to see the progression of methodology, we have also included descriptions/images of strategies your child may have covered in infants. (photo: using grid method to multiply) 3

Glossary of Key Vocabulary We understand that terms and vocabulary may have changed since you were at school yourselves. Here is a brief definition of some of the words that are used in calculation that your child will be taught and use at school. It is not an exhaustive list of mathematical words. Links can be found at the end of this document, to websites and videos that you may find informative and useful. Addition Addition is finding the total, or sum, by combining two or more numbers. Example: 5 + 11 + 3 = 19 is an addition. Algebra The part of mathematics that deals with generalised arithmetic. Letters are used to denote unknown numbers and to state general properties. (The explicit mention of algebra in the primary NC happens in the Year 6 programme of study). Arrays Arrays are objects or shapes arranged in a rectangle. In the array, the answer is always the same (e.g. 2 7 is the same as 7 2) even when you rotate. Bridge to ten A strategy when using numberlines. Adding a number that takes you to the next tens number. Calculation Is the process of adding, subtracting, multiplying or dividing to get an answer e.g. 45-13=32. Chunking Using facts you already know, you can reduce the number of steps used to solve a calculation e.g. instead of counting on in groups of 5 to reach 60, if you know 10x5 you can do one big jump to 50. Then count on 2 more groups of 5 to reach 60. There have been 12 jumps in groups of 5 to reach 60. Column chunking Method of division involving taking chunks or groups or the divisor away from the larger number. Compact Method The shortest route to solve the calculation. Concrete apparatus If your child s teacher refers to concrete apparatus they mean any objects to help children count these are most often counters and cubes (multilink) but can be anything they can hold and move. Dienes (thousands, hundreds, tens and units blocks), Numicon and Cuisenaire rods are also examples of concrete apparatus. Decimal numbers A number that uses a decimal point followed by digits as a way of showing values less than one. A decimal point separates the whole number from the part of the number which is less than one. Decomposition Expanding the numbers and separating them into their components. Used in column addition when carrying and column subtraction; expanding to exchange. 4

Digit A symbol used to make numerals. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numbers. Division Division is splitting into equal parts or groups. It is the result of "fair sharing". e.g. There are 12 chocolates and 4 friends want to share them. If you share the out, each will get 3 chocolates. Division can also be thought of as how many groups of a number are in the total number. e.g. Eggs can be put in boxes of 6. There are 54 eggs altogether. How many boxes will be needed? Divisor The number by which another is divided. Example: In the calculation 30 6 = 5, the divisor is 6. In this example, 30 is the dividend and 5 is the quotient. Difference (find the difference) The result of subtracting one number from another. How much one number differs from another. Example: The difference between 8 and 3 is 5. Estimate To arrive at a rough or approximate answer by calculating with suitable approximations for terms or, in measurement, by using previous experience. A rough or approximate answer. Exchanging Moving a ten or a hundred from its column into the next column and splitting it up into ten ones (or units ) or ten tens and putting it into a different column. Expanded Method Showing each stage and then combining to solve the calculation. E.g. Expanded Multiplication a method for multiplication where each stage is written down and then added up at the end in a column. Formal written methods Setting out working in columnar form. In multiplication, the formal methods are called short or long multiplication depending on the size of the numbers involved. Similarly, in division the formal processes are called short or long division. Geometry The aspect of mathematics concerned with the properties of space and figures or shapes in space. Grid Method The grid method is a way of breaking up numbers into separate units to make multiplication easier. We use an empty number square. To calculate 35 7, the grid looks like this: X 30 5 7 X 30 5 7 210 35 210 + 35 = 245 5

Groups If children are counting in groups, they are counting up or down in the particular number, instead of counting in 1s e.g. counting up in groups of 4 to see how many 4s are in 20. Integer Any of the positive or negative whole numbers and zero. Inverse The opposite operation. Addition is the inverse of subtraction. Multiplication is the inverse of division. Multiple The result of multiplying a number by an integer (not a fraction). Examples: 12 is a multiple of 3, because 4 3 = 12 30 is a multiple of 5, because 6 5 = 30 But 17 is NOT a multiple of 3 Multiplication The basic idea of multiplication is repeated addition. For example: 5 3 = 5 + 5 + 5 = 15 But as well as multiplying by whole numbers, you can also multiply by fractions or decimals. For example 5 3½ = 5 + 5 + 5 + (half of 5) = 17.5 Numberline A line with numbers placed in their correct position. Useful for addition and subtraction, and for showing relations between numbers. Numbers on the left are smaller than numbers on the right. Number sentence Writing out a calculation with just the numbers in a line e.g. 2+4=6 or 35 7 = 5 or 12 x 3 =36 or 32 5 = 27. Partition Separating a number into its place value parts e.g. 157= 100 and 50 and 7. Place Value The value of where the digit is in the number, such as units, tens, hundreds, etc. Example: In 352, the place value of the 5 is tens Example: In 17.59, the place value of the place value of 1 is tens, the 9 is hundredths Product The answer when two or more numbers are multiplied together. Regroup For addition, once the numbers have been partitioned into hundreds, tens and units then add hundreds together, then add the tens to that total, then add the units to that total. 6

Remainder The amount left over after division. Example: 19 cannot be divided exactly by 5. The closest you can get without going over is 3 x 5 = 15, which is 4 less than 19. So the answer of 19 5 is 3 with a remainder of 4. Rounding In the context of a number, express to a required degree of accuracy. Example: 543 rounded to the nearest 10 is 540. Scaling To enlarge or reduce a number, quantity or measurement by a given amount. Statistics The collection, analysis, interpretation, presentation, and organisation of data (previously Data Handling ) Subtraction Taking one number away from another. If you have 5 apples and you subtract 2, you will be left with 3. The symbol of subtraction is Sum The result of adding two or more numbers. Example: 9 is the sum of 2, 4 and 3 (because 2 + 4 + 3 = 9). Times Table facts e.g. 3x4=12, 10x7=70 Children need to be able to recall multiplication and division facts up to 12 x 12. 7

ADDITION Working towards a formal written method By using visual images and objects, your child begins to make sense of combining two groups to make a larger group, to add amounts. They develop mental reasoning and learn mathematical vocabulary as they process what they are doing. Using resources is the best way for a child to understand the concept of number. Manipulatives/objects can be used to support mental imagery and conceptual understanding. Following on from this, number lines are used as an important image to support thinking, and the use of informal jottings is encouraged. Examples using resources can be seen below. Using Numicon to see that addition can be done in any order. Using Numicon to bond to 10. 9 + 1 = 10 1 + 9 = 10 4 + 6 = 10 6 + 4 = 10 Dienes or Base 10 are useful in showing what a one (unit), a ten, a hundred and a thousand look like and how they can be added together and split up to form smaller and larger numbers. 8

Cuisenaire Rods. Although these rods that represent numbers from 1 to 10 can be used as an addition tool, they can also be used with calculations using other operations. Using a variety of resources when dividing. Whiteboard, number line, counters, 100 Square, marbles on plates, beads and straws. Using 'real' coins to make amounts in different ways. Using bundles of straws to see the relationship Using a 100 bead string to see that between 4 + 6 = 10 and 40 + 60 = 100. 40 + 60 = 100 Further examples of resources: Moveable counting objects, coat hangers and pegs, beads, number tracks and lines, number fans, 100 square, Multilink, place value counters and place value cards (to help children move towards regrouping units for tens) arrow cards, straws/objects grouped in tens, etc. resources such as number lines and 100 squares are the next step. Example vocabulary associated with addition. Add, addition, more, plus, make, sum, total, altogether, score, regrouping How many more to make? How many more is than? 9

Moving on to a written method. Below are the key methods your child will have used/be using in school. The updating of the National Curriculum (2014) has meant that there are some changes to the objectives for each year group. The methods we teach are outlined below. Your child may now be introduced to a method of calculating earlier than they would have been previously. Below is an example of using a numberline. Your child may be familiar with this method as it is used in Key Stage 1. There are a variety of types of number lines. They are very useful and are used in all year groups. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 + 5 = 9 Using a number line. Estimate 55 + 37= 60 + 40 = 100 +30 +7 55 85 92 Using a blank number line. To begin with your child can count along on a number line to add two numbers together. Make sure your child starts their number line with the largest number and then add the small number. Partition the number into tens and units to help with adding. 10

Partitioning The next step is to partition to add two numbers together. Make sure your child partitions the numbers into their correct place value, e.g. 47 = 40 + 7 246 = 200 + 40+ 6 47+56 T U 246+175 H T U 4 0 + 7 2 0 0 4 0 6 + 5 0 + 6 + 1 0 0 7 0 5 9 0 + 1 3 = 1 0 3 3 0 0 + 1 1 0 + 1 1 = 4 2 1 Expanded Addition The next step is the expanded method for addition. You child completes complete these calculations in the same way as the previous method. They start with the units and then add the tens. Remember to start with the units Remember to line up the columns T U 5 6 + 4 7 1 3 (6 + 7 = 13) 9 0 (50 + 40 = 90) H T U 2 4 6 + 1 7 5 1 1 (6+5=11) 1 1 0 (40+70=110) 3 0 0 (200+100=300) 4 2 1 (300+110+11=421) Now we have the standard compact method for addition. Make sure when you work through your calculation that you carry the extra 10 or 100 underneath, in the appropriate column. 56 + 47 5 6 + 4 7 1 0 3 1 246+175 2 4 6 + 1 7 5 4 2 1 1 1 When adding decimal numbers make sure your child thinks about the place value of each digit and lines up the appropriate columns. Remember to start with the numbers on the right (tenths or hundredths) 3. 5 4 + 2. 1 7 5. 7 1 1 Remember to line up the decimal points! 5 4. 7 0 You can put + 2. 8 9 a zero here 5 7. 5 9 to avoid 1 confusion. 11

SUBTRACTION 1. To begin with, a number line is used to count on or back to solve subtraction calculations. The smallest number is on the left and the largest is on the right. Add the jumps together to make the final answer, like getting change in a shop. This is called finding the difference. Count on if the numbers are close together. 60-47 = 13 Estimate 60-50 = 10 +3 +10 47 50 60 Count back if the numbers are far apart. 60-13 = 47 47 50 60-3 -10 Count on if the numbers are close together. 203-186 = 17 Estimate 200-180 = 20 +4 +10 +3 186 190 200 203 10+4+3=17 Count back if the numbers are far apart. 203-17 = 186 186 190 200 203-4 -10-3 12

2. The next step is to partition the numbers and write them vertically. 679-135= Estimate 700-100 = 600 H T U 600 70 9-100 30 5 500 + 40 + 4 = 544 First we teach children to begin with numbers where no exchanges are needed. 673-135= 600 60 70 1 3-100 30 5 500 + 30 + 8 = 538 Then once they are confident they move onto exchanging one ten for ten units. 673-185= 500 600 160 70 1 3 Then the next step is to exchange from several columns. - 100 80 5 400 + 80 + 8 = 488 13

3. This is the standard method for subtraction. As with the previous method, your child begins with numbers where no exchanges are needed. Estimate 700-100 = 600 Step 1 6 7 9 Step 2 6 6 7 1 3 Step 3-1 3 5-1 3 5 5 4 4 5 3 8 5 6 16 7 1 3-1 8 5 4 8 8 4. When subtracting decimal numbers make sure your child thinks about the place value of each digit and line them up in the appropriate columns. Estimating is extremely important. 3.54-2.17 54.7-2.89 Estimate 4-2 = 2 3. 4 5 1 4-2. 1 7 1. 3 7 Estimate 55-3 = 52 5 3 4. 16 7 1 0-2. 8 9 5 1. 8 1 A zero is helpful here to avoid confusion. 14

15 9 columns of 5 dots 5 rows of 9 dots MULTIPLICATION Multiplication is repeated addition. 1. 9 x 5 = 45 First use arrays to consolidate understanding of the concept of multiplication. 5 x 9 = 45

2. Then a number line is used to solve multiplication calculations by counting on until the correct amount of multiples, e.g. 14 groups of 5 is reached. The answer is on the number line. 14 x 5 = 70 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 10 x 5=50 4 x 5=20 0 50 70 Using table facts to make bigger jumps is more efficient. 16

The next step is using the grid method. 3. You need to partition both numbers and use them to multiply each other. Then add the totals in each box. 23 x 5 = 115 23 x 52 = 1196 Estimate 20 x 5 = 100 x 2 0 3 5 1 0 0 1 5 Estimate 20 x 50 = 1000 x 2 0 3 5 0 1 0 0 0 1 5 0 2 4 0 6 5 x 20 = 100 5 x 3 = 15 100 + 15 = 115 23 x 5 = 115 50 x 20 = 1000 50 x 3 = 150 2 x 20 = 40 2 x 3 = 6 1000 + 150 +40 + 6 = 1196 17

The next step is using grid multiplication to introduce long multiplication: 18 x 13 = Estimation: 20 x 10 = 200 We introduce long multiplication for multiplying by 2 digits: Multiplication made easy [ish!] https://www.youtube.com/watch?v=t_bnlb2krl4 18

4. The next step is the compact method for short multiplication. Your child needs to multiply the single digit number by the units and multiply the single digit number by the tens (8 x 3 = 24, 20 x 8 = 160). H T U 2 3 x 8 1 8 4 The tens are carried over and put into the tens column. 2 H T U 4 6 x 3 2 9 2 1 1 3 8 0 1 1 4 7 2 1 Here a two-digit number is multiplied by another two-digit number. The method is the same as the short method but calculated in steps. Start with 2 x 6 = 12 and carry the ten Then it is 40 x 2 = 80 (+ 10) A zero is placed in the units column as your child is now multiplying the numbers by a multiple of ten, not a unit. Finally add the two answers together. 30 x 6 = 180 (carry the hundred) 30 x 40 = 1200 (plus the hundred) 19

Multiplying with decimals Estimate 700-100 = 600 3 5 x 6 7 First remove the decimal point (3 5 35). Put numbers into the grid and use grid method to solve, as before. x 3 0 5 6 0 1 8 0 0 3 0 0 7 2 1 0 3 5 Then add all the answers together. Now look back at the original question. In total, how many digits are to the right of the decimal point? 60 x 30 = 1 8 0 0 60 x 5 = 3 0 0 + 7 x 30 = 2 1 0 7 x 5 = 3 5 2 3 4 5 1 3 5 x 6 7 in this case there are 2 digits be the right of the decimal point. So there needs to be 2 digits to the right of the decimal in the answer. 2345 becomes 23 45 Don t forget to put the decimal point back. This is where estimating really helps. 20

DIVISION To begin with your child will use on a number line to solve division calculations. You must start at zero and count on in groups of the divisor e.g. 5, until you reach the target number, e.g. 35. Take care as there could be a remainder! The answer is the number of jumps made. Remember to estimate Division using a Number line 35 5 = 7 1 2 3 4 5 6 7 0 5 10 15 20 25 30 35 Here we are using chunking to solve division calculations using a number line starting at zero. We count up in groups to see how many lots go into the bigger number. We use our times table knowledge to help us. a) 72 6 = 12 10 x 6 = 60 2 x 6 = 1 2 0 60 72 b) 256 7 = 36 r4 r 4 10 x 7 = 70 10 x 7 = 70 10 x 7 = 70 6 x 7 = 42 0 70 140 210 252 256 c) 256 7 = 36 r4 30 x 7 = 210 6 x 7 = 42 r 4 0 210 252 256 For this calculation we have used the same method, but because we have used larger numbers, we have made larger jumps along the number line, making larger chunks in our Calculation. 21

Division using Chunking Chunking is a method used for dividing larger numbers that cannot easily be divided mentally. Chunking is repeated subtraction of the divisor and multiples of the divisor. In other words, working out how many groups of a number fit into another number. The purpose of chunking is for children to be able to think about the relationship between multiplication and division. It involves using rough estimates of how many times a number will go into another number and then adjusting until the right answer is found. Once these skills have been practised, teachers will encourage children to move onto the quicker 'bus stop' division method and continue onto long division, when appropriate. Each chunk is an easy multiple (for example, 100x, 10x, 5x, 2x, etc) of the divisor, until the large number has been reduced to zero or the remainder is less than the divisor. 73 5 = With larger divisors, it may be useful for the child to jot down the multiples of that number at the side, for reference. Also, in the example below, the child has represented the remainder as a fraction by using it as the numerator, over the divisor (denominator) before simplifying the fraction. 386 24 22

Short Division The new National Curriculum states that children must be able to divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context (as a remainder, fraction or decimal) The next step is short division. This is sometimes known as the bus stop method. 471 3 = 157 3 4 7 1 It is easier to think, how many 3s are in 4? But you must bear in mind this is actually 400. First work out the largest number of hundreds that will divide exactly by 3. Carry the remainder to the right. 1 3 4 1 7 1 Secondly work out the largest number of tens that will divide exactly by 3. Carry the remainder to the right. 1 5 3 4 1 7 2 1 Lastly, work out the largest number of units that will divide exactly by 3. 1 5 7 3 4 1 7 2 1 21 divided by 3 = 7 Take care as there might be a remainder! 23

Long Division Long Division [the tricky method!] Following on from this, when appropriate, your child learns long division. Long division breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. As stated, abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Long division is tricky as it relies upon knowledge of: multiplication tables (including related multiplication tables e.g. 3 x 12 = 36, 30 x 12 = 360, 30 x 120 = 3600, and so on) the basic division concept, based on multiplication tables (for example 28 7 or 56 8) basic division with remainders (for example 54 7 or 23 5) Long division is an algorithm that repeats the basic steps of: 1) Divide 2) Multiply 3) Subtract 4) Drop down the next digit. Of these steps, number 2 and 3 can become difficult and confusing to children because they don't seemingly have anything to do with division they have to do with finding the remainder. It is worth bearing in mind that children will not be taught this method until they are ready i.e. until they have the ability to apply what they know, relating to the three facts above. Below is an expanded example of how to do long division. There are links to YouTube videos and websites with further practical demonstrations, should you wish to view them. This explanation is rather long-winded because, in order to fully understand how long division works, one must look at (and understand) each step. Firstly, an example is given where there is a straightforward answer with no remainders. Children will not be taught this until they are ready and can apply what they know. 24

Division example: Key Information: The first digit of the dividend is divided by the divisor The whole number result is placed at the top The answer from the first operation is multiplied by the divisor. The result is placed under the number divided into. The bottom number is subtracted from the top number. Bring down the next digit of the dividend. Divide this number by the divisor. The whole number result is placed at the top. Any remainders are ignored at this point. The answer from the above operation is multiplied by the divisor. The result is placed under the number divided into. Now subtract the bottom number from the top number. Sometimes division problems will not come out evenly, and they will have another number (not 0) at the end. This leftover number is called a remainder and it is written as part of the quotient. See the example above. The red circled number at the bottom of the example is the remainder. This is recorded on top of the division bar with a r in front (25r3) like in short division. 25

Here s a new example: Our answer to this problem is 23 r 1; note that we always write the remainder after the quotient, on top of the division bar. Also notice that our remainder (1) is smaller than our divisor (6). Long Division with Remainders as Fractions The next step is writing remainders as fractions. Instead of writing r and then the number take the remainder and make it the numerator (top number) of a fraction. The denominator (bottom number) is the divisor. Let s look at the following example: Notice that our r is not used at all in front of the remainder when turning it into a fraction. The fraction is recorded instead. 26

Long Division with Remainders as Decimals Another way your child may be asked to express a remainder is in the form of a decimal. When they are asked to express the remainder as a decimal, first complete the division as usual, until reaching the remainder stage. Follow along with this example: The division calculation will keep going. Add a decimal point on the top bar. After the decimal in the divided, add a zero and continue to divide. Keep adding zeros until the subtraction step results in an answer of zero as well. Notice that we added a decimal after the 6 in the dividend, as well as a decimal after the 5 in our quotient. Then, we started adding zeroes to the dividend. This time, it only took us one added zero before our remainder was zero. Now, let s look at a problem where more than one zero is added to the dividend: USEFUL WEBSITES: http://www.mathsisfun.com/long_division.html http://www.wyzant.com/resources/lessons/math/elementary_math/long_division/ long_division_with_remainders https://www.gov.uk/government/publications/national-curriculum-in-england-mathematicsprogrammes-of-study Models and Images documents Primary National Strategy www.ncetm.org.uk National Centre for Excellence in Mathematics. 27

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