Hodge Hill Calculation Policy Workshop: Key Stage 2 Multiplication and Division Multiplication and Division Key Vocabulary Key Vocabulary Progression from Mental Methods to Written Methods for Multiplication l i li i To multiply successfully, children need to be able to: 9 recall all multiplication facts to 10 × 10 9 partition number into multiples of one hundred, ten and one 9 work out products such as 70 work out products such as 70 × 5, 70 5 70 × 50, 700 50 700 × 5 or700 5 or700 × 50 50 using the related fact 7 × 5 and their knowledge of place value 9 add two or more single‐digit numbers mentally 9 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 9 add combinations of whole numbers using the column method (see above). Note: It is important that children's mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication efficient written method for multiplication. Mental Skills Recognise the size and position of numbers Count on in different steps 2s, 5s, 10s Double numbers up to 10 p Recognise multiplication as repeated addition Q Quick recall of multiplication facts p Use known facts to derive associated division facts • Use known facts to generate other facts (e3.g. double the 2 x table to find 4 x table) • Multiplying by 10, 100, 1000 and understanding the effect • • • • • • Year 3 ¾ As the children enter Key Stage 2 they will be able to partition any2‐digit and ill b bl t titi 2 di it d 3‐digit number. They will know x2, x5 g y , and x10 times tables or be able to calculate quickly x3 x4 x6 x7 x8 and calculate quickly x3, x4, x6, x7, x8 and x9.They will be able to multiply by a multiple of 10. ¾The children will be able to multiply by a multiple of 10 using knowledge of place value. 9They will understand when you multiply by 10 the number gets 10 times greater and moves p one place left on the Place Value Grid. Deines apparatus will be used to help the children to visualise this visualise this. ¾Use place value apparatus to support the multiplication of U X TU alongside the grid l i li i f U X TU l id h id method. 9When first introducing the grid method, use 9When first introducing the grid method use resources to show the visual process, as well as recording the workings within a grid as recording the workings within a grid. ¾The children will be able to multiply TU x U using the grid method. h d h d They will begin by partitioning the two digit number into tens and units and then multiply each part units and then multiply each part separately. Then they will add all the answers together to get the the answers together to get the final product. Using the Deines to show how this works visually or by splitting an array of 13x7 into 10x7 and 3x7 array of 13x7 into 10x7 and 3x7 will help the children understand that they can do this and still have y the same answer. ¾Once confident with the layout of the grid and method to follow, the children will be introduced to larger two‐digit introduced to larger two digit numbers and numbers and then three‐digit numbers. ¾They will be confident at applying their mental times‐table knowledge to simplify numbers g g g e.g. thinking of 20x6 as 2x6 and then making their answer ten‐times bigger. They will understand the roll of the placeholder to understand the roll of the placeholder to make numbers ten‐times smaller or larger. Year 4 ¾ The children will be able to multiply any whole number by 10 or a multiple of 10, 100 or a multiple number by 10 or a multiple of 10, 100 or a multiple of 100. They will understand how to They will understand how to move the digits over 1 place to x10 and two places to x100. They will be very secure with place value and be confident to manipulate numbers in manipulate numbers in questions like 40x50 to simplify to 4x5 and then understand they made both numbers ten‐times smaller, so their answer will need to be made ten‐times need to be made ten times bigger twice (one hundred‐times bigger). ¾The children will be able to multiply TU x TU using the grid method. Year 5 ¾By the time children reach Year 5, they ¾B th ti hild h Y 5 th will be confident in using the most g efficient methods for multiplication. They now need to apply these methods They now need to apply these methods to larger numbers and decimal numbers. ¾The children will be able to multiply HTU x TU using the grid method. They will understand the place value of decimal ey u de sta d t e p ace a ue o dec a numbers and know how to move the digits over 1 place to the left to x10 two places to x100 1 place to the left to x10, two places to x100, and three places to x1000. ¾They will understand moving the digits one place to the right is ÷10 etc. They will simplify numbers and then adjust answers to solve numbers and then adjust answers to solve multiplying by decimals. ¾They will use grid method to multiply decimal number by U. Year 6 ¾ By Year 6 the children will be consolidating and refining multiplication methods and and refining multiplication methods and applying them in 2‐step problem solving. 9They will use grid method to multiply decimal 9They will use grid method to multiply decimal number by TU. More Able Year 6 ¾If they are proficient in multiplying any ¾If h fi i i li l i number using the grid method and understand place value and the role it plays, y then they will be shown a more efficient multiplication method ‐ the Column Method for multiplication for multiplication. ¾They will the Expanded Column Multiplication Method to understand the process and make the links to the grid process and make the links to the grid method. Set out in the traditional Set out in the traditional column way making sure the HTU are lined up however HTU are lined up, however breaking down each part of the process separately (as the process separately (as you would in the Grid Method) and showing the Method) and showing the multiplication for each step in brackets in brackets. ¾Once confident and proficient in the expanded column method, and being able to use it to l h d db bl multiply whole and decimal numbers, they will be shown the Compact Column Multiplication Method. Multiplication Method. Set out in the traditional column way making sure the HTU are lined up. 1. Begin with the units 7x5=35, so the 5 goes under the units and the 3 (30)is carried to the tens. 2. Next 7x30=210, but remember to add the 30 you carried onto this total. This answer is written on the same line as the first answer, because this is all part of the 35x7. 3. Now on the line below, 35x30 is going to be solved. First put a placeholder in the units, this is because we are multiplying by a multiple of 10. 4. You need to solve 30x5=150. The 5 (50) goes under the tens column and the 1 (100) is carried over to the ( ) hundreds. 5. Next solve 30x30=900. The 9 (900) goes under the 5 N t l 30 30 900 Th 9 (900) d th hundreds column, but remember to add the carried hundreds onto it. 6. Finally add the answers together, as you would in the column addition method, starting with the least significant digit first. • Progression from Mental Methods to Written Methods for Division To carry out written methods of division successful, children need to be able to: need to be able to: 9 understand division as repeated subtraction 9 estimate how many times one number divides into another estimate how many times one number divides into another ‐ for example, how many sixes there are in 47, or how many 23s there are in 92 9 multiply a two‐digit number by a single‐digit number lti l t di it b b i l di it b mentally 9 subtract numbers using the column method. subtract numbers using the column method. Note: It is important that children's mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division. Mental Skills 9Recognise the size and position of numbers 9Count back in different steps 2s 5s 10s 9Count back in different steps 2s, 5s, 10s 9Halve numbers to 20 9Recognise division as repeated subtraction 9Q i k 9Quick recall of division facts ll f di i i f t 9Use known facts to derive associated facts 9Divide by 10, 100, 1000 and understanding the effect h ff 9Divide by multiples of 10 y p Year 3 ¾As the children enter Key Stage 2 they will ¾As the children enter Key Stage 2 they will know x2, x5 and x10 times tables or be able t to calculate quickly x3, x4, x6, x7, x8 andx9. l l t i kl 3 4 6 7 8 d 9 ¾They will understand division is the inverse of y multiplication and be proficient in using number lines to solve division problems number lines to solve division problems. ¾ The children will understand how to deal with remainders. Grouping Problems 1). A car holds 5 people. How many cars are needed to transport 17 How many cars are needed to transport 17 people? 17 ÷ 5 = 3 r 2 The answer will need to be rounded up so the real life answer to the question is: life answer to the question is: 4 cars will be needed. I can fit 5 chocolates into a box. I have 17 chocolates. How many full boxes will I have? 17 ÷ 5 17 ÷ 5 = 3 r 2 3r2 The answer will need to be rounded down so the real life answer to the l lif h question is: i i I will have 3 full boxes. Sharing Problems Gran gave £17 to be shared between her 5 Gran gave £17 to be shared between her 5 Grandchildren. How much does each child receive? £17 ÷ 5 = 3 r 2 The answer can be broken down to its decimal money form. They need to recognise th the remainder 2 is £2 = 200p. 200 ÷ i d 2 i £2 200 200 5 = 40: 5 40 They will get £3.40 each I share 17 lollies equally between 5 part bags. How many lollies in each bag? y g 17 ÷ 5 = 3 r 2 The answer will need to be rounded down so the real life answer to the question is: 3 lollies in each bag. The children will use number lines in the same The children will use number lines in the same way as before, with repeated addition, but they need to recognise if they can not group they need to recognise if they can not group anymore what ever sized jump needed to get to the number being divided is the remainder. IMPORTANT NOTE: ¾ This method must be introduced in the context of a real life problem. ¾ For example: We have 67 photographs to put in an album. Each page can hold 4 photographs. How p g p g p many pages will we need. This must be physically done with the children before showing them the g recording. ¾The children will use repeated addition, however they will apply their times table g on in ‘chunks’. This knowledge and count makes the number line shorter and therefore the workings quicker to solve They must circle the workings quicker to solve. They must circle the ‘chunks’ and then add them up mentally. Year 4 and 5 ¾By the time children reach Year 4, they will be ¾By the time children reach Year 4 they will be confident in using chunking along a number line and will be ready to transfer this idea into the more formal written method of Chunking. By Year 5 they will be confident in this method and be able to solve more this method and be able to solve more challenging HTU ÷ U problems. Year 6 ¾By the time children reach Year 6; they will be confident in using the most efficient be confident in using the most efficient methods for division. They now need to apply these methods to find decimal l h h d fi d d i l answers. The children will then be consolidating and refining the method and applying it in their 2‐step applying it in their 2 step problem solving. problem solving. ¾The children will use decimal chunks to divide remainders. d Level 5 Children working at Level 5 should also be p expected to: ¾solve division problems involving measures and money and money. ¾use as an inverse operation to check multiplication calculations. ¾convert remainders to decimal remainders ¾convert remainders to decimal remainders. Year 6 More Able • Children, who are confident with the Chunking Method will be shown the Bus stop Method Method, will be shown the Bus stop Method. This method is purely used for efficiency and loses the idea of place value that has been loses the idea of place value that has been focused on previously. It relies heavily on applying times‐table knowledge. It is very p important that the children who are shown this method are completely secure in Place Value and should not be a method learnt by Value and should not be a method learnt by rote. ¾The children will use more efficient methods for division – The Bus Shelter Method. 1. Set out the ‘Bus Shelter’ with the divisor on the outside and the number being divided (the dividend) in the middle (under the (the dividend) in the middle (under the shelter). 2. Start by looking at the most significant number and work out how many times it can go into the number In this example 4 can ‘go go into the number. In this example 4 can go into’ 1 (using repeated addition and times‐ table facts tell us this). So the 1 is carried over to the 3 making 13. 4 goes into 13 three times and 1 is left over. The number 3 is recorded above the shelter and the ‘left recorded above the shelter and the left over over’ 1 is carried to the next number – giving us 14. 3. Now work out how many fours ‘go into’ 14 Again the answer is 3 this is 14. Again the answer is 3, this is recorded above the shelter. However, there are still 2 ‘left over’. 4. With the 2 ‘left over’ we still need to f divide further and a decimal point is added to the number in the shelter along with a place holding zero. The l ih l h ldi Th ‘left over’ 2 can now be carried to the place holder – giving us 20. place holder giving us 20 5. Finally we have a number that is 5 Finally we have a number that is divisible by 4 with and the final answer f p , of 5 is put above the shelter, remembering to bring the decimal point up in front of it.