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Progression in Multiplication Previous Learning Multiplication Tables Children will be familiar with counting in 2s, 3s, 5s, 10s, using counters, numbers lines or a counting stick. They will be aware that they need to learn these patterns and may have already begun to gain instant recall of some of the multiplication tables. (e.g. Two times table, Five times table and Ten times table). If not, we do lots of work at Velmead on this! Progression in Understanding Multiplication Children will have been introduced to multiplication as repeated addition. Drawing a picture can help visualise the problem. e.g. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 e.g. Can you tell me what 4 x 5 = (five seconds to answer) What about 8 x 2 = (five seconds to answer) e.g. Each child has two eyes. How many eyes do five children have? 2 + + 2 + 2 + 2 Children will also understand multiplication as scaling. (a number of times as wide, tall …) Children will be able to draw an array (showing the sum in rows and columns). This develops understanding that 4 x 3 is the same as 3 x 4 They may also have used the number line to carry out multiplication calculations. This is explained in step one of our progression in multiplication. e.g. There are four cakes in a pack. How many in five packs? e.g. The blue ribbon is 5cm long. Find the ribbon that is four times as long. e.g. One sweet costs 4p. How much do three sweets cost? e.g. 5 lots of 4 0 2 Dots or tally marks can also be used. Facts about Multiplication They will know that multiplication can be carried out in any order. (commutative) They will understand that doubling a number is the same as multiplying it by 2, and that halving a number reverses doubling 4 8 12 16 20 e.g. 3 x 4 is the same as 4 x 3 e.g. Double 11 is 22 means 2 x 11. Half of 22 is 11 1 Year Three STEP ONE Children will need to keep using arrays, number lines and pictures to help them. They must learn their multiplication tables in order to have instant recall of them in any order. Multiplication Tables e.g. What is 5 x 6? During Year Three children should learn their 2,3,4,5,6 and 10 What is 3 x 6? times tables facts. Instant recall means knowing the table What is 6 x 4? facts out of order and being able to answer a question within five seconds! It also helps to link these facts to division too What is 24 ÷ 6? and to insert missing numbers too. 4 x ? = 24 Year Three STEP TWO: Number Lines and Arrays Using the Number Line to Show Multiplication as Repeated Addition By doing this, the children learn the patterns of the times tables and can visualise the equal steps. They also use apparatus, such as number beads and number lines so that it helps them learn kinaesthetically. Arrays Arrays are a helpful model/image for developing the idea of commutativity (see below) and an understanding of multiplication and division. They will be taught that multiplication is the inverse of division. They will be reminded that multiplication can be done in any order. Commutativity means that 4 x 3 = 3 x 4 e.g. 6 x 3 = 18 0 e.g. 3 x 4 = 12 12 ÷ 4 = 3 3 6 9 12 15 4 x 3 = 12 12 ÷ 3 = 4 This array shows 4 x 3 = 12 and 3 x 4 = 12 2 18 Year Three STEP THREE: Multiplying Two Digit Numbers by One Digit Numbers Partitioning in Preparation for Multiplying Two Digit numbers by a One Digit number Children will be confident in partitioning numbers and will be introduced to the associative property of multiplication. e.g. Double 6 is the same as double 5 add double 1 12 x 3 is the same as (10 x 3) added to (2 x 3) e.g. 32 x 3 = 96 is the same as 30 x 3 added to 2 x 3 Larger arrays allow demonstration of how a number can be partitioned into tens and ones. This enables children to visualise the image as an aid to mental calculation and is a helpful introduction to the grid method of multiplication. Real Life Problems Children will be given word problems to apply their methods x 30 3 90 90 + 6 = 96 2 6 e.g. Alex has four stickers. Jo has 3 times as many stickers as Alex. How many stickers does Jo have? e.g. A box of biscuits contains twenty four biscuits. How many biscuits in four boxes? Year Four Multiplication Tables Multiplication Tables Children will need to constantly reinforce and revise their instant recall of the 2,3,4,5,6 and 10 times tables to keep their knowledge sharp. They should also be applying their knowledge to deduce other facts. By the end of Year 4 they should also know 7,8,9 times tables too. Constant Practice and Derived Facts e.g. 7 x 8 = ? so I can work out 70 x 8 = ? 56 ÷ 7 = ? so I can work out 560 ÷ 7 = ? 6 x ? = 42 so I can work out 6 x ? = 4200 If 8x 4 = 32, I know that 8 x 8 is double 32 If 8 x 10 is 80, I know that 8 x 9 is 8 less than 80 Year Four STEP FOUR: Multiplication of Two Digit numbers by One Digit numbers Using Arrays Children will continue to use arrays to visualise how numbers can be partitioned in order to carry out multiplication sums. e.g. 18 x 4 can be shown on an array and partitioned into 10 x 4 8x4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . Year Four STEP FIVE: The Grid Method of Multiplication This method builds on a firm understanding of partitioning and arrays. It requires that children are confident with their tables facts to 10 x 10, and also that they can add numbers accurately. NB: This is why it is important to teach children to estimate their answer first, and then carefully check their calculations at the end. Also children must be given numbers that will use the tables facts that they are confident with to start with. Children also need to be confident with recognising factors of a number in order to increase their ability to multiply different numbers. e.g. 20 x 6 is the same as 10 x 2 x 6 or 5 x 2 x 2 x 6 or 5 x 4 x6 e.g. 23 x 7 Estimation: I know that 20 x 7 is 140, and 30 x 7 is 210 so my answer will be between 140 and 210, and closer to 140. e.g. 23 x 65 = Estimation: 20 x 60 is 1200 In teaching the grid method, it is essential to ensure that children have a sound understanding of place value and of factors. X 60 5 20 6 x 2 x 10 x 10 1200 5 x 2 x 10 100 3 180 15 1200 + 180 + 100 + 15 = 1495 (or use the column method) Children may also be partitioning with some calculations. It depends how confident they are with mental calculation. e.g. 47 x 6 = (40 x 6) + (7 x 6) = (240) + (42) = 282 4 Year Four STEP SIX: Working With Decimals Decimals in a Context Children will be working with decimals in the context of money. As using the grid method with decimals requires a huge amount of background work, children will be taught to start with using repeated addition to add decimals. e.g. £2.75 x 3 £2.75 + £2.75 + £2.75 (Use the column method) 2.75 +2.75 +2.75 0. 1 5 2.10 6.00 £8 . 2 5 During Year Four as well, children will be extending their knowledge of place value to include decimals, and it is important that they have a very firm understanding of how multiplying by 10, 100, 1000 affects the position of the number on a place value chart, by moving it places to the left. (Using a place value chart) Make 4 ten times bigger = 40 Make 4 ten times smaller makes 0.4 Making 90 one hundred times smaller makes 0.9 NB: IT IS IMPORTANT that children see this as a move, rather than adding or taking off a 0, as this brings problems when working with decimals! Real Life Problems e.g. A biscuit box holds 74 biscuits. How many biscuits in 6 boxes? Children will apply these methods into real life problems e.g. I buy 5 toys costing £1.40 each. How much change do I get from £10.00? Year Five Multiplication Tables Multiplication Tables Children should now have instant recall of all multiplication facts to 10 x 10 and should learn the 11 and 12 times tables. They should also be confident with their division facts too, and need plenty of practise at deriving facts. e.g. 12 x 11 =? If 7 x 7 is 49, what is 7 x 700 ? 9 x ? = 108 , and 9 x ? = 10800 5 Year Five STEP SEVEN: The Grid Method TO x TO and HTO x TO Children will continue to reinforce their knowledge and understanding of the grid method and they will use larger numbers in their calculations. They will be taught techniques to check their multiplication of multiples of 10. NB: They must be taught to estimate their answers each time and must check their calculations thoroughly e.g. 127 x 54 Estimate: 130 x 50 is half of 130 x 100 = 6500 x 50 4 They will also need to be proficient in column addition. 100 I know that 100 x 5 = 500, so 100 x 50 is 5000 400 20 2 x 5 is 10, so 20 x 5 is 100 and 20 x 50 is 1000 4 x 2 is 8 4 x 20 is 80 7 7x5 is 35, and 7 x 50 is 350 28 Now add the columns using column addition 5000 1000 350 400 80 28 6858 1 Children will progress to using the grid method without the explanations e.g. 127 x 54 X 50 4 100 5000 400 20 1000 80 7 350 28 They may also add the columns across as they work 5000 + 1000 + 350 = 6 3 5 0 400 + 80 + 28 = 508 = 6858 6 Year Five STEP EIGHT: Working With Decimals Children will need a great deal of reinforcement in deriving facts about decimals. This must be linked to place value work on the effects of multiplying by 10, 100, 1000 and dividing by 10, 100 and 1000. Some children may need to see this visually on a place value chart. e.g. 7 x 6 = 42 ….. so what? 0.7 x 6 = 4.2 (7 ÷ 10) x 6 7 x 0.6 = 4.2 7 x (6 ÷ 10) 0.7 x 0.6 = 0.42 (7 ÷ 10) x (6 ÷ 10) Year Five STEP NINE: The Grid Method with Decimals Once children have a better understanding of decimals and place value they can then be taught to use the grid method with decimals. e.g. 54 x 5.6 X 5 0.6 50 250 5 x 10 x 6 ÷ 10 4 20 4 x 6 ÷ 10 50 250 30 4 20 2.4 250.0 20.0 30.0 2.4 302.4 NB Children must be taught to use zeros as place holders and to line up the decimal points carefully when carrying out column addition. 1 e.g Children will progress onto using the grid method without needing the explanation in the boxes X 5 0.6 Children may add across the columns 270.0 32.4 302.4 1 7 Year Five STEP TEN: Vertical Multiplication Partitioning Children will be reminded of partitioning with some multiplication sums. e.g. 97 x 8 = (90 x 8) + (7 x 8) = (720) + (56) = 776 Vertical Multiplication They will be taught to set out multiplication sums in a formal way. e.g. 97 x 8 X For some calculations, children may find it easier to continue using the grid method. Real Life Problems Children will apply these methods into real life problems 97 8 56 7 20 7 76 7x8 90 x 8 e.g. There are 32 children in each class at Velmead. How many children in the school? e.g. Petrol costs £1.23 a litre. How much would it cost to fill a 3 litre can? e.g. I buy 8 tins of soup that costs 96p a tin. How much change would I get from £20? 8 Year Six Multiplication Tables Multiplication Tables and Derived Facts Children will need to know their multiplication tables to 12 x 12. (Even though they only really need tables to 10 x 10 in our metric system, we have decided that it is still good practice for children to know their 11 and 12 times table too.) They will also know division facts to 12 x 12 They will use these facts to derive facts about decimals and multiples of 10, 100 and 1000 e.g. 40 x 6 = 240 240 ÷ 6 = 40 I know that 4 x 8 = 32 0.4 x 8 = 3.2 0.4 x 0.8 = 0.32 Year Six STEP ELEVEN: The Grid Method Children will be confidently using the grid method with whole numbers and will revise the method. NB: It is important that children are taught to estimate and then check their answers carefully. It may help to explain the individual sums to the children to ensure that their knowledge of place value is secure. They will then carry out the sums confidently and use the column addition method to find the answer. e.g. 372 x 24 = 372 x 24 is approximately 400 x 20 = 8000 (4 x 2 is 8, 400 x 2 is 800, 400 x 20 is 8000) Teaching Grid e.g. x 300 20 2 x 10 x 3 x 100 4 4x3x 100 70 2 x 10 x 7 x 10 4x7x 10 2 2 x 10 x 2 2x4 e.g. x 20 4 70 1400 280 2 40 8 300 6000 1200 6000+1400+40=7440 1200+ 280 + 8 =1488 Answer =8928 1 9 Year Six STEP TWELVE : The Grid Method with Decimals to Two Decimal Places Children will need to have carried out a great deal of reinforcement work ensuring that their knowledge of multiplying decimals is sound. They will then be taught to use the grid method to multiply decimals to 2 decimal places This example is one of the hardest examples that they may face, but the principle of the method remains the same. Obviously, most calculations of this nature would be carried out on the calculator, but it is useful for children to know how they could attempt it, without a calculator too. e.g. 5 x 5 = 25, so what … I know that … 0.5 x 5 = 2.5 0.05 x 5 = 0.25 500 x 50 = 5 x 100 x 5 x 10 = 25000 e.g. 34.37 x 29.6 Estimate 30 x 30 = 900 (with teaching explanation) X 30 4 0.3 20 2 x 10 2 x 10 2 x 10 x3x x4 x3÷ 10 10 9 9x3 4x9 9x3÷ x 10 10 0.6 6 ÷ 10 6 ÷ 10 6 ÷ 10 x3x x4 x3÷ 10 10 e.g. (without explanation) X 30 4 0.3 0.07 20 600 80 6 1.4 9 270 36 2.7 0.63 0.6 18 2.4 0.18 0.042 They will use column addition to work out the answer 600 + 80 + 0.3 + 0.07 = 6 8 7. 4 0 270 + 36 + 2.7 + 0.63 = 3 0 9. 3 3 18 + 2.4 + 0.18 + 0.042 = 2 0 .6 2 2 Answer 1 0 1 7. 3 5 2 1 1 1 YEAR SIX Step THIRTEEN :Vertical Method with Two Digit Whole Numbers Children will have been introduced to the vertical method for TU x U and will need reminding of it. 0.07 2 x 10 x 7 ÷ 10 ÷ 10 9x7÷ 10 ÷ 10 6 ÷ 10 x7÷ 10÷ 10 e.g. 97 x 8 Estimate 100 x 8 = 800 97 X 8 56 720 776 7x8 90 x 8 10 They will then be taught to use it for TO and TO e.g. 87 x 26 Estimate 90 x 30 = 2700 87 X 26 42 480 140 1600 2262 6x7 6 x 80 20 x 7 20 x 80 11 Year Six STEP FOURTEEN : Short Multiplication Children will now be taught to carry starting with HTO by O 436 x 8 Estimate 400 x 8 = 3200 436 X 8 3488 3 2 4 Real Life Problems Children will apply these methods into real life problems e.g. Find the cost of 121 bottle of lemonade at 21p each. What change would you get from £50? e.g. There is space in the multi-storey car park for 17 rows of 30 cars on each of 4 floors. How many cars can park? 11 EXTENSION YEAR 6 PLUS: Short Multiplication with Decimals to Two Places Once children have a solid knowledge of place value and of short and long multiplication, they will be introduced to using long and short multiplication with decimals. e.g. 4.92 x 3 Estimate 5 x 3 = 15 4.92 x 3 4.00 x 3 = 1 2 . 0 0 0.90 x 3 = 2 . 7 0 0.02 x 3 = 0 . 0 6 14.76 Children need to use zeros as place holders and must be taught to estimate before they calculate and then check their answers carefully. Long Multiplication with Decimals to Two Places Children will also extend their long multiplication methods to include decimals too. e.g. 34.9 x 23 Estimate 35 x 20 = 700 34.9 23.0 104.7 34.9 x 3 6 9 8 . 0 34.9 x 20 802.7 X 1 1 Children will carry out investigative work which will use some of these methods. e.g. This four digit number is a square number 9__9 Find the missing digits. e.g. A plank of wood weighed 1.4kg. 25cm was cut off its length. The plank then weighed 0.8kg. What was the length of the original plank? 12