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Multiplication and division: mental methods. • Use the relationship between multiplication and division. This includes having instant recall of the multiplication facts to 10 x 10 and knowing these as division facts. All of this is underpinned by a secure knowledge of place value. Understanding what each digit is worth and that we are making a number so many times bigger – this is why multiplication facts are known as ‘times tables’. However, ‘timesing’ is not a verb or a maths word; the maths we are doing is ‘multiplying’! Applying these facts to new contexts e.g. ‘I know that 7 x 5 = 35 so 0.7 x 5 will be 3.5’ or ‘I know that 4 x 7 = 28 so 40 x 7 will be 280’ ‘If 42 ÷ 6 = 7 then 4200 ÷ 6 = 700’ Multiplication and division: mental methods. Understand the term ‘square number’ and know the square numbers to 100. 1² = 1 x 1 = 1 2² = 2 x 2 = 4 3² = 3 x 3 = 9 4² = 4 x 4 = 16 5² = 5 x 5 = 25 etc. Multiplication and division: mental methods. Mental methods for multiplying larger numbers. e.g. multiplying by 50 is the same as multiplying by 100 and halving. Try it both ways: 18 x 50 = (18 x 100) ÷ 2 = 1800 ÷ 2 = 900 18 x 50 = (18 ÷ 2) x 100 = 9 x 100 = 900 By Year 6, children should be developing sufficient number confidence and skills to be manipulating numbers mentally, choosing the most appropriate strategy for the calculation. Multiplication and division: mental methods. How about multiplying by 25? That would be the same as multiplying by 100 and dividing by 4 (which is also the same as halving and halving again) Try it: 14 x 25 (well that seems pretty hard…) 14 x 100 = 1400 1400 ÷ 2 = 700 700 ÷ 2 = 350 ( dividing by 2 again – so that we have divided by 4) (Well, that wasn’t so hard. Was it?) Multiplication and division: mental methods. For our capable mathematicians we can show them more unusual methods that can be helpful: If your multiplication involves a multiple of 5 and an even number, e.g. 35 x 18, then you can manipulate this mentally into a manageable format too. By doubling one number and halving the other, the resulting answer will be correct. Double 35 = 70. Half of 18 = 9. 70 x 9 = 630. So 35 x 18 = 630. (Check it on a calculator if you don’t believe it!) Multiplication and division: mental methods. Mentally multiplying a 2 digit number by a single digit number e.g. 14 x 9. There are a number of possible strategies here, all of which use known multiplication facts in combination. A favourite method is to partition and recombine e.g. Partition 14 = 10 + 4 Multiply each part by 9 10 x 9 = 90 4 x 9 = 36 Recombine the parts 90 + 36 = 126 Multiplication and division: mental methods Multiply or divide a simple decimal number by an integer e.g. 4.2 x 8 or 3.6 ÷ 6. These calculations can be solved using multiplication and division facts. If the decimal point is confusing us, we can solve this by multiplying the number by 10 to begin with. We must remember to divide by 10 again at the end in order to ‘undo’ our process. e.g. 3.6 ÷ 6 3.6 x 10 = 36. 36 ÷ 6 = 6 6 ÷ 10 = 0.6 so, 3.6 ÷ 6 = 0.6 Multiplication and division: written methods. Once calculations become too cumbersome, we must learn to use a reliable written method. For multiplication there are two: The Grid method and the Expanded method. Multiplication and division: written methods. Grid method multiplication. The numbers in the calculation are partitioned and placed on the borders of a grid. Each component part is multiplied by each of the others from the other edge of the grid. The totals are added together. 37 x 46 Partition these: 30 and 7, 40 and 6. Place them on the grid. Write the totals for each multiplication in the spaces. x 30 7 40 (30 x 40) (40 x 7) 6 (30 x 6) (6 x 7) 180 42 1200 280 Finally, add the totals: 1200 + 280 + 180 + 42 --------1702 Multiplication and division: written methods. Common errors with the Grid method: Some children find the construction of a grid fiddly or they do not make the grid large enough to clearly write in the numbers. Errors in transcription can occur when transferring the numbers from the grid ready for addition. Multiplication and division: written methods. The Expanded method, still partitions, multiplies and adds the numbers but the layout is different. 37 x 46 ------1200 (30 x 40) 280 (40 x 7) 180 (30 x 6) 42 (6 x 7) ----------1702 Common errors with the Expanded method: Children forget to carry out one of the multiplication sums. The answers can be fiddly to line up causing errors in addition. Multiplication and division: written methods. Short division involves using known division facts and works from left to right, carrying forward any remainder. Children are expected to interpret the remainder in 3 possible ways: as a ‘remainder’, as a fraction and, on occasions, as a decimal. 4 21r3 8 7 The thought process for this calculation would be something like this: How many 4s in 8 tens? That’s 2 tens so I’ll write that in the tens column on top of the ‘bus shelter’. It goes exactly so there’s no remainder to carry forward. How many 4s in 7? Well, that’s 1 with 3 left over so I’ll write that in the units column on top of the ‘bus shelter’. If there’s 3 out of 4 left over, I could call that 21 and three-quarters or 21.75. Multiplication and division: written methods. ‘Chunking’ (or long division) is used for longer division sums – Warning: it can use up a lot of paper if you work with chunks which are too small! We are going to subtract ‘chunks’ from our number. These chunks are multiples of the divisor. 452 ÷ 13 452 - 130 (13 x 10) --------322 - 130 (13 x 10) --------192 - 130 (13 x 10) --------62 52 (13 x 4) --------10 In total we have subtracted 34 lots of 13 from 452 with 10 left over. Therefore, 452 ÷ 13 = 34 remainder 10. This could also be recorded as 34 and ten thirteenths.