Summary of the Steps for Written Calculation Division 1

Transcription

1 Summary of the Steps for Written Calculation Division 1 Steps 1-14 lead children through the necessary stages for mastering the traditional column method of short division and begin to prepare them for long division. The short division method is used for dividing by one-digit numbers only. Unlike the written methods for addition, subtraction and multiplication where the calculations are worked from right to left, for this method children work from left to right. The answers are also, unusually, written above the question instead of below it (carried digits are in red in this document). Children should always be encouraged to estimate the answer first to give them an idea of how correct their final answer is likely to be. If children are not fully secure with their multiplication tables, they should be provided with multiplication squares when completing column division as this will enable them to master the method without their fluency being hampered by lack of rapid recall. (However, it is obviously vitally important to continue to work on teaching and practising rapid recall of multiplication facts). Step 1: Two-digit one-digit no carrying; eg: 69 3 = Step 2: Three-digit one-digit no carrying; eg: = Steps 1 and 2 introduce and familiarise children with setting out questions vertically on squared paper with the correct digits in each column, working from left to right and writing answers above the question. In terms of teaching, explain to the children that the number on the outside ( the divisor) is trying to get in to the number in the bus-stop (the dividend). So, in the example above, they must think, How many 2s are in 4? The answer to this goes directly above the 4; then, How many 2s in 8? with the answer going directly above the 8; finally, How many 2s in 2? placing the answer above the 2. Ensure the children understand the place value of the numbers they are dealing with and begin by partitioning, if necessary. Step 3: Two -digit one-digit carrying 1 ten; eg: 96 4 = In Step 3 carrying is introduced where there is a remainder after dividing the tens digit by the divisor. This remainder is carried to the right and written next to the units digit (the carried digit is red in this example). Step 4: Two -digit one-digit carrying several tens; eg: 72 4 = In Step 4 the digit carried is more than 1.

2 Step 5: Three-digit one-digit carrying once; eg: = Step 6: Three-digit one-digit first digit smaller than divisor; eg: = In Step 6 the first digit of the dividend is smaller than the divisor and so can be carried to the tens or the first two digits can be taken together and divided, ie: In the first few instances, children can be taught to write a zero above the 3 as the 3 cannot be divided by 5. As children progress, they will realise and understand that writing zero above the 3 is not necessary. Step 7: Three-digit one-digit carrying tens; eg: = In Step 7 the first digit is a multiple of the divisor but the tens digit is not, so carrying must take place from the tens to the units. Step 8: Three-digit one-digit second digit smaller than the divisor; eg: = In Step 8 children will need to write a zero in the answer in the tens column as the digit is smaller than the divisor. Step 9: Four-digit one-digit carrying once, any position; eg: 9,246 2 = 4, Step 10: Three- and four-digit one-digit carrying more than once, any position; eg: 8,736 6 = 1, By now carrying should be becoming an automatic process and errors that occur are likely to be attributable to careless dividing. Step 11: Three- and four-digit one-digit answers with remainders; eg: = 157r r In Step 11 the dividends are not multiples of the divisor, so each answer will have a remainder. At this stage children should just write the remainder as a number, eg: r1. By now, children should be proficient in the short division technique and will realise that the process of division is the same for very large numbers. They can then divide numbers with an infinite number of digits, with or without remainders.

3 Step 12: Four-digit one-digit with fraction remainders; eg: 3,685 3 = 1,228 ⅓ ⅓ This step extends more able children, showing them that remainders can be written as fractions. They should be confident in understanding fractions and realise that, if the whole number remainder is divided by the divisor, then the denominator of the fraction will be the divisor while the numerator will be the remainder. This can be shown pictorially, eg: 7 cakes shared between 3 people is 2r1 but if that remaining cake is shared between the 3 people, they will each get ⅓. Step 13: Four-digit one-digit with remainders as decimals; eg: 7,324 8 = Step 14: Four-digit one-digit with remainders as decimals, 2 or 3 dp; eg: 1,146 8 = Steps continue the method of short division to give remainders as decimals. Children should be helped to appreciate that, when a decimal point is placed at the end of a whole number and zeros are written after it, the number does not change in size-it is the same number. Once children understand this, they can continue the process of short division and give decimal answers.

4 Summary of the Steps for Written Calculation Division 2 Steps lead children through the necessary stages for mastering the traditional column method of long division. It is vital that children learn the process for one-digit numbers first, although long division is used mainly for dividing by two-digit numbers. Step 15: Two-digit one-digit long division; eg: 87 3 = In Step 15, children are introduced to long division. It is vital that children realise there is a repeating cycle of Divide, Multiply, Subtract (DMS) in this process and that, between each round, an extra digit is brought down. Children should be encouraged to repeat the words several times to reinforce this. The example should be worked together, discussing each step and demonstrating the movements; eg: Divide 8 by 3, write the answer (2) above, then Multiply the answer (2) by the divisor (3), then write this product (6) under the first digit (8). Subtract this then bring down the next digit (7 in this example). Now Divide 27 by 3, write the answer (9) above, then Multiply (9 x 3) and write this product (27) underneath and Subtract it. Step 16: Three-digit one-digit long division; = 285 r r Step 16 practises DMS again. In this example: Divide the hundreds digit (8) by the divisor, 3. Answer 2 goes above the hundreds digit. Multiply the 2 by the divisor (2 x 3). The answer 6 goes below the hundreds digit. Subtract this 6 from the 8 to give 2 then bring the tens digit (5) down to give 25. Divide 25 by 3, answer 8 goes above the tens digit. Multiply the 8 by the divisor (8 x 3). The answer 24 goes below the 25. Subtract the 24 from 25 to give 1 then bring the units digit (7) down to give 17. Divide 17 by 3, answer 5 goes above the units digit. Multiply the 5 by the divisor (5 x 3). The answer 15 goes below the 17. Subtract the 15 from 17 to give 2. This cannot be divided by 3 so this is the remainder.

5 Step 17: Four-digit one-digit long division, answers with zeros; eg: 3,625 4 = 906 r r Step 17 includes questions where, when a new digit is brought down, the number created is less than the divisor and so the answer will include a zero. Children often miss recording this zero so answers are incorrect. The worked example should be demonstrated, discussing the importance of writing the zero in the answer before subtracting zero and bringing down the next digit. Step 18: Three-digit 11 long division, no zeros in answer; eg: = 62 r3 6 2 r Now that children have experience of the DMS cycle and have begun to get a sense of the movements involved in the method, they can begin to divide by two-digit numbers. In Step 18 they divide by 11 as most children are very secure in their 11 times table and so their focus can be on continuing to use the long division process. Children should be reminded that when dividing by 11, remainders can be anything up to 10. Step 19: Four-digit 11 long division, with zeros in answers; eg: 4, = 409 r r Zeros are now included and children divide four-digit numbers by 11. As for Step 17, it is vital that children write the zeros into the answers before bringing down the next digit. Children can then start to divide numbers by 12 as they should be secure in this table too.

6 Step 20: Four-digit 13, 14, 15 or 16 long division; eg: 6, = 499 r r x 13 = 130 The process of long division can continue here but it is suggested that the children write the multiples of the numbers 13, 14, 15 and 16, before using these for their long division calculations. Discuss how patterns of numbers in the multiples can be used to help find unknown multiples, eg: adding 13 to the previous multiple, doubling 4 x 13 to find 8 x 13, subtracting from the following multiple. Ultimately, at this stage, children should be able to write their own lists of multiples fairly quickly using a range of appropriate strategies. Step 21 involves looking at the first three digits together, rather than the first two. At this stage, children should be working confidently and accurately. As with Step 20, children should write down the multiples of the divisor, before they begin their calculations. By now, children should be proficient in the long division technique and will realise that the process of long division is the same for very large numbers. They can then divide numbers with an infinite number of digits by any two-digit number, with or without remainders. Where necessary, multiples of the divisor can be written out first. Step 22: Three-digit two-digit long division decimal answers; eg: = x 13 = 13 2 x 13 = 26 3 x 13 = 39 4 x 13 = 52 5 x 13 = 65 6 x 13 = 78 7 x 13 = 91 8 x 13 = x 13 = 117 Step 21: Four-digit 17, 18 or 19 long division, where the first two digits are smaller than the divisor; eg: 1, = 93 r11 1 x 17 = 17 2 x 17 = r11 3 x 17 = x 17 = x 17 = x 17 = x 17 = x 17 = x 17 = x 17 = 170 As in Steps 13 and 14, children are now able to calculate answers with decimals. By now they should be extremely proficient at long division and their answers should be accurate.

7