4. Multiplication and division
"Reeling and Writhing, of course, to begin with and then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision"
Lewis Carrol, 'Alice in Wonderland'
1. The topic can hardly be described as a new element in the Curriculum but changes in emphasis in the Numeracy Project do raise some new issues. The Curriculum calls for children to be able to multiply and divide numbers in specified ranges at appropriate ages but what is not clear is whether it suffices that they are able to perform the operations and arrive at correct answers or whether they are expected to reproduce some standard method and presentation in tests. There are suggestions that more attention should be given to 'informal' methods but what would the reaction be, for example, if a child produced a version of one of the 'trick' methods suggested for mental arithmetic in a written SATS test ?
Advances in technology have made the traditional algorithms for manual and mental arithmetic largely irrelevant to real life, leaving the need to understand the decimal place value system as their main justification in the Curriculum. The emphasis on speed in mental and manual calculation is largely misplaced - the first priority is unquestionably getting the correct answer and being sure it is correct.
The conventional layout of calculations does not make the impact of a place-value system as explicit as it might , presumably in the interest of speed and compactness when they were in frequent use, and some pupils fail to make the connection, leaving the impression that arithmetic consists mainly of arbitrary 'rules' without rhyme or reason.
Examples of alternative (less compact ) presentation schemes are given and also the binary reduction algorithms for multiplication and division which are widely used in computer programs (because they are faster and simpler). These offer a way of enabling pupils who have particular difficulty in memorising multiplication tables or coping with the quirks of positional value to carry out these operations successfully .
| Multiplication | |
Binary multiplication | |
| Hindi algorithm | |
Binary division | |
| Division | |
Hybrid methods | |
3. Multiplication
The most explicit scheme for multiplication is an adaption of the polynomial representation used in advanced number theory eg the number 387, say, is represented as 3×100 + 8×10 + 7×1. In advanced work the form
3x2 + 8x + 7, where x is the number base, would be more usual in the interests partly of making the results general (ie applicable to any base), and partly to make the place value aspects clearer. In the present context introducing yet another new notation would be counter-productive.
With this approach 387× 24, say, gives
| (3×100 | + | 8×10 | + | 7×1) | times | (2×10 | + | 4×1) | |
| 6×1000 | + | 16×100 | + | 14×10 | times | 2 × 10 | |||
| 12×100 | + | 32×10 | + | 28×1 | times | 4×1 | |||
| 6×1000 | + | 28×100 | + | 46×10 | + | 28×1 | adding | ||
| 2×10 | + | 8×1 | carries | ||||||
| 48×10 | |||||||||
| 4×100 | + | 8×10 | carries | ||||||
| 32×100 | |||||||||
| 3×1000 | + | 2×100 | carries | ||||||
| 9×1000 | + | 2×100 | + | 8×10 | + | 8×1 | = 9288 |
Although cumbersome in appearance this approach separates the individual operations and makes no demand on serial recall since the result of each step is written down before the next mental step is attempted. The propagation of carries reinforces the learning of positional value aspects by virtue of the repetition involved in the carry process. The first step also illustrates the associativity property of addition and multiplication in multiplying out the brackets.
4. Hindi Algorithm
An intermediate format, closer to the current convention is
1. Multiply digits of multiplicand by digits of multiplier starting at the high end of multiplier.
2. Enter digit products units at lower right, tens diagonally above to left
3. Add up diagonal columns. Put sums in horizontal SUMS row.
4. For any sums above nine, put the first digit in the next place to the left in the CARRY row and copy the second digit down.
5. Add the first CARRY digit to the next digit in the SUM row.
6. Repeat the Carry step if necessary.
eg 387 × 24
| carries | 0 | 1 | 1 | ||||||
| 6 | 6 | 4 | × 2 | ||||||
| carries | 1 | 3 | 2 | ||||||
| 2 | 2 | 8 | × 4 | ||||||
| sums | 8 | 12 | 8 | 8 | |||||
| carries | 1 | 2 | |||||||
| 9 | 2 | 8 | 8 |
This reduces the demand on instant recall by postponing the 'carry' operations until all the products are completed. The diagonal alignment and addition eliminates the need to displace product rows according to the column value.
This algorithm derives from Indian arithmetic, passed to Europe through the writings of the Arab Al-Khwarizmi (850AD): in medieval Europe the phrase 'dixit Algorismi...' ('Algorismi says...') was the hallmark of mathematical excellence for centuries.
5. Long division
The current conventional format for long division makes greater demands on instant recall, probably accounting for the greater difficulty children seem to experience. The basic algorithmic step for each order of magnitude is longer and more complicated viz
(a) Mentally align the divisor with the dividend so that it is less than the dividend and more than 1/10 of it.
(b) Guess a trial digit as the integer quotient for the leading digits of the dividend.
(c) Multiply the divisor by the trial digit.
(d) Compare magnitudes - if the trial product is greater than the leading digits try again with a lower digit, if less try again with a higher digit.
(e) When the leading digits of the dividend are bracketed by the products from digits differing by 1 enter the lower product below the leading digits of the dividend, then the corresponding trial digit on the 'result' line above the rightmost digit of the 'bracket' value
(f) Subtract the 'bracket' value from the leading digits of the dividend and append the remaining trailing digits.
(g) Repeat the process from (b) until no trailing digits remain. The quotient is given by the top line and the remainder (if any) by the number left after the last subtraction
eg divide 229 by 17
| quotient | 1 | 3 |
first digit above divisor unit divisor moves left 2, back one digit limits 2/2,3/1 try 2 - too big copy down trailing digits digit limts 5/2,6/1 try 4 - too big no trailing digits - remainder |
|||||
| Table | 1 | 7 | / | 2 | 2 | 9 | ||
| 1×17=17 | 1 | 7 | ¦ | |||||
| 2×17=34 | 5 | 9 | ||||||
| 3×17=51 | 5 | 1 | ||||||
| 4×17=68 | 8 |
'Division' questions in Scheme Books and Test Papers sometimes omit to specify which form of answer is required when the result is not an integer ie with remainder, as a proper fraction, or as a decimal fraction. In such cases one might justifiably claim that it is optional, in which case the simplest choice is the remainder form. In this case the basic algorithm step above is repeated until the rightmost digit of the current bracket value reaches the 'units' position, when the final subtraction gives the remainder.
6. When spelled out in detail like this the greater challenge of the mental arithmetic approach to long division is apparent. Since long multiplication is customarily taught before long division difficulties are less likely to arise from the place value aspects than from a prompt recall overload in repeated access to imperfectly memorised multiplication tables during the computation step. Pupils often make a scribbled marginal note of trial products and it might help to encourage them to do this systematically, noting the digit multiplier alongside since these products may be useful in later steps. In extreme cases it may be worth suggesting that they construct a complete 1-to-9 multiplication table for the divisor by repeated addition.
7. Common causes of difficulty are in the first two steps - normalizing the divisor and selecting a trial multiplier. A tip which may be helpful in the first is to mentally move the divisor to the left until it is greater than the dividend and then move back one. A technique used in fast computer programs, (which use bases much larger than ten in the interests of speed and a correpondingly large stored multiplication table) is to take the first digit of the divisor and one or two of the leading digits of the dividend (so as to make it larger than the leading digit of the divisor). The two numbers obtained by adding one to each in turn and dividing by the other (ignoring remainders) bracket the required multiplier, reducing the range of multiples that need to be tested. An alternative to estimating both trial multiplier digit limits is to use the nearest digit to their average as a first guess viz
{2×dividend digit(s)+1]÷{2×divisor digit+1] - in this case 5÷3 and 11÷3 ie 2 and 4 respectively.
8.1 Multiplication
(a) Write down the smaller number at the top of the first column and the larger number at the top of the second column.
(b) Go down the columns dividing the first by 2, ignoring any halves, and multiplying the second by two until the number in the first column is zero
(c) If the number in the first column is odd, copy the number in the second column into the third column: if it is even leave the third column blank
(d) Add up the numbers in the third column.
| 23×231 | | | 34×537 | ||||
| 23 | 231 | 231 | | | 34 | 537 | |
| 11 | 462 | 462 | | | 17 | 1074 | 1074 |
| 5 | 924 | 924 | | | 8 | 2148 | |
| 2 | 1848 | | | 4 | 4296 | ||
| 1 | 3696 | 3696 | | | 2 | 8592 | |
| 0 | carries | 221 | | | 1 | 17184 | 17184 |
| 5313 | | | 0 | carries | 1 | ||
| | | 18258 | |||||
8.2 Division
a) write 1 at the head of a first column and the divisor at
the top of a second column.
b) go down the columns multiplying both by 2 until the number in
the second column is greater than the dividend.
c) write the dividend in a third column opposite the last but one
number in the second column.
d) Starting from this position go up the columns comparing the
numbers in the second and third columns.
i) if the second column is less than the third, copy the
number in the first column to a fourth column. Then take
the second from the third and write the difference above in
the third column.
ii) if the second column is greater than the third, leave the
fourth column blank and copy the third column number to
the place above.
iii) if the numbers are equal the job is done. Go on to e)
e) Add up the numbers in the fourth column - this is the
quotient; if there is a number at the top of the third
column it is the remainder.
eg 229÷17
| Stage 1 | | | Stage 2 | | | Stage 3 | |||||
| 1 | 17 | | | 1 | 17 | 8 | 1 |
| | 1 | |
| 2 | 34 | | | 2 | 34 | | | ||||
| 4 | 68 | | | 4 | 68 | 25 | 4 | | | 4 |
|
| 8 | 136 | | | 8 | 136 | 93 | 8 | | | 8 |
|
| 16 | 268 | | | 16 | 229 | | | 13 |
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| quotient 13 | remainder 8 | ||||||||
9.1 Decimal position value
The above method originated in Egyptian mathematics (around 1600BC) and is in fact very versatile. It works perfectly well with a decimal position-value notation (or hexadecimal, for that matter) and, as might be expected, cumulative symbol systems. It can cope for example with the challenge often cited as an illustration of the superiority of positional value systems (which is indisputable in this particular case) viz long division using Roman numerals, without cheating by converting to Arabic notation.
In the case of a decimal place value system the successive values in Stage 1 are multiplied by 10 instead of 2 but a multiplication table up to 9 times for the divisor is needed. Repeating the example in 5, above (viz 229÷17) by this method demonstrates that the method is essentially identical - in fact reviewing the traditional methods it is seen that the numbers involved are in effect converted to a cumulative symbol form in the process (viz x hundreds y tens z ones)
| remainder | 8 | Table | |||||
| | | 1 | 17 | |||||
| 51 | 3×1 | 3 | | | 2 | 34 | ||
| 1 | 17 | 59 | | | 3 | 51 | ||
| | | 4 | 68 | |||||
| 170 | 1×10 | 10 | | | 5 | 85 | ||
| 10 | 170 | 229 | | | 6 | 102 | ||
| | | 7 | 119 | |||||
| quotient | 13 | | | 8 | 136 | |||
| 100 | 1700 | | | 9 | 153 | |||
In Stage 3 the multiplication table is examined to find the biggest multiple that can be subtracted from the third column, remembering that it must be multiplied by the unit in column1 before comparison: for the bottom row this is 1 x 17. This factor (1) multiplied by the unit from the first column (10) is put in column 4 and the difference, 229-170= 59) above in the third column.This process is repeated moving up the columns until either the difference is 0 or we get to the top, giving the quotient and remainder as before.
In this case only two entries from the multiplication table are used and it is quicker if the factors needed are found mentally so that the whole table need not be calculated.
9.2 Taking up challenge of Roman notation, the previous problem is CCXXIX ÷ XVII
| factor | divisor | dividend | quotient | subtractions | |||
| I | XVII | VIII | I | ||||
| II | XXXIV | XXV | CCXXIX | XCIII | XXV | ||
| IV | LXVIII | XXV | IV | CXXXVI | LXVIII | XVII | |
| VIII | CXXXVI | XCIII | VIII | XCIII | XXV | VIII | |
| XVI | CCLXXII | CCXXIX | XIII | ||||
quotient XIII remainder VIII
Veni, vidi, vici !
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