independent addend (9), put in by the teacher himself. be done with great ease and quickness. This is easily applied to addition of money, thus: £583024 6s. 8d. (1) £416975 13s. 33d. (2) This can Line (1) is any amount, as before; line (2) is formed from it by subtracting the farthing from a penny, the 8 from 11, the 6 from 19, and the others as in simple addition, except that the units figure is taken from 9 instead of from 10 as there required. Simple subtraction. We have now treated of Simple Addition. The next rule is Simple Subtraction. In this rule, as in the other, when the pupil knows the proper table, and its mental application, he has but to become master of two additional principles in order to be able to work any question correctly. He must first learn to arrange the numbers properly, and he must know when to 'borrow,' when to carry,' and when to do neither. 6 In addition, it was immaterial what arrangement of the addends was followed, provided always that units were placed under units, tens under tens, &c.; but in this the pupils will find it easier to place the numbers to be operated upon in a certain relative position. They should place the small number below the other. This the children will have no difficulty in understanding. Borrowing explained. The principle of borrowing may be thus explained, but the explanation should be given of course only at proper times, and when the pupils are fully prepared to understand what is advanced. As a preparatory exercise, pupils should be accustomed to analyse numbers in the following way: 36, for instance, may be written 2(16), the 2 being in the tens' place, and the 16 in the units'. 349 may be written 2(14)9, the 14 occupying the second place, and representing 14 tens. The pupils should prove that these forms represent the same value exactly as the others. When they can do this correctly and intelligently, they should then be called upon to perform subtraction without borrowing,' by simply changing the upper line, in accordance with the models just shown. For instance, to take 198 from 349, they should represent the latter by its equivalent 2(14)9, and when arranged the figures will stand thus:-' 2 (14) 9 1 98 and the subtraction will be '8 from 9 and 1,' '9 from 14 and 5,' ‘1 from 2 and 1,' giving 151 as the true difference. 1 Some may prefer to decompose ing form: the number according to the follow 349=2 21039 SUBTRACTION AND MULTIPLICATION. 147 It must now be explained that the only effect of the changethe reduction of the 3 to 2—was to lessen by one the figure of the answer placed below it, and this would equally have taken place had they increased the figure of the subtrahend by one, instead of diminishing to the same amount the figure of the minuend, for 2 from 3 will be exactly the same as 1 from 2. A child will thus see that there are two methods of arriving at the answer. He should be accustomed to work the questions by both, until he can do so perfectly. He will then be in a position to understand that we reject one plan as being clumsy and awkward, and adopt the other as being free from these defects; that in both we decompose the numbers, but in the usual method of subtraction the change is imagined only, and not really performed. Test questions to be more searching. The test questions on this rule ought to be more searching than they are. The following may be said to be the chief difficulties in simple subtraction: (1) Knowing when to carry; (2) when not to carry; (3) how to deal with ciphers in the minuend or subtrahend, or both; and (4) how to manage in the case of those figures to the left hand of the minuend which may have no corresponding figures in the subtrahend. When children begin to carry, they frequently continue carrying to the end; and when they meet with ciphers, they generally add what is over instead of subtracting it; or else, which they do also in the case of the ending figures having none below them, they bring them down unchanged. Test questions. should therefore contain these difficulties, and questions involving them should be given very frequently during the learning of this rule. Example: 1004307016 Do not let the carrying and not carrying take place at regular intervals, for the children would soon see the regularity, and work in accordance with it. Defects in the working of this rule. In the actual working of this rule the children should avoid expressions like this: '8 from 6, I cannot, but 8 from 16.' They should be required, from the very first, to say '8 from 16' merely, omitting the rest. Multiplication. It is of two kinds. The next rule is Simple Multiplication. It may be divided into two kinds: (1) containing all questions where the multiplier is any number from 1 to 12 inclusive (the multiplication table in ordinary use not extending farther); and (2) all other questions. 1st kind should scarcely require any special explanation. Any child who knows the tables well, and has acquired facility in 'carrying,' should find no difficulty whatever in solving all questions contained in class 1. How to teach the second class. Class 2 should be commenced by solving questions whose multiplier consists of one significant figure with ciphers attached-as by 20, or 200, or 400, &c.-and this they should do by multiplying by the significant part only, and then affixing to the product the proper number of ciphers. They should next learn to multiply by using the parts of the number which are formed by analysis-not the factors. Thus, to multiply by 436, they should multiply by 400, by 20, and by 6, and add the results together. They will then be in a position to understand the ordinary mode of multiplying; but instead of skipping a figure for each place in the multiplier, they should insert the ciphers. Thus, to multiply 3246 by 422, the work should be as follows: Line (1) being the product by 2; line (2) being the product by 20; and line (3) being the product by 400. They will soon see that if the figures are put in their proper places, ciphers are useless, as they do not change the result in adding. And they will thus learn why the ciphers are omitted, and why one place or two places are skipped. Rule about 'skipping.' With regard to skipping,' the best guide to the child is to tell him to place the first figure of the product immediately under the figure by which he multiplies. In the above example, the 2 in line (1) goes under the first 2 of the multiplier, the 2 in line (2) under the second 2 of the multiplier, while the 4 in line (3) goes under the 4 of the multiplier. This rule is peculiarly good in dealing with ciphers. To lessen the chances of error in using it, the figures must be arranged units under units, tens under tens, &c. They ought not to be placed thus, as they too often are: 201 406 Children should know the value of the figure by which LONG AND SHORT DIVISION. 149 they multiply. In all cases the child should know the value of the figure by which he multiplies. He should know, for instance, that in the question above, although he simply multiplies by 4, he in reality is multiplying by 400. It is a good custom to make him, now and then, perform each multiplication separately, and then add the results for the product required. It is a good plan also to get the child to read the several separate products, and then to read their sum. Where this is not attended to, the child will invariably forget that the omitted ciphers should be taken into account in expressing the true value of the figures. Should carry what is over in the memory. In this rule, as in addition, &c., he should bear in the mind the number to be carried. It is unseemly to be putting it down and rubbing it out every moment. May repeat what is carried. In multiplying, he may be allowed to repeat the number to be carried, as it assists the memory. Thus, '9 times 8 are 72-carry 7' (putting down the 2), '9 times 3 are 27, and 7 are 34-carry 3,' &c. Division. Factoring. The last of the four simple rules is Division. With regard to it, teachers often err in allowing their pupils to employ Long Division, when, by the use of factors, they might judiciously shorten the work. Others, again, commit an equally grave error in insisting upon factoring' in the case of all composite numbers. The true plan is to divide by factors when they are not numerous, and when they are easily discovered. Thus, though 432 is made up of 6, 9, and 8, yet the time taken to discover this would be sufficient to do the question by long division; and again, though 3125 is made up of a series of fives, as the termination 25 shows, there would be so many divisions that factoring would be the most tedious course. Again, when factors are made use of, the children should be taught to express the remainders as decimals. It is very easy to show them how to do this, and the results are more correct and the work quicker. The connection between Long and Short Division. The connection between Long and Short Division is scarcely ever perceived by children, or scarcely ever taught to them. On the contrary, they are practically led to look upon these two rules as entirely distinct from each other as much so, in fact, as are the rules of Addition and Subtraction. This error is best avoided by commencing Long Division with a small number as a divisor, such as 6, or 4, or 12; working the same question afterwards by Short Division, and then comparing the two processes. Thus, supposing that we wished to divide thirty-four thousand # five hundred and eighty-eight by six, the work would stand thus by the process of Long Division: 6)34588(5764 45 42 38 36 28 24 4. while it would stand thus by the other process :— 6)34588 Both contain the same processes. The latter process employs apparently, but only apparently, much fewer figures. By Long Division we say 6 into 34 gives 5 (as in Short Division) and leaves 4. This 4 is actually put before the 5 in Long Division, while it is only supposed to be before it in Short Division; and so on with the other operations. Their differences. It follows, therefore, that Short Division does not, as it seems to do, contain fewer processes than Long Division, or any different ones; the only differences between the two consisting in their appearance on the slate, and in the fact that the operations are chiefly mental in the one, while they are actually recorded in the other. The one rule is employed when the carrying and subtracting can be done mentally, the other when they cannot. And as the mind is more exercised in the one than in the other, it is a question whether it would not be found, by the child, to be easier to begin with Long Division than with Short. In this rule, as in multiplication, the children should be called upon to read each result; for instance, in the example in Long Division just given, they should know that the 30 is not in reality 30, but 30,000, &c., and that when multiplying by the 5 they were in reality multiplying by 5,000, the result being retained in its proper place without ciphers. Proof of rules necessary. Children should be accustomed occasionally to prove their work, but only occasionally, and for this purpose they should be taught those methods which are neatest and best. Proofs of addition, The best method of testing the accuracy |