learn a neat way of writing, and the teacher may detect any habits of inaccuracy, slovenliness, or failure to comprehend the arithmetical process.

19

(ii.) Subtraction.-In subtraction it is (or was) a common error to speak of "borrowing," e. g. in subtracting 19 from 41-9 from 41 1, you cannot: borrow 10; 9 from 11, 2; now pay back 1 to the 1 in the lower line; 2 from 4 is 2. This is obviously an 22 incorrect method of reasoning. For if you borrow 10 from 19, you make it 9, and when you pay back the 10 to the 9, it becomes 19 again, not 29. The correct explanation of the process depends upon the truth that, in subtracting one number from another the result is not altered if the same number be added to both. This, therefore, must first be shown to the pupil as follows: "Subtract 5 from 9, what is the result?" 4. "Now add 1 to 5 and also to 9, and subtract 5 and 1 (i. e. 6) from 9 and 1 (i. e. from 10), what is the result?" 4. "Yes, the same as before. Again, if instead of adding 1 to each, you add 2, and then subtract 5 and 2 (i. e. 7) from 9 and 2 (i. e. 11), what is the result?" 4. "The same as before." After this, you add successively to the two numbers 3, 4, 5, 6, etc., and elicit from the pupil that in each case the result of the subtraction is 4, the same as before. "Then it seems that when I am subtracting one number from another, if I add the same number to both, the result of the subtraction is still- -?" The same as before. "Repeat the whole sentence." When I am subtracting, etc. "Try it for yourself, subtracting 5 from 8. Repeat the rule again."

"We have now to subtract 19 from 41, and you will see the use of

tens ones

4

11

41

2

9

2

19

2 22

the rule you have just learned. Can you subtract 9 from 1?" No. "Then we will add 10 ones to the unit column of 41, and afterwards we will add 1 ten to the ten-column of 19; and the result of the subtraction will be the same as before. 9 ones from 11 ones leave?" 2 ones. "Now add 1 ten to the ten-column of 19; what will that make?" 2 tens. "And subtracting 2 tens from 4 tens, we shall have " 2 tens. "The result then is twenty-two."1

Briefly, the process can now be gone through thus: "9 from 1, you cannot; add 10 above; 9 from 11 is 2; add a 10 below; 2 from 4 is 2." Of course in larger numbers the principle is the same; but the teacher had better not risk confusing the child by entering into further explanations. It may be quietly assumed that the same process is to be continued of adding 10 to the top line where needed, and then

1 Another process consists in shifting a ten in the larger number, thus: 9 from 1, you cannot; shift a ten in 41 from the ten's place to the unit's place, making 3 tens and 11 units: 9 from 11, 2; 1 from 3, 2. This depends upon the truth that a number (e. g. 41) is not altered by shifting its parts (e. g. 4 tens and 1 unit; 3 tens and 11 units).

adding 1 to the next figure of the bottom line, by way of compensa

tion.

11111

But if a more than usually quick and intelligent child detects that in larger numbers you are not adding tens, but hundreds and thousands, you may explain the matter further to him thus, by an example, subtracting 9999 from 11111: "9 ones from 1 one, you cannot; add 10 ones to the one 1; 9 ones from 11 ones leave 2 ones.

9999

1112

"But since we added 10 ones to the top ones, we must now add the same (i.e. 1 ten) to the bottom tens; 9 tens and 1 ten make 10 tens; 10 tens from 1 ten you cannot; add 10 tens to the 1 ten; 10 tens from 11 tens leave 1 ten.

"But since we added 10 tens to the top tens, we must now add the same (i.e. 1 hundred) to the bottom hundreds; 9 hundreds and 1 hundred make 10 hundreds; 10 hundreds from 1 hundred, you cannot; add 10 hundreds to the 1 hundred; 10 hundreds from 11 hundreds leave 1 hundred.

"But since we added 10 hundreds to the top hundreds, we must now add the same (i.e. 1 thousand) to the bottom thousands; 9 thousands and 1 thousand make 10 thousands; 10 thousands from 1 thousand you cannot; add 10 thousands to the 1 thousand; 10 thousands from 11 thousands leave 1 thousand.

"But since we added 10 thousands to the top thousands, we must now add 1 ten thousand to the bottom ten thousands; no ten thousands and 1 ten thousands make 1 ten thousands; 1 ten thousand from 1 ten thousand leaves no ten thousands."

If children could thus be practiced in working sums of subtraction aloud, they would be greatly strengthened in the power of realizing the meaning of figures and of reading them into words.

(iii.) Multiplication.-Let the pupil multiply 6 by 4, and note the result (24). Then let him divide 6 into any two parts (1 and 5, 2 and 4, 3 and 3); let him multiply each of the two parts separately by 4 and add the two products (4 and 20, 8 and 16, 12 and 12); and let him thus discover:

Rule I.—When a number has to be multiplied, it makes no difference whether you multiply the whole, or multiply the parts and add the products.

When the child has been led to the discovery of this law by experiments with small numbers, and has learned it by heart, we shall tacitly assume that it holds good for all numbers, and shall proceed to apply it to the multiplication of numbers above 10.1

But first we must practice the child in multiplying tens together.

1 Before beginning, we remind the pupil that twice 6 is the same as six times two: 7 times 4 the same as 4 times 7, and so on; so that, when two numbers are multiplied together, it matters not which is the multiplier and which the multiplied.

"Suppose I have to multiply twice a number by three times the same number. Twice 4 multiplied by three times 4 is?" 8 multiplied by 12, i.e. 96. "Now alter the order of multiplying, and multiply twice 3 by 4 times 4; the result is?" 6 multiplied by 16; I do not know this. "Using the rule just given, you can divide 16 into two parts, 10 and 6, and after multiplying them separately, you can add the results." 6 times 10 is 60, 6 times 6 is 36; 60 and 36 are 96. "The same result as before; so that we see that if we have to multiply twice a number by three times a number, it makes no difference if we first multiply two by three, and then the number by the number, and then multiply the two results.

"Now let 10 be the number, and suppose I have to multiply twice 10 by 4 times 10; then the result will be the same, whether I multiply twice 10 by 4 times 10, or twice 4 by?" 10 times 10. "That is by-?" 100. "And what is twice 4 multiplied by 10 times 10?" It is 8 multiplied by 100 (i.e. 800).

"In the same way twice 10 multiplied by 3 times 10 is the same as twice 3 multiplied by 10 times 10, i.e. -?" 6 multiplied by 100, or

600.

"Hence you see we get a very useful rule."

Rule II.—If you have to multiply a number of tens by another number of tens, we can multiply the two numbers together as though they were ones, and then put hundred after the result.

For example, 3 tens multiplied by 4 tens are 12 hundreds, or 1200; 4 tens multiplied by 5 tens give?" 20 hundreds (i.e. 2000).

Required to multiply 13 by 24.

Here, by Rule I, instead of multiplying 13 by 24, we may multiply 13 first by 4 and then by 20, and, if we add the products, the result will be the same. Again, instead of multiplying 13 by 4, we may first multiply 3 by 4, and then multiply 10 by 4, and the results will be the

same.

We proceed, therefore to multiply 13 by 4 and by 20, and we begin with 4: 13 multiplied by 4 is (by Rule I) the same as 3 and 10 multi

13

24

plied by 4; 3 multiplied by 4 is 12 (i.e. 2 units and 1 ten); set down 2 units, and " carry" the 1 ten; 1 ten multiplied by 4 is 4 tens, which, with the 1 ten "carried," makes 5 tens. 52 Having multiplied 13 by 4, we have now to multiply 13 by 20 (i.e. 2 tens); 13 multiplied by 2 tens is (by Rule I) the same as 3 and 10 multiplied by 2 tens; 3 multiplied by 2 tens is the same as 2 tens multiplied by 3, or 6 tens; set down 0 for the units, and 6 for the tens; 1 ten multiplied by 2 tens is (by Rule II) 2 hundreds; set down 2 hundreds.

260

312

Having now multiplied 13 first by 4 and then by 20, we add the results, 312; and this (by Rule I) is the same as the result of multiplying 13 by 24.

Before passing to any other sums, it will be good practice to multiply in the same way 24 by 13, and to show that the result is the same;

and to multiply 24 by 6 and 7, or by 8 and 5; or to multiply 13 by 12 and 12, or by 10 and 14; and to show that in each case the result is the same.

After all this preliminary training, the teacher may now work the sum above written, briefly thus; "4 ones multiplied by 3 ones is 12 ones; set down 2 ones and carry 1 ten; 4 ones multiplied by 1 ten is 4 tens; set down 5 tens; 2 tens multiplied by 3 ones is 6 tens; set down 6 tens; 2 tens multiplied by 1 ten is 2 hundreds; set down 2 hundreds. Now add."

These and many other sums should be worked by the pupil aloud, the teacher setting down the figures at the dictation of the pupil, who must be trained gradually to increase the rapidity of the process.

But when the pupil is allowed for the first time to set down a sum for himself, great care must be taken not to hurry him, nor to allow him to begin a habit of writing the figures out of the exact vertical columns; and, if possible, the sum should be so simple that he may succeed in his first essay.

The teacher must use some discretion in teaching the above reasoning to children: 1st, he must be perfectly familiar with it himself; 2d, he must be on the alert to detect signs of bewilderment in his pupils; 3d, he must give it up soon, if he finds he does not carry them with him.

Yet even if he does not succeed in making his pupils comprehend the whole of the demonstration, he should keep the form of the demonstration in mind when working a sum aloud for them. For example, after multiplying 13 by 4, he will say, "We have now multiplied 13 by the 4 ones; it remains to multiply 13 by 20, or 2 tens," etc. Thus he will gradually instil into their minds some apprehension of the reasons for the process.

(iv.) Division.—The teacher must be prepared to find the explanation of division more difficult than that of multiplication and subtraction; and none but very easy examples should be given to illustrate it. Perhaps, in the case of a child who is not very quick, it may be better to dispense altogether with the explanation, simply dictating the steps, and trusting partly to the analogy of multiplying, and partly to the inherent proof contained in each example, in the hope that the pupil may gradually be led to an apprehension of the reasons of the

process.

Before beginning the division of large numbers the pupil should be taught to divide, accurately and rapidly, small numbers in which the divisor is not exactly contained, e. g. 73 divided by 9 is-8, and 1

over.

Let the pupil divide 12 by 2, and note the result (6); then let him divide 12 into its parts (taking care that the numbers are even) (a) 10

1 The 0 (to signify that 6 stands for 6 tens, or 60) had better be inserted for some time.

and 2; (b) 8 and 4; (c) 6 and 6; (d) 4 and 4 and 4; (e) 2 and 6 and 4; and let him divide the parts by 2 and add the quotients, (a) 5 and 1; (b) 4 and 2; (c) 3 and 3; (d) 2 and 2 and 2; (e) 1 and 3 and 2. Thus let him discover that in all these cases the quotient is the same as when the whole number was divided, so that he may be led to:

Rule. When a number has to be divided it makes no difference whether you divide the whole or divide the parts and add the quotients.

The pupil must now be reminded of what he has probably already learned on a small scale, when doing little sums that illustrate the division of small numbers, viz., that 6 apples divided by 3 give 2 apples; 6 marbles divided by 3, 2 marbles; 6 tens divided by 3, 2 tens; 6 hundreds divided by 3, 2 hundreds; 6 thousands divided by 3, 2 thousands.

Required, to divide 435 by 3:

3)435 By our Rule, 435 divided by 3 is the same as 400 divd by 100 3, and 30 divd by 3, and 5 divd by 3; or, if we please, it is 40 the same as 300 divd by 3, and 120 divd by 3, and 15 divd by 5 3; or we may divide 435 into any other parts we please, and 145 divide them separately, adding the quotients.

Begin with the hundreds. Threes into 4 hundreds? Not exactly divisible. But 3 hundreds are divisible by 3. We will therefore take away 1 hundred from the hundreds, so as to leave 3 hundreds, and "carry" the 1 hundred (in the shape of ten tens) to the 3 tens, making up 13 tens. Threes into 3 hundred? 1 hundred; set down 1 hundred.

Threes into 13 tens? Not exactly divisible. But 12 tens are exactly divisible by 3. We will therefore take away 1 ten from the tens, so as to leave 12 tens, and carry the 1 ten (in the shape of ten units) to the 5 ones, making up 15 ones. Threes into 12 tens? 4 tens; set down 4

tens.

Threes into 15 ones? 5 ones; set down 5 ones:

3)4734 More briefly we may now dictate a sum of this kind thus: 1000 threes into 4 thousand? 1 thousand and 1 thousand over; 500 set down 1000: threes into 17 hundred? 5 hundred and 2 70 hundred over; set down 5 hundred; threes into 23 tens? 7 8 tens and 2 tens over; set down 7 tens; threes into 24 ones? 1578 8 ones; set down 8 ones.

More briefly still, the next step will dispense with the rows of naughts, by showing that, if we take care to write the thousands and hundreds in the places of the thousands and hundreds, the naughts are

unnecessary.

34. THE TRANSITION TO FRACTIONS.

Before proceeding to Fractions, and indeed before proceeding to Long Division (the explanation for which must be taken upon trust by