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children, as it is far too lengthy to be imparted without confusing them) a good store of easy problems should be worked involving "the first four rules" applied to small numbers and to small sums of money. In the course of these, they must be taught, with care and iteration, the different meanings of an answer in division.

Supposing 12 oranges are to be equally divided: the question may be (1) how many oranges are to be given to each of 4 boys, and in that case I divide the number of oranges (12) by the number of boys (4), and the answer (3) represents the number of oranges. But the question may be (2) how many boys can be sharers, if each is to receive 3 oranges, and in that case I divide the whole number of oranges (12) by the number of oranges in a single share (3), and the result (4) tells you the number of times 3 oranges are contained in 12 oranges; and hence we can infer how many heaps or shares of 3 can be made, and how many boys can share.

Hence our pupils will obtain a useful rule, that:

• When a number of things is divided by a number of the same things (e.g. a number of oranges by a number of oranges, of pence by pence, of boys by boys) the answer is a number of TIMES; but when a number of one kind of things is divided by a number of another kind of things (e.g. a number of oranges by a number of boys, a number of soldiers by a number of regiments, a number of sailors by a number of ships) the answer represents some number of the first kind of things (oranges, soldiers, sailors).

At this stage the Definitions of Multiplication and of Division should be taught and committed to memory, as well as the terms Multiplicand, Dividend, Multiplier, Divisor, Product, Quotient, etc.

The Arithmetical Problems should be varied in every possible way (the numbers being kept small) so as to familiarize the pupil with the different practical applications of Arithmetic. For example, in a certain number of yards how many telegraph posts can be set up? How many revolutions of a wheel can take place? How many sentinels must be posted? How many desks can be placed? How many boys can stand with arms folded? How many with arms outstretched? All these are simply so many changes rung on one simple method of utilizing division.

In order to increase the number of these problems, and to take advantage of the strength of the memory while it is strongest, it is desirable that children should learn the ordinary Tables of "Weights and Measures" (rejecting those which are of no use) before proceeding to fractions.

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At this stage it will be useful to teach the pupil to substitute for "added to," diminished by," "multiplied by," divided by," the signs, +, X,, care being taken that these signs are, from the first, shaped exactly; and in order that the + may be distinguished from the X, let the slope a little, if anything to the left. He may also be allowed to use the sign = to denote "is equal to."

At first it well be well that the pupil, though using these signs for convenience in writing, should orally interpret them by the old terms, "added to," "diminished by," etc., not being introduced to the terms plus and minus till a lesson or two have familiarized him with the written use of these symbols.

The careless use of the sign

must be strictly prohibited. Some boys use it merely as a link to connect together different parts of a problem, thus: "If 12 be multiplied by 3 and then by 4, and the product be divided by 2, what is the quotient?" "12 x 3 = 36 = 36 x 4 = 144 144 ÷ 2 = 72 Ans. Such slovenly statements must be at once branded as "not true." And any pupil who thus abuses the use of the sign = must be condemned to return to the tedious "is equal to" for a week, at least.

35. FRACTIONS.

There are many ingenious methods of showing children the meaning and laws of fractions. Whatever methods may be adopted, the teacher will always bear in mind the principle that the pupil is to be led to the unknown from the known; and that, as far as possible, he is to discover truths for himself.

Break a thin stick into three parts as nearly equal as you can manage. Each of these fragments, you tell him, is a third part, or a "third" of the whole. In Arithmetic, when one whole is thus divided into equal parts, each part is called, not a fragment, but a fraction; but the meaning is the same, viz., "a breaking." How are we to express in writing such a fragment or fraction in Arithmetical writing? We might write it 1÷3; but we prefer to write it, which means 1 divided by 3, or 1 divided into three parts.

Let the child then break a stick into 2, 3, 4, 5, 6, etc., parts, and write down neatly the Arithmetical signs by which he must express these parts, viz., §, 1, 1, 1, 1, etc.

Now what does the lower figure in each case tell us? It tells us the number of equal parts into which one, or unity, is divided. By what names shall we call these equal parts? We will call them a half, a third part, a fourth part (or quarter), a fifth part, a sixth, etc.

Hence we see that the lower number of a fraction always tells us the name of the parts into which unity is divided. Therefore, the lower figure in a fraction may be called the Namer.

Before proceeding further, let the child write down several fractions for himself, e.g.,,, and read them aloud; and let him, after a while, be allowed to drop the word "part" (it being explained to him that this is allowed for the sake of brevity), so that he may now speak of "one twentieth," "one fifty-fifth,” one two-hundred and fortieth," etc.

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Let the pupil for some time call the lower figure of the fraction the Namer, without being allowed to puzzle himself with the less intelligi

ble term (which merely expresses the same thing in a longer word) Denominator.

Next point out that, in breaking a stick or anything else into equal parts; you may take a number of them together. For example, if the stick has been divided into six parts, each of which is called a sixth, you may take 2, 3, 4, or 5 of these together, thus making two sixths, three sixths, four sixths, five sixths, according to the number of parts taken together.

How shall we write down these fractions, say, for example, five sixths? Since we are taking five sixths instead of one sixth, we must write 5 where we wrote 1 before, above the line, to represent the number of the parts, §; and similarly for the rest, , t, t.

Since the upper figure represents the number of the parts taken together, it may be called the Numberer.

This name should be allowed for some time without permitting the pupil to use the term Numerator, which merely expresses the same thing in a longer word.

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Let the pupil take a sheet of note paper, folded in the ordinary way, and having unfolded it, and then refolded it, let him be told to observe that when it is refolded it is folded into half of the whole size.

Now let him fold it a second time into a quarter, then into an eighth, and lastly into a sixteenth of the whole size. Lastly, let him unfold it to the full size, and observe the creases dividing the paper into halves, quarters, eighth parts, sixteenth parts; and let him write down in words how many of the smaller parts are contained by each of the larger parts.

He will find that a half contains two quarters, or four eighths, or eight sixteenths; also that a quarter contains two eighths, or four sixteenths; and that an eighth contains two sixteenths.

Pointing out to him that he may use the term "is equal to " instead of contains," and that he may use the symbol to denote it, you will now bid him write down his discoveries, thus:

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Let him then be asked to find out from his note-paper, and to write down how many eighths there are in three quarters? How many sixteenths there are in three eighths? How many in five eighths?

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Now with the aid of a foot-rule, measure off on a stick, or (better) show your pupil how to measure, a foot divided into inches and halfinches. Carefully avoid using the terms "feet" or "inches," but speak of it as a piece of wood divided into 12 parts, each part again being -divided into 2 smaller parts, so that the whole stick is divided into 24 parts. And bid him write down how many of the twelfth parts are contained in half the stick? How many in a quarter? He will find that:

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How many of the twenty-fourth parts are contained in half the stick? How many in a quarter?

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"Let us now run over our results again. We find that ; how many times is the numberer 2 greater than the Numberer 1?" Twice. "And how many times the Namer 4 greater than the Namer 2?" Twice. "Again † = ; how many times is the Numberer 4 greater than the Numberer 1?" Four times. "And how many is the Namer 8 greater than the Namer 2?" Four times. "In the stick also we find that = how many times is the Numberer 6 greater than the Numberer 1?" Six times. 6. And how many times the Namer 12 greater than the Namer 2?" Six times. "Then we find that, whenever two fractions are equal, if the second Numberer is a certain number of times greater than the first Numberer, the second Namer is also-?" The same number of times greater than the first Namer. Write down this:

1. Rule. Whenever two fractions are equal, if the second Numberer is a certain number of times greater than the first Numberer, the second Namer is the same number of times greater than the first Namer.

Then ask the pupil whether is increased when the Namer and Numberer are both multiplied by 2, by 4, by 6, by 8, by 12. And having shown him, by reference to the above results which he has written down, that is not altered, lead him to the

2. Rule. A fraction is not altered when the Numberer and Namer are multiplied by the same number.

Let us now find out what we have been doing in multiplying the Numberer and the Namer by the same number, and in saying that the fraction is not altered. "What does the Namer name?" The parts into which unity is divided. "Then in multiplying or increasing the number of the Namer, I have increased the-?" Parts into which unity is divided. "I should not say 'increased the parts,' but 'increased the number of parts.' If the fraction is, and if I multiply the Namer by 2, I should not say I increase the part, a third, to the part, a sixth; for a sixth is smaller than a third; but I should rather say I increase the number of parts into which unity is divided from 3 to 6. In reality I diminish the parts (from a third to a sixth), but I increase

the number of the parts (from 3 to 6). Here let me stop to remind you that, when you speak of the number of parts in connection with the Namer, you must always distinguish it from the Numberer. The Namer names the number of parts into which Unity is divided; the Numberer tells you how many of these parts are taken together.

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"Now I resume. In multiplying the Namer 3, I have been increasing a certain number of times-what?" The number of parts into which Unity is divided. "And in multiplying the Numberer by the same number of times-what?" The number of those parts taken together. Then our rule tells us that, when I increase the number of parts into which unity is divided, and increase by the same number of times the number of those parts taken together, the fraction remains-?" The same. "Apply this rule, beginning with a half, doubling the number of parts several times: one half equals two quarters, equals-?" Four eighths, equals eight sixteenths, equals sixteen thirty-seconds, etc.

Since a fraction is not altered by multiplying the Namer and Numberer by the same number, it follows that

3. Rule.-A fraction is not altered by dividing the Namer and Numberer by the same number.

This may be proved to the child by showing (as above) that if you diminish the Namer you increase the size of the parts of unity, and if you diminish the Numberer, i.e. the number of those parts of unity, the same number of times, the fraction must remain unaltered.

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But it would probably be sufficient and more intelligible to illustrate this truth by examples, thus: We have seen that = 2 16, where the Numberer and Namer have been multiplied by 2, by 4, by 8. Reversing these, we see that = † = } = }. Here the Numberer and Namer have been divided by 2, by 4, and by 8, and yet the fraction has remained unaltered in value.

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"Add together a halfpenny and a farthing; what is the result?” Three farthings. "Exactly; and in order to get this result, what did you do to the halfpenny?" I turned it into farthings. "Now add a half and a quarter; what is the result?" Three quarters. "And in order to get this result, what did you do?" I turned the half into quarters.

"Now when you add pence and farthings, or pounds and shillings, or tons and hundredweights, or, generally, a number of things of one name or denomination to a number of things of another name or denomination, you reduce them to the same- -?" Denomination. "Exactly; and you have to do the same thing with fractions; but as the name or denomination of a Fraction depends on its Namer, or

1 By this time the pupil should be introduced to the terms Denominator (for Namer) and Numerator (for Numberer).

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