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ON THE METHOD OF TEACHING
FRACTIONAL ARITHMETIC.

By A. DE MORGAN.

(From the Quarterly Journal of Education, No. X.)

In the preceding article we developed a method of giving the first notions of whole numbers to children. We now proceed to treat the fractional part of arithmetic iu the same manner; premising, however, that on no account should this ground be entered until the pupil has the clearest notions, not only of the method of numeration, but of the first four rules in whole numbers. We do not mean that he should be ready at the solution of questions which involve high numbers, that is, at the mechanical part of the subject; but he should at least be competent to perform any addition or subtraction of not more than four figures, any multiplication of two figures by two figures, and division of three figures by two others.

In treating of whole numbers, where it was sufficient that each one should be like the others, we used marbles or counters: these should now be entirely rejected; the child will be confused by any attempt to divide them into parts, as the whole and its parts will not then be entirely of the same character. So long as nothing more was necessary than to compare one counter with another, all was well, because each unit entirely resembled every other unit; but if we were now to cut these ones into fifths, the fifths would not be of equal dimensions, nor could the child make a one out of any five fifths. Neither will it be sufficient to take any number of balls,

say six, and calling the whole six one, to call each ball one-sixth, since this method, though advisable at a later stage, would only introduce confusion at present. It is, therefore, desirable that the unit should now be perfectly simple, and capable of division into parts exactly like itself; for which purpose we shall adopt length as the object of measurement. We must also observe that the preliminary notions are now to be almost entirely created, while they did exist, though in a rude forın, when the pupil first began the study of whole numbers. He has been accustomed to the consideration of several things of the same kind, but rarely to that of the division of one of these objects into equal parts. His half has, most probably, been merely a division into any two parts whatsoever, and he can accordingly, with perfect consistency, talk of the larger and the smaller half. There is, therefore, more preliminary work, and, as in the case of whole numbers, palpable means of instruction should be adopted. We should recommend the following simple apparatus, making no apology for putting the parent to further expense and trouble: if there be any one of this class who does not think the education of his children the very first object of his life, next to procuring subsistence for them, and arithmetic a most im. portant part of that education, we advise him to laugh at us, and put this paper aside, -we are not writing for him,

Let from twelve to twenty slips of cheap wood be procured, each exactly one foot in length, and about two-tenths of an inch in breadth and thickness. Let the first of these be divided by a line, or a scratch extending all around it, into two equal parts; the second into three equal parts; the third into four equal parts; and so on up to the eleventh, which will be divided into twelve equal parts. These will be enough at first, but we should further recommend several other slips, divided respectively into 36, 60, 84, 90, 100, 120, and 180 equal parts; which numbers are chosen on account of their having a great number of divisors, considering their magnitude. On the small ends of each rod let the number of parts be marked into which the foot is divided. The following is a representation of the rod in which the foot is divided into quarters :

A common pair of compasses will also be necessary. It might be thought that one rod would serve the purpose of four, if the sides were differently divided, but this is not the case, as will be seen hereafter; and also that the halves and quarters might be placed on the same rod, and so on; which, however, would be directly against our object, as tending to something like taking for granted one of the simple propositions, of which we wish to furnish a palpable proof.

These rods having been laid side by side, the pupil is called upon to observe that they are all of the same length, being that which he has used and called a foot, if the suggestion of our last article (page 55) has been adopted; if not, that method should be now put in practice previously to going any farther. He should then have his attention drawn to the fact that each foot rod is divided into equal parts, that is, that the subdivisions of each rod are equal, but that those on different rods are unequal. This he should ascertain by measurement with the compasses. Again, he is to

abserve that two equal parts make the whole foot on the first rod, three on the second, and so on; and that the number of divisions on each rod is marked on its extremity. Let him call this number the denominator of the rod; the meaning of the verb to denominate having been previously explained by familiar instances, Let it now be told him that when three equal lines make up a foot, each is called one third of a foot, and so on; illustrating this phraseology by questions of the follow, ing form :-'Twelve pence make one shilling ; what part of a shilling is one penny ?' until he is ready with the answers. At last, let the half feet be shown him, and the name of one of the parts required ; if he answer one second of a foot, inform him that he is right, but that the words one half' are usually substituted for • one second ;' if he answers one half' of a foot, ask him what it should be, if it were named like the rest ; and thus lead him to the phrase 'one second.' Follow the same process with the words 'one quarter' and * one fourth.'

If any person should think this detail rather minute, we are glad of this opportunity to inform him that it is not every pupil, educated in the ordinary manner, who knows what are the denominators of the fractions one half and one quarter without some reflection, and occasionally a mistake.

The next step should be to establish in the mind of the pupil the order of magnitude in the simple series one half, one third, &c. He is not by any means so familiar with the fact that one fourth is less than one third, as he is with its reciprocal, that four are greater than three; and, to accustom him to feel the first as strongly as the second, he must repeat it often, accom

panied by reference to the simple reasoning which connects the two. He should then be exercised on the different fractions, by such questions as the following: What is meant by one fourth of a foot ?-What is the denominator of this fraction ?-Name other fractions which are greater and less.-How many times is one fourth of a foot contained in one, two, three, &c., feet? or, What is two feet in fourths of a foot, &c. ? The abstract fraction, one fourth, should never be mentioned : it should, at first, always be one fourth of something tangible, and actually present ; next of soine sensible object not present; and lastly one fourth by itself, only when the pupil is able, without hesitation, to answer all the previous questions. Neither should the abbreviation | be yet introduced. We may lay it down as a general rule that no new symbol should ever make its appearance until the words which it expresses are so well understood that the learner can exemplify them at length without an instant's hesitation,

The next step is to habituate the pupil to the more complex fractions—three fourths, two sevenths, &r. This may be done by taking one of the rods, that which shows fifths, for example, holding it so as to let only two appear, and asking the learner to count them. To the question How many are here ? should succeed Two of what?' the answer being two fifths of a foot; and afterwards 'How many fifths are left?' This process having been continued until the learner can show any fraction whatever whose denominator is within the compass of the rods, he should proceed to simple questions of addition and subtraction where the denominators are the same; such as-How many do two sevenths and three sevenths make ? - What remains

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