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10 = .... 2s 10 = .... 3s and .... rem. 10 = .... 4s and .... rem. 10 = .... 5s 11 = .... 2s and .... rem. 11 = .... 3s and .... rem. . . . 89 = .... 9s and .... rem. |
These bonds must be formed before short division can be efficient, are useful as a partial help toward selection of the proper quotient figures in long division, and are the chief instruments for one of the important problem series in applied arithmetic,—"How many xs can I buy for y cents at z cents per x and how much will I have left?" That these bonds are at present sadly neglected is shown by Kirby ['13], who found that pupils in the last half of grade 3 and the first half of grade 4 could do only about four such examples per minute (in a ten-minute test), and even at that rate made far from perfect records, though they had been taught the regular division tables. Sixty minutes of practice resulted in a gain of nearly 75 percent in number done per minute, with an increase in accuracy as well.
(4) The equation form.—The equation form with an unknown quantity to be determined, or a missing number to be found, should be connected with its meaning and with the problem attitude long before a pupil begins algebra, and in the minds of pupils who never will study algebra.
Children who have just barely learned to add and subtract learn easily to do such work as the following:—
Write the missing numbers:—
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4 + 8 = .... 5 + .... = 14 .... + 3 = 11 .... = 5 + 2 16 = 7 + .... 12 = .... + 5 |
The equation form is the simplest uniform way yet devised to state a quantitative issue. It is capable of indefinite extension if certain easily understood conventions about parentheses and fraction signs are learned. It should be employed widely in accounting and the treatment of commercial problems, and would be except for outworn conventions. It is a leading contribution of algebra to business and industrial life. Arithmetic can make it nearly as well. It saves more time in the case of drills on reducing fractions to higher and lower terms alone than is required to learn its meaning and use. To rewrite a quantitative problem as an equation and then make the easy selection of the necessary technique to solve the equation is one of the most universally useful intellectual devices known to man. The words 'equals,' 'equal,' 'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer equivalents should therefore early give way on many occasions to the '=' which so far surpasses them in ultimate convenience and simplicity.
(5) Addition and subtraction facts in the case of fractions.—In the case of adding and subtracting fractions, certain specific bonds—between the situation of halves and thirds to be added and the responses of thinking of the numbers as equal to so many sixths, between the situation thirds and fourths to be added and thinking of them as so many twelfths, between fourths and eighths to be added and thinking of them as eighths, and the like—should be formed separately. The general rule of thinking of fractions as their equivalents with some convenient denominator should come as an organization and extension of such special habits, not as an edict from the textbook or teacher.
(6) Fractional equivalents.—Efficiency requires that in the end the much used reductions should be firmly connected with the situations where they are needed. They may as well, therefore, be so connected from the beginning, with the gain of making the general process far easier for the dull pupils to master. We shall see later that, for all save the very gifted pupils, the economical way to get an understanding of arithmetical principles is not, usually, to learn a rule and then apply it, but to perform instructive operations and, in the course of performing them, to get insight into the principles.
(7) Protective habits in multiplying and dividing with fractions.—In multiplying and dividing with fractions special bonds should be formed to counteract the now harmful influence of the 'multiply = get a larger number' and 'divide = get a smaller number' bonds which all work with integers has been reënforcing.