Page 99 : ARITHMETIC — ARITHMETIC
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At least its earlier use tends strongly to make the work too formal.
There is little object in carrying the multiplication table beyond 10 × 10. In compound numbers reduction “ascending” and “descending” should be confined to numbers of not more than three denominations. The reasons for this are that in practical life we rarely use more than two denominations, as feet and inches or pounds and ounces; and that, if one has learned to perform reduction with two and three denominations, he can easily perform those with more if occasion required.
Quantitative facts are so much more often expressed decimally now than formerly, that much more attention to decimal fractions is in place.
The addition and subtraction of decimals need offer no difficulties. In multiplication the most approved forms are the following:
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Explanation—Since 5 times hundredths are hundredths, the right hand number of the product is placed under hundredths. The rest of the work is identical with that of integers, the decimal point going under the others. | |||||||||||
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Explanation—Since hundredths multiplied by tenths is thousandths, the right-hand figure of the product goes in the thousandths place. | |||||||||||
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Explanation—Since hundredths multiplied by hundredths are ten-thousandths, the right-hand figure of the product goes in the ten-thousandths place. |
Operations with decimals should be limited to fractions having not over three places, and answers need not be carried beyond three places.
Division of decimals should be taught as suggested in the following Austrian method:
Required to divide 6.275 by 2.5—
Old method
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“Point off as many places in the quotient as the number of decimal places in the dividend exceeds that in the divisor.” |
Common Austrian method
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Dividend and divisor having been multiplied by such a power of 10 as makes the divisor a whole number, the decimal point in the quotient simply goes above that in the dividend. |
The following method is recommended for the early work:
2.51 25) 62.75 50 12.75 12.5 0.25 0.25
The entire remainder is brought down each time, and the decimal point is preserved throughout.
In more advanced arithmetic, including the last two or three years of the elementary school, the value of the work must lie largely in the character of the problems, as previously suggested. By the time a child has reached the sixth year of school, he has usually acquainted himself with the various arithmetical processes, and he is now ready for their various applications to actual conditions in life. Correlation with geography, manual training and other studies is, therefore, of much importance.
Percentage, formerly a topic by itself, is merely one phase of decimal fractions, and should be so treated. A large part of business arithmetic involves the finding of per cents, so that the method is continually applied after it is once presented. The treatment of the subject by “cases,” and the learning of definitions of terms like “amount,” “difference” or even “percentage” may be considered obsolete. There is need to know what “per cent”, means, namely “hundredths” (“hundredth” or “of a hundredth,” as in 6%, 1%, %), and there is occasionally some value in using the term “base.” But the two leading problems of the subject are illustrated by two examples not requiring any elaborate vocabulary, namely:
1. 6% of $250 is how much?
2. If 104% of x = $7.28, what does x equal ?
Practical problems in percentage rarely require any other forms.
The explanation of problems should consist of no carefully learned formula, but should be nothing more than an explanation of the steps involved, with the reasons. Some use of the equation, with x to represent the unknown quantity, is fully in place.
In general in the study of arithmetic pupils are tempted to “figure ” too much, and to allow the formal side to dominate the “thinking” side. To overcome this difficulty it is well to have much oral work in the solution of problems, without any figuring. To emphasize the thought side of arithmetic properly, children (1) should often read a problem a second or third time carefully, to get the exact conditions; (2) should then restate the problem in their own words, to make fully sure that they understand its condition; (3) should state
