Arithmetic. (Article. June 1896.)
It is not my purpose to debate in this paper the position that decimal fractions should occupy in ordinary written work. On account of the close relations existing between integers and decimal fractions, some good arithmeticians and teachers have deemed it wise to teach integers and decimal fractions simultaneously. The writer of this article is inclined to believe that decimal fractions are taught most successfully after common fractions have been carefully and thoroughly taught. He believes in making decimal fractions a division of common fractions, and would, to a certain extent, treat them as such. The writing and reading of decimal fractions is in the light of the pupil's knowledge of integers a comparatively simple matter. The laws that hold good in relation to integers hold good in relation to decimals, the decimal point being employed for the sole purpose of indicating the separation of the unit from the fraction. It seems wise in writing a simple decimal fraction to use a cipher to indicate the absence of units. In not a few scientific books, especially in those of European countries, the cipher is thus employed.
In changing common fractions to equivalent decimal fractions, we recall the student's knowledge of changing a common fraction to an equivalent fraction having a required denominator. In the case of changing a common fraction to a decimal fraction, we always ask that the required denominator be ten, one hundred, one thousand, or some power of ten. It is not at all necessary, however, that this method should be pursued. We may multiply the numerator by ten, one hundred, one thousand, divide by the denominator, and by analysis discover that if we multiply the numerator by ten, one hundred, or one thousand, we have a quotient that is ten, one hundred, or one thousand times too large. To correct this quotient divide it by ten, one hundred, or one thousand. The pupil has learned to do this in the division of integers, especially when he was required to divide by ten, one hundred, or one thousand.
In changing a decimal fraction to a common fraction he has simply to express fully in writing what he reads, and reduce said fraction to an equivalent fraction in its simplest form. It seems an imposition upon the reader to mention this very simple explanation, but insomuch as a large number of teachers are not familiar with any explanation we have ventured to explain.
In the teaching of the addition of decimals we have simply to remind the pupil of his experience in adding integers. There his starting point is units; in decimal fractions his starting point is units. Units are placed under units, tenths under tenths, hundredths under hundredths, etc. Unless he is thoroughly trained in the importance of recognizing units as the foundation of writing and reading decimals he will be startled to find that his right hand columns are more or less incomplete. If, however, his training in writing and reading decimal fractions has led him to discover that these blank spaces to the right, here and there, are of no significance he will proceed to add decimal fractions as readily as he ever did integers.
In subtracting decimals nothing need be said in the light of what the pupil has learned in subtraction of integers, together with what has been offered in our discussion on addition of decimals.
In multiplying a decimal fraction by a decimal fraction the careful reader of this series of articles will readily anticipate. Certainly we would not present a simple decimal fraction to be multiplied by a simple decimal fraction. We will first multiply an integer and decimal by an integer; for example: multiply 12.7 by 15. We proceed to multiply 127, instead of 12.7, by 15. We have 1905 for our product. We were told to multiply 12.7, we have multiplied 127, consequently, we have used a multiple that is ten times too large, and have as a result a product that is ten times too large. To correct this product, divide it by ten. This can be done by pointing off units according to what was learned in simple division of integers. We have 190.5 for our product. We then ask the pupil to multiply 12.25 by 1.8. We multiply 1225 by 18 and have 22, 050. We were told to multiply 12.25, consequently, we have multiplied a multiplicand ten times too large. Our product, 22, 050, is, therefore, a hundred times too large. To correct this product divide it by a hundred. Doing this we have 220.50 for a product. We have multiplied by 18, but we were told to multiply by 1.8, consequently, we have multiplied by a number that is ten times too large. Our product, 220.50, is, therefore, ten times too large. To correct this product, divide it by ten. Doing this we have the product of 12.25 multiplied by 1.8, 22.050 or 22.05. After the class has worked, under the eye of the teacher, ten or twelve problems in which both multiplier and multiplicand contain a decimal fraction, the class may be led to generalize as to pointing off. In doing this he discovers that the product contains as many places in the decimal as there are decimal places in both multiplier and multiplicand. Hence, the rule that is usually offered for multiplying a decimal fraction by a decimal fraction.
We select an easy problem in division: divide 2.875 by 2.3. The pupil divides 2875 by 23. We have divided 2875 but we are told to divide 2.875, consequently we have used a dividend that is one thousand times too large, therefore, the quotient is one thousand times too large. To correct this we divide the quotient by one thousand. We now have for the result of dividing 2.875 by 23 = .125; but we were told to divide by 2.3, whereas we have divided by 23, we have a divisor that is ten times too large, and in consequence have a quotient that is one-tenth of the correct quotient. To correct this quotient we multiply it by 10, and have for the result 1.25. After the class has worked a dozen or more of carefully selected problems the rule for pointing off in division of decimals may be deduced. We are aware that difficult cases arise and it is possible that some of our readers would be glad to have these cases presented and explained. We refrain from doing this, because in the more troublesome problems, both dividend and divisor can be changed to a common denominator by annexing ciphers. Having changed the dividend and divisor to a common denominator, we have a problem in one of two forms. It is a problem in which the dividend exceeds the divisor or the converse. It must be either the one or the other. We have shown the reader how easy it is to lead the pupil to master the difficulties in both of these forms.
We have said very little in each of the divisions of our subject-decimals. Intelligent readers do not need to be told that a practical application to the problems of business life would be presented in addition, subtraction, multiplication, and division of decimals. From what we have already written this requirement is obvious. The teacher should now give a large number of miscellaneous problems; in other words, he should pursue practically the same course in cultivating analysis that he pursued in presenting the miscellaneous problems of common fractions. He should especially emphasize in the mental work in teaching the use of hundredths in multiplication and division. This training will pave the way for good work in percentage. It is often wise at this point to give miscellaneous problems involving both common and decimal fractions in all their forms and relations.
We do not present the topic of "circulating decimals". There could be no special objection, as a matter of history, to the presenting of this subject to a class making an exhaustive study of arithmetic and its history. But so far as the majority of students are concerned this subject has no place in modern arithmetic work. The writer of this article would like to have readers ask questions. These questions may be published in Education Extension, or may be sent directly to the author. Questions and answers would make this series of articles much more valuable.