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Arithmetic. (Article. January 1897.)

In the preceding articles no attempt has been made to treat each subject exhaustively. A general plan has been outlined and a few suggestions presented. In treating percentage, we shall make an attempt to proceed more thoroughly. In the beginning of this subject, it is quite important that the teacher recognize the importance of demanding that the pupils pursue the subject of percentage on condition that they have already mastered certain phases of arithmetic. Any boy or girl who is master of common fractions has few difficulties to overcome in percentage. It deals with numbers with special reference to hundredths as a common measure. It is quite true that these hundredths are sometimes converted into equivalent fractions, having their simplest form. If the pupil has made this preparation, if he is thoroughly familiar with fractions, if he is rapid and accurate, he will find percentage and all of its applications a delightful study. In other words, he has no arithmetic proper to learn. The arithmetic of percentage he has already mastered, and now he has simply to make himself familiar with business usages.

In profit and loss he applies his knowledge of fractions to the transactions that involve the making and losing of money, and learned that the key to this subject is cost price. In commission he recognizes the important truth that merchandise is bought and sold under certain conditions for the owner, the owner paying for this kind of labor what he terms a commission. The key to commission is work done. With this in mind he is not misled into supposing that there are two methods for computing commission. In discounting notes and drafts he is introduced to another phase of business that he must master apart from pure arithmetic, a business in which the key is the value of paper when due. In stocks and bonds he has representative values to consider. These representative values are bought and sold in the stock market as the necessaries of life are bought and sold at our stores. The key to stocks and bonds is the faithful consideration of par value. The broker's fee is always estimated on par value.

We give this introduction in order that the teacher may intelligently confront the task that lies before him. In beginning the subject of percentage he has nothing new in arithmetic to teach, but something new in business. In fact, his part of the work ought to be along the line of teaching business usages. Assuming that students have had the necessary preparation, and recognizing the fact that they must get in touch with the business world, the task of pointing out the royal road is comparatively easy.

In the very first lesson, we assume that the pupils are thoroughly familiar with the common aliquot parts of a hundred; for example, the equivalents in hundredths of the following fractions: ½, ¼, ¾,⅛, ⅜, ⅝, ⅞, ⅓, ⅔, ⅙, ⅚, ⅕, ⅖, ⅗, ⅘, 110, 112, 116, 120, 125. But suppose the class do not know these aliquot parts of one hundred. There is but one thing for the teacher to do, and that is to set about teaching these aliquot parts. But the rural teacher says, "In the event that I do this I shall be blamed for not following the book, for not assigning a given number of problems, for not advancing the pupils." We do not advise any teacher to deceive the patrons. After all, tact must be displayed in the home, in the school, yes, everywhere, in order to secure the best results. Assign the minimum amount of written work and make two-thirds of ever recitation oral along those lines that are fundamental, along those lines that lead to a thorough understanding of the subject of percentage. The larger part of the successful teaching of percentage must be oral. There is no escape from this conclusion. In order that there may be no possibility of a mistake at this point we suggest to every live teacher that he precede every case, every variation, in percentage with oral arithmetic drills.

In order to help inexperienced teachers we offer a few words in relation to the best method of presenting the written work. Under no circumstances is it necessary to teach the terms principal, rate, base, amount, percentage, difference. So far as possible we wish the pupils to do what we have already taught them to do. Every so-called principle in percentage has been fully presented in common fractions. Therefore the pupil is led to apply what he already knows. For example, what is 19 per cent of $450? 1 per cent of $450 is 1100 of $450 of $4.50. Since 1 per cent of $450 is $4.50, 19 per cent is 19 times $4.50 or $85.50. The pupil cannot fail to recognize the similarity between this explanation and the explanation for the following: If 3 oranges cost $.12, one orange cost ⅓ of $.12 or $.04. We now have the value of this unit. Since one orange costs $.04, 13 oranges cost 13 times $.04 or $.52. Nothing can be gained by regarding the $450 as the base; on the contrary, much is lost. The ordinary pupil supposes that base is something new, something that he must investigate, something that is subject to new operations, consequently he is prone to ask for a rule and prone to work by a rule. If he is thrown back on analysis, he uses his reason, and quickly discovers that percentage is not a new subject. It is simply an old friend at work in the business world.

Another example: If 14 per cent of my money equals $280, what is my money? If 14 per cent of my money is $280, 1 per cent of my money is 114 of $280 or $20. If $20 is 1 per cent of my money, 100 per cent of my money, or my money, is 100 times $20 or $2,000. Again the student is compelled to recognize the necessity for ascertaining what 1 per cent is.

Another example: $39 is what per cent of $130. 1 per cent is $130 is $1.20. Since $1.30 is 1 per cent of $130, $39 is as many per cent of $130 as $1.30 is contained times in $39, which is 30 times. Therefore, $39 is 30 per cent of $130.

There are no more cases in percentage, notwithstanding the fact that much is said in the arithmetics about ascertaining the principal when the amount and rate are given or when the difference and rate are given. To illustrate: A farmer raised 2040 bu. of wheat this year. This is 2 per cent more than he raised last year, how much did he raise last year? Without going into detail, 2040 bu. of wheat equals 102 per cent of last year's crop. 1 per cent of last year's crop is as many bushels as 102 is contained times in 2040 or 20 times. Hence 1 per cent of last year's crop is 20 bushels. 100 per cent of last year's crop equals 2000 bushels. It simply resolves itself into one of the preceding cases. Let the teacher give his pupils a great variety of problems involving the forms of analysis already given. Then he may proceed fearlessly with any phase of business in which the principles of percentage are involved.

Source: Ferris, Woodbridge N. "Arithmetic." Education Extension. 3:1 (1897).