Arithmetic. (Article. January 1896.)
In our sixth paper we closed with the discussion of addition of fractions. The thoughtful reader has, in a measure, anticipated our treatment of subtraction of fractions. In our first group of problems we suggested subtracting an integer from an integer and a fraction. For example, 219 ¾ - 75 = ? The pupil has had much practice in subtracting integers. In reality, he finds nothing new in the class of problems suggested except that he is required to write a fraction as a part of the remainder. He is delighted to find that he can do this. He is now ready to consider the second group of problems, a group of which both minuend and subtrahend contain a fraction, but the fraction in the subtrahend is of less value than the fraction in the minuend. For example, 195 ¾ - 68 ½ = ? It will be necessary for the teacher to dictate these problems, because the treatment here given is not found in any of the standard text books.
In the third group the minuend consists of an integer, the subtrahend, of an integer and a fraction, the fraction of the subtrahend being of greater value than that of the minuend. It is not necessary to go into the details of how the difficulties involved in this work are to be easily overcome. The difficulties are precisely the same as the difficulties the pupil encountered in studying subtraction of integers. All through fractions, the teacher should endeavor to show her pupil the value of the knowledge he has previously acquired, and endeavor to convince him that there are not so many new things in fractions as he had anticipated.
We are confident that if the four cases in subtraction of fractions are presented in the order indicated, and no succeeding case presented until the preceding one is thoroughly mastered, the average pupil will encounter little or no difficulty. Thus far we have not even so much as mentioned what is commonly presented as the first case in subtraction of fractions. In most arithmetics the pupil is asked to solve such problems as ⅞ - ⅔ = ? In the first place, the pupil has mastered this problem in oral work. He is now ready to use it in its proper place. Its proper place in written arithmetic is, ordinarily, in relation to integers. As in every other division of arithmetic that we have considered, the application of subtraction of fractions should be made to business transactions. We have not thought it necessary to enter into any detailed discussion of the amount of time that should be devoted to the abstract as compared with the concrete.
VII - SUBTRACTION OF FRACTIONS
Multiplication of fractions is a comparatively easy subject. An example illustrative of the first group of problems is the following: 247 - ½ x 48 = ? The pupil's preliminary training in mental arithmetic enables him to handle the difficulty involved in getting 48 x ½. This group of problems he will solve with ease and delight. In reality, there is but little difference between this problem and scores of problems he has worked in multiplication of integers. If the class is slow, if the majority of the pupils have not had the necessary preliminary training, it would be wise to make more cases, so to speak, in the treatment of every subject in arithmetic. Well trained pupils should be able to take as illustrative of the second case, 420 x 7 ½ = ? Pupils who learn fractions with difficulty should be given as a second case such problems as 397 ¾ x 64 = ? The reader will observe that not a small number of the pupils have no difficulty in multiplying ½ by 48, but when they attempt to multiply ¾ by 64 their difficulties frequently assume large proportions. They are often helpless. We suggest that under all circumstances the pupil be led to discover that ¾ x 64 is the same as the product of multiplying 64 by ½ plus the product resulting from multiplying ¼ by 64. This suggestion is worthy of very important consideration. The teacher who has made a close study of the difficulties encountered by most pupils will grant this without hesitation. Granting that the last problem illustrates the second case, our second problem, as given above, would illustrate our third case; and our fourth case would be exemplified in 240 x 37 ⅞ = ? It is needless to say that the suggestion offered in relation to multiplying ¾ by 64 is applicable to getting ⅞ of 240. ⅞ is the equivalent of ½ + ¼ + ⅛. Some teachers will object that there is more work involved by following this suggestion. This is not true, for the simple reason that, as a rule, this work can be done mentally. The habit of performing work mentally gives the pupil a tremendous increase in speed and accuracy, making a positive saving in nine-tenths of all his computations that involve multiplication. It is hardly necessary to suggest that for the next set of problems we would have an integer and a fraction to be multiplied by an integer and a fraction, but in doing this we might make two groups of problems if we liked. To illustrate, we might first take 560 ¼ x 225 ⅓. Pupils who are prepared to do this work should not be allowed the privilege of changing the multiplier and the multiplicand to improper fractions.
It may be wise to hint that there is no special objection to presenting the illustrative problem on the blackboard. In doing this the teacher may "cover up" the fraction in the multiplicand. The pupil will solve this problem because he has already mastered the difficulties involved in it. Then the teacher may uncover the fraction of the multiplicand and lead the pupil to take the finishing step. In the next group the numerator in the multiplier and the multiplicand will be other than a unit.
This is, "by all odds", the most difficult case of multiplication of fractions, and should be treated in harmony with the suggestions offered for teaching the preceding case. If, for any good reason, the teacher thinks that ten or fifteen problems should be solved in which there is no integer in either multiplier or multiplicand, he may present such problems. Strictly speaking, this is not necessary because this class of problems has already been solved in performing the work already outlined, for in the work that he has already done he has been obliged to employ the following method: For example, multiply ⅞ by ⅗. He has obtained ½ of ⅗, ¼ of ⅗, and ⅛ of ⅗. This he has performed mentally and has thought little about what is commonly termed multiplying a fraction by a fraction.
If, however, it seems necessary to present in a formal way this case of multiplying a fraction by a fraction, we offer an explanation somewhat different from that already hinted at. In the previous paper we suggested that the teacher should exercise great care in presenting what are commonly termed the "laws of fractions". This having been done, the solution and explanation of the following problem will illustrate our method. Let the reader bear in mind that we do not claim originality in any of this work. Thus far in fractions we have discarded the ordinary graphic language that is used by most teachers in presenting these cases. The business man has lost nothing in mental discipline by discarding this language. The teacher loses nothing, the pupil gains. We are aware that those who believe in presenting the subject of algebra in the grades below the grammar school will take exception to this view. That we will not discuss here.
⅘ x 3⁄ 7 = ? We will first multiply ⅘ by 3. According to a preceding principle, to multiply a fraction by an integer we multiply the numerator or divide the denominator. In this instance we would multiply the numerator, 4, and we obtain 12⁄ 5. We have multiplied ⅘ by 3, but we were told to multiply ⅘ by 3⁄ 7, therefore, we have used a multiplier that is 7 times too large. This gives a product that is 7 times too large. To correct this, divide the product by 7. According to a preceding principle to divide a fraction by an integer, we must divide the numerator or multiply the denominator. In this instance we will multiply the denominator. Doing this we have 35 for a new denominator, and ⅘ multiplied by 3⁄ 7 equals 12⁄ 35.