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The last principle includes several principles of useful application, either implied or stated explicitly in the discussions upon number and its development.

(a) The order of numerical factors may be changed. (6) Multiplying a factor by any number multiplies the product by the same number. (©) Dividing a factor of any number divides the product by the same number. (d) Multiplying the dividend by any number multiplies the quotient by that number. (e) Dividing the dividend by any number divides the quotient by the same number. (f) Multiplying the divisor by any number divides the quotient by the same number. (9) Dividing the divisor by any number multiplies the quotient by the same number. (k) Multiplying or dividing both divisor and dividend by the same number leaves the quotient unaltered. (2) All these principles are necessarily involved in the principles of number as already unfolded. The following is worthy of attention: In an aggregate whose terms contain multipliers and divisors, the multiplications and the divisions are to be performed BEFORE the additions and the subtractions are made.

CHAPTER XII.

MEASURES AND MULTIPLES.

GREATEST COMMON MEASURE.—The pupil who has been led to have a clear idea of number—who has been tanght to look upon the unit as the measurer—will find no difficulty in mastering greatest common measure. With all the preliminary notions he is familiar, and it will be an easy matter to pass to the formal process.

While in the illustrations given in this chapter we generally use the pure number symbols, it must be borne in mind that here, as everywhere in number and numerical processes, the idea of measurement is to be kept prominent, especially in the introductory lessons. A common factor is a common measure—a unit of measure that is contained in two or more quantities an exact number of times. A common multiple is a definitely measured quantity, which can be measured by two or more quantities, themselves measured by units of the same kind and value as those of the given quantity. The teacher should see to it, then, that all his illustrations and examples deal with the concrete ; that the measuring idea be kept prominent from first to last.

Easy Resolution into Factors.—Taking the number 15, the learner sees that it can be considered 3 fives, or 5 threes; the five or the three is a measurer or measure of 15, and the equation 15 = 5 X 3 puts in evidence the fact that 5 and 3 are measures or factors of 15. Taking 35, he sees the significance of the equation 35 = 5 x 7. He further notes that 5 is a measure of each of the numbers 15 and 35, and is therefore a common measure. If, next, the numbers 12 and 18 are taken, he will see that all the ineasures of 12 are

1, 2, 3, 4, 6, 12; and that all the measures of 18 are—

1, 2, 3, 6, 9, 18. Then it will be seen that 1, 2, 3, 6 are common measures of 12 and 18, and that while there are several such measures, there is one that is the greatest—the one that will be called the greatest common measure. Before any process is taught the class should be exercised in the working of easy examples, both mental and written; being asked to find common measures, and the greatest common measure of 16 and 24, of 24, 36, 48, etc. An additional interest will be secured by proposing some simple practical problems.

It will be better, before beginning the ordinary formal treatment, to have exercises in finding the greatest common measure, by resolving the numbers given into their simple factors. It would be necessary, then, to recall or develop a certain fundamental principle. The

. 2)60 division 3130 is to be interpreted, first, that 60 is 30

10 twos, and, next, that the 30 twos are 10 three-twos or 10 sixes; and thus that if a number contains the factor 2, and if the quotient contains the factor 3, the number itself contains the factor or measure 6. Then, since

108 = 2 X 2 X 3 X 3 X 3,

and 72= 2 X 2 X 2 X 3 X 3, we may see that all the single common factors are 2, 2, 3, 3; and that, therefore, 2 X 2 X 3 X 3, or 36, is the greatest common measure. Practice on this method will find a place : the pupil has a new interest, and the teacher can take advantage of it to secure further training in number and in the elementary processes.

The General Method.But soon it will be found that this method is limited, as its successful application depends on the pupil's ability to discover a factor. An example, such as, Find the greatest common measure of 851 and 1073, we may suppose to have been given the class, and found beyond their present power of factoring. The reason for the failure will be manifest to them—their inability to find any factor of either number. The need for some new, or, it may be, extended method, is felt; and this need is the teacher's opportunity for introducing the more powerful method, and for the development of it he has his class in a state of healthy, natural, unforced interest.

The Fundamental Principles.-To develop the method, it would be well to turn aside from the example attempted and give attention to certain facts upon which the method is based. Taking for illustration the numbers 21 and 35, we see, as before, that 21 is 3 sevens and 35 is five sevens. Thus, if 21 is added to 35 we shall have 3 sevens, and 5 sevens or 8 sevens; the seven being the unit of measure, or measurer. Similarly, if 21 is subtracted from 35 the result А

will be 2 sevens. Further, if to 21 is added 3 times 35, we have 3 sevens and 3 times 5 sevens—that is, a certain number of sevens. This is seen to be true for any number of times seven, any number of times eight, or nine, . . . etc. Actual measurements will make the principle still clearer. Thus, if A B and C D have a common measure, it must measure A B exactly, and C D exactly :

в с E D and measuring off on CD a part C E= to A B, the common measure must measure C E exactly, and therefore ED exactly, because it measures the whole of CD; but E D is the difference of the quantities, etc. In the same way E D may be measured off on A B, and the same reasoning will apply. Thus the pupils are led to see certain general principles, and to see them in their generality.

1. From the fact that if we take the sum or the difference of 21 and 35—that is, of 3 sevens and 5 sevens —or the sum or the difference of any number of times 21 and any number of times 35, we are sure to have a number of sevens (seven representing any measured quantity whatever), it is plain that any number which measures two numbers will measure their sum or their difference, or the sum and also the difference of any of their multiples. The pupils can be got to develop the general form of this principle. If c is a common measure of a and 6, so that a= mc, and b = nc, then a +6 =mc+nc, etc.

2. Because the common measure of two numbers measures their sum, and because the minuend, in a sub

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