677. Teachers should carefully avoid giving unnecessary "rules." There is no good reason why an average pupil should not be able to determine for himself how to ascertain what part of $15 a man has spent when he has spent $5. While the introduction of fractions into such an example makes it more difficult for the scholar to give the answer off-hand, his instruction up to this time should have taught him that the same process is to be employed. A pupil should be required to depend upon himself to at least a reasonable extent.

678. As a preliminary to the work in denominate numbers in the next three pages, the teacher should place on the board a few such examples as the following, to which the scholars should give answers at sight:

Nearly every member of the class will be able to obtain the results in a moment, without any suggestions from the teacher. If the examples are left on the board, the pupils can refer to them for aid in working some of those found in the text-book.

The teacher that wishes to develop power in her scholars should be careful not to give a particle more assistance than is necessary. She should permit the children to deduce from the above examples the rules necessary to solve the others, being patient if the pupils are somewhat slow in doing this work. When, however, a circuitous method has been employed, she should lead the class to see how the work can be improved by the use of a shorter way.

680. It may be necessary to take up again, for purposes of review, the preliminary exercises of the previous chapter. See Art. 569, pp. 56 and 57.

681. As the table of square measure is not introduced until the next chapter, it will be necessary to reduce to yards the dimensions that are given in feet or inches.

No. 15. 8 pieces, each 36 yd. long and 3 yd., or § yd. wide. No. 18. See Arithmetic, Art. 818, problem 20. A modification of this diagram, showing four squares instead of four rectangles will be the drawing required, except that the squares above and below need not necessarily occupy the positions there indicated.

ΧΙ

NOTES ON CHAPTER EIGHT

With this chapter begins the regular work in decimal fractions, and the pupils should now be taught the principles underlying the various operations.

685. While pupils may know that 23 means that 23 is to be divided by 8, it may be well to lead them again to see that

is the same as 34, or of 3. After they understand that every common fraction may be considered an "indicated division," they will understand that the decimal fraction obtained by performing this operation is the equivalent of the common fraction whose denominator is used as a divisor and whose numerator is used as a dividend. See Arts. 563 and 564.

686. As previous work in decimals has been confined chiefly to three places, some review and extension of the notation and numeration exercises of Arts. 547-551 may be necessary.

687. After writing each of these decimals in the form of a common fraction, a scholar should be able to determine at a glance whether or not it can be reduced to lower terms. This reduction is possible when the decimal is an even number or terminates in a 5.

While it is inadvisable to waste time in calculating the greatest common divisor, pupils should be encouraged to use large divisors; 4 rather than 2, when possible, and 25 rather than 5.

688. The common fractions contained in these exercises are such as do not require much calculating to change them to deci

mals. The scholars should be able to write the numbers in vertical columns directly from the text-book, making the necessary reductions mentally.

In reducing

sider it of, or

to a decimal, it may be easier for some to con

of .25. The reduction of is simplified by multiplying each term by 2, making it, or .46, instead of dividing 23 by 50, etc., etc.

690. Nos. 62, 64, 66, and 68 may be worked by using the common fraction given, and also by reducing this to a decimal before performing the multiplication.

691. See Art. 563, p. 55, and Art. 616.

692. The teacher should not permit the employment of long division in these examples. In No. 92, the children can see that changing the dividend to .18756 divides it by 100, and that .187563 is the same as 18.756 by 300. See Arithmetic, Art. 668.

694. Ciphers at the right of a decimal should be rejected, excepting, perhaps, the final O in cents. See Nos. 3, 4, and 10.

2. $.95 x 7.6.

3. $2.80 x 48.6.

4. $21.30 x 39.25.

5. $.68 x 18.75.

6. $22 × 108.745.

7. $.75 x 148.6.

+

8. $.13 (2376 ÷ 12).
9. $35 x 4.5.

10. $13.50 x [(28 × 12) + 144].

While it is inadvisable to confuse children by too many short methods in the earlier stages, they should be encouraged in examples like the foregoing to use as a multiplier the number that will make the work easier, and to employ a common fraction instead of a decimal whenever the use of the former would lighten their labor. In No. 1, for instance, the result is obtained with fewer figures by multiplying 24.4 by 61, instead of 6.25 X 24.4.

703-704. See notes on previous special drills, Arts. 286 and

350.

705. See Arts. 528 and 649. In multiplying 46 by 331, divide 46 by 3, which gives 15, and substitute 33 for the fraction in the quotient, thus obtaining the result, 1533.

708. Use first as "sight" problems, if the pupils find the numbers too large to be carried in the mind. By degrees, however, they should acquire the power to solve problems of this kind without seeing the figures, especially when the operations are not numerous or involved.

709. In such examples as Nos. 1, 9, 10, 11, and the like, many children fail to comprehend the form of analysis generally given. While they get some facility in applying the method, they do not understand the underlying principle. In finding a number, of which is 180, they learn to divide by 5 and to multiply the quotient by 6, and to repeat the customary formula, without knowing the reasons for the different operations. There are only four fundamental processes in arithmetic, and children should be taught to determine for themselves which to use in a given example that is within their experience, rather than to depend upon a rule which they do not fully understand, and which they are likely to forget or to misapply. See Art. 635. A few diagrams are here introduced, to be used by the teacher that does