(or parts) cost 20 cents, what will 1 third (or part) cost? A diagram similar to that given in Arithmetic, Art. 636, may help others to understand the method.

Other successful teachers think the written work is benefited by treating these examples as problems in division. They lead their children to determine in each case what operation is involved, by requiring them to consider what they would do if the fraction were a whole number. In No. 1, for example, the cost of 16 balls at $3 each would be $3 x 16. In No. 2, the pupil would say, "If I paid $12 for base-balls at $3 each, the number of balls would equal 123. I must, therefore, divide." He mentally inverts the divisor, &, then cancels, etc.

636. The scholars should be allowed sufficient time to work these out in their own way.

=

cost of a pound.

639. No. 4: 241÷3 427-497. Some pupils will see that time is lost in No. 6 by finding the No. 7 is an example in division: 1÷2 10 ÷ 15 = 10 ÷ 15, etc. In No. 16, 36 hats will cost 3 times $7.

642. See Art. 546.

649. In multiplying by 25, the pupil is generally told to annex two ciphers and to divide by 4. In mental work especially, the annexation of the ciphers confuses some scholars by giving them a larger dividend than is really required. The product of 25 times 19 may be obtained more easily by taking one-fourth of 19, or 44, and changing this quotient to 475, than by finding one-fourth of 1900. In No. 9, the pupil should see that at $100 per bbl., the pork would cost $5600, and that at $12.50 per bbl. (of $100), it would cost of $ 5600.

650. In No. 1, divide 837 by 4, and for the 1 remainder affix 25 to the quotient. In No. 4, annex two ciphers to the quotient of 508 by 4. In No. 9, affix 250 to the quotient of 837 ÷ 4.

In multiplying 6281 by 121, No. 18, divide 6281 by 8, obtaining 785, and annex 12 for the 1 remainder, making the result 78,512

654. When the divisor is a whole number, time should not be wasted in changing a mixed number dividend to an improper fraction. Nos. 64, 65, and 66 resemble those already worked. In No. 69, after obtaining the quotient 14, there will be a remainder 21⁄2, which is changed to 1⁄2 and divided by 5, giving as the result. In No. 69, the remainder, 54, is changed to 4, which gives when it is divided by 8.

656. Some mistakes would be avoided if pupils would learn to ask themselves if the answer they have obtained is a reasonable one. Permit the scholars to work out all these examples without giving them a rule for "pointing off."

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.

UNIVERSITY

NOTES ON CHAPTER SEVEN

OF CALIFORNIA

67

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 3÷4, 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