VIII

NOTES ON CHAPTER FIVE

348. The denominators of fractional multipliers have heretofore been factors of the multiplicands, and the latter have been, as a rule, small numbers. With the introduction of larger numbers and the occasional use of multiplicands that are not multiples of the denominators of the fractions in the multiplier, it becomes necessary to furnish pupils with a general method of dealing with this class of examples. (See Arithmetic, Art. 347.)

349. In multiplying 27 by 131, some pupils may be tempted to follow the rule, and to multiply 27 by the numerator 1. In the first few examples this may be permitted, but the scholars should soon be taught to discontinue the practice, and to divide the multiplicand without rewriting it. (See Art. 178.)

350. In adding 56 and 17, the pupil should first combine 56 and 10 to make 66, and then add 7. (See Art. 286.)

351. Children taught subtraction by the "building-up" method will ascertain how many must be added to 19 to make 66, by saying 19 and 40 are 59, and 7 are 66; or 19 and 7 are 26, and 40 are 66. While the second plan is easier in some respects, it gives the 40 and the 7 of the result in the reverse order, which makes it necessary for the pupils to transpose them. In this respect, the first plan is more satisfactory.

When the other method of subtraction is practiced in slate work, 66 is first diminished by 10 and then by 9. To find the difference between 94 and 76, the pupil takes 70 from 94, leaving 24, and from this remainder takes 6.

352. In multiplying 24 by 4 the pupil begins at the tens. Four times 20 are 80, to which is added 4 × 4, making 96.

353. While nearly the whole class will learn to give answers mentally to the previous combinations, it may be necessary to use the division drills as "sight" work chiefly.

359. See Art. 319.

362. Oral problems involving several operations, or those of an unfamiliar type, should be solved from the book as "sight" work, and should be followed later on by similar questions answered without seeing the numbers. No. 5 is of the second kind; and it might be well to place it on the board, writing " 2 thirds and "1 third" to express the parts, instead of employing the fractional form or that given in the book. In No. 7, the quotient of 60 40 will be expressed by 14, instead of the 128 obtained by writing the remainder over the divisor. No. 5 should not be made an excuse for teaching a method of obtaining the cost of the whole when that of a part is given.

+

These examples are introduced to give variety to the work, to lay a foundation for subsequent systematic treatment of problems of this kind, and to give a pupil an opportunity to use his thinking powers. The way to deprive them of value is to "explain how they should be done, or to require from the scholars too much analysis.

"

363. If the school does not own these measures, the teacher should endeavor to secure the loan of a quart, a peck, and a bushel, for a few hours, at least. Sawdust could be used to show pupils that the peck contains eight quarts, etc.

364. While many of the problems of this article resemble the previous oral problems, it may be advisable to solve a number of them as "sight" work, changing the numbers when necessary. The first may be read "How many 200-lb. barrels can be filled from 6,000 lb.?" In the second and third, the fractions may be omitted. The cost of the calico and of the ribbon in No. 4 may be made 10. Nos. 5 and 6 need no change, perhaps.

370-372. Do not waste time by endeavoring to use these examples to explain "carrying" or the local value of digits.

374. The answers should be written directly from the book. Do not permit scholars to copy the examples on their slates.

377. First, perform operations on the quantities enclosed within the parentheses.

(a)

?

384. Very little preliminary explanation will be needed. Place (a) on the blackboard, and ask a 125 pupil to write the missing number 632 in its place, one figure at a time,

99 beginning with the units' figure.

(b)

?

125

632

1000

Have another pupil work (b) in the same way. Nos. 1 to 5 may be used as a class exercise, each pupil writing only the answer on his paper, the examples being placed on the board.

385. In many German schools, children are not permitted in long division to write the partial products. Examples 6-23 are given to train pupils to omit these products when the quotient contains but one figure. After a few of them are worked on the board, the answers to the others may be written by all the pupils, as suggested in the preceding article. In writing the answers, the pupils should first set down the quotient figure, then the divisor as the denominator of a fraction, and lastly the remainder as a numerator. (See Art. 563, p. 55.)

386-388. These examples should be placed on the board, and the pupils should write the results one figure at a time.

397-401. See Art. 321.

405-406. See Arts. 306 and 307.

407. Prove the correctness of the grand total by comparing the total of the 6th column with that of the 11th row.

412. Permit the pupils to use their own method of working these examples, and avoid giving unnecessary assistance.

413-414. Example 1 should be omitted where pupils do not receive marks that are thus averaged. No. 2 may also be omitted if the word "average" is not understood by the pupils.

424-426. See Arts. 286-290, page 34.

429. In Examples 1, 2, 5, 9, etc., it will hardly be necessary to inform the pupils that 1 is not considered a factor of a number.

IX

NOTES ON CHAPTER SIX

The previous work in mixed numbers should make the pupils reasonably familiar with the addition and subtraction of fractions having small denominators. In this chapter, the work is extended to cover the addition and subtraction of fractions whose common denominator is determinable by inspection. For the present, the teacher should be satisfied if her pupils acquire reasonable facility in performing the various operations, even if they are unable to formulate, in the language of experienced. mathematicians, the reasons for the different steps. The children should be required to use correctly and intelligently such technical terms as are required by the work of the chapter; but they should not be compelled to memorize any definitions that convey to them no meaning. They should incidentally learn what is meant by numerator, denominator, common denominator, multiple, etc., by hearing the teacher employ these words from time to time, rather than by commencing with what is to them an unintelligible jumble of words.

451. While systematic work in fractions belongs properly to the next chapter, the teacher should not hesitate to call 12, 15, etc., "improper fractions," and to ask a pupil to state how they are changed to whole or to mixed numbers.

453. Do not, for the present, formulate the rule for changing a fraction to an equivalent one with higher terms.

458. The meaning of "lowest terms is given in No. 6. Leave the rule for the next chapter. After a pupil has rea