given by pupils, even of high-school classes, to simple problems. When the early arithmetical instruction is largely given to work in the fundamental processes, the teacher should make liberal use of oral problems, to give the requisite knowledge of number that will enable a pupil to know when his answer is very much out of the way. Systematic instruction in finding "approximate" results is supplied in later chapters.

239. These examples are intended to lead up to finding the difference between a whole number and a mixed number.

240. Pupils will find little difficulty in working out these examples if they are left to themselves.

241. When the addition and the subtraction of mixed numbers containing halves are readily performed, the teacher will find comparatively little trouble with the work under Arts. 241-245. Encourage pupils to make diagrams; or, if necessary, to divide circles into quarters, and to use these parts in performing the required operations with the fractions.

To find, for instance, the sum of 4+, it may be advisable to permit some scholars to arrange the six quarter-circles in such a way as to make a whole circle and a half-circle.

246. As children are more accustomed to dealing with halves and quarters than with thirds, a little more illustrative work may be needed in Arts. 246–250, than was required in the previous work in the addition and the subtraction of mixed numbers.

VII

NOTES ON CHAPTER FOUR

253-258. In the last chapter, pupils were required to add. only fractions containing the same denominator; in this chapter, an addition or a subtraction example may contain fractions whose denominators are different. For the present, however, it will not be necessary to call attention to the need of reducing fractions to a common denominator. The average scholar can solve these examples without assistance, if he has been able to work out those found in Chapter III.

259. While these problems are becoming more difficult, they are still well within the powers of a pupil that is really anxious to solve them. When, however, they are found to be beyond the capacity of many members of the class, the teacher may first use them as "sight" problems, with some slight changes in the figures.

If, for instance, after a pupil that reads the first from his book declares that he is unable to obtain the answer mentally, the teacher may give it as follows:

A sailor has 10 yards of cloth. He uses 4 yards for a coat and 2 yards for a vest. How many yards has he left?

In the second, 1 pounds may be substituted for 1 pounds; in the third, 3 packages instead of 4; 20 dozen in the fourth, instead of 3 dozen.

Slate work on these problems should not be permitted until so many have been solved in this way that the pupil has time to forget what operations have been used in each. This will

require him to study the conditions of the different problems, instead of relying upon his memory.

266. When the formal analysis of oral problems is made a feature of the work, it is important that the statements be not so long as to be tedious.

In the first, for example, the following would be sufficient, after the pupil has stated the problem:

"If 8 ounces of tea cost 40 cents, 1 ounce will cost 5 cents, and 5 ounces will cost 25 cents."

While the customary order has been followed in the systematic treatment of the various topics, pupils are called upon in the earlier chapters of Mathematics for Common Schools to solve many problems that are frequently deferred in other books to a later stage of their arithmetical instruction. While scholars readily solve this class of problems, they are not always able to state in technical language the reasons for the various processes employed in obtaining the answers. A child who sees that

division is used to ascertain the number of ten-cent pies that can be purchased for forty cents, cannot be made to understand thus early in his school life that the same process is used to find what part of such a pie can be bought for five cents. A correct statement by the pupil of his method of reaching the result, should usually be accepted as satisfactory. Even in the more simple questions, set forms of analysis should be carefully avoided.

268. To prevent misunderstanding, parentheses have been employed even when not required by arithmetical usage. The quantities within the parentheses must be added, multiplied, etc., before being operated upon by the quantity outside. The third example becomes 30 × 3; the fourth, 80÷4; the fifth, of 80; the eighth, 707, etc.

269. These may be used as slate examples, if they are found too difficult for "sight" work.

271. Some of these questions may not require the use of a pencil; Nos. 6, 7, 8, 11, and 19, for instance.

272. The answers to the first ten examples should be given at sight.

273. Use 49 to 57, inclusive, as "sight" examples; also as many as possible of those in the next section.

8|0)434|1

274. When the divisor ends in one or more ciphers, the latter are set off by a vertical bar, and also a corresponding number of figures from the right of the dividend. To keep the pupil from omitting these figures from the remainder, it is advisable to require him to write the partial remainder as above, before he begins to divide. Then, using 8 as a divisor, he writes the quotient figures in their places, and completes the partial remainder by prefixing 2 to the 1 that was originally brought down.

8|0)434|1

54금

It being the usual practice in abstract examples in division to refrain from reducing the fractional part of the quotient to lowest terms, the above

80)4340 543

8|0)434|0 5420

method may be used in examples where both the divisor and the dividend terminate in a cipher. Some teachers prefer, however, in this case, to cancel the cipher in each, and to give the quotient of 4340 ÷ 80 as 54.

277. Employ in "sight" work.

278. The foot-rule and the yardstick should be used by the children. They should ascertain, for instance, the length of their slates in inches, the length of the blackboard in yards or in feet, the height of the blackboard in feet, the dimensions of the room, etc.

280. It will be sufficient to accustom pupils to placing the product by the tens' figure one place to the left without giving the reason therefor. Neatness in the arrangement of the work, and the careful writing of figures, will prevent some mistakes.

4)12361

30901

282. In short division, the scholar has been taught to place the first figure of the quotient under the last figure of its partial dividend, and to write under each succeeding figure of the dividend its corresponding quotient figure. When his work is neatly arranged, he seldom omits ciphers, nor does he often obtain two quotient figures from one partial dividend.

2030 21)42631

42

To obtain the benefit of this experience, the pupil should be taught in long division to write the quotient over the dividend. By doing this, he will not be tempted, as are some beginners that place the quotient at the right, to give 23 as the answer to the above example; nor will he be likely to think that 252 contains 21, 111 times. This last result is obtained by assuming that the second partial dividend, 42, contains the divisor 1 time, with a remainder of 21. This latter is then made a partial dividend, Iwith the above result.

63

63

1

30

285. While the pupil may write 16 as a multiplier in the 5th problem, he should be required to multiply by 30, in order to shorten the work. The multiplication by 30 should be performed, also, without rewriting the numbers so as to place 30 under 16.

× 16

286-290. The special drills will be found of great value in giving pupils a knowledge of numbers; and many oral problems employing these and similar combinations should be made by the teacher. Oral problems containing large numbers should, as a rule, require but one operation for their solution.