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Suggestions

Art. 1-19. The teacher must exercise discretion in use of definitions. Those in Section I. need not all be committed at once, as some of them are given in the body of the work as needed.

55. Addends, although a comparatively new term, is not used without authority.

61. In Mental Exercises, use the model best adapted to each pupil. If deemed best, give the younger pupils the mental problems as dictation exercises on the slate.

73. Refer to definitions (4 and 5), or explain fully concrete and abstract.

78. As early as possible, it is well to show that the placing of the subtrahend under the minuend, is a matter of convenience.

89. When multiplying we may regard both terms as abstract, and then attach to the product such name as the nature of the question demands. Thus, since 1 bbl. flour costs $8, 9 bbls. cost 9 times as much, or $72. 9×8=8x9= 72. The answer must be dollars. Therefore, $72, answer.

93. Prob. 10 may be contracted thus:

374781

1402

749562 5246934 525442962

First multiply by 2; then multiply this product by 7, placing the right-hand figure under 4, and the remaining figures in order to the left; for 2 × 7 14. This is given merely as a sample. Once show the pupil how to lighten his labor, and generally he is not slow to take advantage of any short process; and will soon reap great benefit. Try Prob. 21.

Many operations in multiplication may be solved by such devices as this:

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95. When ciphers are on the right of significant figures, either in the multiplicand or multiplier, or both, they should not be considered until the significant fig ures are multiplied together, when the ciphers must be annexed to the product

102. EXAMPLE SECOND. This may also be solved thus: If each of two persons receive $1, to divide $10 equally between them, each must receive as many times $1 as $2 is contained times in $10, or 5 times $1 = $5. This relieves us of the Inconsistency of calling the divisor an abstract and the dividend a concrete number.

Or, since taking one-half of $10 is the same as multiplying $10 by 1, we have the multiplicand and product of the same kind (89).

106-108. Show that the divisor may be written in any other convenient place as well as at the left or right of the dividend.

PROBLEMS. The earnest teacher will not fail to supply the student with abundant examples. Pp. 73-78 are thought to afford a fair variety of such examples as the pupil needs to make him thoroughly familiar with the principles and operations already discussed.

112-119. Show the strong resemblance of a whole number, or Integer, and a decimal; and exhibit in the strongest light the importance of the decimal point.

120. The Second Method of Numeration and Notation possesses such great advantages over the first method that we wonder at the limited use of the former.

129-138. Too much care cannot be exercised in teaching "Multiplication of Decimals" and "Division of Decimals "; and no part of the Arithmetic will better repay this care; for, pupils once thorough in these, will move along easily and rapidly.

138. Reducing dividend and divisor to the same denomination before dividing has the advantage of clearness, but is sometimes inconvenient in practice. Thus: Divide 1728 by 1.2. Annexing .0 to 1728, we have 1728.0, dividend and divisor, both tenths, and the quotient a whole number, 1440. Dividing .0001728 by 1.2 by this method, however plain it may be, is a clumsy performance.

167. In finding the factors of a number, the facts in this article may be greatly extended, as for example: FOURTH. Any even number, the sum of whose digits is divisible by 3, has 6 for an exact divisor; FIFTH. Any number, whose two righthand figures are divisible by 4, has 4 for an exact divisor, etc., etc.

183. Special Rules on p. 148 are given, that teacher and pupil may have both variety and choice. We prefer, however, the GENERAL RULE.

184. The same remarks apply to RULES on pp. 154, 155.

185. The pupil should be able at once to change a decimal into a common fraction, or a common fraction into a decimal.

186, 187, are eminently practical and should be thoroughly mastered.

192-232 embrace the tables used in Denominate Numbers, and are placed together as a matter of convenience for easy reference. The exercises which follow are placed under their appropriate headings, so that a table, or a convenient number of tables, with the exercises belonging to each, may be readily assigned as a lesson. 193. The Table of Federal Money, together with definitions, are here inserted to preserve the uniformity of the system.

The representations of the coins of the United States are all that are coined at the present time. Of the coins of Great Britain, Germany, and France, only a lim

ited number are inserted. It is advisable, when possible, for the teacher to exhibit the actual coins. The same is true of all the weights and measures represented.

210, 211, 216. Show how square and cubic measure stand related to Long

measure.

211. We see no good reason why a pupil should wait until he studies a treatise on surveying before he knows how a township is subdivided.

By means of cut, p. 192, the teacher may suggest, and the pupil solve, a great number of interesting problems. By means of Section Maps of the Western States, the pupil may locate the principal cities, towns, etc.

214. The edge of a cube is one of its dimensions. The edge of any solid is a line formed by the meeting of two adjacent faces of that solid.

245. Although three methods are given, whether the pupil shall study them all at once should be determined by his stage of advancement.

246, 257, are entirely practical, and while intended for pupils in general, are especially useful for those who have no prospect of taking a course in Mathematics, or who cannot even find the time to go as far as Mensuration proper, as treated in the last pages of this work. Much is given here in a convenient form that is not easy to find in any one book.

248. Nearly every pupil in the Western States will be interested in knowing the manner of dividing lands in his own county.

257. Prob. 4. Since when the sides were 15 inches high, the wagon-bed held 50 43 cubic feet; in order to hold 50 cubic feet, it must be of 15 inches high, or 43 172 inches. Hence the depth is increased 23 in.

Probs. 13, 17 and 18 may be solved in the same manner.

258. This may be illustrated by the change apparent in a watch, in going from San Francisco to New York, New York to Liverpool, Liverpool to Canton (China), Canton to San Francisco; illustrating that a good time-keeper will lose a day in going round the world eastwardly; and, in like manner, going westwardly will gain a day.

259. The word measure is here used in its general sense; for although 25 consists of a number of parts, yet, for the purpose of measuring, it is a unit. For example: How many centals in 2050 lbs. of wheat? Here we divide 2050 by 100, the number of lbs. in one cental, and 100 lbs. is regarded as the measuring unit.

286. P. 285, Ex. Equimultiples of numbers are the products of those numbers by a given number. Thus: 7 x 5 and 8 × 5 are equimultiples of 7 and 8. 291, 292. Note especially the difference between Rate per cent. and Rate. 305. In Higher Mathematics and Applications, Formulas are deemed invaluable. Why not in Arithmetic ?

319. Solution of Ex. 1, in Review Problems: To make 25%, the selling price must be of cost, or of 24c. = 30c. But 30c. is 163% less than asking price. Since 163% = }, } − } (= {) of asking price, must equal selling price, and of asking price

equals of selling price; therefore, § of asking price equals of selling price. of 30c. 36c., asking price.

353. One method of computation well taught is worth more than all the others poorly taught; and all special methods should be omitted with beginners.

P

396. Ex. 5 is solved by formula × 100% = R).

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Thus B =

403-412. The five cases correspond to the five cases of simple interest.

900,

426. Both Rules and Formulas are omitted in computations of Stocks and Bonds, because they are the simplest applications of percentage.

Prob. 17. ANOTHER SOLUTION: Dividing the given by the required rate % gives the rate. .75 x 500 375; or, $375 is the price.

427. Letters of Credit, that is, written orders on which partial payments are made at sight, are issued to travellers in all or nearly all civilized countries; thus affording great security in the transportation of necessary funds for travelling expenses. The value of such letters, of course, is reckoned according to the principles governing the calculation of exchange.

436. The table on p. 366 will suggest to the ingenious teacher a great variety of interesting problems.

461. Pupils should also be exercised in solving Problems in Partnership and Bankruptcy by Distributive Proportion, pp. 285-288.

484. Prob. 1. Had Simpson paid the stipulated sums at the times agreed upon, the party from whom he bought would have had at the end of 15 mo. the interest of $500 for 15 mo.; of $600 for 9 mo.; and of $700 for 3 mo.; or, at 6%, he would have had $37.50 + $27+ $10.50 = $75 interest. Simpson then should pay him $2700 ($500+ $600 + $700 + $900) at such a time before the end of 15 mo., that at the end of the 15 mo. its interest at 6% would equal $75. In 12 mo. $2700, at 6%, will gain $162; and will gain $75 in of 12 mo. = 5; mo. Simpson, therefore, should pay $2700 in 15 mo. -55 mo. = = 9; mo., Ans.

485. Let the pupil work a number of examples by selecting both the first and the last dates as focal dates.

509. In the applications of Square and Cube Roots, no demonstrations are attempted.

531. Prob. 1: As the first payment is made at the beginning of the first year, and the tenth payment at the beginning of the tenth year, the entire 10 payments are made within 9 yrs, and 1 da. We must, therefore, regard the annuity as having 10 yrs. to run.

567. It is a matter of regret that the Metric System has not come into general use in the United States.

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