SUGGESTIONS TO THE TEACHER.

1. The questions in the smaller sized type are designed for mental and oral exercises.* Although these exercises are more numerous and extensive than are to be found in other similar works, it is hoped that the teacher will not limit his pupils to these exercises, but that he will extend them as far as in his judgment may be practicable.

2. Exercises marked thus, II., are rather more difficult than others, and may be omitted by beginners, until the book, or portions of it, are reviewed.

3. The importance of frequent reviews cannot be overestimated by the teacher. By such reviews, the pupil will acquire that familiarity with first principles, and that facility in performing arithmetical operations, which are necessary to render his progress both rapid and pleasant, and which can be acquired in no other way. Both teacher and pupil should aim to be thorough. With this end in view, the teacher will not confine himself to the exercises prepared by the author, but will extend them till the end is attained; for no author can anticipate the precise number of exercises each pupil will need upon any one principle before he will be prepared to advance to another.

4. The pupil should be required to prove his work to be correct, as far as practicable. For example: All his operations in Division should be proved by Multiplication; those in Reduction Ascending, by Reduction Descending, and the reverse, when the pupil has progressed far enough to be able to do it. Operations in Proportion should be proved by Analysis, &c. The pupil's progress by this method will be apparently slow at first, but the facility and correctness which he will acquire in this way, will render his future progress far more rapid and satisfactory than it would be without such training.

5. The teacher should ever bear in mind that all the topics treated of in arithmetic are not of equal importance to every pupil, and that he should adapt his instructions, as far as practicable, to the peculiar wants of the pupil. The scholar whose opportunities for learning arithmetic are very limited should be exercised very thoroughly in the elementary rules, and in their application to as great a variety as possible of common business transactions. He should be encouraged "to make up questions" for himself, and solve them, and every method should be used to render the knowledge he may acquire most useful to him when his short term of pupilage shall have expired. He who is intended for the counting-room should be carefully drilled in Practice, Percentage, Equations, Accounts Current, &c., Art. 100 to 125. While the future mechanic should be as thoroughly drilled in the

*The only exceptions are on pages 55 and 151.

Square and Cube Root, and their application to a great variety of practical questions, in Mensuration and the Mechanic Powers. And he who has sufficient time to devote to study should be made familiar with all the topics treated of, and thus be better fitted to engage in any occupation than he can be whose attention has been confined to only a part of these topics.

The "practical application" of the Square and Cube Root, contrary to the practice of authors generally, is, in this book, postponed till the study of Geometrical Definitions and Mensuration has been commenced. The reason for this is, that such "practical questions" are little else than arithmetical puzzles to the pupil who has no knowledge of the principles of Mensuration; -puzzles which the teacher is generally obliged to solve for the pupil, not only the first time he goes over them, but at every subsequent review of them; at least, such has been the author's experience. Whereas, if taken up in connection with the subject of Mensuration, where, indeed, they belong, no such difficulty exists.

The beginner in arithmetic should be taught to call the sign of addition and, and the sign of subtraction less. Thus 8+6=14; and 15-510, should be read 8 and 6 are 14, and 15 less 5 is 10.

The following answer to the question, “How much and what kind of assistance shall I give my pupils while they are pursuing their studies?" is submitted for the consideration of the teacher.

"If the proposition of the text-book is not understood by a pupil, he should be required to point out definitely to the teacher what it is which he does not understand, and then, not before, the teacher may give him the help he needs. If a pupil complains of not understanding the meaning of the text-book, the teacher should, generally, require him to read the passage aloud, telling him to stop when he comes to a word or expression which he does not understand. In four cases out of five, the difficulty will vanish without a word of explanation from the teacher. It is important also to require the pupil, in most cases, so to frame his question that it may be answered by yes, or no. So that, instead of saying, 'Please tell me what this means; or, How shall I perform this question?' he shall say, 'Does this mean so and so?" or, 'Is this question to be solved by such a rule?' or, 'Is it similar to such a question?' The answer to such questions may be yes, or no; but more generally it should be, "Why do you think so?" By such means the pupil will be trained to a careful study of principles, and will learn not to depend upon his teacher to remove every little difficulty."

KEY

TO THE

AMERICAN

COMMON-SCHOOL ARITHMETIC.

BY RUFUS PUTNAM,

PRINCIPAL OF THE BOWDITCH (ENGLISH HIGH) SCHOOL, SALEM, MASS.

BOSTON:

TAPPAN, WHITTEMORE & MASON,

114 WASHINGTON STREET.

1849.

Entered according to Act of Congress, in the year 1849, by

RUFUS PUTNAM,

In the Clerk's Office of the District Court of the District of Massachusetts.

ADVERTISEMENT TO THE KEY.

THIS Key is not intended to give the pupil who may use it any assistance in solving the questions in the Arithmetic, except what is to be found in the answer alone. Those teachers who may wish their pupils to have the answers to the questions while preparing their lessons can have the Key bound in the same covers with the Arithmetic; and those who prefer that the pupil should not have the answers can have the Arithmetic without the Key; so that, in ordering the book, if the Arithmetic and Key are both wanted, it should be so stated,.otherwise the Key will not be sent.

The author intends to publish. as soon as it can be prepared, an Appendix to the Key, for the use of teachers only; containing, 1st. Solutions, so far as may be desirable to assist the teacher in examining the work of his pupils. 2d. Such explanations and suggestions as the author mav think proper to make in reference to the topics treated of in the Arithmetic. 3d. Additional questions, in most cases similar to those in the Arithmetic, to be dictated by the teacher to the pupil at each recitation, so as the more carefully to test his knowledge of the subject.

ERRATA.

In the Arithmetic, page 89, 3d line from the top, "12" should be 72; page 135, example 3, "1845" should be 1847; page 197, 5th line from the top, for "4564," as printed in the first issue, read 5064. No other. errors have been noticed that may not be easily corrected by the reader.

Stereotyped by

HOBART & ROBBINS;

NEW ENGLAND TYPE AND STEREOTYPE FOUNDERY,

BOSTON.

KEY.

SECTION I..

NUMERATION. [ART. 9-12.]

ART. 9. (1.) Ten. (2.) One hundred. (3.) One thousand. (4.) Ten thousand. (5.) One hundred thousand. (6.) One hundred and one. (7.) One hundred and eighty. (8.) One thousand and one. (9.) One thousand and twelve. (10.) Two thousand and eighty-four. (11.) Seven thousand, eight hundred and four. (12.) Ten thousand and one. (13.) Thirty thousand, eight hundred and five. (14.) Thirty-eight thousand and fifty. (15.) Thirty thousand and eighty-five. (16.) Five hundred thousand and five. (17.) Three million, fifty thousand, six hundred and one. (18.) Eight hundred and fifty million, one hundred and sixty thousand, eight hundred and four.

II. (19.) 4 bil., 16 mil., 80 thous. and 900. (20.) 1 tril., 851 bil., 608 mil., 90 thous., 504. (21.) 50 tril., 80 bil., 90 mil., 607 thous. and 10.

10. (1.) 84. (2.) 904. (3.) 940. (4.) 1,001. (5.)1,010. (6.) 1,101. (7.) 80,008. (8.) 80,080. (9.) 80,088. (10.) 575,637. (11.) 8,000,035.

(12.) 34,034,034. (13.) 500,000,050. (14.) 6,000,006,600. II. (15.) 15,000,000,000,040,008,040. (16.) 60,000,000,000,000,090,000,000,003,000. (17.) 15,000,000,000,008,000,000,000,400,008,000,000.

12. (1.) Five tenths. (2.) Five hundredths. (3.) Thirtyseven thousandths. (4.) One thousand and eight ten thousandths. (5.) Five hundred and one, and 8 hundredths. (6.) Seven thousand one hundred and eight ten thousandths. (7.) Eight thousandths. (8.) 1,001 millionths. (9.) 16,- and 16 hundredths. (10.) 180, and 18 thousandths. (11.) 7,—and 16 hundred thousandths. (12.) 30,175,- and 4 hundredths. (13.) 301,680, and 45 hun