t PART II. Tār Key contains remarks on the principles employed and illustrations of the manner of solving the examples in each section. All the most difficult of the practical examples are solved in such a manner as to show the principles by which they are performed. Care has been taken to select examples for solution, that will explain those which are not solved. Many remarks with regard to the manner of illustrating the principles to the pupils are inserted in their proper places.} Instructers who may never have attended to fractions need not be afraid to undertake to teach this book. The author flatters himself that the principles are so illustrated, and the processes are made so simple, that any one, who shall undertake to teach it, will find himself familiar with fractions before he is aware of it, although he knew nothing of them before; and that every one will acquire a facility in solving questions, which he never before pos. sessed. The reasoning used in performing these small examples is precisely the same as that used upon large ones. And when any one finds a difficulty in solving a question, he will remove it much sooner, and much more effectually, by taking a very small example of the same kind, and observing how he does it, than by recurring to a rule. The practical examples at the commencement of each section and article are generally such as to show the pupil what the combination is, and how he is to perform it. This will learn the pupil gradually to reason upon abstract numbers. In each combination, there are a few abstract examples without practical ones, to exercise the s learner in the combinations, after he knows what these combinations are. It would be an excellent exercise for the pupil to put these into a practical form when he is reciting. For instance when the question is, how many are 5 and 3 2 Let him make a question in this way; if an orange cost 5 cents, and an apple 3 cents, what would they both come to ? This may be done in all cases. The examples are often so arranged, that several dend on each other, so that the preceding explains the ollowing one. Sometimes also, in the same example, there are several questions asked, so as to lead the pupil gradually from the simple to the more difficult. It would be well for the pupil to acquire the habit of doing this for himself, when difficult questions occur. The operations can be illustrated by counters, or marks on the blackboard, according to the necessity of the pupils. These illustrations will be less necessary as the pupils advance in the work; but a frequent reference to them throughout most of the book will be useful in fixing more clearly in mind the principles involved in the operations. The book may be used in classes where it is convenient. The pupil may answer the questions with the book before him or not, as the instructer thinks proper. A very useful mode of recitation is for the instructer to read the example to the whole class, and then, allowing sufficient time for them to perform the question, call upon some one to answer it. In this manner every pupil will be obliged to perform the example, because they do not know who is to answer it. In this way it will be best for them to answer without the book. It will often be well to let the elder pupils hear the younger. This will be a useful exercise §. them, and an assistance to the instructer. t SECTION I. A. THIs section contains addition and subtraction The first example may be solved by means of beans, peas &c., or by means of the blackboard. The former method ( is preferable, if the pupil be very young, not only by the examples in the first part of this section, but by o: examples in all the sections. The pupil will probably solve the first examples without any instruction. B & C. The articles B and C contain the common addition table as far as the first 10 numbers. In the first the numbers are placed in order; and in the second, out of order. - o The pupil should study these until he can find the answers readily, and then he should commit the answers to memory. D. In this article the numbers are larger than in the preceding; and, in some instances, three or more numbers are added together. In the abstract examples, the numbers from one to ten are to be added to the numbers from ten to twenty. * E. This article contains subtraction. F. This article is intended to make the pupil familiar with adding the nine first numbers to all others. The pupil should study it until he can answer the questions very readily. G. In this article all the preceding are combined together, and the numbers from 1 to 10* are added to all numbers from 20 to 100, and subtracted in the same manner. 18. 57 and 6 are 63, and 3 are 66, and 5 are 71, and 2 are 73, less 8 are 65. H. This article contains practical questions which show the application of all the preceding articles, 6. 87 less 5 are 82, less 8 are 24, less 6 (which he kept himself) are 18; consequently he gave 18 to the third boy. * Figures are used in the key, because the Instructer is supposed to be acquainted with them, They are not used in the first of the book, be: cause the pupil would not understand them so well as he will the words. SECTION II. THIS section contains multiplication. The pupil will see no difference between this and addition. #. is best that he should not at first, though it may be well to explain it to him after a while. A. This article contains practical questions, which the pupil will readily answer. 1. Three yards will cost 3 times as much as 1 yard. N. B. Be careful to make the pupil give a similar rea: son for multiplication, both in this article, and elsewhere. 11. A man will travel 4 times as far in 4 hours as he will in 1 hour. 15. There are 4 times as many feet in 4 yards as is 1 yard, or 4 times 3 feet. B. This article contains the common multiplication table, as far as the product of the first ten numbers. The pupils should find the answers once or twice through, until he can find them readily, and then let him commit them to memory. C. This article is the same as he preceding, except in this the numbers are out of their natural order. D. In this article, multiplication is applied to practical examples. They are of the same kind as those in article A of this section. 12. There are 8 times as many squares in 8 rows, as in 1 row. 8 times 8 are 64. 18. There are 6 times as many farthings in 6 penca, as in 1 penny. 6 times 4 are 24. 17. 12 times 4 are 48. 28. There are 3 times as many pints in 3 quarts as in 1 quart 3 times 2 are 6. And in 6 pints there are 6 times 4 gills or 24 gills. 28. In 8 gallons there are 12 quarts, and in 12 quarts there are 24 pints. 81. In 2 gallons are 8 quarts, in 8 quarts 16 pints; in 16 pints 64 gills. 16 times 4 are 64. 85. In one gallon are 32 gills; and 32 times 2 cents are 64 cents. Or, 1 pint will cost 8 cents, and there are 8 pints in a gallon. 8 times 8 are 64. 38. They will be 2 miles apart in 1 hour, 4 miles in 2 hours, &c. SECTION III. A. THIS. section contains division. The pupil will scarcely distinguish it from multiplication. It is not important that he should at first. The pupil will be able to answer these questions by the multiplication table, if he has joi it to memory thoroughly. B. In this article the pupil pbtains the first ideas of fractions, and learns the most important of the terms which are applied to fractions.” The pupil has already been accustomed to look upon a collection of units, as forming a number, or as being itself a part of another number. #. knows, therefore, that one is a part of every number, and that every number is a part of every number larger than itself. As every number may have a variety of parts, it is necessary to give names to the different parts in order to distinguish them from each other. The parts receive their names, according to the number of parts which any number is divided into. If the number is divided into two equal parts, the parts are called halves; if it is divided into three equal parts, they are called thirds; if into four parts, fourths, &c.; and having divided a number into parts, we can take as many of the parts as we choose. If a number be divided into five equal parts, and three of the parts be taken, the fraction is called three fifths of the number The name shows at once into how many parts the number is to be divided, and how many parts are taken. The examples in this i. are so arranged that the - *As soon as the terms applied to fractions are fully comprehended, he operations on them are as simple as those on whole numbers. 10 |