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THE Key contains an explanation of the plates, and the manner of using them. The manner of solving the examples in each section is particularly explained. 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 possessed.

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 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. 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 depend on each other, so that the preceding explains the following 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 plates should be used for young pupils, but they are not necessary for the older ones. The plates for fractions however will frequently be useful to these. The first plate need not be used much, after the pupil is familiar with the multiplication table. 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 for them, and an assistance to the instructer.

Explanation of Plate I.

This plate, viewed horizontally, presents ten rows of rectangles, and in each row ten rectangles. In the first row, each rectangle contains one mark, each mark representing unity or one. In the second row each rectangle contains two marks, in the third, three marks, &c. The purpose of this plate is, first, to represent unity either as a unit, or as making a part of a sum of units. Secondly, to represent a collection of units, either as forming a unit itself, or as making a part of another collection of units; and thus to compare unity and each collection of units with another collection, in order to ascertain their ratios. All the examples, as far as the eighth section, can be solved by this plate. The manner of using it is explained in the key for each section in its proper place. The pupil, if very young, should first be taught to count the units, and to name the different assemblages of units in the following manner : The instructer showing him the first row, which contains ten units insulated, requests the pupil to put his finger on the first, and say one; then on the second and say, and one are two, and on the third and say, and one are three, and so on to ten ; then commencing the row again, let him continue and say, ten and one are eleven, &c.

After adding them, let him begin with ten, and say, ten less one are nine, nine less one are eight, &c. Then taking larger numbers, as twenty or thirty, let him subtract them in the same manner.

Next let him name the different assemblages, as twos, threes, &c. Afterwards, let him count the number of units in each row.

Note. The sections, articles, and examples, are referred to by the same marks which distinguish them in Part I.

SECTION I.

A. This section contains addition and subtraction. The first examples may be solved by means of beans, peas, &c. or by plate 1. The former method is preferable, if the pupil be very young, not only for the examples in the first part of this section, but for the first examples in all the sections. The pupil will probably solve the first examples without any instruction. Examples in addition and subtraction may be solved by plate I. as follows. How many are 5 and 3?* Select a rectangle containg 5 marks, and another containing 3 marks, and ascertain the number of marks in both. How many are 8 and 61 Select a rectangle contain. ing 8 marks, and another containing 6 marks, and count them together. How many are 17 and 52 Keeping 17 in the mind, select a rectangle containing 5 marks, and add them thus: 17 and 1 are 18, and l are 19, and I are 20, and 1 are 21, and I are 22. o If you take 4 from 9, how many will remain Select a rectangle containing 9 marks, and take away four of them. 18 less 5 are how many ? Keeping 18 in mind, select a rectangle containing 5, and take them away I at a time. In this manner all the examples in this section may be solved. 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. The pupil should study these until he can find the answers readily, and then he should commit the answers to memory.

* Figures are used in the key, because the instructer is supposed to be acquainted with them. They are not used in the first part of the

book, because the pupil would not understand them so well as he will

the words,

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 ail others. The pupil should study it until he can answer the questions very readily.

• * G. In this article all the preceding are combined

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