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mencing 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 thom 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 I. · 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 con taining 5 marks, and another containing 3 marks, and ascertain the number of marks in both,

How many are 8 and 6 ? Select a rectangle contain

.

Pigures 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 books, because the propil would not understand then so well as be will the wonin

ing 8 marks, and another containing 6 marks, and count them together.

How many are 17 and 5 ? Keeping 17 in the mind, select a rectangle containing 5 marks, and add them thus : 17 and 1 are 18, and I are 19, and 1 are 20, and 1 are 21, and I are 22.

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 1 at a time.

In this manner all the examples in this section may be solved.

B&C. The articles B and C contain the cominon 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,

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. 37 less 5 are 32, less 8 are 24, less 6 (which he kept himself) are 18; conscquently he gave 18 to the third boy.

SECTION II.

This section contains multiplication. The pupíl will see no difference between this and addition. It 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. T'hree yards will cost 3 times as much as 1 yard.

N. B. Be careful to make the pupil give a similar reason for multiplication, both in this article, and elsewhere.

This question is solved on the plate thus ; in the second row, count 3 rectangles, and find their sum, 2 and 2 are 4 and 2 are 6.

11. A man will travel 4 times is far in 4 hours a3 he will in 1 hour. In the third :ow count 4 times 3, and ascertain their sum.

16. 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 pupil should find the answers once or twice through, until he can find them readily, and then let him commit them to memory.

43. 6 times 3. In the third row count 6 times 3, and then ascertain their sum, 3 and 3 are 6, &c.

59. 7 times 9. In the ninth row count 7 times 9, or 7 rectangles, and ascertain their sum. 9 and 9 are 18, &c.

C. This article is the same as the 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.

13. There are 6 times as many farthings in 6 pence, as in 1 penny. 6 times 4 are 24.

17. 12 times 4 are 48.

Note. When a number is taken more than 10 times, as in the above example, after taking it 10 times on the plate, begin at the beginning of the row again, and take enough to make up the number.

23. 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 3 gallons there are 12 quarts, and in 12 quarts there are 24 pints.

31. In 2 gallons are 8 quarts, in 8 quarts 16 pints; in 16 pints 64 gills. 16 times 4 are 64.

35. In one gallon are 32 gills ; and 32 times ?

eents 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.

Though the pupil will be able to answer these questions by the mæltiplication table, if he has committed it to memory thoroughly; yet it will be better to use the plate for some time.

9. As many times as 3 dollars are contained in 15 dollars, so many yards of cloth may be bought for 15 dollars. On plate I, in the third row, count fifteen and see how many times 3 it makes. It is performed very nearly like multiplication.

B. In this article the pupil obtains 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. He 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

* As soon as the terms applied to fractions are fully comprehended, the operations on them are as simple as those ou whole nuanberk.

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