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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 com
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 them in Part 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 anv instruction. Examples in addition and subtraction may be solved by plate I. as follows. How many are 5 and 31* 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
* 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 *...* the pupil would not understand them so well as he will the Words,
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 l are 18, and I are 19, and 1 are 20, and 1 are 21, and 1 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 1 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 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.
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.
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 S 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; consequently he gave 18 to the third boy.
This section contains multiplication. The pupil 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. Three yards will cost 3 times as much as I 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 as far in 4 hours as he will in 1 hour. In the third row count 4 times 3, and ascertain their sum. 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 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 cre 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 galions 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 2