but one unit and but one place of units in numbers; but they are simply different orders of numbers, signifying different and certain aggregations or sums of units. The number 1,000 is not one unit, but one thousand units. Figures and units are very different things. One is a unit, 2 is not, though standing in the place of units. In the numbers 1, 10, 100, 1,000, removing the figure I to the left does not change the order of units which it represents, but the number of units, just as removing the same figure I to the right (as .1, .01, .001) expresses a greater division of the unit. Doubtless much confusion grows out of the common use of fraction and part as synonymous; but in arithmetic they are not so, for a part of a number may not be a fraction at all-as 4, which is a part of 8, is not a fraction; and a fraction may include several parts, as 34, which includes three of the four equal parts into which the unit has been divided.

Whatever may be the cause, the error seems to be a growing one, and I cannot but regard it as a serious one, and I would be very glad to see it checked and corrected.

A. A. MACDONALD.

Outline of Primary Arithmetic.

BY S. T. PENDLETON.

II.

ADDITION. Here, as in substraction, multiplication, and division, we get the scholar to make the table. We commence with what the scholar knows, counting objects; having taught him that + means "and," and = means "are," and 3+2=5, reads 3 and 2 are 5. Thus :

We then get him to make and memorize the table (written as above without the marks), in order, forward, backward, out of order, skipping, putting hand over, &c.

For little beginners the teacher says: "Start with 9 and count 2 more; as, 9, 10, 11; 9 and 2 are II"; and so for other figures. This is a very good plan :

To test him, write on the board

and pointing at the different figures in any order, get the scholar to give the sum of the figure and 2 or without pointing, let each scholar in succession say 9 and 2 are 11, 4 and 2 are 6, &c,; and sometimes only the sums, 11, 6, 2, 10, &c., around the class, starting at different figures at different rounds.

Also, as another method of test and drill, call out 4 and 2, 8 and 2, 5 and 2, 9 and 2, &c., and let the scholar write the sums on the slate, and to prevent copying hold his slate to his breast, so as not to be seen, or put a book over each sum as written, or let the scholars sit or stand a yard apart, &c.

We also teach the addition of 2 not only to single figures, but also to numbers of 2 figures. This is very readily taught upon the principle or key 22+2=24, because 2 and 2 are 4; 65+2=67, because 5 and 2 are 7, &c. The scholars catch this right off without any large number of exercises or combinations in which some teachers are apt to get "swamped." This is necessary, because in adding a column of figures we have to add 2 not only to single figures, but also to numbers of 2 figures. It is necessary in multiplication, as 9 times 6 equal 54, and 54 and 4 we carry equal 58.

We next give columns to add, orally or on slate, containing i's and 2's only; as

Also add orally and quickly, so as not to have time to count; thus, from bottom 5, 7, 8, 10, 12, 13, and from top 3, 5, 6, 8, 9, 11, 13.

We can then also add by 2's to 100, beginning with o, or 1, merely as a help and practice in addition.

I

I

1 2 2 2 2 2

I

REMARK.-Some teachers have made the mistake here of supposing that adding by 2's, 3's, 4's, &c., up to a hundred, was the final object; whereas the tables are the chief thing, and all operations in arithmetic depend on them. Some again have supposed that teach

ing the numerical frame was a chief thing-when it may or may not be used merely as one of the helps.

We do exactly the same with the 3rd column in the addition table; and then the 2nd and 3rd mixed up, getting the scholar to make the table; as

2

And so for each other column in the addition table. But notice the following triangle; which, upon the principle of 3+2=2+3, will shorten the work by half, always drilling in each column, be

2

132

331

ginning with the 2nd, thus: 3+2, 2+3, 8+2, 2+8, 5+2, 2+5, &c. Show the principles by making the marks | | | | |, and pointing first at the 3 marksand then at the 2 marks, and asking how many; and then pointing first at the two marks and then at the three marks, and asking how many :

8+4

8+5

8+6

8+7

8+8

8+2 8+3
9+2 9+3 9+4 9+5 9+6 9+7

9+8 9+9

Then we need begin each column only with the number of the column added to itself; as the 3rd column, 3+3, 4+3, &c.; the 4th column, 4+4, 5+4, &c.; so that when we get to the 9th column, the scholar knows all except 9+9, because 8+9=9+8 in 8th column; 7+9=9+7 in 7th column; 6+9=9+-6 in 6th column; 5+9=9+5 in 5th column, &c.

It is well, after getting through the 5th column in the addition table, to work out examples of several columns, first containing 1's and 2's; as—

and then examples with 1's, 2's and 3's; I's, 2's, 3's and 4's; &c.,

2 2 O 2 2

I 2 I 2 I

2 2 II 2

I 2 2 2 2

I 2 2 2 2

2 2 2 2 I

Also give examples of this sort to test knowledge of table:

5 1974 8 3

62

2 2 2 2 2 2 2 2

7439 26 15

3 3 3 3 3 3 3 3 &c., to add.

Also another excellent plan is first to give two numbers of several figures each to add, then three numbers, then four numbers, &c. You can get them from Pendleton's Arithmetic Cards. This plan drills a class up to perfect work.

Notice that one special difficulty is when you sometimes carry and sometimes do not carry.

Also from the very first, and all along, give practical questions, mental and written, bringing in questions of every-day life; as. If your mother gives you three cents and your father four cents, how many cents will you have?

John gives Mary 2 apples, Susan gives Mary 3 apples; how many apples has Mary in all? How many are 2 and 3? Each left scholar takes two grains of corn, and then each right scholar gives each left scholar 8 grains of corn; how many grains of corn has each left scholar in all? Also such exercises as, What two numbers make 10? Answer, 8 and 2, 7 and 3, 5 and 5, 6 and 4, &c.

The difficulty with beginners in writing down the right-hand figures under each other, may be gotten over by getting them to draw lines separating the periods; as,

The dictating of examples to add of unequal numbers, and numbers containing o's, is the hardest exercise and last step; as,

This is a good practice. Ask the class to add all the numbers from 13 to 23; or any two numbers. You will know the answer without adding

II being more than the difference between the two numbers, 13 and 23.

23486
7 3 4

894

67

9,846

2684026

30004

67

8200

200480

13+23

-XII

2

Also get different examples with the same answers, by changing the

order of the numbers; as,

The teacher can examine more quickly, while the scholars need not know that the answers are the same.

Keep up quick oral adding: as, 8, 11, 20, 27; carry 2; 6, 12, 17, 23. Make plain figures.

Object Lessons.

[NOTE. This lesson was recently given to a class of children seven years of age, and the answers below are the ones which those children gave.]

Object.—To cultivate perception, memory, and language.

Point. To develop idea of, and teach parts and uses of parts of a thimble.
Method.-Teacher calls the attention of the class, and shows them a thimble.

"What have I here?" "A thimble."

"What then, is our lesson to be about?"

"What word shall I write on the board?"

"A thimble."

"Thimble."

Teacher has the children spell the word thimble, if possible; if not, the teacher writes the word, and the class spell after it is written.

"What have I on the board ?" "Thimble."

"How many think I have a thimble on the board?"

The children will see the point, and some bright child will say "The word thimble." Teacher has the class tell her that she has the word thimble on the board. "That is right; now we will talk a little about the thimble. How many ever used a thimble?" Children raise hands.

"For what are thimbles used?" "To sew with."

"How do we use a thimble to sew with?" "We push the needle through the cloth with the thimble."

"Yes, but why do we need a thimble with which to push the needle?" "We should hurt the finger sometimes if we did not use the thimble." "Then what does the thimble do for the finger?" finger."

"The thimble protects the

"Yes, that is true; now who will come and find some part of this thimble,—some part that protects the finger?" Children raise hands. Teacher selects a child who comes and finds the sides.

"Who can come and find just the same part ?" Another child finds the same part. "What part have they found?" "Sides."

* Pendleton's Arithmetic Cards, however, will give any number of examples for drill in addition.