30. How many times one are four times two? Count off four times two: now how many have you? Answer, eight. Then four times two are eight. Well, how many times one are two times four? Count the first parcel of four; this is one time four; count on five, six, seven, eight; now we are through the second parcel of fours; how many are there? Answer, eight. Right, the same as the other; two times four are eight, and four times two are eight this proves the work to be done right.

31. Let us make a new lesson. How many times one are three times three? Look at the three parcels of three: one, two, three, four, five, six, seven, eight, nine. Ans. 9.

32. How many times one will four times three make? Three, and three are six, and three are nine, and three are twelve. Answer, twelve.

33. How many times one are there in four times four? Look at the parcels of four, place your finger down at the end of the fourth parcel; now count how many there are till coming to your finger. Sixteen. Right. Then four times four make sixteen, or sixteen times one.

34. How many times one are two times eight: Look at two parcels of eight: count them: how many do they make? Sixteen. Right. Then two times eight are equal to four times four. Can you tell me how many eight times two will make? Count off eight parcels of two: how many do they make? Sixteen, the same as two times eight. That is right; figures will prove one and the other when reckoned right. Four times four are sixteen, two times eight are sixteen, and eight times two are sixteen.

35. I wish to know how many times one three times five will make? Count off the first, second, and third parcels of five: how many? Fifteen. Well, how many times one will five threes make? Count off five parcels of three. Five threes make fifteen also, the same as three fives. All right.

36. DIVISION.

Now we will try to perform Division with our checkerboard.

How many times two are contained in eighteen? Place your finger at the end of eighteen on the range of twos; how many times two are there in that space or range till coming to your finger? Answer, nine.

37. Count off thirty on the parcels of five: How many times five in thirty? Answer, six.

38. How many times eight in forty? Count off forty on the parcels of eight: how many times eight? Five.

Count off forty-eight on the same range. How many times eight? Six times.

39. In forty-nine, how many times seven? Count off forty-nine on the range of sevens; how many times seven are there? Seven times. Then seven times seven are forty-nine: all right.

40. I have sixty-four chestnuts, and wish to divide them among eight boys; how many times eight are there in sixty-four, and how many will each boy have? Count off sixty-four on the range of eights: how many times eight are

there? Eight times.

Then each boy must have eight. Give eight to each one and see if the operation will prove in the result.*

41. Suppose I have forty-two walnuts, and wish to divide them equally among six boys; how many can each one have? Count off forty-two on the range of sixes: how many times six are there? Seven.

42. Nine cakes cost forty-five cents: how much was that apiece? Count off forty-five on the range of nines; how many times nine are there? Five times. Right, the cakes cost five cents each.

43. I have fifty-six pencils; how shall I divide them that seven boys may have an equal number each? Count off fifty-six on the range of sevens: how many times seven are there? Eight times. Very well, each boy must have eight pencils. But, here comes little George Sprightly; you all consent that I divide them again and give him an equal share? Agreed.

will

Now there are eight of you: how many must each one have? Count off fifty-six on the range of eights: how many times eight are there? Seven. Right, each one must have seven pencils, because seven times eight are fifty-six.

44. These questions are only a small number of specimens for the Teacher; he may exercise his ingenuity in forming different lessons, and of such variety, as his judgment will direct from time to time for the benefit of pupils. Exercising questions similar to the following, may be * Here explain result, and use no technical terms without explanation.

given out occasionally to advantage. As, four more three, that is, four added to three, equal how many?

45. Four, more three, more two, equal how many?

46. Four, more three, more two, less one; that is, one subtracted, equal how many?

Six, more three, more, two, less one, equal how many times five?

47. Eight, multiplied by two, equals how many? Eight, multiplied by three, less four, equals how many times ten?

48. Four, more the half part of two, equal how many ?

49. Fifteen, less five, gives a certain sum for a remainder: how many times five equal that remainder?

50. The third of twelve, more the one half the third of twelve, equal how many?

51. The third of forty-eight, and half the third of fortyeight, are equal to how many times six? The fourth part of eight, more the third of fifteen, more the half of eighteen, equal how many times the half part of eight? 4.

52. The younger classes, under the inspection of some one from a senior class, may be exercised with such simple, easy, and diverting questions, and save much time, which otherwise would be worse than lost; for this employment at home and at school, will often keep them out of mischief; it will exercise their mental faculties, and teach

them how to distinguish truth from error; it will facilitate an attainment of the knowledge of the multiplication table suspended by the wall, and it will aid in expediting, that which is already an expeditious mode of teaching and learning arithmetic.

53. The Teacher may use his ingenuity in contriving such lessons as will suit every class designed for this mode of working, and compose the latter part of lessons progressively easier or plainer than the first, that the inexperienced of the class may not grow discouraged, by being unable to answer questions as well as the more knowing. This variation in the lessons cannot be justly construed as partiality, because these exercises are separate from the common tasks in school.

54. If business be so pressing that the Teacher cannot attend to these simple, though in the sequel, important manœuvres, let him call on some of the able monitors to prepare a lesson on paper, for any certain class or number of classes who are designed for an airing, and give directions to their overseers how to proceed.

55. As to a general rule for pastime, I chuse to be liberal, especially in the summer season, and endeavour not to confine them in school, except as a punishment for crimes more than two hours at a time; and wish to come short of that, if business render it convenient. When they first approach that delightsome and healthy element, the open air, let cach class form separately in a ring, with their overseer in the centre; he will then read to them the first question to be solved, two or three times, very slowly and distinctly; he will also inform them that in case No. 1,