Original Problems.

To be Composed by the Pupils for the Class.

60. 1. Give the prices of things which you have bought, or which have been bought of you, supposing yourself to be a salesman in a store, at a fair, etc., and ask the cost. (What must you give besides the prices?)

2. Problems almost without number can be made about the number of square feet or yards in the floors, ceilings, blackboard, wall-maps, etc., of the school-room, giving the pupils time to measure for themselves.

Note. The pupils proposing such questions should first take the necessary measurements. Let them take the nearest whole number of feet, inches or yards, according to the length of the line measured.

3. Give the number of rows of seats and the number of grown people that can sit in one row, and require the class to find how many can be seated in any church or hall you may name.

4. Borrow a tape-line, measure for yourself, and give the distance from one telegraph-pole to another, and tell the number of poles between any two places you may name; then ask how many feet or yards, from one place to the other.

5. Ask how many trees in Mr. -'s orchard, after telling the class how many rows there are, and how many trees in a row. How many hills of corn in a field, etc.

6. Require to know about how many apples there are in a wagon-load of 50 bushels, say of greenings or any common fruit sold at market. If the class can't tell how it may be done without counting all the apples, tell them.

7. Give such problems as these, to be done in the shortest possible time, changing the numbers from those given here : What is the difference between 8 and 9 times 562? 35 and 45 times 976? Between 132 and 232 times 78? The sum of 36 times and 64 times 84? (Be sure that you see the point yourself.)

1. Write the letter a twenty-four times on slate or paper. How many times 12 a's are there in the 24 a's? How many times 6 a's? How many times 4 a's ?

2. At 94 apiece, how many lead-pencils can be bought for 724 ? At 124 apiece? At 84? At 64 ?

Suggestion. If a boy had 72 one-cent pieces, and knew no more of arithmetic than how to count, he could divide the 72 pieces into lots of 94 each, and thus find that for 724 he could buy 8 tops at 94 apiece, or as many tops as there are times 9¢ in 724.

3. With 42 ears of corn, how many horses can be fed if 6 ears are given to each? (Make 42 marks, and divide into groups of 6.)

4. In the same way, show how many dozen there are in 84, in 60, in 96, in 48. (Twelve single things make a dozen.)

5. How many top-strings can be cut from 48 feet of twine, if the strings are made 6 ft. long? If 8 ft. long?

61. A thorough knowledge of the multiplication table supersedes the necessity of marks, or other counters, except for purposes of illustration; for, if we know that twice 12 are 24, we know equally well that in 24 there are two 12's.

Note 1.-For practice at this point let a table, like the one suggested on page 53, be written on the black-board, the first column being omitted. Then a pupil pointing successively to each number in a column, and knowing that the number at the head is one factor, he announces the other; thus under 7 he announces 10, 6, 7, etc., as rapidly as possible.

Note 2.-Here and elsewhere let counting be absolutely prohibited, except in the way of illustration.

ORAL EXERCISES.

62. Tens and Units.-Caution. In the following exercises do not repeat 3 in 18 six times, 3 in 30 ten times, but knowing that you are to tell how many times 3 there are in 18, 30, 27, etc., say at once 6, 10, 9, etc.

Note. The illustrative exercises introduced here and elsewhere are intended only as examples of what should be done in this direction. Problem after problem should be illustrated till the learner feels that in dealing with figures he is dealing with representatives of number, and that the arithmetical processes only indicate what a person ignorant of arithmetic might do to solve similar problems. Let the work be actually performed whenever possible. Labor impresses its lessons more deeply than observation. Mere instruction is not to be compared with it.

63. Hundreds. -How many times 9 in 270?

Write the letter c 27 times in a line, and mark them off into groups of 9 each; thus,

c c c c c c c c c, c c c c c c c c c, c c c c c c c c c. Now, think how many c's there would be in ten such lines. How many times 9 c's. If you can not think the answers to these questions without writing the ten lines, write them and count.

64. Thousands.-How many times 4 in 3600 ?

Write the letter e 36 times in a line, and mark them off into groups of 4 e's each; then, think how many e's there would be in 10 such lines; in 100. How many groups of 4 in one line? In 10 lines? In 100 lines?

27. How many times 10 in 80? 110? 70? 120? 60? 90? 20? 28. How many times 11 in 22? 121? 55? 132? 88? 66? 33? 29. How many times 12 in 84? 144? 72? 108? 60? 96? 132? Applications.-30. How many dozen in 720? 600? 8400? 13200? 31. An army of 96000 men is marching 12 in a rank. How many ranks are there?

32. If pork is worth $7 a cwt., how many cwt. can be bought for $6300? $7700 ? $5600 ? 840 ?

33. If a sheet of paper is folded so as to make 8 pages, how many sheets will be needed for 1600 pages? 5600 ?

34. If a man walks at the rate of 4 miles an hour, in how many hours will he walk 48 miles ? 3600 miles? 400 miles?

ILLUSTRATIVE SLATE EXERCISES.

65. Thousands, and lower orders.-35. How many balls of twine can be bought for $2.73, or 2734, at 74 a ball? (How many times 74 are there in 2734.)

Make ten lines of 27 dots each, and beneath them make 3 single dots. Divide each full line of dots into groups of seven, in this way:

When the above work is neatly done, copy the following paragraphs, carefully filling all the blanks:

3. The ten 6's over and the 3 in the short line beneath make

together dots. Out of 63 dots we can make

7 dots each. Hence there are times 7 cents in 2734.

groups of

times 7 dots in 273, and

Therefore, at a ball, we can get

balls of twine for 2734.