The rule for such bonds is, of course, to form them strongly enough so that they work quickly and accurately for the time being and facilitate the bonds that are to replace them, but not to overlearn them. There is a difference between learning something to be held for a short time, and the same amount of energy spent in learning for long retention. The former sort of learning is, of course, appropriate with many of these propædeutic bonds. The bonds mentioned as illustrations are not _purely_ propædeutic, nor formed _only_ to be transmuted into something else. Even the saying of 'and' in addition has some genuine, intrinsic value in distinguishing the process of addition, and may perhaps be usefully reviewed for a brief space during the first steps in adding common fractions. Some such propædeutic bonds may be worth while apart from their value in preparing for other bonds. Consider, for example, exercises like those shown below which are propædeutic to long division, giving the pupil some basis in experience for his selection of the quotient figures. These multiplications are intrinsically worth doing, especially the 12s and 25s. Whatever the pupil remembers of them will be to his advantage. 1. Count by 11s to 132, beginning 11, 22, 33. 2. Count by 12s to 144, beginning 12, 24, 36. 3. Count by 25s to 300, beginning 25, 50, 75. 4. State the missing numbers:-- A. B. C. D. 3 11s = 5 11s = 8 ft. = .... in. 2 dozen = 4 12s = 3 12s = 10 ft. = .... in. 4 dozen = 5 12s = 6 12s = 7 ft. = .... in. 10 dozen = 6 11s = 12 11s = 4 ft. = .... in. 5 dozen = 9 11s = 2 12s = 6 ft. = .... in. 7 dozen = 7 12s = 9 12s = 9 ft. = .... in. 12 dozen = 8 12s = 7 11s = 11 ft. = .... in. 9 dozen = 11 11s = 12 12s = 5 ft. = .... in. 6 dozen = 5. Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents, one dollar," and so on. 6. Count by 15s to $1.50. 7. Find the products. Do not use pencil. Think what they are. A. B. C. D. E. 2 × 25 3 × 15 2 × 12 4 × 11 6 × 25 3 × 25 10 × 15 2 × 15 4 × 15 6 × 15 5 × 25 4 × 15 2 × 25 4 × 12 6 × 12 10 × 25 2 × 15 2 × 11 4 × 25 6 × 11 4 × 25 7 × 15 3 × 25 5 × 11 7 × 12 6 × 25 9 × 15 3 × 15 5 × 12 7 × 15 8 × 25 5 × 15 3 × 11 5 × 15 7 × 25 7 × 25 8 × 15 3 × 12 5 × 25 7 × 11 9 × 25 6 × 15 8 × 12 9 × 12 8 × 25 State the missing numbers:--