Numbers like 7⁄8, 1⁄5, 2⁄3, 3⁄4, 11⁄8, 7⁄6, 1⁄3, 4⁄3, 1⁄8, 1⁄6 are called fractions.
Numbers like 5¼, 73⁄8, 9½, 164⁄5, 3157⁄8, 11⁄3, 12⁄3 are called mixed numbers.
(17) The terms numerator and denominator are connected with the upper and lower numbers composing a fraction.
Building this somewhat elaborate series of minor abilities seems to be a very roundabout way of getting knowledge of the meaning of a fraction, and is, if we take no account of what is got along with this knowledge. Taking account of the intrinsically useful habits that are built up, one might retort that the pupil gets his knowledge of the meaning of a fraction at zero cost.
KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES
Consider next the knowledge of the subtraction and division 'Tables.' The usual treatment presupposes that learning them consists of forming independently the bonds:—
| 3 − 1 = 2 | 4 ÷ 2 = 2 |
| 3 − 2 = 1 | 6 ÷ 2 = 3 |
| 4 − 1 = 3 | 6 ÷ 3 = 2 |
|
. . . |
. . . |
| 18 − 9 = 9 | 81 ÷ 9 = 9 |
In fact, however, these 126 bonds are not formed independently. Except perhaps in the case of the dullest twentieth of pupils, they are somewhat facilitated by the already learned additions and multiplications. And by proper arrangement of the learning they may be enormously facilitated thereby. Indeed, we may replace the independent memorizing of these facts by a set of instructive exercises wherein the pupil derives the subtractions from the corresponding additions by simple acts of reasoning or selective thinking. As soon as the additions giving sums of 9 or less are learned, let the pupil attack an exercise like the following:—
Write the missing numbers:—
| A | B | C | D |
| 3 and ... are 5. | 5 and ... are 8. | 4 and ... are 5. | 4 and ... are 8. |
| 3 and ... are 9. | 3 and ... are 6. | 5 and ... are 6. | 1 and ... are 7. |
| 4 and ... are 7. | 4 and ... are 9. | 6 and ... are 9. | 6 and ... are 7. |
| 5 and ... are 7. | 2 and ... = 6. | 1 and ... are 8. | 8 and ... are 9. |
| 6 and ... are 8. | 5 and ... = 9. | 3 and ... are 7. | 3 + ... are 4. |
| 4 and ... are 6. | 2 and ... = 7. | 1 + ... are 3. | 7 + ... are 8. |
| 2 and ... are 5. | 3 and ... = 8. | 1 + ... are 5. | 4 + ... are 9. |
| 2 and ... = 8. | 1 and ... = 4. | 4 + ... are 8. | 2 + ... are 3. |
| 3 and ... = 6. | 2 and ... = 4. | 7 + ... are 9. | 1 + ... are 9. |
| 6 and ... = 9. | 3 and ... = 8. | 2 + ... = 4. | 3 + ... = 6. |
| 4 and ... = 6. | 6 and ... = 7. | 3 + ... = 8. | 5 + ... = 9. |
| 4 and ... = 7. | 2 and ... = 5. | 4 + ... = 5. | 1 + ... = 3. |