In regard to the remaining five, it is the half of ten, and by cutting the long rod in two, that is dividing ten by two, we would have five; ten divided by two equals five. The written record of all this reads:
|
10 − 4 = 6 10 − 3 = 7 10 − 2 = 8 10 − 1 = 9 | 10 ÷ 2 = 5 |
Once the children have mastered this exercise they multiply it spontaneously. Can we make three in two ways? We place the one after the two and then write, in order that we may remember what we have done, 2 + 1 = 3. Can we make two rods equal to number four? 3 + 1 = 4, and 4 - 3 = 1; 4 - 1 = 3. Rod number two in its relation to rod number four is treated as was five in relation to ten; that is, we turn it over and show that it is contained in four exactly two times: 4 ÷ 2 = 2; 2 × 2 = 4. Another problem: let us see with how many rods we can play this same game. We can do it with three and six; and with four and eight; that is,
| 2 × 2 = 4 | 3 × 2 = 6 | 4 × 2 = 8 | 5 × 2 = 10 |
| 10 ÷ 2 = 5 | 8 ÷ 2 = 4 | 6 ÷ 2 = 3 | 4 ÷ 2 = 2 |
At this point we find that the cubes with which we played the number memory games are of help:
From this arrangement, one sees at once which are the numbers which can be divided by two—all those which have not an odd cube at the bottom. These are the even numbers, because they can be arranged in pairs, two by two; and the division by two is easy, all that is necessary being to separate the two lines of twos that stand one under the other. Counting the cubes of each file we have the quotient. To recompose the primitive number we need only reassemble the two files thus 2 × 3 = 6. All this is not difficult for children of five years.
The repetition soon becomes monotonous, but the exercises may be most easily changed, taking again the set of long rods, and instead of placing rod number one after nine, place it after ten. In the same way, place two after nine, and three after eight. In this way we make rods of a greater length than ten; lengths which we must learn to name eleven, twelve, thirteen, etc., as far as twenty. The little cubes, too, may be used to fix these higher numbers.
Having learned the operations through ten, we proceed with no difficulty to twenty. The one difficulty lies in the decimal numbers which require certain lessons.