It would be well now to give children some practice in counting backwards, and in rapid viva voce addition, which the exercises in analysis of numbers will have made easy. E.g., 15 + 7, the number naturally falls apart into 5 + 2, and we get 22; 29 + 7, it falls into 6 + 1, at the next step into 3 + 4.
Multiplication.We should next proceed to continued addition or multiplication. Many children come to school not knowing that multiplication is continued addition, and still fewer have any idea that division is continued subtraction. In entrance papers I have had sheets covered in reply to such questions as “How often can 19 be subtracted from 584?”
A few multiplications should be worked with real things. Thus, we have to give to 5 people 3 buttons each. We arrange them in parcels of 3 and add 3 to our pile five times. Now, if we have 15 and want to know how many times we can take away threes, we find we can do it five times over; this is subtraction or undoing the addition. It is the same as making little parcels of 3 each, and so continued subtraction is called division. Some continued addition sums should be given, thus: Find 4 times 891.
| 891 |
| 891 |
| 891 |
| 891 |
| 3564 |
It will be easily seen that such sums are done much more quickly if we know by heart how much 4 nines come to, and how much 4 eights; and so people learn their addition tables by heart, and children make them out for themselves thus, generally up to 12 times, some learn up to 20 times. Here is part of 7 times worked out:—
| 7 | times | 1 | = | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | |||||
| 7 | „ | 2 | = | 14 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | |
| 7 | „ | 3 | = | 21 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | ||
| 7 | „ | 4 | = | 28 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | |||
| 7 | „ | 5 | = | 35 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | ||||
| 7 | „ | 6 | = | 42 | 7 | 7 | 7 | 7 | 7 | 7 | |||||
| 7 | „ | 7 | = | 49 | 7 | 7 | 7 | 7 | 7 | ||||||
| 7 | „ | 8 | = | 56 | 7 | 7 | 7 | 7 | |||||||
| 7 | „ | 9 | = | 63 | 7 | 7 | 7 | ||||||||
| 7 | „ | 10 | = | 70 | 7 | 7 | |||||||||
| 7 | „ | 11 | = | 77 | 7 | ||||||||||
| 7 | „ | 12 | = | 84 | |||||||||||
The signs × and ÷ may now be given. All tables should be written out and learned, and it is well to say both ways, 6 × 7 = 42, and 7 × 6 = 42. There are certain numbers that are easily remembered, others in which children habitually make mistakes: it is a waste of time to hear the tables therefore all through after a time, but these difficult ones, 7 × 8, 6 × 9, 11 × 11, etc., should be insisted on; then, finally, the whole heard through, and any about which there is the slightest hesitation asked for daily. If children can learn up to 20 times without much trouble, it is an advantage.
We could next point out that this continued addition is called multiplication, and all the numbers made up by continually adding threes would be called multiples of 3, i.e., many times 3. So 12 would be a multiple of 2 or 3 or 4.
Then examples should be worked, but here let me say that at the early stages concrete examples should abound. Many good books there are containing miscellaneous examples of concrete quantities, such as, There are 319 fruit trees planted in each field for making jam, and there are 12 fields; how many fruit trees? Or, 7 labourers have to be paid on Saturday £17 each; how much will they get in 12 weeks?
When children know the effect of pushing numbers to the left, multiplication by two figures will be easy, but the child should be accustomed to write at the end of each row the real sum, thus: 73 × 25:—