16 + 18 - 21 + 22 24.
If we have larger numbers, the pupils must never be allowed to make a number of long-division sums, but work thus: 7230 + 346 + 11621. They would factorise and put down 72 × 5 × 23 + 32 × 23 + 113 × 3 × 3 × 23. To get the common denominator we see we must multiply the first by 3 × 3 × 3; the second denominator by 5 × 3 × 3 × 3, the third by 5 × 2:—
7 × 3 × 3 × 3 + 3 × 5 × 9 + 11 × 5 2 × 5 × 23 × 3 × 3 × 3.
I have not given a complete exposition, but touched on what seems essential as regards the method and the order of teaching, derived from my experience of children’s difficulties, some will think, I fear, at unnecessary length.
In regard to the later rules for decimals, I need only make two remarks: that the points should be always removed from the divisor, e.g.:—
·000035 ÷ 5·9623 = ·3559623.
and the point put in as soon as we reach the decimal fraction. In working circulators it is well for a time to express the equations thus: ·32̇94̇ = No.
| 10,000 No | = | 3294·294, etc. |
| 10 No | = | 3·294, etc. |
| 9990 No | = | 3291 |
| ∴ No | = | 3291 |
| 9990 |
Proportion.As regards proportion, I need add little. But there is one vexed question: Shall we let children work by the unitary method? I think not, at least not those who are likely to go on to mathematics. We cannot get the thought of proportion too ingrained, and the unitary method evades it.
In compound proportion I would make pupils work out the double process in detail, and then with factors only, e.g.:—