Let there be plenty of such questions as these: What is the effect of increasing the numerator or the denominator? Of doubling each? Of halving each? Notice that most things grow larger the larger the number, but with a fraction the larger the denominator the smaller the pieces. Children should not have books giving explanations. They must discover these by the dialectic process, and then in their own words answer questions, and sometimes explain every step in the sum they are working. All we require in books are well-chosen examples. Those who have not taught, have no idea how hard children find it to get really hold of the nature of a fraction. Homely illustrations should not be spared. For instance, there are two ways of getting much cake. To take many pieces, that is have a large numerator,—or to look out the biggest piece, that is have a small denominator.
Multiplication and division by integers.We are now ready for multiplication and division by integers. Take 5⁄12. There are two ways of making the fraction twice as large, that is by taking twice as many pieces, that is 10⁄12, or twice as large pieces, 5⁄6. The shortest way must always be insisted on. Similarly, 4⁄5 may be divided by 2 in two ways. Many examples should be worked out in detail thus:—
3⁄14 × 7 ÷ 3 ÷ 4 × 5 ÷ 8.
3⁄14 × 7 = 3⁄2; 3⁄2 ÷ 3 = 1⁄2; 1⁄2 ÷ 4 = 1⁄8; 1⁄8 × 5 = 5⁄8; 5⁄8 ÷ 8 = 5⁄64.
Nearly all children will write thus: 3⁄4 × 7 = 3⁄2 ÷ 3, etc., and leave the whole unreadable.
Next should come the proposition 7 is 8 times as large as 7⁄8. (Some pupils might be ready to use letters by this time, a is b times as large as ab. The teacher must be on the watch for such.) It is very difficult for young children to see this, and also that 7⁄8 is the same as 7 ÷ 8. This should be illustrated by drawings in a variety of ways.
By fractions.On that would follow multiplication of fractions by fractions, which is explained as making a mistake and correcting. Thus if we have to multiply 5⁄7 × 2⁄3, we know how to multiply by 2, so we do that first: 5⁄7 × 2 = 10⁄7. But we have multiplied by a number three times too large; to correct the mistake, we must divide by 3; 10⁄7 ÷ 3 = 10⁄21. Similarly, we explain division. Not until some sums have been worked in detail should pupils be allowed to get hold of the rules. They should work with factors only, whenever possible.
Reduction.Now we might return to the subject of multiples and measures. We have 16⁄24. We want to have it in its simplest form. We divide it into factors: 1624 = 2 × 82 × 12; 2 is a common measure of both; the 2 above makes the fraction twice as large, the 2 below twice as small, so both may be taken out. But we might have said 1624 = 8 × 28 × 3; 8 is the largest number that will measure both, so it is called the greatest common measure. I think it better not to give the ordinary rule for finding G.C.M. until its proof can be given algebraically. It is very seldom that children will fail in the attempt to analyse numbers, and so find out all their common measures.
G.C.M. and L.C.M.The common rules should now be given for finding at sight when a number is commensurable by each digit, though the reason of these rules will not perhaps appear yet. These children know at a glance whether a number can be measured by 2, 4, 8, 3 or 9, and remove the common factor.
Suppose we have 80089009, we cannot see a common factor, but we can proceed to break it up, one being commensurable by 8 and the other by 9. Then we get 8 × 10019 × 1001, and the greatest common measure comes to light. We see that the numerator of 11762205 is commensurable by 4 and 3, i.e., by 12, the denominator by 9:—