| 1909 | ||
| 23 | ) | 43921 |
| 23 | ||
| 209 | ||
| 207 | ||
| 221 | ||
| 207 | ||
| 14 | ||
Factors, measures, multiples.Here, while continuing to work many miscellaneous examples, it may be well to interpose some useful exercises on matters interesting and yet puzzling to children, on factors and measures of numbers, and primes and squares. If they get quite familiar with factors, they will not have such difficulty as they do when they come upon the whole set at once: factors, common factors, measure, common measure, G.C.M., multiple, common multiple, L.C.M.
Let us bring out the box of buttons once more and arrange the numbers, finding the factors. 1, 2, 3 have only the number itself, and so these are called primes, because they have no other factor than 1, the first number.
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But 4 is not only 4 × 1, it is 2 × 2, and we may notice that the dots form a square—it is a compound number. 5 is again a prime; 6 can be arranged in three ways—in a row of ones, in three rows of 2 or two rows of 3, but these are the same if we look at them a different way round, i.e., 2 × 3 is 3 × 2. 7 is a prime, but for 8 we can have 2 × 4 and 4 × 2, which are the same. 9 is again a square number; it has no factors except 3. Here we might give the expressions 22 for 2 × 2, 32 for 3 × 3 and 33 for 3 × 3 × 3.
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We might go on to pick out all the primes by what is called the sieve of Eratosthenes, and to give all squares and cubes, say up to 100. Sometimes we speak of measure of numbers; 4 can be measured into rows of twos, 6 into rows of twos or threes, so 2 is said to be a common measure of 4 and 6.
After working some examples in factors and measures, it will be well to leave the matter, returning to the subject later. I should pass over for girls the wearisome exercises in weights and measures, bills of parcels, etc., very slightly. These things belong to the shop rather than the school, and waste the time that should be given to learning principles.
Vulgar fractions.We may proceed at once to fractions. In nothing is the advice Festina lente more valuable than now. Once give the children a clear idea of what a fraction is, how the two numbers represent respectively the size of the pieces and the number taken, and all will be easy. They are already familiar with 1⁄2d. and 3⁄4d., so we can get from them that the lower figure stands for the number of pieces into which the penny is divided, and that the figure above shows the number of pieces taken. Many fractions should be drawn by the children—5⁄6 of a line, a circle, a square, etc. The fraction may be written thus: 56 numberernamer,
5 gives the number of pieces taken; is numberer or numerator; 6 gives the number of pieces into which the whole is cut,
the size, the name, the denominator.