first numbers. We take a sheet of paper ruled in squares and shade the first square of the first line, then the first two squares of the second line, the first three of the third, etc. (Fig. 7). The whole number of squares shaded in this manner represents visibly the sum of the first whole numbers up to any one we may choose—to 7 in the figure. If we give this paper to the child and ask him to return it, he will
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| Fig. 7. | Fig. 8. | |
very easily perceive that the figures formed by the white and the black squares are alike. The number sought for will therefore be equal to half the sum of the squares—that is, in the present example
1 + 2 + 3 + 4 + 5 + 6 + 7 = (7 X 8):2 = 28,
we can prove by reasoning that if n be taken to represent the last number we shall have for the sum
S = n(n +1)2
I introduce this formula to define my thought better, but one can make the child perceive the numbers that are wanted without writing down a single character.
Somewhat similar is the method of finding the sum of the odd numbers. For this it will be enough to take our square-ruled sheet of paper and shade the first square on the loft, then the three squares around it, which will form with it a square (1 + 3 = 4); continuing thus we obtain, as the figure readily shows (Fig. 8), a square formed of a series of shaded zones, representing the series of odd numbers, the examination of which will illustrate the property to the child.
In another direction it is possible to give the child algebraic ideas much beyond anything we would imagine. Suppose, for example, we want to give him a conception of addition. He easily realizes
