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We have test sheets where the numbers from 1 to 100 are arranged in rows of 10, forming a square. Here the child's exercise consists in underlining, in different squares, the multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10. The numbers so underlined stand out like a design in such a way that the child easily can study and compare the tables. For instance, in the square where he underlines the multiples of 2 all the even numbers in the vertical columns are marked; in the multiple of 4 we have the same linear grouping—a vertical line—but the numbers marked are alternate numbers; in 6 the same vertical grouping continues, but one number is marked and two are skipped; and again in the multiples of 8 the same design is repeated with the difference that every fourth number is underlined. On the square marked off for the multiples of 3 the numbers marked form oblique lines running from right to left and all the numbers in these oblique lines are underlined. In the multiples of 6 the design is the same but only the alternating numbers are underlined. The 6 therefore, partakes of the type of the 2 and of the 3; and both of these are indeed its factors.
VI
SQUARE AND CUBE OF NUMBERS
Let us take two of the two-bead bars (green) which were used in counting in the first bead exercises. Here, however, these form part of another series of beads. Along with these two bars there is a small chain:
By joining two like bars, the chains represent 2 × 2. There is another combination of these same objects—the two bars are joined together not in a chain but in the form of a square:
