Equivalent, Identical and Similar Figures
First Series of Insets: Squares and Divided Figures. This is a series of nine square insets, ten by ten centimeters, each of which has a white foundation of the same size as the inset.
One inset consists of an entire square; the others are made up in the following manner:
| A | square | divided | into | two equal rectangles |
| " | " | " | " | four equal squares |
| " | " | " | " | eight equal rectangles |
| " | " | " | " | sixteen equal squares |
| " | " | " | " | two equal triangles |
| " | " | " | " | four equal triangles |
| " | " | " | " | eight equal triangles |
| " | " | " | " | sixteen equal triangles |
The child can take the square divided into two rectangles and the one divided into two triangles and interchange them: that is, he can build the first square with triangles and the second with rectangles. The two triangles can be superimposed by placing them in contact at the under side where there is no knob, and the same can be done with the rectangles, thus showing their equivalence by placing one on the other. But there also is a certain relation between the triangles and the rectangles; indeed, they are each half of the same square; yet they differ greatly in form. Inductively the child gains an idea of equivalent figures. The two triangles are identical; the two rectangles also are identical; whereas the triangle and the rectangle are equivalents. The child soon makes comparisons by placing the triangle on the rectangle, and he notices at once that the small triangle which is left over on the rectangle equals the small triangle which remains uncovered on the larger triangle, and therefore that the triangle and the rectangle, though they do not have the same form, have the same area.
This exercise in observation is repeated in a like manner with all the other insets, which are divided successively into four, eight, and sixteen parts. The small square which is a fourth of the original square, resulting from the division of this latter by two medial lines, is equivalent to the triangle which was formed by dividing this same original square into four triangles by two diagonal lines. And so on.
By comparing the different figures the child learns the difference between equivalent figures and identical figures. The two rectangles are the result of dividing the large square by a medial line and are identical; the two triangles are formed by dividing the original square by a diagonal line, etc. Similar figures, on the other hand, are those which have the same form but differ in dimension. For example, the rectangle which is half of the original square and the one which is half of the smaller square—that is, an eighth of the original square—are neither identical nor equivalent but they are similar figures. The same may be said of the large square and of the smaller ones which represent a fourth, a sixteenth, etc.
Through these divisions of the square an idea of fractions is gained intuitively. However, this is not the material used for the study of fractions. For this purpose there is another series of insets.
Second Series of Insets: Fractions. There are ten metal plates, each of which has a circular opening ten centimeters in diameter. One inset is a complete circle; the other circular insets are divided respectively into 2, 3, 4, 5, 6, 7, 8, 9, and 10 equal parts.