The children learn to measure the angles of each piece, and so to count the degrees. For this work there is a circular piece of white card-board, on which is drawn in black a semicircle with a radius of the same length as that of the circular insets. This semicircle is divided into 18 sectors by radii which extend beyond the circumference on to the background; and these radii are numbered by tens from 0° to 180°. Each sector is then subdivided into ten parts or degrees.
The diameter from 0° to 180° is outlined heavily and extends beyond the circumference, in order to facilitate the adjustment of the angle to be measured and to give a strict exactness of position. This is done also with the radius which marks 90°. The child places a piece of an inset in such a way that the vertex of the angle touches the middle of the diameter and one of its sides rests on the radius marked 0°. At the other end of the arc of the inset he can read the degrees of the angle. After these exercises, the children are able to measure any angle with a common protractor. Furthermore, they learn that a circle measures 360°, half a circle 180°, and a right angle 90°. Once having learned that a circumference measures 360° they can find the number of degrees in any angle; for example, in the angle of an inset representing the seventh of the circle, they know that 360° ÷ 7 = (approximately) 51°. This they can easily verify with their instruments by placing the sector on the graduated circle.
These calculations and measurements are repeated with all the different sectors of this series of insets where the circle is divided into from two to ten parts. The protractor shows approximately that:
| 1/3 | circle | = | 120° | and | 360° | ÷ | 3 | = | 120° |
| 1/4 | " | = | 90° | " | 360° | ÷ | 4 | = | 90° |
| 1/5 | " | = | 72° | " | 360° | ÷ | 5 | = | 72° |
| 1/6 | " | = | 60° | " | 360° | ÷ | 6 | = | 60° |
| 1/7 | " | = | 51° | " | 360° | ÷ | 7 | = | 51° |
| 1/8 | " | = | 45° | " | 360° | ÷ | 8 | = | 45° |
| 1/9 | " | = | 40° | " | 360° | ÷ | 9 | = | 40° |
| 1/10 | " | = | 36° | " | 360° | ÷ | 10 | = | 36° |
In this way the child learns to write fractions:
1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10
He has concrete impressions of them as well as an intuition of their arithmetical relationships.
The material lends itself to an infinite number of combinations, all of which are real arithmetical exercises in fractions. For example, the child can take from the circle the two half circles and replace them by four sectors of 90°, filling the same circular opening with entirely different pieces. From this he can draw the following conclusion: