1 = 1/2 + 1/2 = 2/2 = 1.
All these substitutions may be expressed in figures thus:
5/10 + 1/4 + 2/8 = 1/2 + 1/4 + 1/4 = 1/2 + 2/4 = 1/2 + 1/2 = 2/2 = 1.
This is one means of initiating a child intuitively into the operations used for the reduction of fractions to their lowest terms.
Improper fractions also interest them very much. They come to these by adding a number of sectors which fill two, three, or four circles. To find the whole numbers which exist under the guise of fractions is a little like putting away in their proper places the circular insets which have been all mixed up. The children manifest a desire to learn the real operations of fractions. With improper fractions they originate most unusual sums, like the following:
[8 + (7/7 + 18/9 + 24/2) + 1] =
8
[8 + (1 + 2 + 12) + 1] =
8
8 + 15 + 1 = 24/8 = 3.
8
We have a series of commands which may be used as a guide for the child's work. Here are some examples:
—Take 1/5 of 25 beads
—Take 1/4 " 36 counters
—Take 1/6 " 24 beans
—Take 1/3 " 27 beans
—Take 1/10 " 40 beans
—Take 2/5 " 60 counters
In this last there are two operations: