Suppose we have selected a set of 12 objects. One-fourths of 12 are 3. Three times of three are nine. Thus, according to the first meaning of three-fourths, three times one-fourths of 12, we get nine as an answer. Now, to apply the second meaning, i.e. one-fourths of three sets of 12 objects we take three sets of 12 objects. This will in all make 36 objects. If we divide 36 objects into four equal parts, each part will have nine objects. This again shows that any meaning of fraction gives the same result.
Usually when we present the outcome of three-fourths using papers or objects, the two meanings become clear to the child instantly. The child's immediate reaction is "three times one-fourths of a thing equals one-fourths of the same three things". This clarity makes learning of various operations on fractions very easy.
However, once the child gets used to the different meanings of any fraction using concrete models, it should be taught to present fractions using appropriate drawings. Ability to represent addition, subtraction, multiplication and division of fractions in pictorial form brings in accuracy and clarity in the calculations, which is not possible in the calculations done using only rules. Moreover, drawing adds fun to learning fractions.
We now know that three-fourths can be represented using three squares of equal sizes drawn touching each other horizontally and then taking one-fourths of these three squares together. However, it involves a lot of labor, and children will have difficulty in dividing these three squares together into four equal parts. Therefore, we tell them to use the other method of representing three-fourths. We tell them to draw a single square. Then we tell them to divide that square into four parts using vertical lines. Then we ask them to color the three consecutive vertical parts of the squares beginning from the left-hand. This is shown below:
This way of representing any fractional number brings in some discipline required in the advanced work on fractions. Let children draw several pictures representing fractions like two-eighths, four-eighths, six-sixteenths, ten-thirtyseconds, etc.
Let children practice representing fractions like two-thirds, four-fifths, six-elevenths, etc., where the denominators are odd. Approximately equal number of odd or even portions should be acceptable as far as children's correct understanding of fractions through drawings gets reflected.
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