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We were discussing how we could bridge the concept of whole numbers with the fractional numbers.
When we talk about 3 on a number line, we really take 3 times "unit 1" on the number line. In this way we refer to a distance from point 0 to 3 on the number line, which is thrice the distance from point 0 to 1. See the figure below:
Whereas when we refer to 1/3 on a number line, we really take 1/3rd of "unit 1" on the number line. To get 1/3rd of "unit 1" we choose two points between 0 to 1 on the number line in such a way that we get three equal parts of unit 1. Then the distance from 0 to first point to its right on the number line is 1/3rd. See the figure below:
Here children also should be reminded about the similarity of the fractional number with the division of whole numbers. We have shown them that the division "4 ÷ 2 " can also be written as 4/2. In the article Introducing division (I) we have discussed two meanings of division. For example, 4/2 can be interpreted as "if 4 objects are divided into two equal parts, each part will have two objects" or as "if each time two objects are taken away from the four objects to form a group, two such groups can be formed". Both the interpretations will give us the answer as two. However, when we talk about 4/2 as a fraction we have to show the learners that it is 4 times ½ (or half). We can give them many halves of squares of same sizes. Let them take four halves and find out how many whole squares could be formed. Later they can be shown to represent 4 times ½ on the number line as well. Those interested in understanding and teaching fractions should read Some remarks on teaching of fractions in elementary school and Chapter 2: Fractions (Draft) (June 20, 2001; REVISED September 3, 2002) both by Prof. H Wu of Department of Mathematics of University of California, Berkeley. Both offer very good insights into the teaching of different aspects of fractions.
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