Addition of Fractions 1

Let us think of a word problem for which the result is 1/2 + 1/3 which we know equals 5/6. How do we interpret the result as a physical event?

Each of these fractions is related to a "whole". Can we think of each of the wholes being different? A typical situation could be ½ of a banana & 1/3 of a mango. In real life, we can easily place half a banana & one third of a mango in the same plate. But can we mathematically interpret the result of this addition in terms of bananas & mangoes? Can we relate 5/6 to the plate having both banana & mango pieces?

A little thought would reveal that we cannot mathematically interpret this operation and its result unless both the fractions are of the "same" whole. In the above problem, both the fractions ½ & 1/3 should be of either a banana or a mango or a bowl of a dessert made of bananas & mangoes.

Rephrasing the Problem

Hence one valid word problem for the above fraction addition could be - Mother gives ½ of a banana to one child and 1/3 of the same banana to another child. What fraction of the banana has she used up? An extension to the problem is how much fraction of the banana is still remaining with her?

The above problem can be rephrased in a general sense as below.

What fraction of a whole do you get, when a fraction of that whole is added to another fraction that same whole?

The same idea would also be applicable for subtraction problems.

The same idea also exists, though we may not realize it, even for addition & subtraction of whole numbers. When we say 4 + 5 we are implying that 4 and 5 of the same type. If we have 5 apples & 5 oranges, while adding, we refer to both as "fruits".

Implications of the "same Whole"

The implication here is that the "whole" should be such that both ½ & 1/3 can be "taken out" of that whole.

It is also clear that we can take "any entity" as a whole for doing the above problem, so long as that whole is such that we can take a and a ½ & 1/3 from the same whole.

Difficulty with Continuous Representation

Using the idea of "taking out" or "giving out" creates a practical problem if we use continuous representations of fractions for an explanation.

If we use paper where we have to either shade or cut out a or , then the original whole gets "destroyed or obscured" and the students would have difficulty in visualizing the second fraction which also has to be cut or marked out from the same whole.

Another problem would be to represent situations where the sum of the fractions is more than a whole or results in an improper fraction. An example is ½ & ¾ which total to 5/4 or 1 ¼.

For such situations, understanding would be easier if we use discrete representation of fractions.