3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
d) Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
In the previous post, I discussed a common misconception that might arise from comparing fractions with the same numerator. In this post, I want to discuss a different part of this standard: "Recognize that comparisons are valid only when the two fractions refer to the same whole." I know the authors were thinking about situations like the following:
In this situation, 1/3 looks greater than 1/2. Thus, in order to compare these fractions, we must have the same whole. But, when we compare the two shaded parts above, are we comparing fractions, or are we comparing fractional pieces? Are fractions and fractional pieces the same? I tend to think not. Even in the picture above, when we are comparing "fractions" we are comparing the act of partitioning a whole into 2 or 3 equal parts and taking one of them. We are comparing the actions, not the results of the actions. The question becomes, how can we compare two actions and say one is "greater" than the other. In a way, we cannot compare "sizes" of actions. However, one way we can compare may be to look at the effects when the actions are applied to the same whole. Another way is to compare equivalent actions. So, taking 1 of 2 equal parts is the same as taking 3 of 6 equal parts while taking 1 of 3 equal parts is the same as taking 2 of 6 equal parts. Then, we can compare the number of parts we take.
Fundamentally, I think we should not be using drawings like the one above when we are comparing fractions. We compare fractions because they are numbers, and fractions as numbers always refer to the whole of 1. If fractions are numbers, then the whole is always the same. Therefore, if we are comparing fractions as numbers, perhaps the model we should be using is a number line.