4.NF.1 in the Common Core says, "Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions." When I first saw the expression (n x a)/(n x b), I thought that was odd. Usually I see math textbooks write this relationship as a/b = (a x n)/(b x n). Then I realized that the authors of the Common Core were trying to be consistent with the way they write multiplication expressions. "n x a" means "a is multiplied by n." Since we describe the process of creating equivalent fractions as "multiplying both the numerator and the denominator by the same non-zero number," it does make sense to write the expression as (n x a)/(n x b).
However, as I was thinking about how we might explain the process of equivalent fractions, I realized something else. One way, we can explain the process goes something like this - using 2/3 as an example. 2/3 is made up of 2 1/3-units, which is one of 3 equal parts of 1. So, if you use a diagram and a number line, it looks something like these:
Now, if we partition (split) each 1/3-unit into 4 equal pieces, we will have partitioned 1 into 12 equal parts, or 1/12-units. Pictorially, it will look like these:
Now, each 1/3-unit is made up of 4 1/12-units. So, 2 1/3-units are made up of 2 sets of 4 1/12-units. So, the number of 1/12-units in 2/3 is 2 x 4, and the number of 1/12-units in the whole is 3 x 4. So, 2/3 = (2 x 4)/(3 x 4), which is consistent with the conventional notation, a/b = an/bn.
I've been thinking about how we can use (n x a)/(n x b), but I haven't been successful, yet. I wonder if this is another instance where our language suggests the order of multiplication expression should be (group size) x (# of groups).
