## Tuesday, November 17, 2015

### Thou Shall Not Reduce Fractions

With the increased use of the word "reduce" in the sustainability and recycling movements there is even more reason now not to use the term in reference to fractions.

If we use the word "reduce" to change 4/8 to 1/2 we are giving students two clues to make them think that 1/2 is smaller than 4/8. The word "reduce" clearly means to make smaller while the two numbers in the 1/2 fraction are clearly smaller than the two numbers in the 4/8 fraction. We are, in fact, giving chidlren two clues to make them think 1/2 is smaller than 4/8. To students who have good fraction sense this is probably not a major issue but to the many students who struggle to learn fractions this metaphorical use of the word "reduce" can cause untold misconceptions and misunderstandings.

I have been recommending the use of the term rename or regroup to describe changing a fraction from one form to another. In my math and Diversity grad course last night one of the students suggested using the term "synonym". At first, this seemed quite inappropriate since synonyms do not have exactly the same meaning while 1/2 is exactly the same as 2/4 or 3/6. Then we started to think about about the use of the term and it began to seem more plausible The discussion went something like this.

While two  synonyms may  not exactly be the same it is really the context that requires the use of one over the other. For example, if you walked into a small grocery store and asked for a dozen big eggs you would get a funny look, perhaps, from the store keeper. The usual word to use when referring to eggs is large, a synonym of big. Now apply the same reasoning to fractions. 3/6 is a synonym for 4/8. Even though they are exactly the same size, given that they both refer to the same whole, each one is more appropriate in certain contexts. If you were adding 1/2 + 1/6 it would be more appropriate to rename the 1/2 as 3/6, or use 3/6 as a synonym for 1/2. On the other hand if you  were subtracting 1/8 from 1/2 it would be much more appropriate to use 4/8 as a synonym for the half in just the same way that it is more appropriate to use the word "large" when referring to eggs rather than the synonym "big".