Factors and Primes: Oh really?

Once in awhile a middle school teacher in my district asks me questions that make me blush…at my own hesitations.  Today Evan (fake name) asked, “Could you help me with the conceptual understanding of factors?  …I would like to make sure my instruction/definition of factors is a fit for any level of mathematics.”

Evan had found contradictory definitions.  The common theme, though, was that factors are numbers (or symbols).  So what about fractions?  I have, on a rare occasion, “factored out” a fraction:  1/4(3x-2).  Then I thought about…

since (y2/3 + x2/7) (y2/3– x2/7) = ( y4/3 – x 4/7 ), then (y2/3 + x2/7) is a factor.

I think our utilitarian definition of “factor” can get clouded because a “prime” is defined as an integer and composites are defined with respect to primes.  In lower grades, students typically only factor composites to get integers.   The bottom line is that we can factor any number and any number can be a factor, but primes are formally defined only on integers.

While it is incorrect to think of factors only as integers, one might think twice about teaching fractions as factors unless students are very comfortable with progressions of  area models.  To that end, I would propose vertical alignment.  Alignment with area models would also help students to be able to understand that fractions can be units…blush?

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