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Often I like to make a subject more accessible by breaking it down into more manageable chunks. Sometimes these chunks have established mathematical names. For instance, before teaching the Pythagorean theorem, I would like to investigate right triangles with a perpendicular drawn from the right angle vertex to the hypotenuse. The geometric mean of two numbers appears again and again in such calculations. I think it is worthwhile to introduce the term “geometric mean” at this point. This prepares students beautifully for understanding the Pythagorean theorem – you can have them “solving” right triangles using similarity before they have seen the theorem, and the theorem follows by just doing the same stuff to a general triangle.
The concept of “geometric mean” is not in the standards, but I believe that it is of great utility in getting students to develop an appreciation for the Pythagorean theorem. Is it “Common core aligned” to introduce this operation? More generally, if a concept of great utility connects to a standard, but is not explicitly mentioned, may it be included in “Common core aligned” material?
This might be answered by the Publishers Criteria here: http://www.achievethecore.org/files/1413/6545/4893/Math_Publishers_Criteria_HS_Spring_2013_FINAL.pdf
Some comments though . . . one thing that I notice in discussions of curriculum and instruction is that a given topic can be taught in different ways. Just saying that a given topic is included isn’t necessarily evidence that something (e.g., curriculum materials) is standards-aligned or not. Obviously, you’re thinking of a particular approach rather than just a topic. The question might become “How does this approach fit with the standards?” I’d suggest thinking of “standards” (plural) rather than just the Pythagorean theorem or just standards that involve the Pythagorean theorem.
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