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I am posting this question for a high school teacher friend (I teach middle school). A-APR.4 mentions proving polynomial identities and A-SSE.2 mentions seeing structure. Evidently it is quite a debate with the high school folks as to whether or not the patterns for factoring sum of cubes and difference of cubes are included in these. I think it is, and that having students look at expressions like x^6 – y^6 as either a difference of cubes or a difference of squares can lead to some really interesting mathematical conversations.
I try to err on the side of saying that the standards cannot list every possible polynomial identities and that we should think about the ones that are the most useful. Others want to limit the identities to only those explicitly listed in the CCSS (such as the difference of squares).
What do you think? Would sum/diff of cubes only be allowed for classroom discussion or would it be reasonable to expect students to know and use a sum/diff of cubes factoring on an assessment?
I think your example of looking at $x^6-y^6$ in two different ways is an excellent example of seeing structure in expressions, and it is no more complicated than the identity mentioned in the “for example” part of A-APR.4. It’s certainly very reasonable for classroom discussion. As for assessments, I don’t know what limits the assessment consortia will set on types of identities. I hope we don’t end up with some long list of identities students have to memorize. In some strange way that can work against seeing structure, because the list becomes the object instead of the expression. But you are quite right that it does not make sense to limit to only the identities explicitly list. That would be a strange way to interpret A-APR.4, for example, which only lists one identity as an example. It would be odd if that one identity made the list but difference of cubes did not.
I lead my students in a discussion of factoring sums and differences of cubes by comparing similarities with factored differences of squares. For example, it is easy to see why $2x-3y$ might be a factor of $8x^3 – 27y^3$. Rather than memorize the pattern for the other factor, my students divide out $2x-3y$, generating the other factor.
[2013-12-06: Typo corrected]
Thanks for the replies! I had to move up to high school at the semester break, and I jumped straight into Algebra II.
When we got to this identity, we discussed it. We even explored the graph of y = x^3 + c. Since the kids already did some with roots in the fall, they were able to make some great connections with the structure and the graph.
y = x^3+c has only 1 real solution (aka x-intercept)
And we know it factors as (x+c)(x^2-x+c^2). Since we only see one x-intercept, it makes sense that the second factor is a quadratic with non-real roots. That’s part of the reason why we know the factoring pattern is as simplified as it can be.
It was a great conversation. Hooray for structure!An update on the sums & differences story.
A few days after we talked about the graph of y = x^3 +c (which, if c is negative covers both factoring), I was working with a student on some factoring questions. The student came to x^3 – 125. In the air above his paper he moved his finger in the shape of the cubic parent function (as if tracing it in his mind). Excited by this, I asked him to explain.
He said, “Well, the graph of y = x^3 -125 would have a negative y-intercept down here and its x-intercept would be over here at positive 5. That means that the first factor has to be (x-5).” I asked him to explain if it worked for y=x^3+125 as well, and he explained that it did because the root is at -5 which makes the factor in x^3+125 be x+5.
I was so excited by this connection that the student made! I told him that he was really making some good connections and understanding and that he should share his explanation with the class. For this shy kid who usually is middle-of-pack gradewise, that was a real boost to his confidence. He explained it well in class. When someone asked if it worked for things like 8x^3 – 125, I suggested that everyone think about it and explore that on their own. We discussed that one briefly another day.
Had to share that story. I love those kinds of connections.
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