Friday, February 02, 2007

Next Lesson: Sum & Difference of Cubes

This lesson will start with the new weekly quiz + notes check. If students' are unorganized or have incomplete notes, they will lose points. I just did this in my reglar algebra 2 class this morning, and only a handful of people got full credit on the notes check. While that is discouraging, it also shows how critical it is that they get continually assessed on short bursts of organization (they only needed to show me a binder with 2 days worth of notes, and an index with 2 entries).

After this, there will be some basic direct instruction on factoring polynomials with the difference and sum of cubes patterns (which I'm not exactly clear why this is worthy of a state standard, but there you go..). The only interesting part of this is that I will use the visual model shown in this post to help them see where the pattern comes from. You can think of a^3 - b^3 as the big cube's volume minus the small cube's volume. This volume is equal to the volumes of shapes I + II + III. You can easily get expressions for their volumes, and then factor out the common (a - b) factor from each term to derive the (a - b)(a^2 + ab + b^2) pattern.

That segues nicely into the next piece, which is factoring cubic polynomial by grouping. Students already learned this method for factoring quadratics, so this piece should go pretty smoothly.

I hope to get through the instruction piece fast enough to allow a good chunk of time for students to just practice using these techniques. They definitely have troubles applying the patterns to expressions like 125x^3 - 64y^3, or even worse, when you need to factor out a common monomial first. I'm trying to coach them to always write the problem in the form ( )^3 + ( )^3 first, so they can clearly see what the values for a and b are.

Myrtle Hocklemeier said...

Oh that is interesting stuff actually.

I just got finished reviewing/relearning this concept on my own from an older algebra book. The neat thing for me was that understanding that a^x - b^x always has a factor of a - b segueyed into finding the limit of a geometric series.

It wasn't until I figured out the general statement for factoring cubes and higher powers that I appreciated how finding the limit of the sum of an infinite geometric series "worked."

Dan Greene said...

I agree that these patterns can lead to interesting stuff, but I don't see why it needs to be covered in algebra 2 per se. There is so much to get done; I think that the standards should focus on the foundational algebra concepts needed for success in higher level math. The sum and difference of cubes patterns can only be used to solve a very specific kind of problem - I think it should be an optional topic at this level, not something that we are held accountable for on the STAR tests.