Rationalizing the denominator

[tomergal]

I’m wondering what’s the Standards’ view on “rationalizing the denominator.” On the one hand, it kind of falls within N-RN (specifically N-RN.2) and is generally a nice manipulation to practice when working with radicals. On the other hand, the motivation behind this procedure is pretty vague for high school students, and they don’t really get to use it anywhere else.

[lhwalker]

I’ve been on a safari looking for that answer, too! I’m thinking rationalizing goes with N.CN.8(+) which one wouldn’t know is in the Algebra 2 standards unless they read Appendix A, pages 8 and 38. I suspect the writers felt it was best not to teach “rationalizing for the sake of rationalizing.” Some of us think radicals in the denominator look cute the way they are, so why…? On the other hand, what are students to do when their answers don’t look like the ones at the back of the book? I am hoping Dr. McCallum or someone else can answer that.

The importance of N.CN.8 was not obvious for me. I now think it is important because students need to connect polynomial identities (specifically difference of squares and perfect square trinomials) from real number quadratics to complex quadratics. Along with that connection comes the idea of “conjugates” that eliminate middle terms when they are multiplied. Of course, that means N.CN.8 connects with rationalizing. However I’m thinking that rationalizing is only done to effect division with two complex numbers. How else can we get an a+bi number from dividing two a+bi numbers? Quotients of complex numbers is only in N.CN.3 which is in the “fourth course” of high school math. So for N.CN.8 I think we only need to connect polynomial identities. We won’t need to rationalize with conjugates since we won’t be dividing complex numbers.

While I was out hunting, I realized that (I think) A.SSE.2 is where we get perfect square trinomials (along with difference of squares).

[tomergal]

Thanks for the response!

I agree that RtD (“rationalizing the denominator”) is simply an application of the “difference of squares” pattern, which is also applied when we divide complex numbers using conjugates. I think we should definitely stress this fact, that these seemingly different things are all applications of the same algebraic tool.

However, while RtD is nice to practice, it doesn’t serve any useful purpose like conjugates do in complex number division. Therefore, I’m unsure whether we should teach it to students as something they need to know and do. Maybe it suffices to let students extend expressions such as (√6+√5)(√6-√5)=6-5=1, or factor expressions as in x-1=(√x+1)(√x-1), to show how the pattern is applicable even in cases where the terms aren’t perfect squares.

Let’s wait to hear from Prof. McCallum on how he perceives the status of RtD in the curriculum.

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