Proving Radical Simplification

Training them to beg for proof
I wrote a post a couple of weeks ago about proving things in my calculus class.  I aim to create the same ethic of proof in my Math 9 class.  Some of them have definitely bought into the need to prove everything we use, but since it's a bit more mixed class, some of them wish that I would just give them formulas.

Just before vacation, we were working on the Pythagorean Theorem (I'll put up another post soon on how we proved this without any algebra using "I Notice, I Wonder" thank you @maxmathforum!).  I gave a quiz last week on the Pythagorean Theorem, and the kids totally rocked it, better than any other quiz so far this year.

How it fits in
All of this is part of the unit that is leading up the grand programming project TI-83 Programming Project.  We are working toward the distance formula next week, then midpoint.  We have already discussed slope of parallel and perpendicular lines.

The investigation
So, the next couple of days after break are devoted to simplifying radicals (i.e. =  ).  Before break, students worked through the Discovering Algebra investigation to see this visually:

The activity guided them to see than = .  This process works great to be able to "see" the what we are doing.  Next, I want to guide them to the more traditional method of simplifying roots (i.e. = = ).  Also, this visual method only works for roots that are sums of perfect squares.

The proof
So, last night I was thinking about how to connect the visual activity to a more classical simplifying strategy.  I don't really know quite yet--maybe the visual one is just to get students thinking about how it's possible to write roots in multiple ways, and then we go into a more classical approach.

My students don't yet know about fractional exponents, and I'd like to be able to prove this without them.  Several of you (including @adkpiper, in person) suggested that I just square both sides of the equation above to show that it's true.  How simple!  It's great to have a community of folks who will play with the problems that I get stuck on, no matter how simple they actually are.

@jaz_math pic.twitter.com/ptUpM4dokD
— Hollis (@adkpiper) November 30, 2013

So elegant and totally appropriate for ninth graders without a great grasp on proof or complicated exponent rules.