Square Root Fun

Today and tomorrow, we're spending Geometry class time exploring the geometric nature of square roots. Well, that's a loose description, anyway. It's a transition topic and something I loved from last year, so we're doing it.

Today my kids constructed using protractors and rulers a Spiral of Theodorus (see above). They started with the smallest right triangle measuring 1cm in each leg. In the margins they showed all the Pythagorean math work, which helped them practice a gazillion times how square root and square cancel each other out, and helped them see why $\sqrt{5}^2 + 1^2 = c^2$ means that $c = \sqrt{6}$, for example.

Then, we simplified things like $\sqrt{9}$ and $\sqrt{16}$ to verify with a ruler that our integer lengths matched what we saw on the ruler. And we simplified expressions like $\sqrt{12} = 2\sqrt{3}$ to make sure that we were able to compare the two lengths (as measured with rulers) and verify one length to be double the other. All in all, it was a productive, visual, and relatively pressure-free practice of their arithmetic skills without using the calculator.

Then, I took the opportunity to quickly explain visually what square root means and why it's not correct to simplify as such: $\sqrt{a^2 + b^2} = a + b$. I told the kids that it's one of those common mistakes in algebra that you can avoid if you have a geometric understanding of what's going on.

I drew this on the board and we discussed it briefly.*
It was immediately evident to the kids that you can't simplify as described above, because the area you have is "too small" to be broken up nicely into two sides of lengths $a + b$ each. I re-emphasized that if you have a geometric understanding, it helps you out a lot of times with algebraic concepts!

I'm excited about tomorrow! I will be doing more square root investigation with my kids... this time on Geoboards! My worksheet is based off of this lovely activity, but I re-worded/scaffolded it a bit more and added a few questions. I am, in fact, getting videotaped for this lesson, which makes me OH SUPER NERVOUS!!

Anyway, I am cherishing these last weeks with my kiddies. June is coming way too fast. I am running out of time to play Geometry with them! :(

*PS. I guess I wasn't being totally accurate in the second diagram. Should have used abs(a) as the square root value. Oops.