#MTBoSBlaugust Day 2: Similarity in Right Triangles

As I write this, I'm currently sitting in the public library- my second home for the summer- waiting for my next tutoring student to show up. I looped for about 30 minutes to find parking, so I've dug in with my Algebra II books for the better part of the day and am happily working away on my plans for the coming year. 

But what am I tutoring? Geometry. Always Geometry. 

Geometry is so full of definitions and theorems and can be taught in a way that requires an unrealistic amount of memorization....to the point where students write it off without ever getting to know it. I was one of those students. It wasn't until I taught the course, saw the conceptual structure lying below the surface, and understood that I could eliminate so much memorization if I focused on a few key concepts that I started to appreciate it. 

As a Geo teacher, one of the topics I remember feeling the least satisfied with the resources I found to help my students with this was similarity in right triangles. Most textbooks would have a picture like this, showing the similarity between the triangles:

Followed by a picture like this, with the geometric mean formulas to be memorized:

And then we would wonder why students can't remember the very important leg and altitude rule. Where the heck did these come from? What did they mean? Can anyone explain what x and y are anyway? 

When I first taught Geometry, a teacher told me they'd discovered the KEY to teaching this topic! Get out your colored markers, folks. 

Depending on what you knew and what you wanted, you'd color code either the altitude or the leg one color. Then, you'd color the 2 pieces of the hypotenuse that touch that color....so for an altitude, it would be either side of it and for a leg, it would be the piece closest to the leg and then the whole hypotenuse. Then, you set up a proportion where the thing that is it's own color repeats. Boom! You always get the right proportion!

This worked for some kids, but was no where near the level of conceptual understanding I expected from my kids typically and was unsatisfyingly difficult when the topic we were really talking about was just good old similarity! 

One of the joys of tutoring to me is seeing the approaches other teachers have towards specific skills. I can share my best strategies with the kids, but sometimes they wind up sharing a strategy with me too. This one- simply drawing a table to organize- was a real eye opener for me and made the topic a "no brainer" for most of my kids. 

We start the way I would have anyway, with doing a hands on exploration that you can find here:

Right Triangle Similarity Activity

This was created by my amazing coworker, not by me! But I LOVE it and can't imagine teaching this topic without it anymore. Students cut out the pieces of the triangle and orient them in the same direction, writing both proportional relationships and similarity statements between each triangle.  Then, there are coordinating cards to go with each of the questions on the exploration. Students are then asked to hypothesize about any patterns they see (which was admittedly very hard for my kids this year, but easier for many of the honors teachers). 

Since these could be flipped, turned, rotated, and mapped onto each other, this activity also hit on the idea of rigid motions (which I always love!) and got kids thinking about corresponding parts. from there.

And now for the stolen part (and a huge thank you to whatever teacher my tutoring student had that taught them this). Have students create a table for each triangle with the 2 legs and hypotenuse. When set up correctly, BAM! A wild proportion will appear! (This Pokemon reference may someday get old, but I'm hanging on to it for dear life until then).  Here's 2 examples:

No more special "rules" kids need to memorize. No more "oh yeah you just color this one and this one" and then color the wrong ones. Just a simple understanding of corresponding parts and proportional relationships.

Should this have been a "duh" idea for me??? Maybe. But it's made teaching this topic so much easier for me and for kids, so I'm passing it along in case anyone else hasn't used this idea before.