Triangle Congruence Theorems- Why Not ASS?

First of all, kids, yes. We are going to write the word ASS. We are going to write it repeatedly. I can insist you write it as SSA, but we all know that isn't going to happen OR help you remember it nearly as well. Get your giggling out now. 

We've been digging into triangle congruence proofs in Geometry and I wanted to make sure were digging into WHY, not just HOW. 

We started examining why the Side-Side-Side Criteria works, inspired by this post from Math Giraffe. I displayed each step one at a time so groups couldn't rush through and to make it a bit of a game. I offered candy to any group that could create a triangle that was different from the first on they created. Any group who felt like they had done it presented their ideas to the class and we examined each one. Something crazy happened....we could ROTATE, REFLECT, or TRANSLATE them all to be on top of the others. This was handy, as we'd defined congruence using the existence of a rigid motion between the 2 shapes that would match one onto the other. 

I knew this trick wouldn't work again (you can't fool kids out of candy twice). So for round 2, we started comparing SAS with ASS. This time, I went pasta! I literally woke up in the middle of the night thinking about this and wrote up my directions when I got to school on the board, so no file to share there. Here's the gist:

1) Each group gets a piece of poster paper & folds it into 4 sections 

2) Have kids break pasta into 4 pieces of 2 lengths (you can specify what lengths with your kids or just let them choose.....I think I did 4 4" pieces and 4 5" pieces). 

3) In each section on the paper, have kids draw an given of the same size. I did 30 degrees and had kids extend it out to the edge of that section. 

4) Challenge the students to use one of each length of the pasta to create 2 different triangles if the angle given is included between the 2 sides. (They can't. SAS for the win here.) 

5) Challenge the students to use the remaining pasta to create 2 different triangles if the angle given is NOT included. Don't let them glue until they have tried a bunch of different options- some will try to say this is impossible too. 

6) Have students summarize the difference between the 2 scenarios and name them as SAS and ASS on their poster. 

7) DECORATE! You have awesome new posters! 

Here's some of the resources I've used throughout the unit. I'll continue to add if I see anything else unique: 

• Triangle Congruence Start Up Worksheet from the wonderful Math Teacher Mambo
• Triangle Congruence Warm Up that I made to help differentiate between possible ways to prove triangles congruent. 
• Triangle Proof Planning Guide that I created this as a triage for some kids who were struggling with the basics, but wound up being really helpful as a conversation piece for lots of my students.

I haven't taught triangle congruence in 5 years, so I'm slowly starting to get back into the groove. So many things I'll change for next year!