Transitioning to Transformations

Last week I gave my students a task from Illustrative Mathematics called Similar Triangles as an introduction to the concept of Angle-Angle Similarity.  Here's how it played out...

A brief disclaimer is that although I have been teaching Geometry all year, I have not made the complete transition to teaching it through Transformations.  I have done a few things here and there, but more than anything it was to try tasks and strategies out for myself and to challenge my students.  They have had some experience with transformations and each time we do something with them, the students seem to grasp the concepts quite well.  When doing the AA task, I encouraged my students to use patty paper as a tool for doing transformations.  Also, this task was their first introduction to dilations.

I started by drawing 2 equiangular triangles on the board (see picture below) and asked the students if the 2 triangles were similar.  A lot of them said NO because there wasn't a "nice" number to multiply 3 by to get 8. :) So, I wrote the equations from the picture and we came to the agreement that 8/3 or 3/8 would have done the trick, depending on which way we were scaling/dilating.  They seemed to find this pretty magical...this should be an indication of the level of number sense my students are working with. :)

So, as they worked on the AA task, they did great with the translating and rotating.  I wrote up a list of requirements (see picture below) and they were able to articulate them quite well.  

They did great with the "vagueness" that I had been struggling with.  So, when it came to dilating Triangle ABC (I didn't worry too much about the prime notation for this task), they knew that they had to make it bigger, but they didn't necessarily know by how much.  This is where I wrote the new equation from the picture below (AB x ? = DE) and they were actually able to connect it back to the 3 and 8 sided triangles.  The magic (or madness for some of them) continued.  Ultimately, they were able to say that you needed to multiply it by DE/AB in order to dilate the smaller triangle to the larger triangle, and I was pleasantly surprised.

Another issue that I'm having is, Where do I go from here?  I guess I'm still having difficulty wrapping my head around the fact that THIS is the new definition of similarity, but I'm getting there.  Pulling it all together and not seeing these tasks as individual exercises is something that I'm still working on.  I appreciate your feedback.  Please forgive my vagueness or lack of precision in notation, but I think that mostly comes from trying to meet my students where they are, and let's be honest...where I am, too.

Thanks for reading!