## Reflection: Coherence Triangle Similarity Criteria - Section 2: Establishing the AA Similarity Criterion

For me, the advent of the CCSS has totally changed by perspective on how students should experience school mathematics. In the past, I thought my job was to share the the knowledge of mathematics. The goal was basically for them to learn mathematics. My goal now is more for them to do mathematics. This change in perspective has forced me to continually ask myself, what is it that mathematicians do.

This lesson provided a cool opportunity for students to do some of what mathematicians do. We got to create our own corollary. To establish the AA similarity criterion, one would really need to show that whenever AA exists, there is guaranteed to be a series of rigid transformations and dilations that will map one triangle onto the other (You can see this type of proof in AA Similarity.) But, in our case, students had already labored hard to make similar arguments for SSS, SAS, and ASA congruence criteria. These arguments were some of the toughest things all year for students to wrap their brains around, so they definitely would appreciate not having to go through that whole ordeal again.

But lest students feel that we were cutting corners, I had to be clear that we were doing what mathematicians would do. I had to be careful to ensure that they understood what we were up to. We knew, and had already proven, that if SAS, SSS or ASA congruence exist, then the triangles could be mapped onto each other through rigid transformations. They also needed to understand that we were trying to prove that AA, for example, would ensure that two triangles could be mapped onto each other through rigid transformations and/or dilation. And here is where the corollary comes in. If we can establish SAS, SSS, or ASA through dilation, then that guarantees that there exists a series of dilations and rigid transformations that map one triangle onto the other.

I felt this was one of the moments when students really got to experience what is like to create and work within an axiomatic system. They were able to connect what we were doing in this lesson to a lesson that took place long before this one, and they got to experience how the work we did in that earlier lesson could actually be referenced and leveraged to make our work in this lesson a lot easier.

Coherence: Making our own Corollaries

# Triangle Similarity Criteria

Unit 6: Similarity
Lesson 4 of 8

## Big Idea: Enough is enough...but what is enough? In this lesson, students will determine the criteria that are sufficient for determining triangle similarity.

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Standards:
Subject(s):
Math, Similarity and Congruence, reasoning and proof
75 minutes

### Anthony Carruthers

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