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Starting Congruent Triangles

Every year I reassure my geometry students that once we get to congruent triangles, that they will feel more confident about the course. They tend wax nostalgic over predictability of solving equations in algebra and lament the feelings of confusion they experience when learning how to use upwards of 30 theorems and definitions. Learning how to use all these rules in 6 weeks is hard. Students start to feel better we we begin triangle congruency proofs because the goal of each proof is more predictable. Compiled with the fact that those 30 rules are more rehearsed and deductive structure no longer feels like foreign language, their comfort and confidence increase.

As eager as I was to get to this part of the course (despite preaching daily on effort and learning from mistakes), I didn't want to just hand them the triangle congruency theorems. I have done just that though in years past, using my excuse of only having 135 forty minute classes each year to relieve my guilt of not doing many activities involving inquiry. No longer! I fit it in and my classes still learn the content and IMO, are more engaged and have a deeper understanding of the content.

Last year I saved Kate's activity for introducing congruent triangles. This year I assigned this exploration using math open ref designed to get them thinking about what it means for triangles to be congruent. Then on day 1, we started Kate's exploration activity with students cutting out the triangles in groups. Drawing and measuring is always more difficult than I anticipate, which is why I feel it is so important to do. When students interact with a shape "for real" it gives them time to think more about how the sides and angles come together to form a triangle. As they completed their triangles, I posted them on the board.

 I wish I had more time on day 1 (I tried to squeeze in a 10 minute skill check on the same day) as I didn't get to check each one and send it back if they didn't follow directions. I also didn't get to discuss their predictions.

 Day 2, students used this form to go on a gallery walk - well sort of, I placed the triangles too close together. As they studied each group they were to decide if the shortcut worked. This took about 15 minutes and led in nicely to a class discussion on SAS/SSS/ASA, a sample proof and a discussion on why SSA isn't a thing.