Success after Chaos

It is important for Newbies, Oldbies, and Inbetweenbies to look for resilience in ourselves. How do we handle a dismal result from a well-planned lesson?

Blame and negative talk to ourselves are not useful or productive. So acknowledge it didn't go so well, and tweet out to your PLC, share with a trusted colleague, sleep on it, check the MTBoS Search Engine, take a run or a bath, or both!

I wanted my students to discover which triangle congruence shortcuts worked and which didn't. Last year I had good success with lesson inspired by the original CPM Math 2 curriculum, where the students cut out various triangles, we put them up on the board and decided which were the same and which were not. Problem was, not enough accuracy with protractors and notion that vertices had to be the endpoints of the triangles, so sometimes non-congruent triangles ended up in the SSS category for example.

So this year I started with a lesson inspired by one I found on James Rahn's website: Discovering Ways Triangles are Congruent. I wanted to use Patty Paper to try to get a grip on accuracy. The idea was to create triangles from all the permutations of six parts of a triangle, each of three vertices and each of three lengths. Problem was there were too many triangles (9). The students didn't like making a triangle out of one single angle, or just an angle and a side.

Most students used the arrows at the end of the rays as endpoints and closed the triangle, so many of the triangles looked the same and plus, each student had about a zillion pieces of patty paper to try to keep organized and categorized. There wasn't enough time to debrief and I wasn't prepared to store the bazillions of pieces of patty paper for the students.

After reaching out to MTBoS community and reading a January 2015 post by Kate Nowak on the same topic, I re-organized thanks to Kate's helpful recording sheet and tried again the next day. I focused on what I wanted the students to specifically see, and decided to concentrate on the triangles given three pieces of information.

This time, instead of every student doing every iteration of  a triangle with the six pieces from James' lesson,  I gave the groups of 4 and 5 students the same 6 pieces of information as they saw previously, and using Kate's recording sheet, specifically asked the students to use the patty paper to create one each per group of very specific triangles in specific orders. For example, create a triangle using side AC, then, CB, then, side BA. ENDPOINTS MUST BE VERTICES AND ANGLE VERTICES MUST LINE UP WITH SEGMENT ENDPOINTS!

I told the students they could divide the work up anyway they wanted...each take a triangle to create, work all as a team, as long as when they were finished they had 5 triangles, on 5 separate pieces of patty paper, labelled for the corresponding letter with the directions for the triangle. You have 20 minutes, GO!

I put 5 pieces of colored paper on the board and had a member tape up their triangles with the ones that were built the same way.


Geez, what a relief! The students saw which shortcuts worked and which didn't right away! What a difference. We then went to Kate's Chart,
 Called the short cut SSS and showed the markings on the triangles in the recording sheet. I had this foldable printed for the students from the day before: and had the students cut up the recording sheet and add it to the foldable.

The take away was really this: students were successful
because they already had experience getting messy with material.  That was a huge epiphany for me. They needed that experience of being familiar with the pieces. Of having been given the opportunity to become familiar with a new concept. When I explained that I wanted to try the activity again approaching it another way because the first time didn't have the results I wanted and admitted it was a flop, the groaning quickly disappeared and the students were open to trying it again. I even played music this time.

Next time I will run through the entire activity with just two items, an angle and single side, let all the students show their work on the board and let the groups only look how congruent triangles can be made with three aspects of a triangle. And I will allow for nearly the entire block, I will will definitely play music!

On the second day, I walked one class through HL, but this too I want to change already. I really want the students to work out why HL works from the given shortcuts that work.

How do you handle AAS (vs. ASA) and HL? I kinda really like HL, I have to admit.

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