Analyzing Triangle Congruence with AngLegs

I’ve tried to explain why AngLegs are a must have for high school geometry and should not only be considered a tool for younger students. Here is an example of how I find them indispensable in teaching triangle congruence. This lesson is adapted from MARS: Evaluating Conditions for Congruency.


“Ok class, you are sitting in pairs and at each table is a bag of AngLegs. On the board I have written the question we are trying to answer for each of the scenarios I will present.”

2016-02-26 15.31.182016-02-26 15.41.52

“I made a triangle that includes a blue AngLeg. Can you make a different triangle that also has a blue AngLeg?”

“Make one. Hold it up.”

“How do you know that these triangles are different?”

“So, you are saying that keeping one side the same does not mean that the triangles must be congruent.”


Next, lets look at Card 3:

Capture

 

“I made a triangle out of a blue, a purple and a yellow AngLeg. Can you make a different triangle using the same three AngLegs?”

“What if you put them in a different order? …or move the purple between the blue and the yellow? Are you sure they have to be the same? How can you tell?”

Student:  “The triangles still fit perfectly on top of each other”.

Student: “If the three sides on one triangle are the same as three sides of another triangle, then the triangles must be the same.”


Look at Card 7:

Capture“How can you tell if an angle is the same in two different triangles?”

Student: “They fit perfectly on top of each other!”

“Is there a way to make these triangles so that they are not congruent?”

Student: “No way. These have to be the same”

Student: “Wait! I made two different triangles with all three angles the same and one side the same.” Does this count? Look!”

Student:”If two angles are the same, then the third angle always has to be the same because they add up to 180 degrees!”

“So what is the conclusion for this one?”

Student:”The triangles can still be different sizes, but their angles are all the same.”


“For the remainder of this class period, individually analyze card 5 and any other card, so 2 additional cards. Take good notes and write down your conclusions for tomorrow, where you will be randomly assigned a partner to complete the triangle activity. ”


From here the lesson continues as described in the SHELL center teacher guide linked above and described further in a blog post here.

Removing the hurdle of constructions allows students to focus on the learning goal for this activity: determining the minimum information required to guarantee triangle congruence, and what congruence means. It also connects nicely to congruence proofs through transformations as students are physically checking of the triangles fit on top of each other.

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10 thoughts on “Analyzing Triangle Congruence with AngLegs

  1. I love the conjectures students were able to make through the experience of this task. I’m definitely going to share it with 7th grade teachers at my school. This is very adaptable, thanks for sharing.

  2. I used patty paper for this lesson for the first time this year and it was WAY better than when I tried using straws. I like this idea a lot. I have some AngLegs but I was only thinking about using them for triangle inequalities. Do you have any ideas for that? How many sets of AngLegs did you end up needing?

  3. Update on how I used this lesson with my 10th geometry: I paired students, gave each group a set. Asked the to just play with them for a few minutes and tell each other what they noticed, wondered. I listened. We took a few moments to jot down those things on a poster (I use the lined poster paper that has a sticky bar across the top) and put the list up at the side. Then I put up the first ‘condition’ on the wall and said go (made it a little competitiony). They were clumsy with it at first, so I asked questions about their models. The BEST thing about this activity (I used it as review for the end of course test) was the conversations it produced, and the evidences each student used to argue for congruence or no, for similar instead of congruence. As the used the small protractor, they improved their angle measuring skills. Finally, it let me see exactly who had some misconceptions that I addressed further, by changing up the pairing throughout the event, speed dating every two cards or so.

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