Geometry : Unit #7 - Discovering and Proving Triangle Properties : Lesson #9

Triangle Properties, Triangle Congruence, and Proof Review

Objective: Students will be able to apply and prove triangle properties.
Standards: HSG-CO.C.10 MP3 MP6
Subject(s): Math
60 minutes
1 Return Group Tests - 15 minutes

I pass back the group tests so my students can see how they performed and so they can continue to deepen their understanding of properties of triangles, triangle congruence, and proof writing.  While students review their work and my feedback, I encourage them to make sense of their errors and ask questions to clarify their confusion—this includes wondering about other possible proof paths they might have considered before, as well as seeking feedback from other groups and myself. 

My goal in this whole-class discussion is to make sure every student has a clear understanding of my expectations so they can be successful, which is why I ground this discussion in the “features of high-quality proofs,” a list of criteria which have been posted on the whiteboard throughout the unit.  I use some of the whole-class discussion time to highlight the most commonly missed problems on the assessment.  I focus the discussion on identifying faulty assumptions and gaps in reasoning, as these are the main problems that persist when students have difficulty determining whether pairs of triangles are congruent or writing proofs.     

Learning from Feedback

I wanted to push my students to make sense of the feedback I had given them on their group tests so they could improve their understanding of triangle properties and proof.  The group test was, after all, fairly challenging and seemed like a promising site for some rich learning.  After giving groups time to review their work and my written feedback, I addressed any remaining questions students still had about their work.  Afterward this discussion, I asked students to share round-robin style in their groups about something they thought they could take away from my feedback so they could improve their work moving forward. 

I selected one student from each group to share out either something he/she individually wanted to work on or something else that had emerged in the group.   Here are some examples that students shared out loud with the whole class:

-“We really need to check that the CORRESPONDING sides and angles are congruent.  There was a problem where we saw ASA in one triangle and AAS in another, but that’s not enough information to show BOTH triangles are congruent to each other because it’s not like we really knew if all the corresponding parts were congruent.”

-“We need to explain ideas more specifically, like not just say ‘alternate interior angles’ but ‘alternate interior angles are congruent because they were on parallel lines.”

-“Just because we are given two sides are congruent doesn’t mean we can say that the two sides are parallel.” 

Having students share out about their own big take-aways was far more powerful than me summarizing the patterns I had noticed in my students’ work.  My students’ insights also showed me that they were able to take ownership of their understanding in a way that was authentic and relevant for them.  

2 Proofs Review - 20 minutes

This activity features two levels of proof—foundation and application/extension—and will require students to take risks by exchanging their proofs with a “critical friend” with whom they will exchange feedback (more on this in the next section of the lesson).  I explicitly tell students that the individual assessment will feature proofs from both levels, which is why they are getting “leveled” practice during this lesson. 

For this activity, students choose the proofs they would like to work on, knowing it is in their best interest to choose proofs that offer an appropriate level of challenge to them since the best kind of feedback is the kind that pushes them further.  I find that this type of choice works well in this context. 

3 Peer Assessment via Critical Friends - 15 minutes

There are three rounds to the Critical Friends Protocol (CFP). The CFP gives students the opportunity to engage in peer assessment and to revise their own work.  I pair students up so they are working with an appropriate friend.  The job of a Critical Friend is to give honest and critical feedback that will help their friend grow in their understanding and proof writing abilities.  

Once the students are in pairs, they each select 1-2 proofs to exchange, then critical friends complete the protocol: 

  1. Write directly on your critical friend’s work when making suggestions or asking questions, putting check marks or stars next to statements/reasons with which you agree and x’s next to anything with which you disagree.  When possible, offer compliments and/or make suggestions to improve work. (6 minutes)
  2. Discuss with partners (4 minutes)
  3. Silent time to fix own work (2 minutes)

I use this protocol because it requires students to carefully construct their arguments and to critique the reasoning of others in a structured way (MP3). I find that the formal structure enables trust and promotes respectful interaction.

4 Homework - 5 minutes

For tonight's homework, students will write A+ proofs for the leveled proof practice activity in today's classwork in addition to completing Proof Challenge 2.