Group Assessment: Triangle Congruence and Proof

As we near the end of this unit on triangle congruence and proof, students must be able to:

1. Self-monitor their proof writing
2. Assess the logical flow and cohesiveness of their own proofs

In my experience, checking one’s own reasoning is a difficult thing for students to do. So, to begin today's lesson I ask students to analyze a sample proof (Warm-Up: Critique Someone Else's Reasoning), checking for the following features of high-quality proofs:

• Did the "student" start with what was given?
• Did the "student" end with what they were trying to conclude?
• Did the "student" include EVERYTHING needed to draw that conclusion in an order that makes sense?

After students check for those features, I ask them to identify changes that can be made to improve the proof,  explaining why/how those changes improved the quality of the proof.

Since today’s warm-up is an individual endeavor, I want students to share their work in a structured way because I want to develop students’ habits of mind when they write proofs. I follow up the individual work with one or more of these interventions.

• Students decide on a grade to assign the original proof and a reason for giving the grade—the group cannot move on to discussing the second proof until they come to an agreement. 
• For the second problem, I have students pass their papers around so they all have the opportunity to see others’ proofs, which can give them concrete ideas for how to improve their own proof writing, as well as receive feedback on their own proof.  
• I ask groups to choose the proof they believe to be the clearest and most concise, and to be prepared to display it under the document camera.  I want at least two different students’ proofs to be presented so that we can have a rich whole-class discussion about the extent to which these proofs fulfilled the features of high-quality proofs.

During this group assessment students work on two types of problems. 

The first type of problem requires students to analyze given diagrams of pairs of triangles to decide whether there the triangles are congruent.  This kind of problem is great for student discussion because students must use the given information to draw other conclusions before using one of the triangle congruence shortcuts to draw a conclusion about the triangles. 

The second type of problem is the writing of a proof.  Students will have three proofs to write together, which will give them the opportunity to share ideas about possible proof paths to take.  Working together on these proofs will also give them the opportunity to check for the features of high-quality proof, which we discussed in the warm-up.  

When students complete the Group Assessment I distribute the Proof Challenge 1, a set of proof writing challenges that I will ask all of my students to work on for homework this evening. I find that this type of assignment works well after a Group Assessment, because students have the opportunity to extend themselves individually.