Angle Relationships Review and Focus on Justification

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Students will be able to justify their reasoning about angle relationships.

Big Idea

By collaborating on the Collective Proof, students will bring individual pieces of reasoning to write their first lengthy proof.


10 minutes

In this Warm-Up, students solve two problems that ask them to find the indicated angle measure and to justify their reasoning.  As I circulate the room, I encourage students to justify their reasoning by using precise mathematical vocabulary.  The ultimate goal of the warm-up is to give students an opportunity to practice using the correct words to support their ideas, which they need to do on the Collective Proof, a task they will work on later in the lesson.

Reciprocal Teaching

10 minutes

I have students work in pairs in this Reciprocal Teaching activity.  Each person has silent time to solve his/her own problem.  Then, when both partners are done, they take turns teaching each other about their problem and justifying their work by using correct angle vocabulary.

While pairs work, I circulate the room looking for exemplary student work--that is, work shown clearly and correctly, correct use of angle vocabulary, given information marked clearly in the diagram.  I select students to project their work on the document camera so other students in the class have a clear picture of what they should strive for.  Because the diagrams for the Reciprocal Teaching activity are largely blank, I look out for students who have introduced their own variables into the diagram to help make their reasoning clearer and easier to follow.

Collective Proof (Group Justification)

50 minutes

In the Collective Proof, students each tackle one part of a larger proof.  The practice writing a small justification, exchange papers to get feedback from another member of their group, and then then bring together their best proof writing to try to write a full, coherent, clear proof (MP3).

On the whiteboard, I make my expectations clear to students by writing out "High Quality Justifications Include..." and showing them specific ways they should attend to precision (MP6):

  • State the given information you need to draw logical conclusions
  • Re-state what you need to show (the goal of the proof)
  • Put a "because" on it (or a "therefore," or a "thus") or use "If...then..." language throughout the proof
  • Check that there is always be a "why" for every "what" you state
  • Check that your proof reads well (clear, concise, not gaps in logic) 

Resource Citation: I want to acknowledge Shira Helft, math teacher at Gateway High School in San Francisco, who shared this "collective proof" task with me.


Text Messaging

25 minutes
After this intense focus on justification, I find that students appreciate the chance to wind down the lesson with some "regular," yet challenging, problems to solve.  In Text Messaging, students work silently in pairs, problem solving together and recording their thoughts by writing their ideas and questions in their own color (MP1).  As students work through the problem and text each other their ideas and questions, I circulate the room, noting strategies that students try out like extending the lines to look for and make use of structure (MP7).
Requiring students to work silently and to write out their thoughts forces students to slow down and make their thinking visible to others.  I make sure to model how the Text Messaging activity should go by having a student come up to the document camera to work out the beginnings of a sample problem with me silently.  The sample problem I typically use is one where I show a line, r, parallel to a line, s, with a given 108o angle and angle w given as consecutive interior angles.