## Collective Proof - Section 3: Collective Proof (Group Justification)

# Angle Relationships Review and Focus on Justification

Lesson 5 of 6

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

#### Warm-Up

*10 min*

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.

#### Resources

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#### Reciprocal Teaching

*10 min*

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.

#### Resources

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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.

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#### Text Messaging

*25 min*

**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**).

*r,*parallel to a line,

*s*, with a given 108

^{o}angle and angle

*w*given as consecutive interior angles.

#### Resources

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review