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# Triangle Properties, Triangle Congruence, and Proof Review

Lesson 9 of 10

## Objective: Students will be able to apply and prove triangle properties.

#### Return Group Tests

*15 min*

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.

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#### Proofs Review

*20 min*

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.

#### Resources

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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:

- 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)
- Discuss with partners (4 minutes)
- 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.

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#### Homework

*5 min*

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.

#### 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

- LESSON 1: The Language and Properties of Proof
- LESSON 2: Triangle Sum Theorem and Special Triangles
- LESSON 3: Triangle Inequality and Side-Angle Relationships
- LESSON 4: Discovering Triangle Congruence Shortcuts
- LESSON 5: Proofs with Triangle Congruence Shortcuts
- LESSON 6: Triangle Congruence and CPCTC Practice
- LESSON 7: Proving Isosceles Triangle Conjectures
- LESSON 8: Group Assessment: Triangle Congruence and Proof
- LESSON 9: Triangle Properties, Triangle Congruence, and Proof Review
- LESSON 10: Discovering and Proving Triangle Properties Unit Assessment