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# The Language and Properties of Proof

Lesson 1 of 10

## Objective: Students will be able to apply properties of equality to write proofs.

Today is the first day of my unit on **triangle congruence and proof**. Over the last few weeks, I have used Warmups to give students time to practice proof writing. I continue this practice today, giving students a short proof through which they exercise their proof writing skills

Today's Warm-Up proof requires students to use either the substitution property or transitive property to come to the correct conclusion. I ask my students to complete this task because my students often struggle with the kind of language required to complete a rigorous proof. I give them the chance to struggle through as motivation for “the language and properties of proof,” in the next section of this lesson.

During the Warm-Up today we will also revisit the properties of parallelograms. Students will write many proofs over the next few weeks that will require them to apply or prove theorems about parallelograms. As students finish writing their proofs, I like to give them a parallelogram and ask them to solve for the unknown angle measures --this gives students the opportunity to recall properties they discovered in the introduction unit for the course (see Investigating Special Quadrilaterals).

#### Resources

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For this activity, I assign each group a term or property from The Language and Properties of Proofs to look up in their books. Groups write out a definition and draw an example that illustrates the term or property. I tell students that their Recorder/Reporter should be prepared to share out their group’s work on the document camera.

After groups have had sufficient time to complete their portion of the work, I call Recorder/Reporters up to share their work. Students in the audience record the definition and draw examples on their own note taker. If students have questions about the term or property, I try to let the presenter answer or defer to someone in their group to answer before providing clarity or another example because it is good for students to engage in a productive struggle and wrestle with the precise meaning of words (**MP6**).

Resource Citation: I want to acknowledge Cathy Humphreys, a colleague and mentor, who shared this task with me.

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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