##
* *Reflection: Rigor
Properties of Parallelograms and Special Parallelograms - Section 1: Warm-Up: Logical Road Map

There are times when I wonder whether discussing proofs (or big problems, for that matter) are really worth taking up a significant amount of class time. Today's discussion was a reminder of how valuable this time is for my students. Even during the discussion, I could sense growth (maturation?) in their willingness to:

- Persist in the face of challenge
- Appreciate different approaches

Similar to the way in which I facilitated the discussion around Three polygons meet at Point B, today I sequenced student presenters in a particular order. Again, I used the subjective criteria, "I think this strategy will support the greatest number of students’ work as they try to deepen their understanding."

In today's lesson, the high leverage student strategy involved using:

**the base angles of the isosceles triangle****corresponding angles****alternate interior angles**

Since I have been promoting the flexible use of properties of special quadrilaterals in this unit, I also chose to promote the idea of using **properties of isosceles trapezoids**. This approach was relatively novel (and only arose in one of my classes). Nonetheless, I thought it deserved public validation.

When the use of properties of isosceles trapezoids did not emerge in the other classes, I decided to treat this idea as an original scholarly contribution. I directed my students to focus on a diagram of the task, which I had projected using a document camera. Then, I silently numbered and labeled the “work” asking students to discuss reasons why the work made sense (or not) in small groups. (I was essentially asking students to re-imagine the parallelogram and isosceles triangle as forming an isosceles trapezoid). After approaching the idea in this way, it proved to be quite popular. It became a catalyst for building students’ interest in the reasoning of others and for motivating students to supply an argument explaining the contribution made by their own work. Our discussions of diagrams became much more engaging!

*Constructing Viable Arguments*

*Rigor: Constructing Viable Arguments*

# Properties of Parallelograms and Special Parallelograms

Lesson 6 of 9

## Objective: Students will be able to apply angle relationships and properties of isosceles triangles and trapezoids in a proof.

## Big Idea: In the Logical Road Map, students will make sense of multiple pathways to writing a proof.

*90 minutes*

#### Warm-Up: Logical Road Map

*25 min*

I like using the Logical Road Map for today’s lesson because we will spend time and detail on justification and proof writing. There are several ways to write the proof for the logical road map—many students will use corresponding and alternate interior angles to write the proof. Since this unit is on discovering and proving polygon properties, I make sure to look out for students who use properties of parallelograms and isosceles trapezoids to write their proofs, having them display their work under the document camera for the class to see. Viewing and discussing alternative proofs enables students to make connections among concepts (**MP3**).

#### Resources

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It is important for me to give adequate time to review last night's homework assignment:

*Prove that if a trapezoid has congruent base angles, it must be isosceles.*

These are often the most difficult proofs for my students. Students start with seemingly nothing (no diagram, for example), but they are required to prove a rather important idea.

I like to have at least two student volunteers present their proofs (or ideas for how to write the proof) to the whole class. Ultimately, someone will see that one of the ways to write the proof is directly related to the Logical Road Map Warm-Up. Ideally, others will see that adding **auxiliary lines **(two perpendiculars to the bases) can allow them to use properties of rectangles to write the proof, or that extending the sides of the trapezoids that are not the bases can allow them to use properties of isosceles triangles to write the proof.

The Common Core calls students to construct viable arguments and critique the reasoning of others, which is why it is important to allow students to discuss multiple ways to write the proof (**MP3**) and to give feedback to improve the quality of these proofs, which requires them to attend to precision (**MP6**).

#### Resources

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After beginning the class with a focus on proofs, we take brief reminder notes on the properties of parallelograms and special parallelograms. My students often need a quick review of these properties. Today, they will apply the properties to solve leveled problems in a Group Challenge during tomorrow's lesson.

Next, I introduce the Parallelogram Properties Umbrella to students. Since classification and differentiation often pose challenges to students, this is a good time to give students an umbrella **graphic organizer** to help them organize their thinking hierarchically. The use of a graphic organizer better enables my students to see the relationships that exist within the parallelogram family. Encouraging students to see rhombuses as equilateral parallelograms, rectangles as equiangular parallelograms, and squares as both equilateral and equiangular parallelograms, is an important learning objective at this stage of my course.

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I like to close this lesson with the Determining Parallelograms worksheet. The task is tricky. It requires students to apply all properties of special parallelograms and to think about proofs before actually writing them. I find that this worksheet often helps my students to self-assess their understanding of parallelogram and special parallelogram properties.

By the time that they complete this worksheet, my students prove several theorems about parallelograms. One of the problems gives them the opportunity to prove that **if a quadrilateral has one pair of congruent, parallel sides, it must be a parallelogram**.

#### 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: Review Angle Relationships to Begin Conjecturing About Polygons
- LESSON 2: Interior and Exterior Angle Sum of Polygons
- LESSON 3: Going Deeper with Interior and Exterior Angles
- LESSON 4: Focus on Justification: Interior and Exterior Angles
- LESSON 5: Kite & Trapezoid Properties and Midsegments of Triangles & Trapezoids
- LESSON 6: Properties of Parallelograms and Special Parallelograms
- LESSON 7: Proofs: Properties of Parallelograms
- LESSON 8: Review: Polygon Properties
- LESSON 9: Discovering and Proving Polygon Properties Unit Assessment