##
* *Reflection: Diverse Entry Points
Proving Properties of Special Parallelograms - Section 2: Guided Practice

While the resource for this section is has blanks for students to fill in, there are multiple correct answers that can be used to fill in the blanks. Teaching this lesson several times now has taught me to be prepared for these and also to be on the lookout for students who share some of the alternative solutions.

For example, when proving that the diagonals of a rhombus each bisect a pair of opposite angles, students have chosen at least two options for which pair of triangles to prove congruent. One pair requires them to use reflexive property (a relatively simpler proof) and the other requires using the fact that diagonals of a parallelogram bisect each other.

During the share out, I make sure to bring out these options and their comparative merits. Then I ask students to discuss which they prefer and why. In general, in these discussions I emphasize that there are always multiple pathways and that we should be aware of our options and strategic in making our choices.

*Multiple Pathways*

*Diverse Entry Points: Multiple Pathways*

# Proving Properties of Special Parallelograms

Lesson 4 of 8

## Objective: SWBAT prove the distinguishing properties of rectangles and rhombuses.

#### Activating Prior Knowledge

*20 min*

In the previous lesson on Special Parallelograms, we explored the properties that distinguish rectangles and rhombuses from other parallelograms. In this lesson, we'll prove these properties. My students have prior knowledge from the preceding lesson and an analogous lesson, Proving Properties of Parallelograms, that will help them in this lesson. The purpose of this section is to surface that prior knowledge.

I start by guiding students through the graphic organizers in Part 1 of Activating Prior Knowledge: Proving properties of special parallelograms. I do this using a **think-pair-share **protocol. For example, I might say (and also write the bolded material on the board):

Students, without looking at your notes, take 60 seconds to **write the definition** of rectangle in the space provided, **mark the diagram** with the information contained in the definition, and **write what has to be true** about the diagram **using mathematical notation**.

After 60 seconds I might add, "A-B partners, I want you to exchange papers then come to agreement on how to correctly complete the graphic organizer for the definition of rectangle." Finally I might show my correct answer under the document camera or (time permitting) have a randomly selected student come up to share what they've written.

Next I remind students how we proved properties of parallelograms by proving triangles congruent and then using CPCTC. I explain that we'll be using a similar strategy to prove properties of special parallelograms. Having provided this background, I ask students to start on Part 2 of the handout. As in Part 1, I use a **think-pair-share** protocol. However, for this part, I allocate more time for students to share their explanations. I want to make sure that students have adequate time to practice their reasoning as they prepare to write proofs in the next section.

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#### Guided Practice

*25 min*

In this section of the lesson, I guide students through the Guided Practice: Planning proofs of special parallelogram properties handout. Again, I use a think-pair-share protocol to structure the students' work.

So, again, I guide students through the handout using the think-pair-share format, clarifying, modeling, and engineering classroom discussions as opportunities arise. In the next section, my plan is for students to use the completed handout from this section as a documented plan for the proofs they will be writing.

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#### Cooperative Proof Writing

*30 min*

By the time we reach this section, I've done my best to make sure that students are prepared to prove the distinguishing properties of rectangles and rhombuses. In this section, students will be working with a partner to write the proofs. The guidelines for this activity are laid out in the Cooperative Proof Writing: Special Parallelogram Properties handout.

Before letting students loose on the activity, I take some time to go over the Proof_Writing_Protocol so that students understand the protocol. I impress upon students that using the protocol is mandatory. I explain that its intent is to make sure that the work is shared equitably within each group. I also remind students of the importance of following and understanding the reasoning of others. To practice this, the protocol requires each partner to echo the other's reasoning and to supply reasons for the other's statements.

Finally, I explain to the students that after this activity, they will be asked to complete similar proofs on their own and should therefore pay attention to each proof as a whole, and not just the parts for which they are responsible.

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#### Formative Assessment

*50 min*

At this point, students have had enough experience with the proofs to reasonably expect that they could write the proofs on their own. In order to determine the extent to which students are actually able to write the proofs, I give them the Formative Assessment: Proving properties of special parallelograms. I give the students 7-8 minutes per proof so it takes 40 to 50 minutes to administer.

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Parallelogram Definition and Properties
- LESSON 2: Proving Properties of Parallelograms
- LESSON 3: Special Parallelograms
- LESSON 4: Proving Properties of Special Parallelograms
- LESSON 5: Proving that a Quadrilateral is a Parallelogram
- LESSON 6: Using Coordinates to Prove a Quadrilateral is a Parallelogram
- LESSON 7: Trapezoids
- LESSON 8: Quadrilaterals Transfer Task