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.
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.
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.
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.