Proving that a Quadrilateral is a Parallelogram
Lesson 5 of 8
Objective: SWBAT prove the converses of the parallelogram property theorems.
While I want students to know the criteria that are sufficient to prove that a quadrilateral is a parallelogram, it is more important to me that they can use Geometry to prove that these criteria are sufficient. This is the actual practice of Geometry. In this section, my goal is for students to understand the rationale behind the proofs so that their proof-writing is based on a strong conceptual foundation, not on mere memorization.
The Ways to Prove a Quadrilateral is a Parallelogram activity is designed to be a guided interactive reading. In general, I select students randomly to read a small chunk of text at a time. I insert pauses as needed to discuss important concepts that should not be glossed over. For example, I pause for a think-pair-share on the meaning of sufficient in the context of this lesson. I also pause to think-pair-share whenever there are blanks on the handout for students to fill in. Additionally, after we have finished reading about a particular method, I ask each A-B pair to take turns summarizing the rationale to each other or, if they can't, to explain what is unclear. Then I have students share with the class what is unclear so that I can clarify these things.
Since this is a reading activity, I am intentional about developing the skills and strategies required to effectively read a mathematics text. These include re-reading, making connections between narrative and diagram, and understanding the sequencing of statements. These strategies are particularly useful for reading the rationale for Method 3. See the following video for an example of how I model the strategies.
In the previous section, students learned the rationale for three of the four proofs we'll be writing. In this section, we will be writing formal proofs. The general flow for this section is:
- Have students create a diagram that represents the given information,
- Brainstorm to develop a plan for the proof, and
- Write the proof.
To begin, I give students a copy of the Guided Practice: Proving a quadrilateral is a parallelogram activity. I explain that we'll be writing four proofs that quadrilaterals are parallelograms and that these four proofs will differ only in terms of the information that is given. I explain that in general we prove a quadrilateral is a parallelogram by showing that it satisfies the definition of parallelogram, i.e., that it has two pairs of parallel sides.
Starting with #1, I direct students to read the given information and then to create a diagram that represents what is given. Then I give students one minute of individual reasoning time to think of a plan for the proof. Next, I begin calling on randomly selected students to help brainstorm components of a plan for the proof. As we are brainstorming, I create a visual flow diagram on the whiteboard that represents the plan for the proof.
After that, A-B pairs take turns explaining one step in the flow diagram at a time. Finally, the students begin to write the proof. As they are writing, I go around looking for student exemplars. I'm looking for one exemplar that is essentially correct but may be less than thorough or difficult to follow, and another which is very thorough, precise and easy to follow. I would then discuss the merits of each exemplar with the intention of going over the logic of the proof (twice) while pushing students to be thorough, precise and clear.
I lead students through #2 and #3 in similar fashion, except I only choose student exemplars that are thorough, precise and clear.
I like to see what students can do without my help on #4. I have them draw the diagram and then I give them a minute to think. Next, I have them share their ideas with their A-B partners. Then I tell them to try writing the proof on a separate sheet of paper. I've found that students are more willing to risk being wrong when they work on a paper that they know they can crumple and throw away. Depending on how things are going, I may give a hint, e.g., "What do we know about the sum of the interior angle measures of a quadrilateral?" or "What are the ways we can prove two lines are parallel?". In any case, I walk the room moving students forward without telling them what to do. When at least 85 percent of the students have a completed proof, I look for two or three exemplars that are thorough, precise and clear and these students come up to the front to present their proofs.
The purpose of this section is to have students reflect on what they've accomplished and to make sure they understand what they've done. Students complete the Closure: Prove a Quadrilateral is a Parallelogram handout independently. They should have all handouts from the lesson organized on their desks so that they can refer to them.
Item 1 asks students to put into words the theorems they proved in the Guided Practice activity. Students know they have written proofs, but they may not realize that they have proven theorems that they will be able to use in future proofs.
Item 2 asks students to create a flow diagram to represent the general sequence of steps used to prove theorems 1,2 and 3. Getting students to generalize (MP8) is one of my broader goals for the course. I ask my students to generalize on a regular basis and in this case I would expect them to come up with something that communicates:
- Use the givens to prove two triangles congruent.
- Use CPCTC to establish that alternate interior angles are congruent.
- Use the converse of the alternate interior angles theorem to establish that lines are parallel.
- Use the definition of parallelogram to establish that the quadrilateral is a parallelogram.
Finally Item 3 asks students to write a new proof. They should be able to complete the proof if they can transfer what they learned from the proofs they wrote in the lesson. So this is a check for understanding that lets me know how successful the lesson has been with individual students.