In this lesson, students will be using triangle congruence to prove the properties of parallelograms. The purpose of this section is to bring prerequisite knowledge to the forefront. There are three instructional goals for this opening section:
1. Get students to recognize the triangles that can be proven congruent when the diagonals of a parallelogram are drawn.
2. Make sure that students can identify the hypothesis and conclusion implicit in each property (these will be the givens and what we're proving, respectively).
3. Refresh students' memories of triangle congruence proofs and Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
The resource for this section is the APK_Prove Properties of Parallelograms handout.
After passing out the papers and getting the class focused, I give the students 30 seconds to read and follow the directions for item #1. Next, I have each student check with their A-B partner to make sure that they have done it correctly. Finally, I show the diagram on the document camera.
For item #2, I show the following frame on the document camera:
Triangle pairs that appear to be congruent
_________ and ___________
Then I give students 3 minutes to work with their A-B partners to complete item #2. Partner A begins by giving a triangle pair, and then partner B agrees or disagrees. Then the roles alternate until the item is completed.
After 3 minutes, I show the correct answers on the document camera. As I’m doing this, I emphasize the importance of naming the corresponding vertices of the congruent triangles in the same order.
Item #3 is intended to reinforce the logic inherent in the four properties and to bring the properties back into students’ memories. I don’t want to spend too much time on this item so I show the answers for the first property immediately. I then give students one minute to write the hypotheses and conclusions for the remaining three properties.
As I show the answers on the document camera, I communicate to students that it is important to know these hypotheses and conclusions because they will be the givens and what we are trying to prove later on in the lesson.
Moving on to item #4, I begin by explaining that we are trying to prove two parts of two triangles congruent. I remind students that we have done this in the past by first proving that the triangles are congruent and then showing that their corresponding parts must be congruent. Having focused students on the goal of proving the triangles congruent, I have them mark the diagram with any congruencies that are implied by the givens or the diagram. I then show how I’ve marked my diagram and explain why. Next I ask student to quickly discuss with their partner which triangle congruence criterion we can use to prove the triangles congruent. Finally I ask student to write the proof. As students are writing, I begin to reveal the steps of the proof I’ve written, one step at a time (lagging behind where they are), so that students can compare what they’ve done to what I’ve done.
In this section of the lesson, we'll be using triangle congruence to prove three of the parallelogram properties. To a novice, it is not at all intuitive (nor should it be) why we would use triangle congruence to prove properties of parallelograms. So my goal in this section is to explicitly model my thinking and decision-making for students so they can pattern their own after mine. For more on why I do this see my video on Modeling Expert Thinking:
Video Source URL:
The primary resource for this section is the Guided Practice_Proving Parallelogram Properties handout. As seen in the video, I start by assuming full control for item #1 on the handout. Then I gradually release control to students as they assimilate the expert strategies that I have modeled for them.
For example, after I have modeled Proof #1, students usually have some insight on how to approach Proof #2 (an analogous proof). Prior to writing Proof #2, students are asked:
I give students one minute to think of their responses to each of these questions, and then I engineer structured talk around these two questions. For example, I might say, "A partners, tell your B partner what you think our givens are...Then B partners, say whether you agree or disagree and why." For the second question, I might say, "A partners, say which triangle pair you would work with and why....B partners, echo what you heard your A partner say then ask if you heard them correctly."
At this point, I insert myself back into the conversation. I review the givens and what we are trying to prove. Then I clarify that we need to prove two different pairs of triangles congruent in order to show that both pairs of opposite angles are congruent. I then talk about how we will prove that triangle ADC and triangle CBA, for example, are congruent. I emphasize that we can use opposite sides of a parallelogram are congruent as a theorem now that we have proved it. Then I would ask students to explain to their partners how we can show that triangle BCD is congruent to triangle DAB to make sure they have processed the explanation I have just given. So now that we have the givens, what we're trying to prove, and a plan of attack, I ask students to work independently to write the proof that Angle A is congruent to Angle C. I walk the room checking in on individual students and looking for things that may need clarification. As I see that students have forged ahead, I return to the document camera and begin to reveal the proof one step at a time (lagging behind where students are) so that they can compare what they've done to what I've done.
In item #3, I'm showing students an alternate way to prove opposite angles congruent. First I state the given:
ABCD is a parallelogram (students write this in the space provided on the handout).
Then I state what we're proving:
Angle B congruent to Angle D (students write)
Then I show myself drawing diagonal segment AC on the diagram. Next I talk about how I intend to use the Third Angles Theorem:
I will need to show that two angles in triangle ADC are congruent to two angles in triangle CBA so that I can say Angle B is congruent to Angle D.
With that goal in mind, I mark the diagram to show the two pairs of parallel sides. I pause for a minute to allow students to think then discuss how we can establish the two angle congruencies we need. Then I call on non-volunteers to share their plan for the proof. As students explain their plans, I'm careful to re-voice, revise and paraphrase, as needed, so that the whole class gets it. When I feel that students have what they need to write the proof, I release control to the students and they write the proof. As before, I lag behind the students and reveal the steps of the proof one at a time so that they can check their work and ask for clarification if they need it.
Item #4 is the proof that diagonals bisect each other. There are several things in my decision-making process writing this proof that I want to reveal to students. For this reason, I do some front-loading before I have them write the proof.
While I am talking I always make sure to jot down on the board or document camera the ideas we've been brainstorming so that we have a record. When I've explained everything I want to explain, I then have students describe the overall plan for the proof to their A-B partner. After this, I have the A-B partner echo back what they've heard. Finally, it's time for students to actually write the proof.
In this section, the goal is to gauge the extent to which students have assimilated the learning from the previous sections. Independent Practice_ Proving Parallelogram Properties asks students to replicate the proofs of parallelogram properties from scratch. In order to vary the task slightly, they are asked to write two of the proofs in paragraph form. Finally students are asked to explain why it was not necessary to prove that consecutive angles in a parallelogram are supplementary.
In this activity students apply knowledge and strategies developed in the first half of the lesson. Before asking my students to begin this task, I reflect momentarily on their preparation: "Are we where we need to be for students to work independently?" For this topic, I am generally able to assume that my students are prepared to succeed.
When the 20 minutes for the Independent Practice has elapsed, I collect the papers and make sure that students are aware of the homework assignment for the night (Later, I will look at the papers in order to get some good formative assessment data). Then, I make a closing statement such as:
Today we used a lot of what we learned about congruent triangles and parallel lines cut by transversals to prove some important properties of parallelograms. Isn't it cool how those things we learned became immediately useful in helping us to learn and prove these new things? This is how I want you to experience Geometry. We are building a body of knowledge, establishing truths that lead to more and more truths. The knowledge keeps expanding. Remember we started the year with just a point...and look where we are now...pretty impressive.