Using Coordinates to Prove a Quadrilateral is a Parallelogram

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SWBAT use four different methods to prove that a quadrilateral defined by given vertex coordinates is a parallelogram.

Big Idea

Impostors beware! In this lesson students learn to distinguish the real parallelograms from the pretenders.

Activating Prior Knowledge

25 minutes

So far in this unit, students have learned and proven properties of parallelograms. They have also proven the criteria that are sufficient for proving a quadrilateral is a parallelogram. In this lesson, students will be using the criteria to establish that quadrilaterals are parallelograms. 

I like to get the students primed for the type of thinking they'll be required to do in the lesson. To do this, I use the Measure Properties of Parallelograms activity in which students will use a ruler and protractor to verify parallelogram properties. My main goals for this activity are for students to:

  1. recall the four properties of parallelograms,
  2. get used to the idea of producing evidence that a figure has a particular property, and
  3. select appropriate tools to gather the desired evidence.

To get started, I give students one minute to fill in the four properties of parallelograms in the designated spaces on the handout. This is an individual task and I ask students not to use their notes. I give students 2 minutes to exchange papers with their A-B partners, give feedback, and make corrections. 

Next, I give an overview of the activity. I explain that we will be gathering evidence to support the fact that these quadrilaterals are parallelograms. I also emphasize the importance of making precise measurements and documenting these measurements using precise notation (MP3).

I then direct students to find the place on the handout where they wrote the property "Opposite sides of a parallelogram are congruent." I give them 3 minutes to collect and document evidence. Next I show a model response under the document camera that exemplifies the level of precision I want. Once I've modeled a response, I give students 10 minutes to complete the rest of the activity independently. As they are working, I walk around making sure that students are measuring the right things and that they are documenting these measurements precisely.

Later in the lesson, students will be using coordinate geometry to gather evidence that a quadrilateral satisfies the definition of  a parallelogram and possesses the properties of parallelograms. As a primer, I lead them through the Activating Prior Knowledge_Verifying Parallelogram Properties activity. By the end of the activity, I want every student to know that coordinate geometry uses:

  1. slopes to determine parallelism,
  2. distance formula to establish congruent segments, and 
  3. midpoint formula to determine if a segment has been bisected. 

I give students 15 minutes working in pairs to complete the activity. I remind them that each of the three items on the handout asks them to make a final conclusion. I explain that for each item they must write a clear concluding statement that references the evidence they've gathered.

As students are working, I reveal the answers one at a time, being careful to lag behind where I see the majority of the students are on the handout.

Independent Practice

35 minutes

In this section of the lesson, students will be using coordinate geometry to prove that a quadrilateral is a parallelogram. In the previous section, students were oriented to the coordinate geometry methods they will need to use in this section. So this is an opportunity to apply what they have just learned.

I give students 15 minutes to work independently on the Using Coordinates to Prove that a Quadrilateral is a Parallelogram handout. Before we start, I remind students of the following:

  1. They are only to use coordinate geometry.
  2. They are to show all calculations using precise notation (e.g. Slope of segment AB = 2/5 as opposed to slope = 2/5 or just 2/5)
  3. They are to write a short narrative after each method explaining how their work proves that the quadrilateral is a parallelogram.

Then students begin working. As they are working, I walk around the classroom providing guidance to ensure that students are producing quality work that is correct. As I walk around, I am also looking for students with exemplary work who might present to the class.

When the 15 minutes have elapsed, I announce that I will be selecting four students to present. I give students 5 minutes with their A-B partner to check their work and rehearse their presentations. 

Finally, I call the students up one at a time to present. As they present I give specific praise related to the aesthetics and organization of students' work, their use of academic language, and the thoroughness of their narratives.

An example of a thorough narrative is:

"If a pair of opposite sides in a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. Segment AB and segment CD have the same slope so they are parallel. The segments also have the same length so they are congruent. ABCD is a parallelogram because segment AB is parallel and congruent to segment CD."

Exit Ticket

5 minutes

To close today's lesson I give students a chance to reflect on what they've done. I have students write a paragraph responding to the following prompt:

What was the main goal of the lesson today? Generalize about how you used coordinate geometry as a tool to achieve the goal.