This Warm-up prompts students to compare the graphs of four quadrilaterals. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Invite each group to share one reason why a particular set of three go together. Record and display the responses. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology they use, such as “equilateral” or “diagonal,” and to clarify their reasoning, as needed. Consider asking:

• “How do you know _____?”
• “What do you mean by _____?”
• “Can you say that in another way?”

If possible, leave the list of responses displayed until the end of class. Students will return to these images during the Lesson Synthesis.

Students are presented with four points and are asked to fully describe the quadrilateral that has those points as its vertices. Slopes will show that all pairs of adjacent sides are perpendicular, making the shape a rectangle. Then students will use the Pythagorean Theorem to calculate both the area and the perimeter of the quadrilateral.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Students observe that the angle formed by connecting endpoints of a diameter to a third point on the circle appears to be a right angle. They confirm that this is true for a few particular points, and they then write a conjecture. This conjecture will be generalized in an upcoming unit. The term "chord" will be defined in a subsequent unit. It is not necessary to define it here. As students make a conjecture and later listen to each other's reasoning about triangles and quadrilaterals, they create viable arguments and critique the arguments of others (MP3).

In the digital version of the activity, students use an applet to make their conjectures about the right angle that is made from two endpoints of a diameter and another point on the circle. The applet allows students to move the third point around on the circle, calculate the slopes of the chords, and calculate the product of the slopes. 

This activity works best when each student has access to devices that can use the embedded applet, because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the Activity Synthesis.