Lesson 14

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.

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn‘t belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as equilateral or diagonal. Also, press students on unsubstantiated claims.

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 4 points and asked to fully describe the quadrilateral with those 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 area and perimeter.

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

Make graph paper available to students who would like to use it.

Some students may state that the quadrilateral is a rectangle simply because it looks like one. Remind these students that we need to back up our reasoning with mathematics. Suggest students review their reference charts for definitions and properties of rectangles.

Invite students to share their reasoning for each question. Highlight students who carried information from one question to the next, such as recognizing that in a rectangle, opposite sides have equal length, so they only need to calculate 2 distances (rather than all 4).

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, then write a conjecture. This conjecture will be generalized in an upcoming unit.

Monitor for different ways that students word the conjecture to highlight during the discussion. Listen for terms such as diameter, endpoints, right triangle, and right angle. The term chord will be defined in a subsequent unit. It is not necessary to define it here.

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 synthesis.

Ask previously identified groups to share their conjectures. Challenge students to decide whether all the conjectures are saying the same thing, and if any of the conjectures need to be refined. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

There are many ways to correctly word the conjecture. As students work to refine the wording, be sure that the word diameter is included, that it’s clear that 2 points are at the endpoints of the diameter, and that the third point is clearly somewhere else on the circle. Students may describe the result as 2 segments that form a right angle or 3 segments that form a right triangle.

Once the conjecture is finalized, ask students if they have proven it (they have not; they’ve just shown it’s true for a few particular cases). Tell students that we will look at a more general case of this assertion in an upcoming unit.