In this Warm-up, students use 1-inch strips with evenly spaced holes and metal paper fasteners to recall the relationship between parallelograms and rectangles. This prepares students to prove that all rectangles are parallelograms. Before students begin working, they are explicitly asked to determine which mathematical statement(s) they agree with. Making a reasonable conjecture is often an important aspect of making sense of problems (MP1).

Ask again, “Which statement(s) do you agree with?”

• All rectangles are parallelograms. (True)
• All parallelograms are rectangles. (False)

Add the following definition to the class reference chart, and ask students to add it to their reference charts:

A rhombus is a quadrilateral with four congruent sides. (Definition)

Invite students to use their strips to hold up first an example and then a nonexample of a rectangle and a rhombus. If needed, remind students that the fasteners don’t need to go in the last holes.

In a previous lesson, students matched diagrams to conjectures. In this lesson, they identify the given information and the statement to prove in the conjectures. For example, “all rhombuses are parallelograms” might turn into “show that in quadrilateral , with segments , , , and congruent, is parallel to and is parallel to .” Deciding what precisely to show in order to prove a conjecture is not trivial. Monitor for students who try to prove other properties of the shapes are true rather than using the definition, such as trying to prove a figure is a rectangle by proving that opposite sides are congruent.

Part of being a mathematician is reading and reviewing other people’s proofs. Reviewers look at completed proofs to understand them and how they may be useful, and they look at draft proofs to help improve them. Students continue to practice the skill of reviewing proofs in this activity.