Lesson 10

Throughout this lesson and in several subsequent lessons, students will make conjectures and prove properties of quadrilaterals. This activity gives students a chance to experience a dynamic tool: 1-inch strips cut from card stock with evenly-spaced holes and metal fasteners.

Students might not realize that braces go diagonally from vertex to vertex, and they may think they go through the midpoints of opposite sides. Clarify by tracing around the endpoints to demonstrate how to visualize the quadrilateral.

Invite a student to name or describe a quadrilateral they made, without describing how they arranged the braces. Give students 1 minute to think, ask questions, and arrange their braces, then have them hold up their idea of how to make a quadrilateral of the same type. Allow the student who suggested the quadrilateral type to give feedback. 

Students will sort different images during this activity. A sorting task gives students opportunities to analyze representations closely and make connections (MP2, MP7). Students interpret the model in a situation and then check if their results make sense by applying a triangle congruence theorem.

Students will identify triangles that are congruent based on the triangle congruence theorems. Focusing on triangles \(KNL\) and \(MNL\), monitor for students who: 

• think there is not enough information
• use the Isosceles Triangle Theorem or use the Angle-Side-Angle Triangle Congruence Theorem without finding the measure of the third angle
• use the Pythagorean Theorem to use the Side-Side-Side Triangle Congruence Theorem

Encourage students to use the braces and fasteners from the warm-up to model.

Arrange students in groups of 2. Tell them that, in this activity, they will sort some cards into categories of their choosing. When they sort the figures, they should work with their partner to come up with categories.

Pause the class after students have sorted the cards into categories of their choosing. Select groups of students to share their categories and how they sorted their cards. Choose as many different types of categories as time allows. Attend to the language that students use to describe their categories and figures, giving them opportunities to describe their figures more precisely. Highlight the use of terms like rigid, congruent, and ambiguous. After a brief discussion, invite students to complete the remaining questions.

In the isosceles triangle image, students will need to use either the Triangle Angle Sum Theorem or the Pythagorean Theorem to figure out that if they know two pairs of corresponding angles or sides are congruent in this right triangle, the third pair must be, too. Students may struggle because we don’t have specific measures for the third angle or side, so support them to think about “If we knew these angles were, say, \(90^{\circ}\) and \(60^{\circ}\), could the third pair of angles have different measures?” or “If we knew these sides were, say, 4 cm and 5 cm, could the third pair of sides have different measures?”

Select students to share their different reasoning about triangles \(KNL\) and \(MNL\):

• who thought there was not enough information
• who used the Isosceles Triangle Theorem or used the Angle-Side-Angle Triangle Congruence Theorem without finding the measure of the third angle
• who used the Pythagorean Theorem to use the Side-Side-Side Triangle Congruence Theorem

If not mentioned by students, discuss how the Triangle Angle Sum Theorem and Pythagorean Theorem can help us prove that a third side or angle is congruent even if we don’t know the specific measurements.

Ask students, “How could you make the structures that are flexible into rigid ones?” (Add a diagonal brace that would decompose the shape into triangles.)

In this partner activity, students take turns thinking about which diagrams (on the cards from the previous activity) support different proof statements. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). Monitor for students who confuse the given information with what is to be proven, for example, by matching the rhombus with “A quadrilateral with perpendicular diagonals that bisect each other is equilateral.”

Students will use these diagrams to write proofs in the cool-down.

Students should still have access to the cards from the previous activity.

Keep students in groups of 2. Tell students that for each statement, one person finds a diagram and explains why they think it could be used to illustrate the given/proven statement. Their partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next statement, the students swap roles. If needed, demonstrate this protocol before students start working. Allow students to use their hole-punched strips and metal fasteners, if desired, during this lesson as well.

Draw students’ attention to figure \(ZIAB\). Check that students understand why it matches “Opposite angles in an equilateral quadrilateral are congruent.” and not “A quadrilateral with perpendicular diagonals that bisect each other is equilateral.”