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Unit 2, Lesson 10

Practicing Proofs

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Integrated Math 1

10.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Copy (from Blackline Masters)

Brace Yourself! Long Strips

Brace Yourself! Short Strips

To Gather

1-inch strips cut from card stock with evenly spaced holes Metal paper fasteners

Activity Narrative

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: metal fasteners and 1-inch strips cut from card stock with evenly spaced holes.

Launch

Give students 2–3 minutes to explore the questions on their own, then pause the activity.

Explain how diagonal braces are used in carpentry. When carpenters want to ensure that something they are building that is supposed to be rectangular doesn’t slant, sometimes they use diagonal braces, which are pieces of wood or metal that go from opposite corners of the rectangle. Ask students if they’ve ever seen this, and ask how the braces help the structure keep its shape. (Four edges are flexible, but the diagonal forces a certain angle, so now the shape is rigid.)

Explain that they will be making different configurations of braces to see what kinds of shapes can be built around them. Demonstrate choosing two strips, attaching them at one point with the metal paper fasteners, arranging them at a desired angle, and then tracing around them to create a quadrilateral.

<p>Picture of strips making a cross to form a kite.<br>
</p>

Activity

None

What can you do with the braces and fasteners your teacher will give you?

What different ways can you arrange them?

What different quadrilaterals can you make by changing the braces?

Keep track of your findings.

Activity Synthesis

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.

10.2

Activity

Instructional Routines

Card Sort MLR2: Collect and Display

Materials

To Copy (from Blackline Masters)

More Practice Seeing Shortcuts Cards

To Gather

1-inch strips cut from card stock with evenly spaced holes Metal paper fasteners

Activity Narrative

Students will sort different images during this activity. A sorting task gives students opportunities to analyze representations closely and make connections (MP2, MP7).

Students will identify triangles that are congruent based on the triangle congruence theorems. Focusing on figure , monitor for students who:

Think there is not enough information.
Use the Angle-Side-Angle Triangle Congruence Theorem.
Use the Side-Side-Side Triangle Congruence Theorem.

10.3

Activity

Instructional Routines

Take Turns

Materials

To Gather

1-inch strips cut from card stock with evenly spaced holes Metal paper fasteners

Activity Narrative

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.”

Launch

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 or 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 1-inch strips and metal fasteners during this activity as well.

10.4

Activity

Optional

Instructional Routines

None

Materials

To Gather

Geometry toolkits (HS)Scissors

Activity Narrative

This optional activity provides students with an opportunity to apply triangle congruence theorems in context. Students watch a video which shows a method for solving a problem in woodworking. They are then asked to justify why this method works and extend their thinking to determine if it would also work in other scenarios.

Launch

Arrange students in groups of 2. Play the video of cutting wooden planks to fit together. After playing it once, ask students if they have ever done woodworking or needed to solve a similar problem. Play the video again, this time asking students to record the process.

A video showing two pieces of wood being cut to fit into a corner.

Representation: Access for Perception. Students may benefit from watching the video more than once. Provide students with three screenshots from the video: the first line drawn, the second line drawn, and the final result.
Supports accessibility for: Language, Attention

Lesson Synthesis

Invite students to choose one of the statement and diagram pairs from the matching activity to prove. Give students 2–3 minutes to copy the diagram from the matching card and annotate their diagrams with additional information they can prove is true. Arrange students in groups based on the statement they chose. Give students another 2–3 minutes to talk through their ideas for the proof. They will summarize the proof in the Cool-down.

to access Cool-down.

Student Lesson Summary

To prove that segments or angles are congruent, we can look for triangles that those segments or angles are part of. Can the triangles be proven congruent? Are the segments or angles corresponding parts of congruent triangles? Does that help prove the conjecture?

To prove that the triangles are congruent, we can look at the diagram and given information. Think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then check if there’s enough information to use the Side-Side-Side, Angle-Side-Angle, or Side-Angle-Side Triangle Congruence Theorem.

Here's an example: As part of a proof to show that a quadrilateral with four congruent sides has parallel opposite sides, we need to show that a diagonal splits the quadrilateral into two congruent triangles.

First, sketch a diagram to see what is given, and look for congruent triangles. Since this is about a quadrilateral, adding a diagonal auxiliary line to make triangles will be helpful.

<p>Quadrilateral A B C D with 4 congruent sides. Diagonal A C is drawn.</p>

Because all the sides of the quadrilateral are congruent, and the triangles formed by the diagonals share a third side, we can use the Side-Side-Side Triangle Congruence Theorem to prove that triangles and are congruent.

Launch

Encourage students to use the 1-inch strips and fasteners to model the figures on the cards.

Arrange students in groups of 2 and distribute pre-cut cards.

Give students 2–3 minutes to sort the cards into categories of their choosing.
Pause the class after students have sorted the cards.
Select groups to share their categories and how they sorted their cards.
Discuss 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.

MLR2 Collect and Display. Circulate to listen for and collect the language students use as they complete the Card Sort. On a visible display, record words and phrases, such as “corresponding sides,” “congruent angles,” or “perpendicular.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading

Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on their own progress. For example, ask students how comfortable they are describing the characteristics of the figures.
Supports accessibility for: Organization, Conceptual Processing

Activity

None

Your teacher will give you a set of cards that show different figures. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.
Sort the cards by rigid vs. flexible figures.
State at least one set of triangles that can be proved congruent using the:
1. Side-Angle-Side Triangle Congruence Theorem.
2. Angle-Side-Angle Triangle Congruence Theorem.
3. Side-Side-Side Triangle Congruence Theorem.

Student Response

to access Student Response.

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Audience

Assessment Access

Lesson 10

Practicing Proofs

Warm-up

Standards Alignment

Building On

7.G.A.2

Addressing

Building Toward

HSG-CO.C.11

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSG-CO.B.8

Addressing

HSG-MG.A.3

Building Toward

HSG-CO.C.11

Standards Alignment

Building On

HSG-CO.B.8

HSG-CO.C.11

Addressing

Building Toward

Standards Alignment

Building On

HSG-CO.B.8

Addressing

HSG-MG.A.1

HSG-MG.A.3

Building Toward