Lesson 5

Points, Segments, and Zigzags

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5.1: What's the Point? (5 minutes)

CCSS Standards


Building On

Addressing

Warm-up

This activity invites students to think carefully about the definition of congruence. Students can use any rigid motion to demonstrate all points are congruent.

Student Facing

If \(A\) is a point on the plane and \(B\) is a point on the plane, then \(A\) is congruent to \(B\).

Try to prove this claim by explaining why you can be certain the claim must be true, or try to disprove this claim by explaining why the claim cannot be true. If you can find a counterexample in which the “if” part (hypothesis) is true, but the “then” part (conclusion) is false, you have disproved the claim.

Student Response

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Activity Synthesis

Invite students to share. If students only suggest translation, ask if it is possible to use another transformation. If rotation is not mentioned by students, there is no need to continue to push for that solution.

5.2: What's the Segment? (20 minutes)

CCSS Standards


Addressing

Routines and Materials


Instructional Routines

Required Materials

Activity

Students disprove a conjecture that all segments are congruent, introducing them to proof by contradiction. Then, students work to prove segments of the same length are congruent. This proof leads directly to the triangle congruence proofs in subsequent lessons.

Launch

Display the conjecture and invite students to prove or disprove it.

“If \(AB\) is a segment in the plane and \(CD\) is a segment in the plane, then \(AB\) is congruent to \(CD\).”

Give students 1 minute of quiet work time. Invite them to refer to the instructions in the warm-up for how to prove or disprove an if-then statement.

Select a student who has drawn a counterexample to share. Invite a student to rephrase the conjecture so it will always be true. (If \(AB\) is a segment in the plane and \(CD\) is a segment in the plane with the same length as \(AB\), then \(AB\) is congruent to \(CD\).) 

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof of the conjecture: If two segments have the same length, then they are congruent. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the directed line segment of the translation?”, “What is the center and angle of rotation?”, and “How do you know that these points will coincide?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use precise language to describe the sequence of rigid motions that takes segment \(AB\) to \(CD\).
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch take time to review the following terms from previous lessons that students will need to access for this activity: coincide, translation, reflection, rotation.
Supports accessibility for: Memory; Language

Student Facing

Prove the conjecture: If \(AB\) is a segment in the plane and \(CD\) is a segment in the plane with the same length as \(AB\), then \(AB\) is congruent to \(CD\).

Student Response

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Student Facing

Are you ready for more?

Prove or disprove the following claim: “If \(EF\) is a piece of string in the plane, and \(GH\) is a piece of string in the plane with the same length as \(EF\), then \(EF\) is congruent to \(GH\).”

Student Response

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Anticipated Misconceptions

Students may forget to specify that the points will coincide since the segments are the same length. Remind them of the counterexample in the launch.

Activity Synthesis

Invite students to share pieces of the proof until the whole class agrees that the proof is sufficiently detailed and convincing. Help students determine when they should refer to rays versus segments to solidify the idea that the segments aren‘t congruent until they have used the fact that they are the same length.

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

If two segments have the same length, then they are congruent. (Theorem)

\(AB = CD\) so, \(\overline{AB} \cong \overline{CD}\)

Congruent line segments A B and C D.  

5.3: Zig Then Zag (10 minutes)

CCSS Standards


Addressing

Routines and Materials


Instructional Routines

Required Materials

Activity

In this partner activity, students take turns using their new theorem about congruent segments to prove figures made of congruent segments are congruent. As students trade roles listening and explaining their thinking, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Launch

Arrange students in groups of 2.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students develop the mathematical language of geometric proofs. After students write their proof that figure \(ABCD\) is congruent to figure \(EFGH\), invite them to create a visual display of their proof. Then ask students to quietly circulate and read at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their proofs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to convey the idea that a transformation exists to take a segment to a congruent segment. Also, listen for and amplify the language students use to justify why each of the points in the figures will coincide after the sequence of rigid motions.
Design Principle(s): Cultivate conversation
Representation: Access for Perception. Read the statements about the two zigzags aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language

Student Facing

\(\overline{QR} \cong \overline{XY}, \overline{RS} \cong \overline{YZ}, \angle R \cong \angle Y\)

Figures Q R S and X Y Z, with different orientations. Angles R and Y each have 1 tick mark. Line segments Q R and X Y each have 1 tick mark. Line segments R S and Y Z each have two tick marks.
  1. Here are some statements about 2 zigzags. Put them in order to write a proof about figures \(QRS\) and \(XYZ\).
    • 1: Therefore, figure \(QRS\) is congruent to figure \(XYZ\).
    • 2: \(S'\) must be on ray \(YZ\) since both \(S'\) and \(Z\) are on the same side of \(XY\) and make the same angle with it at \(Y\).
    • 3: Segments \(QR\) and \(XY\) are the same length, so they are congruent. Therefore, there is a rigid motion that takes \(QR\) to \(XY\). Apply that rigid motion to figure \(QRS\).
    • 4: Since points \(S'\) and \(Z\) are the same distance along the same ray from \(Y\), they have to be in the same place.
    • 5: If necessary, reflect the image of figure \(QRS\) across \(XY\) to be sure the image of \(S\), which we will call \(S'\), is on the same side of \(XY\) as \(Z\). 
  2. Take turns with your partner stating steps in the proof that figure \(ABCD\) is congruent to figure \(EFGH\).
Figures A B C D and E F G H

Student Response

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Activity Synthesis

Invite students to write sentence frames for the new transformation they used in their proofs. Add them to the display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. You will add to the display throughout the unit. An example template is provided in the blackline masters for this lesson.

Transformations:

  • Segments \(\underline{\hspace{1in}}\) and \(\underline{\hspace{1in}}\) are the same length, so they are congruent. Therefore, there is a rigid motion that takes \(\underline{\hspace{1in}}\) to \(\underline{\hspace{1in}}\). Apply that rigid motion to \(\underline{\hspace{1in}}\).

Lesson Synthesis

Lesson Synthesis

Display two congruent zigzags with several segments each. Invite students to contribute one statement at a time to prove the zigzags are congruent. Refer them to the sentence frames when they get stuck.

5.4: Cool-down - Circular Logic (5 minutes)

CCSS Standards


Building On

Addressing

Cool-Down

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Student Lesson Summary

Student Facing

If 2 figures are congruent, then there is a sequence of rigid motions that takes one figure onto the other. We can use this fact to prove that any point is congruent to another point. We can also prove segments of the same length are congruent. Finally, we can put together arguments to prove entire figures are congruent.

These statements prove \(ABC\) is congruent to \(XYZ\).

Figures A B C and X Y Z, with different orientations. Angles B and Y each have 1 tick mark. Line segments A B and X Y each have 1 tick mark. Line segments B C and Y Z each have two tick marks.  

  • Segments \(AB\) and \(XY\) are the same length, so they are congruent. Therefore, there is a rigid motion that takes \(AB\) to \(XY\). Apply that rigid motion to figure \(ABC\).

  • If necessary, reflect the image of figure \(ABC\) across \(XY\) to be sure the image of \(C\), which we will call \(C'\), is on the same side of \(XY\) as \(Z\).

  • \(C'\) must be on ray \(YZ\) since both \(C'\) and \(Z\) are on the same side of \(XY\) and make the same angle with it at \(Y\).

  • Since points \(C'\) and \(Z\) are the same distance along the same ray from \(Y\), they have to be in the same place.

  • Therefore, figure \(ABC\) is congruent to figure \(XYZ\).