Lesson 11

Defining Reflections

11.1: Which One Doesn’t Belong: Crossing the Line (5 minutes)

Warm-up

This warm-up prompts students to compare four figures. 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. 

Launch

Arrange students in groups of 2–4. Display the figures 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 why a particular item does not belong, and together, find at least one reason each item doesn't belong.

Student Facing

Which one doesn’t belong?

Figure 1

Vertical line l intersects line A B.  Line AB is slanted downward and to the right. The segment between point A and line l is labeled 4, the segment between line l and Point B is labeled 6.

Figure 2

Vertical lines l intersects horizontal line A B. The segment between point A and line l is labeled 4, the segment between line l and point B is labeled 6. 

Figure 3

Vertical line L intersects horizontal line C C prime. On line C C prime, two tick marks labeled 5, on either side of line L.

Figure 4

Line l is the perpendicular bisector of AB, each half of AB is 7.

 

Student Response

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Activity Synthesis

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 “reflection.” Also, press students on unsubstantiated claims.

11.2: Info Gap: What’s the Point: Reflections (20 minutes)

Activity

This is the first info gap activity in the course. See the launch for extended instructions for facilitating this activity successfully.

This info gap activity gives students an opportunity to explore properties of reflections before officially defining them.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Info Gap cards.

Launch

Tell students they will continue to study transformations, only now without a grid.

This is the first time students do the Information Gap Cards instructional routine, so it is important to demonstrate the routine in a whole-class discussion before they do the routine with each other.

Explain the info gap routine: students work with a partner. One partner gets a problem card with a question that doesn’t have enough given information, and the other partner gets a data card with information relevant to the problem card. Students ask each other questions like “What information do you need?” and are expected to explain what they will do with the information. Once the partner with the problem card has enough information to solve the problem, both partners can look at the problem card and solve the problem independently. This graphic illustrates a framework for the routine:

Info Gap Flowchart

Tell students that first, a demonstration will be conducted with the whole class. As a class, they are playing the role of the person with the problem card while you play the role of the person with the data card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question. 

Display an image of the task statement (the collection of points, along with the steps for the person with the problem card and data card) along with the question:

Triangle \(TDG\) has been reflected so that the vertices of the image are labeled points. What is the image of triangle \(TDG\)?

Ask students, “What specific information do you need to find out what the image of the triangle is?” Select students to ask their questions. Respond to each question with, “Why do you need that information?” Once students justify their question, only answer questions if they can be answered using these data.

Data Card

  • The image of \(A\) is \(G\) and the image of \(G\) is \(A\)
  • The image of \(D\) is \(D\)
  • The image of \(N\) is \(N\)
  • The image of \(V\) is \(I\) and the image of \(I\) is \(V\) 
  • The image of \(L\) is \(Q\) and the image of \(Q\) is \(L\) 

Explain that if the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, “I don’t have that information.” Ask students to explain to their partner (you) how they used the information to solve the problem. (Since \(D\) and \(N\) are taken to themselves by the reflection, the line \(ND\) must be the line of reflection. Since the image of \(T\) is a labeled point, the point \(J\) is the only point that makes sense as its image when reflected across \(ND\). The image of triangle \(TDG\) when reflected across line \(ND\) is triangle \(JDA\).)

The fact that the image and original figure overlap might be confusing for students. If not mentioned by students, ask them to reflect each point individually and then look at the final result.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles. Encourage students to annotate their diagram.

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving ________. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity. 
Supports accessibility for: Memory; Organization

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

  1. Silently read the information on your card.
  2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
  3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
  4. Read the problem card, and solve the problem independently.
  5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to answer the question.
  2. Ask your partner for the specific information that you need.
  3. Explain to your partner how you are using the information to solve the problem.
  4. When you have enough information, share the problem card with your partner, and solve the problem independently.
  5. Read the data card, and discuss your reasoning.
14 points on a circle.

Student Response

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Activity Synthesis

The purpose of discussion is to emphasize that the line of reflection seems to be the perpendicular bisector of segments that connect the original figure to the image. 

Display the conjecture that the set of points that are the same distance from two given points is the perpendicular bisector of the segment connecting those two points. Ask, “What should we expect to see if we made segments connecting points to their images?” (We should expect to see that the line of reflection is the perpendicular bisector of all segments that connect points to images.) Ask students to verify this experimentally by drawing segments that connect points to images from their data card and the line of reflection.

Here are some additional questions for discussion. Choose based on time and students’ understanding from the previous lesson:

  • “What kinds of questions were the most useful to ask?” (What is the image of this point? What points do not move?)
  • “Were there any questions you weren't sure how to answer?” (What is the line of reflection? I didn't have that exact information, but I could figure it out from the points that didn't move.) Note that the person with the data card should just be providing information, not making assumptions. But it's okay to be helpful by saying, “I don't have that information; I only have information about the images of points.”
  • “How do you know if a point is on the line of reflection?” (Reflections leave points on the line of reflection fixed where they are.)
  • “What do you notice about points and their images?” (The original point and image are on opposite sides of the line the same distance apart. If one point is taken to another point, then the second point is also taken to the first point.)
  • “How can you test whether a point and its image are the same distance away from the line of reflection?” (If we construct a circle centered at any point on the line of reflection that goes through the original point, it should also go through the image.)

11.3: Triangle in the Mirror (15 minutes)

Activity

In this activity, students use what they know about constructing perpendicular lines to determine where a reflection should take a point. This activity strengthens students’ understanding of reflections without reference to a coordinate grid. Recognizing reflection takes a point to another point the same distance from the line, and that the distance between a point and a line is measured on the perpendicular line, leads to a rigorous definition of reflections that they will use to prove theorems in the next several lessons.

Monitor for strategies students use to determine the image of point \(C\), including:

  • Constructing circles centered around 2 or more points on \(m\) that go through point \(C\). The point \(C'\) will be the common intersection of all these circles because the points on \(m\) are each the same distance from \(C\) as they are from \(C'\).
  • Constructing a line perpendicular to \(m\) going through \(C\), then marking a point that is the same distance away from \(m\) as \(C\) is.

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

Launch

Tell students that they may use the reflect tool to check their answer, but the annotation and instructions should only use straightedge and compass moves.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step.
Supports accessibility for: Organization; Attention

Student Facing

Kiran started reflecting triangle \(CDE\) across line \(m\). So far, he knows the image of \(D\) is \(D’\) and the image of \(E\) is \(E’\).

  1. Annotate the diagram to show how he reflected point \(D\).
  2. Use straightedge and compass moves to determine the location of \(C’. \) Then lightly shade in triangle \(C’D’E’\).
  3. Write a set of instructions for how to reflect any point \(P\) across a given line \(\ell\).
  4. Elena found \(C’\) incorrectly:

    Elena's Diagram

    Triangles C D E and C prime D prime E prime.
    Elena is convinced that triangle \(C’D’E’\) “looks fine.” Explain to Elena why her \(C’\) is not a reflection of point \(C\) across line \(m\).

Student Response

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Launch

If students have access to dynamic geometry software, suggest that it might be a helpful tool in this activity. Ask students to use the GeoGebra Constructions tool, or navigate to this URL: geogebra.org/m/VQ57WNyR. Since the purpose of this activity is to explore properties of reflections, GeoGebra Geometry is not an appropriate tool, while GeoGebra Constructions is.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step.
Supports accessibility for: Organization; Attention

Student Facing

Kiran started reflecting triangle \(CDE\) across line \(m\). So far, he knows the image of \(D\) is \(D’\) and the image of \(E\) is \(E’\).

  1. Annotate Kiran's diagram to show how he reflected point \(D\).
  2. Use straightedge and compass moves to determine the location of \(C’. \) Then lightly shade in triangle \(C’D’E’\).
  3. Write a set of instructions for how to reflect any point \(P\) across a given line \(\ell\).
  4. Elena found \(C’\) incorrectly. Elena is convinced that triangle \(C’D’E’\) “looks fine.” Explain to Elena why her \(C’\) is not a reflection of point \(C\) across line \(m\).

Kiran's Diagram

Triangle C D E to the left of line m. On the right of line M, points E prime and D prime. Points E and E prime equidistant to line m. D and D prime are equidistant to line m.

Elena's Diagram

Triangles C D E and C prime D prime E prime.

Student Response

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Student Facing

Are you ready for more?

  1. Using your response from question 2 (with the correct location of \(C’\)):
    1. Draw the line \(CC’\).
    2. Reflect triangle \(C’D’E’\) across line \(CC'\).
    3. Label the image \(C’’D’’E’’\).
  2. Find a single rigid motion that takes \(CDE\) to \(C’’D’’E’’\).

Student Response

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Anticipated Misconceptions

If a student has trouble getting started, suggest connecting \(E\) to \(E’\) and asking what they notice about the distances to the line. If they are still stuck, ask them to mark some point on the line of reflection and think about the distance from that point to \(E\) and \(E’\).

Activity Synthesis

The important idea for discussion is that line \(m\) is the perpendicular bisector of the segments connecting each point in the original figure to its image. Remind students that the distance between a point and a line is the perpendicular distance, so for a reflection to give points the same distance away, they need to use a perpendicular.

Ask students to share their strategies for locating \(C’\). If not mentioned by students, discuss the strategy of constructing a line perpendicular to \(m\) going through \(C’\) and marking the point that is the same distance away from \(m\) as \(C\).

Lesson Synthesis

Lesson Synthesis

Explain to students that they have used tools to explore reflections, but to be able to prove whether the conjectures they have made are true, they need to have a precise definition of reflection. This definition will help them explain why a reflection guarantees one point will be taken onto another. Ask students to add this definition to their reference chart as you add it to the class reference chart:

Reflection is a rigid transformation that takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line.

Reflect _(object)_ across line _(name)_.

Reflect $A$ across line $m$.

Line m intersects segment A A prime at a right angle and tick marks indicate congruent halves.

11.4: Cool-down - What Went Wrong? Reflection (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Think about reflecting the point \(A\) across line \(\ell\):

Line L, facing downward and to the right. Point A sits to the left of the line.

The image \(A’\) is somewhere on the other side of \(\ell\) from \(A\). The line \(\ell\) is the boundary between all the points that are closer to \(A\) and all the points that are closer to \(A’\). In other words, \(\ell\) is the set of points that are the same distance from \(A\) as from \(A’\). In a previous lesson, we conjectured that a set of points that are the same distance from \(A\) as from \(A’\) is the perpendicular bisector of the segment \(AA’\). Using a construction technique from a previous lesson, we can construct a line perpendicular to \(\ell\) that goes through \(A\):

Line L, facing downward and to the right. A perpendicular line passes through Line L and Point A.

\(A’\) lies on this new line at the same distance from \(\ell\) as \(A\):

Perpendicular lines and a circle

We define the reflection across line \(\ell\) as a transformation that takes each point \(A\) to a point \(A’\) as follows: \(A’\) lies on the line through \(A\) that is perpendicular to \(\ell\), is on the other side of \(\ell\), and is the same distance from \(\ell\) as \(A\). If \(A\) happens to be on line \(\ell\), then \(A\) and \(A’\) are both at the same location (they are both a distance of zero from line \(\ell\)).

Line l is the perpendicular bisector of AA’.