Lesson 11

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. 

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.

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.

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:

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:

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.

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.)

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).

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 \(E’\).

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 \(C\).