Lesson 14

Defining Rotations

14.1: Math Talk: Comparing Angles (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for using tools to compare angles. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to compare or accurately reproduce angles to draw rotated figures. 

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion. It is okay if you don’t get through all 4 problems.

Student Facing

For each figure, which pair of angles appears congruent? How could you check?

Figure 1

3 angles. Angle A B C opens to the right, angles D E F and G H L open up.

Figure 2

3 angles. Angles M Z Y and P B K open up, angle R S L opens to the right.

Figure 3

Identical circles. Circle V with central angle GVD opens to the right, circle J with central angle LJX  opens to the left and circle N with central angle CNE opens up.

Figure 4

A figure of 3 circles. H. B. E.

Student Response

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Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate  \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to  \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

Note that measuring angles is not the same as proving they are congruent, due to the limitation of measuring tools. Make sure all students are comfortable using a protractor to measure angles.

If not mentioned by students, there is no need to bring up arcs, chords, or congruent triangles. If students measure chords or arc lengths to compare angles, emphasize that the circles used to define the arcs or chords must be the same size.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

14.2: Info Gap: What’s the Point: Rotations (15 minutes)

Activity

This info gap activity gives students an opportunity to determine and request the information needed to define a rotation.

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).

Look for students who do not specify the angle of rotation, the direction of rotation, or the center of rotation to compare to correct rotations during discussion.

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

Problem Card 1

Triangle \(DCI\) has been rotated so that the vertices of its image are labeled points. What is its image?

Data Card 1

  • The angle of rotation is \(HLI\).
  • The rotation is counterclockwise.
  • The center of rotation is \(L\).
  • The image of \(C\) is \(G\).
  • The image of \(I\) is \(H\).
  • The image of \(L\) is \(L\).
  • The image of \(U\) is \(C\).
  • The image of \(V\) is \(D\).

Problem Card 2

Several points have been rotated around a labeled point. Precisely describe the rotation.

Data Card 2

  • The angle of rotation is \(HNW\).
  • The center of rotation is \(N\).
  • The image of \(F\) is \(H\).
  • The image of \(H\) is \(W\).
  • The image of \(J\) is \(B\).
  • The image of \(N\) is \(N\).

Launch

Conversing: This activity uses MLR4 Information Gap to give students an opportunity to determine and request the information needed to define a rotation. 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.
Several points: B, C, D, F, G, H, I, J, L, N, U, V, W.

Student Response

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Anticipated Misconceptions

Students may need to be reminded of the tools in their geometry toolkits, such as tracing paper, straightedges, and compasses.

Students may try to estimate the measure of the angle of rotation. Remind them they can name an angle with 3 letters.

Activity Synthesis

The purpose for discussion is to emphasize that an angle, a direction, and a center of rotation are all needed to precisely describe a rotation. If there were any responses that used the wrong center of rotation or direction of rotation, display these next to correct responses. Without revealing who is correct, ask students from these groups to reconcile these differences.

14.3: Turning into Triangles (15 minutes)

Activity

The goal of this activity is to practice applying the definition of rotation. In this activity, students rotate a segment around its endpoint to create an isosceles triangle. By rotating a segment around one endpoint, students can observe that the distance between the center of rotation and a point is the same as the distance between the center of rotation and the image of that point. This observation is later captured in the formal definition of rotation.

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

Launch

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to aide 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

  1. Draw a segment. Label the endpoints \(A\) and \(B\).
    1. Rotate segment \(AB\) clockwise around center \(B\) by 90 degrees. Label the new endpoint \(A’\).
    2. Use the Polygon tool to draw triangle \(ABA'\).
    3. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
  2. Draw a segment. Label the endpoints \(C\) and \(D\).
    1. Rotate segment \(CD\) counterclockwise around center \(D\) by 30 degrees. Label the new endpoint \(C’\).
    2. Rotate segment \(C’D\) counterclockwise around center \(D\) by 30 degrees. Label the new endpoint \(C’’\).
    3. Use the Polygon tool to draw triangle \(CDC’’\).
    4. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.

Student Response

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Student Facing

Are you ready for more?

You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a regular polygon: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.

Student Response

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Launch

Briefly demonstrate how to create an angle of a given measure with a protractor and ask students to use their protractors along with you.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5). You may wish to demonstrate using GeoGebra Geometry in Math Tools.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to aide 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

  1. Draw a segment. Label the endpoints \(A\) and \(B\).
    1. Rotate segment \(AB\) clockwise around center \(B\) by 90 degrees. Label the new endpoint \(A’\).
    2. Connect \(A\) to \(A’\) and lightly shade in the resulting triangle.
    3. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
  2. Draw a segment. Label the endpoints \(C\) and \(D\).
    1. Rotate segment \(CD\) counterclockwise around center \(D\) by 30 degrees. Label the new endpoint \(C’\).
    2. Rotate segment \(C’D\) counterclockwise around center \(D\) by 30 degrees. Label the new endpoint \(C’’\).
    3. Connect \(C\) to \(C’’\) and lightly shade in the resulting triangle.
    4. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.

Student Response

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Student Facing

Are you ready for more?

You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a regular polygon: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.

Student Response

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Anticipated Misconceptions

Students may not notice any properties other than that rotation preserves distances to the center of rotation. Encourage these students to use tracing paper to make other conjectures about side lengths or angles.

Activity Synthesis

Display an isosceles triangle \(ABC\) with base \(BC\).

Here are some key conjectures and observations students should come away from the discussion with: 

  • Rotations preserve the distance to the center of rotation.
  • The angle bisector of angle \(A\) is also the perpendicular bisector of the base (\(BC\)) of an isosceles triangle.
  • The two base angles (angle \(B\) and angle \(C\)) of an isosceles triangle are congruent.
  • An isosceles triangle where angle \(A\) is 60 degrees is also equilateral.

It is not necessary at this time to formalize the language of these ideas. This will come later as students prove some of these statements in a subsequent unit.

Lesson Synthesis

Lesson Synthesis

Explain to students that they have developed conjectures involving rotations, but to be able to prove whether the conjectures they have made are true, they need to agree on a definition of rotation. This definition will help them explain why a rotation guarantees one point will be taken onto another.

Remind students that to describe a rotation, they need to specify a center, an angle, and a direction (clockwise or counterclockwise). Ask students to add the definition to their reference charts as you add it to the class reference chart:

Rotation is a rigid transformation that takes a point to another point on the circle through the original point with the given center. The two radii to the original point and the image make the given angle.

Rotate _(object)_ (clockwise or counterclockwise) by _(angle or angle measure)_ using center _(point)_ .

(Definition)

Rotate $P$ counterclockwise by $a^\circ$ using center $C$.

Rotation of point P.

The angle \(PCP'\) measures \(a^{\circ}\) and \(P'\) is counterclockwise around the circle from \(P\). If the direction were clockwise instead, then \(P'\) would be clockwise around the circle from \(P\). Notice that if \(P=C\), then the rotation sends \(P\) to \(P'\) on the circle of radius zero, and so points \(P\), \(C\), and \(P'\) are in the same place before and after the rotation.

Ask students, “This definition mentions a circle but there’s actually no circle in the picture. What about the picture indicates that a circle is involved?” (The distances \(CP\) and \(CP’\) are equal, so each is a radius for a circle centered at \(C\).)

14.4: Cool-down - What Went Wrong? Rotation (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

A rotation is a transformation with a center, an angle, and a direction (clockwise or counterclockwise).

Here is how a rotation with a center point \(C\), an angle that measures \(t\) degrees, and a counterclockwise direction transforms a point \(P\):

  • The rotation sends point \(P\) to a point \(P'\) on the circle of radius \(CP\).
  • The angle \(PCP'\) measures \(t\) degrees and \(P'\) is counterclockwise around the circle from \(P\).
Two images of angle t degrees. First shows center C, and t degrees angle, congruent line segments PC and P’C and a counterclockwise rotation taking P to P’. Second shows angle t degrees.

If the direction were clockwise instead, then \(P'\) would be clockwise around the circle of radius \(CP\). If \(P\) and \(C\) are in the same place, then the rotation sends \(P\) to \(P'\) on the circle of radius zero, and so points \(P\), \(C\), and \(P'\) are all in the same place.