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Unit 1, Lesson 14

Defining Rotations

Geometry

14.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Geometry toolkits (HS)Protractors

Activity Narrative

The purpose of this Warm-up is to elicit strategies and understandings students have for using tools to compare angles. These understandings 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

Provide access to protractors and geometry toolkits.

Activity

None

Which pairs of angles appear congruent? How could you check?

<p>6 angles. Angles A B C and R S L open to the right, angles D E F, G H L, M Z Y, and P B K open up.</p>

Student Response

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Building on Student Thinking

Activity Synthesis

Ask students to share their strategies. Record and display their responses for all to see.

Note that because of the limitation of measuring tools, measuring angles is not the same as proving they are congruent. Make sure all students are comfortable using points to name angles.

14.2

Activity

Instructional Routines

MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

What's the Point Rotations Cards

To Gather

Geometry toolkits (HS)

Activity Narrative

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

The Information 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).

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

14.3

Activity

Optional

Instructional Routines

Draw It

Materials

To Gather

Protractors

Activity Narrative

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).

Lesson Synthesis

Explain to students that they have developed conjectures involving rotations, but to be able to prove whether the conjectures that 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). Add the following definition to the class reference chart, and ask students to add it to their reference charts.

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, the one from the center to the original point and the one from the center to the image, make the given angle.

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

(Definition)

<p>Rotation of point P.</p> — Rotate counterclockwise by using center .

The angle measures , and is counterclockwise around the circle from . If the direction were clockwise instead, then would be clockwise around the circle from . Notice that if , then the rotation sends to on the circle of radius zero, so points , , and 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 and are equal, so each is a radius for a circle centered at .)

to access Cool-down.

Student Lesson Summary

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 , an angle that measures , and a counterclockwise direction transforms a point :

The rotation sends point to a point on the circle with a radius of length .
The angle measures and 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.</p>

If the direction were clockwise instead, then would be clockwise around the circle of radius . If and are in the same place, then the rotation sends to on the circle with a radius of 0 units, so points , , and are all in the same place.

Launch

Tell students they will continue to study rotations. Display the Information Gap graphic that illustrates a framework for the routine for all to see.

Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it. There is an extra set of cards available for demonstration purposes.

Arrange students in groups of 2 or 4. If students are new to the Information Gap routine, allowing them to work in groups of 2 for each role supports communication and understanding. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem and instruct them to switch roles.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity, or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

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 problem card:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”
Continue to ask questions until you have enough information to solve the problem.
When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read the information on your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

<p>Several points: B, C, D, F, G, H, I, J, K, L, M, N, U, V, W.</p>

Student Response

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Building on Student Thinking

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.

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Audience

Assessment Access

Lesson 14

Defining Rotations

Warm-up

Standards Alignment

Building On

Addressing

HSN-Q.A.3

Building Toward

HSG-CO.A.4

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

8.G.A.1

Addressing

HSG-CO.A.4

Building Toward

Standards Alignment

Building On

8.G.A.1

Addressing

HSG-CO.A.4

Building Toward