Lesson 14

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. 

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.

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.

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:

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.

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.

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).

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.

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.