Exploring Rotations in the Plane Extension Activity

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To begin conceptual development of how rotations are the basis for many other geometry concepts.

Big Idea

Use compound rotations to develop a conceptual understanding of regular polygons, rotational symmetry, interior angle sums, and area.

Warm-up: Lesson Opener

15 minutes

This activity is so long that it can easily take two or three days to complete if you really consolidate the students' thinking by using several mini wrap-ups.  Therefore, the opening is short.  I suggest simply clarifying the learning intentions:  We will be applying rotations in the plane to several additional geometry concepts that you will study further in high school geometry.  


Pass out the activity and point out the directions next to question one - measure all the angles and side lengths of the triangle before beginning the rotations.  I recommend to students that they write these measurements on the diagram itself because this information will be so useful during later questions at the end of the activity. Then allow students to complete all the rotations and additional questions about naming the polygon and deciding if it is a regular polygon.  I like to play inspirational music while my students are working.  I played the song "You Spin Me Round" Chipmunk version, because it is cleaner. While students are working, move about the room formatively assessing and assisting students as they work . A good question to ask as students are working on the fourth or fifth rotation is, "Which parts of the triangle are you rotation as you move the triangle using tracing paper?  Do these parts of the triangle ensure you have rigid motion?"  Students usually figure out the pattern after the first few rotations and will begin to only rotate two sides and the included angle  It is a good time to make a connection to side-angle-side congruence postulate that will be a part of high school geometry.

You probably already know the groups to visit first to ensure students are working productively with a little additional help.  While moving about the room be choosing and conferencing with your expert groups who will present to the class or making your experts groups if none exist without your help.  One key area of focus is the question about regular polygons.  You want to know that your experts will answer this question about what it means to be regular in a productive fashion.  Then during group discussion for the mini wrap-up, you know what information will be shared when you ask students to respond.

Continuing the Activity

30 minutes

After about 12 minutes of work on rotation number one, call the class together and allow a group to place their paper under the camera for all groups to review and check their own work.  Discuss the measurements of the triangle and then review the name of the composite figure.  Spend some time discussing what it means to be a regular polygon and how students know this hexagon is regular.  Some example responses could be that a regular polygon has congruent sides and a rotation is a rigid motion meaning the triangle does not change size or shape as it rotates about the center point (all rotated triangles are congruent).  You could ask what happens to a point when it is on the point the rotation is moving about?  (Answer: it stays in the same location).   

Next, allow students to complete the second composite rotation and series of few questions.  Move about the rooms as before and decide on which experts will present during the mini-wrap up of the second rotation.  When asking students about the composite figure and is it regular, make note of different explanations.  It is good to have more than one perspective presented during the wrap up time.  Stop the class after about 10 minutes of work and allow one to two groups to put work under the document camera and discuss how they rotated the triangles, what type of triangles were being rotated (isosceles), what type of composite figure was formed, and is the figure regular and why.  Now is a key time to discuss why the students have so far been given an equilateral and an isosceles triangle to rotate.  What would happen if the triangle were scalene?  (Answer: the edges would not overlap neatly to form a composite polygon if the left and right sides of the triangle were not the same length).

Wrapping Up the Lesson

5 minutes

The final wrap up is short since you already pulled the class together to wrap up the learning after the hexagon rotations and the pentagon rotations.  You might simply want students to brainstorm around these questions:  What does it mean to be a regular polygon?  What type of motion created regular polygons?  Is there anything special about the triangles used to create regular polygons?  What is a compound figure? If students did not have time to complete the third and final compound rotation, then assing the rotation for homework so students are prepared to begin answering the "thinking" questions that follow the rotations.  I believe it is important to ask students to work on the challenging questions while in class so they have a partner to help them talk and work through the process.