Geometry : Unit #4 - Defining Transformations : Lesson #4

Defining Rotation

Objective: SWBAT explain that in rotations, all points are rotated through congruent angles with congruent radii.
Standards: HSG-CO.A.4 MP6 MP8
Subject(s): Math
60 minutes
1 Activating Prior Knowledge - 15 minutes

Where We've Been: Students have been doing activities similar to the one for todays lesson. So far, we've defined translations in terms of congruent line segments with the same slope, and we've defined reflections in terms of segments and their perpendicular bisector.

Where We're Going: Students will be using the definitions we've developed to verify that a particular transformation has occurred or to solve problems related to transformations.


Since we have been doing this type of activity for the past couple of lessons, students should be ready to jump right in at the beginning of class. I do need to make sure that they haven't forgotten how to use a protractor to measure and create angles before we start, though. So the task for this section of the lesson is:

 Use a protractor to create a 72 degree angle. Then exchange papers with a  neighbor so that they can measure your angle and verify that it measures 72 degrees. Then exchange again with a new person and do the same thing. (MP6)



2 Self-Guided Activity - 45 minutes

To begin, I pass out the Defining Rotations Activity resource and students get right to work on it. The task is intended to be self-guided. My job is to coach and facilitate.

A couple of foreseeable issues:

  1. When students measure their angles they should draw the rays of the angles lightly, preferably in four different colors, so that the graphs don't get overly convoluted.
  2. Regarding item #4, some students may not come up with the formula right away. This is ok...they can move on to #5 and find the distances using the distance formula if they like. Hopefully, after a few times doing it this way, these students will start to recognize some regularity in their calculations and see that there is an easier way. Then they can come back and complete #4. This is a good time to have a conversation about MP8 as I explain in the video.
Targeted praise on exemplars communicates expectations

I choose student exemplars, partly because I want to show the class correct answers. But beyond that, I show student exemplars because students are more likely to believe that they can achieve things that they see their peers have been able to achieve. 

So it is important to deal with these opportunities to inspire students to aspire to the level of their peers very intentionally. I try to explain one aspect of this in the video that follows.



3 Closure - 15 minutes

Once students have completed the activity I’ll want to make sure they have the important concepts as I intended. In this section of the lesson I recap while observing student body language and class conversation informally.

First, we discuss how we know that the transformation in this activity was a rotation.

Next I explain (and write on the whiteboard) that in order to define a particular rotation precisely, we need to specify three things:

1. The center of rotation

2. The angle of rotation

3. The direction of rotation (by convention, positive angles of rotation indicate counterclockwise rotation and negative angles indicate clockwise)


Having established that, I speak precisely about the rotation that took place in the activity:

It was a 90 degree rotation counterclockwise about the origin. I point out that if anyone had noticed, we could have determined this just by noticing the rule (x,y) ---> (-y,x) [which students have learned in an earlier lesson on transformations as functions]

Directing students' attention to item #3, I point out how every angle formed by a point on the pre-image, the center of rotation as vertex, and the corresponding point on the image was 90 degrees. That means that every point on the pre-image got rotated 90 degrees.

Moving our attention to item #5, I point out how we found that the distance from any point on the pre-image to the center of rotation was the same as the distance between its corresponding point on the image and the origin. I emphasize (and write on the whiteboard) how this means that after a rotation, all points will still be their original distance from the center of rotation.

Finally I explain how in a rotation each point on the pre-image travels on a path called an arc, which is part of a circle's circumference. How much of the circle's circumference the point travels is determined by the angle of rotation (360 degrees is a full circle so the 90 degree rotation in this example was a quarter of a circle.) and the radius of the circle is the distance between the original point and the center of rotation.

I check for understanding by asking "Do all points in a rotation travel the same distance? Explain." I have students do a Think-Pair-Share to answer this question. When we share out after the Think-Pair-Share, I make sure that everyone goes away knowing (and writing in their notes) that all points move through the same angle but do not travel the same distance. Points originally farther away from the center of rotation travel along bigger circles (therefore farther), and points that are originally closer to the center of rotation travel along smaller circles. As a final check, I ask for volunteers to answer the question "What do you suppose would happen to a point that was originally on the center of rotation?"

After the check for understanding, if at least 85% of the students seem to "get it", it's time to put the concepts into action by rotating a figure around a specific center of rotation using a specified angle of rotation. For that, I use the Define Rotations Independent Practice Task, which I explain and demonstrate in the Independent Task video.