Verifying Transformation Definitions
Lesson 5 of 7
Objective: SWBAT use dynamic geometry software to verify the definitions of transformations
Activating Prior Knowledge
In theory, students have learned definitions of translations, reflections and rotations in terms of line segments, perpendicular lines, circles, arcs, and angles during the previous three lessons: Defining Translations, Defining Reflections, and Defining Rotation. In reality, though, many students have merely been exposed to these definitions. They haven't necessarily internalized the definitions yet. This is another opportunity for students to do some elaborative rehearsal in order to get these definitions into long-term memory.
In the three lessons preceding this one, students crafted and rehearsed elevator speeches summarizing the defining properties of translations, reflections and rotations. In this section of the lesson, I want students to continue rehearsing these definitions so that they will eventually be able to recall them effortlessly.
I use a Think-Pair-Share format as follows:
Think: I tell the students to scan through their memory banks and see how much of their Translations Elevator Speech they can remember without looking at their notes (2 minutes). Then I have them look at their notes to see how well they recalled the speech. At this time, they should also be practicing the correct version of the speech in their minds (2 minutes).
Pair: Next, I have the students partner up and take turns reciting their elevator speech to one another (2 minutes).
Share: Finally, I call on one or two non-volunteers to share their elevator speech with the entire class. This gives me an opportunity to correct, clarify and elaborate as needed.
I repeat this Think-Pair-Share cycle for Reflections and Rotations, then I begin to segue to today's lesson. I say to the students:
The things we said in our elevator speeches are conjectures based on the limited number of examples we've seen so far. Today, we will use the power of dynamic geometry software to create more cases so that we can strengthen our belief in our conjectures. But if we aren't very familiar with our conjectures, we won't know what we're doing when we get to the computer. So that's why it was important for us to take this time to make sure that we are familiar with our conjectures.
For this section, we travel to the computer lab and students get right to work on the Verifying Transformations student resource. My job during this exercise is quality control. I walk around looking at the definitions students have written. By this time, I have given them the definitions at least twice and they have applied the definitions during hands-on activities. Therefore, they should have a record of the definitions as well as a point of reference for what the definitions mean. All this to say that when students have an off-center or incomplete definition, I refer them back to their notes to come up with a better definition.
The second thing I'm checking for is whether students have correctly interpreted the "asymmetrical pentagon" direction. Students often think that pentagons must look a certain way, and that they must be convex. I urge them to play around a little bit to investigate all of the cool variations of pentagons there are. See Assorted Pentagons
As I walk around, I'm also asking students about their processes for verifying the definitions of the respective transformations. This allows me to see if students understand what they are doing, but it is also my way of getting students to pre-write. They will, after all, have to explain their process in writing.
With regard to feedback on their processes, I'm mainly looking to ensure that students have carried out the transformations successfully, that they are measuring the correct things (slopes, angles, distances, etc.) and that they examine multiple cases in order to conclude that there is a general phenomenon at play. I also challenge assumptions that students seem to make such as: the origin is the only possible center of rotation.
Finally, I try to get each student some feedback on their first round of writing so that their second and third attempts will be informed by the feedback. I read their explanations right there on the spot. The motto is: revision until exemplary. If it's not in final product form, no problem, just take it back and work on it some more. Eventually we'll get it right.
As a quick closure activity, I have students write an exit ticket. The prompt is as follows:
1. Explain what you did in this lesson and why you were doing it.
2. Why do you think the choice was made to do this activity using the dynamic geometry software rather than using pencil and paper? Be specific and support your answers with examples. (This gets students to think along the lines of SMP 5: Use tools strategically)