Mapping with Reflections

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Objective

SWBAT identify a series of at most three reflections that maps a figure onto a congruent figure.

Big Idea

Three degrees of separation? In this lesson, students learn that any series of rigid transformation can be duplicated with at most three reflections.

Activating Prior Knowledge

20 minutes

Where We've Been: We've just finished a lesson on Mapping and Congruence in which we defined congruent figures as figures that can be mapped onto one another through a series of rigid transformations.

Where We're Going: We'll soon be looking at congruent triangles using the rigid transformation definition to establish triangle congruence criteria.

So in this section, I will develop the concept of mapping a point onto another point using reflection. Students have had experience with reflecting points across lines in previous lessons, so the goal here is just to have them realize what they already know.

The resource for this section is Activating Prior Knowledge: Mapping with Reflections. The students will have to determine the reflection line that maps one point onto another using construction and using coordinate algebra. 

Students may be a little rusty on the rules for performing the perpendicular bisector construction, but that doesn't mean they can't at least figure out that they need to construct the perpendicular bisector. So I keep copies of the instructions for basic constructions (perpendicular bisector, angle bisector, copying a segment, copying an angle) handy so that I can hand students whichever one they think they need.

If students are stuck on the coordinate algebra, which we've done before several times, I assist by suggesting that they draw a diagram first. Then I make sure they know the relevant formulas. Past that, they are left to collaborate with their classmates.

After 15 minutes or so, I review the correct results.

 

Concept Development

15 minutes

In the last lesson, students have just started working on mapping one figure onto another through a series of transformations. In this lesson, we will develop the idea that no matter what, we can use a series of reflections to map a figure onto a figure to which it is congruent.

 

To develop this idea, I perform a demonstration, the basic essence of which is captured in the following screencast.

 

Guided Practice

40 minutes

Once we've illustrated that the end state of any series of rigid transformations can be replicated by a series of at most three reflections, I model for students how to document that series of reflections.

The following screencast will give you an idea of what the demonstration is like:

 

The item in the video is the first item from the GP_Reflection Mapping resource. I model the first item using Sketchpad on the projector, as shown in the video. For the second item, I call on students to give me explicit non-ambiguous directions regarding what step I should take next. I require them to speak in complete sentences using correct vocabulary (MP 6). By the time we reach the third item, I start to select random non-volunteers and we pass the wireless mouse to that person so that they, themselves, can execute the next step. As they execute the step, they must explain what they are doing, again in non-ambiguous terms using complete sentences. See the Pass the Mouse Video Description for more information.

Independent Practice

40 minutes

For this section, we travel to the computer lab so that every student can have extensive practice executing the steps on Sketchpad. Each student receives the Independent Practice_Reflection Mapping resource. There are various correct answers to each item depending on which corresponding vertices students decide to map, and in which order they choose to map them. This is a good way to make sure that the whole class is not copying from one another.

My job during this phase of the lesson is to be a constant presence as students are working at their workstations. By going around to each student and asking them to explain what they are doing, I get a good sense for how well each student is working, who the overachievers are, and who the loafers are.