SWBAT draw and identify images of figures over composites of two reflections.

Students discover that consecutive reflections results in a translated or a rotated figure.

12 minutes

Before students enter, I project the Model Apt (Launch) image on the SmartBoard. Once all the students are settled in, I address my learners, and point out that they are looking at the floor plan of a new apartment complex being built. Since students may not be familiar with floor plans, I tell the class that engineers make a floor plan of the structure being built. I point out that they are viewing an entire floor as if it had no ceiling and we were looking from above. Then, I state:

**Every apartment will be the mirror image of the one next door and the one across the hall. Note that 27 apartments make up the complex but only one has been completed (top left). It is the model apartment that will be shown to prospective buyers. How many apartments will have the same orientation and rigid position as the model apartment?**

I don't rush to call on any student who may quickly put his hand up, in order to give everyone time to think of what is being asked and figure out the answer. I ask if any student wants me to restate the question and I do. Not everyone can listen and pick up what is being asked with the same ease, so I always try to repeat oral questions or statements and make sure everyone understands what is being said.

**Teacher note**: The correct answer is 9 apartments.

10 minutes

Today, I use the Launch task to transition directly into the presentation of new information. Once a student comes up with the correct answers to the Launch Problem, I ask how he/she obtained their answer. It is typical for a student to explain the answer in the following way:

In the first row, the model apartment repeats at every other image from left to right and that this occurs in the 3rd row as well.

I keep probing students to tell me what they see until I get someone to say that the model apartment occurs every 2 reflections and that this is like a translation of the model.

I always ask where else in the real world have they seen something like these repeated reflections. Common responses are:

- Barber shop or beauty parlor mirrors
- Kaleidoscopes
- Carpet or rug design patterns.

I then tell the class to take notes while I write on the board:

When a transformation is followed by another transformation, like in the apartment floor plan where a reflection followed another, we say that the transformations have been composed. The result of the consecutive transformations is called the composite.

In the apartment building floor plan, what kind of transformation maps the first image onto the composite image or final image?

I then ask, "Will this always be the case?" The remainder of today's lesson focuses on this question. We will focus on this in tomorrow's activities as well.

25 minutes

I tell the class that in this activity we will construct composite reflections of a figure ourselves. Through this activity we will discover if and when we get a result that can be described as a translation.

Then, I hand each student a copy of the Composite Reflection Task, a piece of tracing paper, and a ruler. I ask that they use only pencil for the task. Working along with their elbow partners is fine as long as each student performs the task themselves. See Composite Reflection Ans for an image of what a student's finished work should look like.

See the Composite reflections.mp4 video on construction of composite reflections over parallel lines.

10 minutes

After students have completed their drawing task I ask the class what they understand the task's purpose to be. I expect students to say that in order for the final image to be a translation of the first, the composite reflections should be over two parallel lines. I ask the class to identify the real function of the reflecting lines in a floor plan diagram. I want to make sure that my students think withing the context and recognize that the role that the corridors are playing in the activity.

The Exit Slip is to be done with their partners and discussed as a pair. I will ask the students to hand me their Exit Slip before they leave the room. I like to give students 10 minutes at the end of class to think and to answer the questions.

I will check each Exit Slip before the next day and begin the next lesson with a discussion of the questions.

Almost all students quickly become proficient at drawing reflection images. Some will probably recognize and ask if reflections can be performed using technology. I believe that we should encourage this. In the next lesson I will leverage technology to help students see the composite reflections over parallel and intersecting lines.

A natural extension is to draw images of images and this can easily be done with sketchpad or other program. As an extension, students can use sketchpad to reflect a scalene triangle, or any other figure, over parallel lines.

Teachers sometimes ask students to erase the intermediate image in composition of transformations. I'd ask students to do so when they see that the images overlap too much and erasing the middle image helps. Otherwise, I find it unnecessary. In the second questions of tonight's Homework, I make erasing the intermediate image optional. For this assignment, I ask students to use rulers and pencils only, especially for question one, where they may have to measure. Like with all hands-on construction, precision is important.