Where We've Been: Students have been introduced to rigid transformations. We have described translations as slides, rotations as turns, and reflections as flips. We've also introduced important terminology like vector, center and angle of rotation, and line of reflection.
Where We're Going: Now we're taking a second pass at rigid transformations, this time defining them more rigorously.
In this particular lesson, students will need to use the midpoint and slope formulas correctly. They will also need to recall what a reflection is, write equations of lines, and recall the slope criteria for perpendicular lines.
To get students primed for the lesson, I use the APK_Define Reflection resource. I give students five minutes to work on it independently. When the five minutes have elapsed, I go over all items in detail. My goal as I do this is to clarify concepts and model precise language usage and work documentation.
At the outset of this section of the lesson, I create test-like conditions. I let the students know that this is an independent activity. I prompt them to "get into the zone". I tell them we are practicing a very important habit of mind called "sustained focus and concentration" which is essential for complex problem solving (MP1)
I provide each student with a ruler, a sheet of grid paper, a calculator, and the Define Reflections student resource. Once I've done that, my job is to maintain order in the room and to make sure students are reading and following the directions.
When students are confused, I try my best not to think for them, but rather to ask them open-ended questions that will move them forward. For example, students often get stuck on item #4, which asks them to observe something about the collection of midpoints of the segments joining corresponding vertices on the pre-image and image. My first question to students is "What collection of midpoints is being referred to?" Next, "Have a look at that collection of points and tell me what about the geometry of this collection of points seems significant or worth noting?" When students say, "They all form a straight line" I say, "Isn't there a geometric term for that?"...."Oh! they're COLLINEAR!" [mission accomplished].
About half-way through the lesson, I pose the question to the class, "So what type of transformation do you guys say this is?" Students tend to say rotation and/or refection. I don't give the answer. Instead I call on a student to argue that it is a rotation. Then I call on a student to argue that it is a reflection. Next I ask each of these students if, after hearing the other argument, they want to change their minds. If this doesn't resolve it, I have the students critique the other side's argument and explain why it cannot be true (MP3). This class discussion tends to get students talking and collaborating. I allow this transition to occur since students have had ample time to generate their own independent ideas about the lesson.
In terms of managing the logistics of the exercise, it helps to have fine-tip colored markers on hand so that students can spruce up their final products and, more importantly, make their preimage and image distinct from the segments connecting the corresponding vertices.
Students tend to make at least one mistake when calculating slopes and midpoints. The nice thing about this exercise is that they have the graph to give them feedback on the correctness of their answers. For example, in step 4, when students calculate and plot the midpoints, they should be able to recognize when the plotted point does not appear to be the midpoint of the segment. Similarly, when calculating slopes, if all but one of them come out to be -2/3, I ask, "Based on the graph does it appear that one of the slopes is different than all the rest?"
For completed work samples, see Student Work_Define Reflection.
The purpose of this section is to make sure that everyone gets the important take-aways from the lesson. I choose to collect the papers before this section of the lesson to prevent last minute copying.
The activity requires students to do a TON of work (or so they think). Still, even if they have done all of the work, it is still likely that they don't fully understand the significance of what they've done. So there are some concepts that I want to make sure to highlight during this closure section.
First, the idea that whenever we have a reflection, the points on the reflected image are the same distance from the line of reflection as the points on the preimage. We know this because the line of reflection went through the midpoint of each segment that joined corresponding points on the preimage and image.
Next the idea that the line of reflection is actually the perpendicular bisector of all the segments joining corresponding points on the preimage and image. Knowing this would allow us to figure out the coordinates of any figure reflected over any line. It would also allow us to determine the equation of the line of reflection as long as we have at least one ordered pair on the preimage and its corresponding point on the image. Students will have experience with this in the follow-up to this lesson.
For more talking points on the activity, see Define Reflections_Teacher Notes.
As an at-home assignment, I have students craft their reflections elevator speech. For a video example see the Sample Reflections Elevator Speech.