Where We've Been: We've been exploring transformations. Specifically we've been defining them in rigorous terms.
Where We're Going: We'll soon be introducing the concept of mapping a figure onto another. We'll begin exploring this by showing that any isometry can be thought of as the composition of at most three reflections.
In this lesson, students will get to know more intimately what it means to reflect a figure across a line. We'll be doing this without the aid of dynamic geometry software so lots of prior knowledge is needed, both geometry and algebra.
So the APK_Reflecting Across a Line resource is really meant to be a mini-lesson as opposed to a warm-up or do now.
To start off, students have a mini whiteboard and a marker. Rather than giving the students a copy of the handout, I personally prefer to have them see the items on the document camera or projector. That way, I can control the pacing and flow of the exercise.
I begin with some confidence building (problems 1 and 2): Students should be able to find the midpoint of a segment with no problem. But let's be sure. Showing problem 1, I announce, "20 seconds on the clock, markers ready...here's the prompt, go!" After 20 seconds have elapsed, I ask to see all student whiteboards. I scan the boards, give immediate feedback on any errors I catch. Thumbs up for people who got it right. Then I quickly show the process for getting the right answer. Same deal for problem 2.
Now that students have gotten into a groove using the midpoint formula, and they're feeling pretty confident, I throw the curve ball (problem 3). Inevitably, some students simply plug the available coordinates into the midpoint formula. One of the essential understandings for this lessons is that formulas only give outputs that are as good as the inputs we feed into them. Bogus in, bogus out, so to speak. So I use this as a teachable moment vis a vis how to read and understand a problem (MP1). Students to take notes. I start with a visual representation of the problem. I identify my known values. Then I declare my unknown variables. Finally, substituting knowns and unknowns into the midpoint formula, and making use of structure (MP7) I solve for the unknowns. [educreation]
Some students get it the first time around. Some need to have it explained more than once. In any case, I continue explaining until everyone gets it. Now that students think they've got it, I say, "Ok, let's look at another one, just to make sure everyone can do it on their own." Intentionally I've written the next problem (problem 4) to see if they blindly apply the technique we just learned without reading the problem. Sure enough, some students do. Another chance to talk about the importance of reading the problems.
Then we look at problem 5. Hopefully, students have started reading the problems by now. I check student whiteboards to make sure.
Problem 6 is the second problem that has asked us to find the endpoint of a segment given the midpoint and one endpoint. Thinking out loud, I announce, "We might have to do this business of finding an endpoint lots of times this year, and maybe even in the future...It would be good if we had a quick and dirty way to find the coordinates of the endpoints without having to create and solve an equation every single time...that gets old, you know?" So I show how we derive the endpoint formula from the midpoint formula. Students record the process in their notes, and now this is a viable formula for us to use whenever we need it [A good example of MP8]. And I'll see if they know how to use it when I check student whiteboards (giving them only 15-20 seconds) for problems 6 and 7.
I directly model problems 8 and 9 by showing every step and thinking aloud [see educreation]. At best, I have to shake the rust off of the point-slope formula. And for many of the students I teach, they have only understood the formula at a plug and chug level, not conceptually. So I have three take-home lessons about it: 1. It's a tool used to write the equation of a line; 2. It requires the slope of the line as an input; 3. It requires a point...not just any point...but a point that the line actually goes through.
Problem 9 involves a seminal concept about the graphical solution to a system of equations. It also deserves focused attention to its conceptual underpinnings [see educreation].
I start this section by handing each student the GP_Reflecting Across a Line resource. "Ok", I say, "Everyone try number one...use any method you like"
After about 3 minutes, I have students share their methods and answers with their elbow partners. During this time, I walk the room a couple of times to identify some students whose methods I would like to share with the class. So I call these students up to the document camera to explain their process. My job during the share-outs is to make sure that students have explained their processes fully.
The purpose of this first problem was establish that there are some reflections across lines that lend themselves to a purely graphical solution method. We will also see, and this is an essential understanding, that graphical methods sometimes provide only an estimate of a solution and an analytical method is needed.
So we move on to problem 2. Again, I let students loose to try the problem. I give them 5 to 10 minutes depending on the extent to which they maintain their engagement with the problem. Some students will want to stick with a graphical method. This is ok. Other students guess. This is fine, too. Regardless of the method students use, I want an answer from everyone. Once I get everyone's answer written on their whiteboards, I let them know that I have a method for determining the correct answer. So I direct them to put pencils down and watch and listen as I model the process for reflecting the point. The process is:
1. We have the coordinates of A and we want the coordinates of A'.
2. We know that Segment AA' is perpendicular to the reflection line and intersects the reflection line at its own midpoint.
3. I draw a diagram to visually represent the facts mentioned thus far.
4. I find the equation of line AA' using the point-slope formula.
5. I find the midpoint of Segment AA' by finding the point of intersection between line AA' and the reflection line.
6. Now having the coordinates of A and the coordinates of the midpoint of segment AA', I use the endpoint formula we created earlier in the lesson to find the coordinates of A'.
This is a long problem, and my message to students, which I repeat throughout the lesson, is "Math requires sustained focus and concentration."
Once I have the well-organized, worked example on the board or document camera, I give students time to copy it into their notes. I also take the pulse of the room and determine whether students need a slight brain break before starting on problem 3.
Considering an arbitrary point (x,y), and performing algebraic manipulations using it, is new territory for my students at this point, and its an important competency for the analytic geometry learning progression that I have planned in this course. So, being that my students are novices at this point, I show #3 as a worked example. Not to worry, they will have a chance to transfer what they learn in the project that follows this lesson. Here is the process for the worked example:
1. Declare the arbitrary point A (x1, y1), A', and the midpoint of segment AA' (xm, ym)
2. Conclude that line AA' has slope 1/2 and it passes through the point (x1,y1)
3. Use point-slope to find the equation of line AA' in terms of x1 and y1.
4. Find the coordinates of the intersection of line AA' and the reflection line, i.e., the midpoint of segment AA', in terms of x1 and x2.
5. Find the coordinates of A' using the endpoint formula we created in the previous lesson.
Again, taking a pulse on the room, I decide whether I need to explain again, or whether there are certain parts of the process for which I need to do some deeper check for understanding. When it appears we're ready to move on, I give students the opportunity to copy the worked example into their notes.
Then we see if they can use the formula we created to solve problems 4 and 5. We can check to see that the reflection line is the perpendicular bisector of segment JJ' and segment BB' as a way to instill confidence that the formula actually works.
Finally, I give students the directive to write their reflection narratives.