We spent a second day on reflections in our Rigid Motions unit.
We had started by really thinking about what a reflection buys us mathematically, and so we were ready to generalize what happens when we reflect an image over the lines y=x and y=-x and over the lines y=a and x=b.
I used a Numerical Input Quick Poll to let students explore the reflection of a point over the line y=x. The graph side of the page was dynamic. So students could grab and move point R, observe what happens to R’, and then make a conjecture about what would happen if we reflected the point (13,-8) about the line y=x (note that the point was purposefully not within the given window). Students moved to pencil and paper to reflect a triangle about the line y=x and were ready to generalize what happens when we reflect the point (x,y) about the line y=x: (x,y)→(y,x).
When we started thinking about a reflection about the line y=–x, I asked students to predict their generalization before interacting with the TNS document.
One student said that (x,y)→(–y,–x). Did anyone have a different conjecture? Another student wanted to use words instead of symbols for her conjecture: x and y are going to flip and be opposite in sign. We went through the Quick Poll, reflected a triangle about the line using pencil and paper, and generalized the results.
Next up: reflecting a triangle about the line y=1. The goal was to eventually generalize our results for reflecting the point (x,y) about the line y=a. We started on paper. Students could pretty easily get the coordinates for reflecting the triangle about the line, and then they began to look for regularity in repeated reasoning. Can you find a pattern for what happens?
I’m glad that I read Smith & Stein’s “5 Practices for Orchestrating Productive Mathematics Discussions” this summer: anticipate, monitor, select, sequence, connect. I have talked in detail about them in a previous post.
Sequencing was important here. I had some students who generalized the reflection of the point (x,y) about the line y=a. But I had some students who generalized the reflection of the point (x,y) about the line y=1. And I had some students who were making sense of the relationship between the coordinates of the image, the pre-image, and the equation of the line who were thinking more geometrically and quantitatively about distance and were not quite ready to reason abstractly. I didn’t start by calling on the students who had generalized completely. And I might have had I not been deliberately engaged in the 5 Practices. We started with the students who began to articulate what was happening geometrically. The x-coordinates of the pre-image and image are equal. The distance from the pre-image point to the line of reflection has to equal the distance from the line of reflection to the image point.
F.J. shared his work. He began to reason abstractly and quantitatively, using Y for the y-coordinate of the pre-image, L for the y-coordinate of a point on the line of reflection, D for the distance between the pre-image and the line, and then y’ for the y-coordinate of the image. He worked out how to get y’ in two steps.
Then B.E. shared what his group had worked out. They focused on the pattern when reflecting a point about the line y=1. What was happening every time? What is the same going from 5 to –3, from 4 to –2, and from 2 to 0? They tried it going from their pre-image to the image. They tried it going from the image to the pre-image. They generalized that result to be y’=–y+2.
So what happens if we are reflecting a point about a horizontal line that is different from y=1? M.A. wanted to share what her table discussed. They had generalized the result to be (x,y)→(–y,–y+2) for y=1, but upon further reflection decided that (x,y)→(–y,–y+2a) would work for any line y=a.
Did anyone else get that result? R.E. said that their group used the expression 2a–y instead, so we talked about whether those were equivalent. He actually knew they were equivalent, but I think he really wanted me to take a picture of his work to show on the board 🙂
So what did we learn?
I sent two Quick Polls to formatively assess student learning.
The first was a “Drop Points” poll where students placed a point at the image of the point (–4,3) about the line y=–2.
The second was a “Numerical Input” poll where students entered the coordinates of the image of the point (3,–4) about the line y=7. About half of the students used their general rule to get the coordinates of the image, and the others sketched a graph of the point and the line to get the coordinates of the image.
So I’ll be honest. After all of this work, I’m not actually sure it is important for students to know a rule for reflecting a point about the line y=a or x=b. What is important is the work the students did to come up with a rule. Within a minute of being asked to explore a rule, M.A. said that she couldn’t do it. M.A. is not used to having to make sense of problems and persevere in solving them. She usually figures out how to solve problems quickly. She isn’t used to being challenged to think in her mathematics class. The bell rang during our work the first day, and so we left class without a rule (easing the hurry syndrome), taking time to think more about it before reconvening for the next class. M.A. came back to class the next time and announced that she had come up with a rule.
We could take a look back through the Standards for Mathematical Practice and find evidence of most of them in this exploration. Ultimately, that is what is important to me…and not that students remember that they can use (x,y)→(–y,–y+2a) to reflect a point about the line y=a.
And so the journey continues…