What’s My Rule

Instead of having a whole lesson of What’s My Rule explorations, we are adding one exploration to each bellringer during our unit on Rigid Motions. From yesterday:

Students move point Z and observe how W follows. Z is mapped to W according to some rule that the students are trying to determine

I’ve written about this exploration before, so I want to focus on what was different this year.

Students constructed viable arguments and critiqued the reasoning of others. We are learning how to attend to precision, so we were lenient in giving credit to responses for which the oral explanation helped us make sense of the written explanation.

One team wrote that if you added something to Z and subtracted something from W, then the points would map onto one another. I wouldn’t have worded what they were trying to say like they did. But they were getting at some important mathematics. Ultimately, they were trying to convey that Z and W are the same distance from the origin. We constructed a circle with the origin as the center and Z as one of the points on the circle and noticed that both Z and W always lie on the circle.

A few other students said that the rule was to reflect Z over the line y=x to get W. Does that always work? We looked back and decided it wasn’t always true. When is it true? When does (x,y)→(-x,-y) also represent a reflection about the line y=x?

Others thought that the rule was to reflect Z over the line y=-x to get W. Does that always work? We looked back and decided it wasn’t always true. When is it true? When does (x,y)→(-x,-y) also represent a reflection about the line y=-x?

Other students noticed that we could describe the rule using a rotation of Z 180˚ about the origin.

No one noticed that we could reflect Z about the x-axis and then about the y-axis. So what happens when no one notices something we want them to notice? I could have moved on. It wouldn’t have been detrimental to my students learning of mathematics if they didn’t know that. But I didn’t. Instead I asked whether there was a reflection that we could use to map Z onto W. I gave students just seconds to think alone and then time to talk with their teams. I monitored their team talk. 5 teams said that we could reflect Z about the perpendicular bisector of segment ZW to map Z onto W. Yes. Not what I was expecting … but absolutely true.

One team said that we could reflect Z about the line y=x and then about the line y=-x to map Z onto W. Oh…we can reflect Z about y=x and then y=-x? How can you show that?

What happens when you reflect (x,y) about y=x? (y,x)

What happens when you reflect (y,x) about y=-x? (-x,-y)

Is there another sequence of reflections that will map Z onto W?

Reflecting about y=-x and then y=x.

Is there another sequence of reflections that will map Z onto W?

Teams worked together – and after another few minutes, they figure out that reflecting about y=0 and then x=0 would work. Or reflecting about x=0 and then y=0.

And then we were called to the cafeteria for school pictures.

And then a student came up to me in the line for school pictures and asked whether there would be an infinite number of pairs of lines about which we could reflect Z onto W.

Are there an infinite number of pairs of lines that will work?

What relationship do the pairs of lines have that we found?

y=x and y=-x; y=0 and x=0

What is significant about the pairs of lines?

After a few more questions, the students around us in line for pictures noted that the lines are perpendicular.

So if we reflected Z about y=2x, then about what other line would we need to reflect Z’ to get W?

I will be the first to admit both that of course all of this makes sense mathematically, and also that I’ve never thought about it before. And so the journey continues … ever grateful for the students with whom I learn.

 

Thanks to Michael Pershan for sharing Transformation Rules.

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