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.

howardat58

August 21, 2014 at 10:27 am

I did like the rule given by one of them: -Z = W

Clearly looking forward to the study of vectors!

jwilson828

August 21, 2014 at 7:13 pm

Yes! This activity has so much content-wise!

Andrew

August 23, 2014 at 9:01 am

This is a nice idea. I find that my students get comfortable with describe the transformations verbally or with patty paper, but the transition to the more algebraic-looking rules is usually a pretty rocky one, not one that they all make, I’m sorry to say. While we aren’t an Nspire-using school, this idea might get stolen in some form or fashion.

jwilson828

August 23, 2014 at 9:18 am

Good! It is like a puzzle for students – they end up trying to figure it out, whether they think they like to do math or not.