5 Practices for Orchestrating Productive Mathematics Discussions might be the book that has made me most think about and change my practice for the better in the past 10 years.

At the beginning of our second day on dilations, I asked students to work on this.

Because of the 5 Practices, I pay attention differently when I walk around and monitor students working. I know that I looked for different student approaches before I read the book, but I didn’t consciously think about selecting and sequencing them for a whole class discussion. I often asked for volunteers. And then hoped that another student would volunteer when I asked who worked it differently [who had actually worked it differently and correctly].

I asked a few questions of students while I was monitoring them to clarify what they were doing and selected and sequenced a few to share. The student work above looks similar at first glance, but there are subtle differences in their thinking that make important connections about dilations.

TM shared first. She used slope to find the vertices of the image. She went down 1 and to the right 3 from C to X, and then because of the scale factor of 2 went down 1 and to the right 3 from X to get to X’. She went down 3 and to the right 2 to get from C to Z, and then went down 3 and to the right 2 from Z to get to Z’.

JA shared next. He focused on the line that contains the center of dilation, image, and pre-image. He knew that X’ would lie on line CX and that Z’ would lie on line CZ.

MB shared next. He also used slope, but a bit differently from TM. He noticed “down 1 and to the right 3” to get from C to X and so because of the scale factor of 2 then did “down 2 and to the right 6” from C to get to X. He noticed “down 3 and to the right 2” to get from C to Z and so then did “down 6 and to the right 4” to get from C to Z’.

I had not seen additional methods while monitoring. This exercise didn’t take too long, and so I didn’t get around to everyone. [This is where Smith & Stein’s advice about keeping a clipboard to pay closer attention to whom you check in with and whom you call on helps so that you aren’t checking in with and calling on the same few every time you have a whole class discussion.] I hesitated before I asked, but I did then ask, “did anyone find X’Y’Z’ a different way?” [This is also where I am learning to trust my students to recognize when their method is different.] TC raised his hand. I treated C as the origin and used coordinates. He shared his work and showed that the coordinates of X (3, -1) transformed to X’ (6,-2) with a dilation about the origin for a scale factor of 2.

And so the journey continues, thankful for friends like Gail Burrill [one of my voices] who recommend authors like Smith and Stein to help me think about and change my practice for the better, making me feel like a conductor rehearsing for a beautiful, exciting mathematics masterpiece …

howardat58

December 21, 2016 at 10:13 am

Here is another one, as when I initially thought about dilations when I was a kid it seemed that when a hole was made in a sheet of metal the hole got smaller after a dilation.

Will they think it’s true or false, and with reasons ?

howardat58

December 21, 2016 at 10:16 am

And another one.

Dilate a circle…..(pick a centre of dilation, not the centre of the circle)

Travis M Bower

December 21, 2016 at 10:24 am

While I/we constantly proclaim multiple representations, I often forget about the sub-mult. rep with Geometry view and Graph (coord.) view. The contributor and the rest of the class love it when another method is presented–one they understand, but did not consider in the time allotment….and the teacher, too.

Trever Reeh

January 4, 2017 at 2:40 pm

Love this activity, I will definitely use this in a week when we cover dilations. Thanks!!

jwilson828

January 4, 2017 at 6:15 pm

Great! I’m glad it is helpful & would love to know if your students use any different approaches!