Hinge Questions: Dilations

Students noticed and noted.

I wanted to be sure that they could answer a dilations question based on their observations. I had two questions premade in my set of Quick Polls. Which question would you ask?

In the past, I would have asked both questions without thinking.

I am learning, though, to think more about which questions I ask. If we only have time to ask a few questions, which questions are worth asking?

From slide 34 in Dylan Wiliam’s presentation at the SSAT 18^th National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

I decided to send the second poll. I decided that if they get that one right, they can both dilate a point about the origin and pay attention to whether they are given the image or pre-image. If I had sent the second poll, I wouldn’t know whether they could both do and undo a dilation.

Next we looked at this question.

Students worked on paper first.

Then some explored with technology.

What do you want your students to know about the relationships in the diagram?

What question would you ask to see whether they did?

I asked this question to see what my students were thinking.

And so the journey to write and ask and share and revise hinge questions continues …

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Hopewell Geometry – Similarity

A while back we gave our students the Mathematics Assessment Project assessment task called Hopewell Geometry.

I have just finished reading Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions, and so I have been thinking a lot about sequencing.

Students are given a set of Hopewell Triangles (along with a historical explanation, which you can see at the link above).

And they are given a diagram with the layout of some Hopewell earthworks.

The first question is for students to explain which triangle is similar to Triangle 1. Some student responses are below. With which student explanation would you start in a class discussion? How would you sequence the student explanations? Are there any you would be sure to include? Some you would leave out?

Student A

Student B

Student C

Student D

Student E

Student F

Student G

Student H

Student I

Student J

Student K

Student L

Student M

Student N

Student O

Student P

I have to say that the most thrilling responses are those that justify the similarity of the triangles through dilations and scale factor. Teaching CCSS-M Geometry this year has forced me to change how we talk about similarity and congruence. And while we still discuss similarity postulates such as AA~, SSS~, SAS~ and congruence postulates such as SSS, SAS, ASA, our focus has been on talking about congruence of figures through rigid motions – and similarity of figures through a dilation and if needed, rigid motions. I will be even more comfortable having the transformational geometry-congruence-similarity discussions next year than I was this year. And I will be even more comfortable with sequencing the student work in our classroom discussion so that students can make connections between the different ways to justify similarity of figures.

And so I look forward in great anticipation as the journey continues …