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Carrying a Figure Onto Itself + #ShowYourWork

G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our content learning goal for the day: I can map a figure onto itself using transformations.

Our practice learning goal: I can attend to precision.

Combining those, we were working on: I can show my work.

Do your students know what you mean when you ask them to show your work?

Jill Gough has written a transformative leveled learning progression for showing your work. This was our first day in geometry this year to focus on it.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem.

For this task, our focus was on describing clearly the transformations that would carry a rectangle or equilateral triangle onto itself so that a partner could follow the steps.

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Which of the following is clear?

Reflect ABCD about a line through the middle of the rectangle.

Reflect ∆ABC about its center.

Rotate ∆ABC 60˚.

Reflect ABCD about the perpendicular bisector of segment AB.

Rotate ∆ABC 180˚ about point A.

Students set to work individually, paying attention to their language. I walked around to see what they were writing.

I noticed MR’s first, which said, Translate ∆ABC using vector AA. As I looked more closely, I realized that she was mapping the triangle on the left side of the page onto the triangle on the right side of the page, but even so, she had come up with a remarkably trivial solution, had she been mapping the triangle onto itself.

The next student that I saw had rotated ∆ABC 360˚ about point A.

And then the next student that I saw had dilated ∆ABC about point A using a scale factor of 1.

I decided at this point that perhaps a class discussion was in order to limit additional trivial solutions to this task. So we talked about transformations that will, of course, map the figure onto itself, such as rotating the image about one of its vertices 0˚ or 360˚, and also, really, are simple and not very interesting.

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And then I let them work some more. The idea was for them to write a transformation or sequence of transformations and have their partner try it, following their directions exactly. The partner helped revise the directions as needed if the directions didn’t work the first time.

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Instead of selecting particular students to share their work with the whole class, I asked students to write at least one set of their successful mappings in a shared Google Doc so that they could see multiple solutions to both the rectangle and the triangle.

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Thanks to the leveled learning progression, I think we are off to a good start practicing “show your work”, as the journey continues …

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Posted by on September 7, 2015 in Geometry, Rigid Motions

 

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Carrying a Figure onto Itself

CCSS-M-G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

By the end of our course, we want students to be able to say, “I can map a figure onto itself using transformations”.

 

In our lesson on Reflections, we asked about reflecting a rectangle onto itself.

What do we need to do a reflection?

An object and a line.

What line do we need to reflect rectangle ABCD onto itself?

 

We talked about the difference between drawing a line and constructing a line.

How many lines will work?

What is the significance of the lines?

 

Students constructed lines different ways. Some students used the midpoints of the opposite sides to create a line.

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Some students constructed the perpendicular bisector of a side to create a line.

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We made Abby the Live Presenter. She changed the size of the rectangle to show us that using the perpendicular bisector always works.

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Several students thought that the diagonals of the rectangle could be the lines of reflection. We made Max the Live Presenter, and he showed us what happens when we reflect a rectangle about its diagonal.

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How many ways can you reflect a regular hexagon onto itself?

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The next day, we explored rotations.

How many ways can you rotate an equilateral triangle onto itself?

Where is the center of rotation?

What is the angle of rotation?

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If we change the size of the equilateral triangle, does the rotation still work?

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One student defined the rotation angle using three points instead of an angle measure. How should you arrange the points of the angle to rotate the triangle onto itself?

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We are now towards the end of the unit. In class yesterday, I asked students to write down any two ways to transform the rectangle onto itself. After a minute, I asked them to look back at what they had written. Have you attended to precision? If you said to reflect, have you described what is the line of reflection? If you said to rotate, have you described what is the center of rotation? Several students rotated the rectangle 360˚ or 720˚ or -360˚ about any point on the rectangle. I guess it’s not completely trivial to recognize that the rotation will work about any point (and not just a vertex), but I asked them to use an angle measure that wasn’t a multiple of 360.

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Students revised their work and then shared with their team. One team member told another that hers was not going to work. They called me over to mediate, which reminded me again how good it is for students to have dynamic action technology in their hands. Try it and see. I don’t have to be the judge … students can use the technology to test their conjectures. MJ wanted to reflect the rectangle first about a diagonal and rotate about the midpoint of the diagonal.

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By the time we made MJ the Live Presenter, she had decided to reflect the rectangle first about a diagonal and then reflect it about the perpendicular bisector of the diagonal.

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As I continued to monitor students working, I saw several who used a sequence of transformations to map the rectangle onto itself. Our standard specifically says to use reflections and rotations, but I asked BB to share her work. She reflected the rectangle about one of its sides and then translated it using a vector equal to the side perpendicular to the first side. Will that always work?

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Another student found the intersection of the perpendicular bisectors of the sides and rotated the rectangle 900˚ about that point. Why does 900 work?

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And so the journey to ease the hurry syndrome continues, often spending 20 minutes on what I had planned to take 5 …

 
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Posted by on August 29, 2014 in Geometry, Rigid Motions

 

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