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MP6 – Mapping a Parallelogram Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

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

We continued working on our learning intention: I can map a figure onto itself using transformations.

Perform and describe a [sequence of] transformation[s] that will map parallelogram ABCD onto itself.

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This task also requires students to practice I can look for and make use of structure. What auxiliary objects will be helpful in mapping the parallelogram onto itself?

The student who shared her work drew the diagonals of the parallelogram so that she could use the intersection of the diagonals as the center of rotation.

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Then she rotated the parallelogram 180˚ about that point.

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Could you use only reflections to carry a parallelogram onto itself?

You can. How can you describe the sequence of reflections to carry the parallelogram onto itself?

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How else could you carry a parallelogram onto itself?

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Posted by on September 22, 2016 in Geometry, Rigid Motions

 

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MP6 – Mapping a Figure Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3:

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our learning intention for the day was I can map a figure onto itself using transformations.

Performing a [sequence of] transformation[s] that will map rectangle ABCD onto itself is not the same thing as describing a [sequence of] transformation[s].

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We practiced both, but we focused on describing.

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I asked the student who listed several steps to share his work.

  1. rotate rectangle 180˚ about point A
  2. translate rectangle A’B’C’D’ right so that points A’ and B line up as points B’ and A. [What vector are you using?]
  3. Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

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What if we want to carry rectangle ABCD onto rectangle CDAB? How is this task different from just carrying rectangle ABCD onto itself?

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What about mapping a regular pentagon onto itself?

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Many students suggested using a single rotation, but they didn’t note the center of rotation. How could you find the center of rotation for a single rotation to map the pentagon onto itself?

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This student used the intersection of the perpendicular bisectors to find the center of rotation, but didn’t know what angle to use for the rotation. How would you find an angle of rotation that would work?

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What can you do other than a single rotation?

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This student reflected the pentagon about the perpendicular bisectors of one of the side of the pentagon.

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The descriptions students gave made it obvious that we needed more work on describing. The next day, we took some of the descriptions and critiqued them. Which students have attended to precision?

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It’s good work to distinguish precision from knowing what someone means as we learn to attend to precision. And so the journey continues …

 
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Posted by on September 21, 2016 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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