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Category Archives: Rigid Motions

The Magic Octagon – Dan’s, Andrew’s, and mine

I had saved Andrew’s post in my folder for a recent lesson, which was about Dan’s video.

We paused halfway in, and students decided where it would be. They answered a Quick Poll to let me know, and by the time they had all answered, some had changed their minds.

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We quickly looked at the responses, and they decided using time would be easier to decipher than some of the other descriptions.

I sent a second poll. I waited for everyone to answer, even the ones who wanted to take their time thinking about it.

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And then we continued to watch.

We paused for the last question, they discussed with their team, and then we finished watching.

Good conversation. But we didn’t get to the sequel proposed by one of Andrew’s students: If the front side arrow is pointed at 5:00, would the other arrow point at 5:00, too? Why or why not?

So I emailed that question to my students.

  • Yes, the two points move like opposite hands on a clock moving closer to each other and overlapping at 5:00. At about 11:00 they would overlap again. Otherwise, there is no overlap.
  • They would be at 5:00. This is because when he flips the magic octagon, the back arrow also flips, causing the new time to be 3:00 instead of 9:00. This means that if you were to find a line of reflection, you could flip the octagon on that line and the arrow would always land right where the previous one did. If this was on transparent paper, you can see that if one arrow points to 5:00, then the other one would be pointing at 7:00. But if you were to flip the octagon on the reflection line which intersects 12:00 and 6:00, then you would continuously get 5:00 because of the reflection.

As I got the responses from students, I realized that I wished I had asked a different question. While I did include why or why not, and it was obvious from the responses that students didn’t just answer yes or no, I wish I had asked “At what time(s), if any, are the front side and back side arrows at the same time?”

I am reminded of something I can no longer find that I read in a book. A group of teachers observed a “master” teacher for a lesson and then went back to their own classrooms to teach the lesson. The teachers asked the same questions that the master teacher asked; however, the lessons didn’t go as hoped. The teachers were not asking questions based on what was happening in their own classrooms; they were asking questions based on what had happened in the other classroom.

I love reading blog posts and learning from so many mathematics educators. They give me ideas that I wouldn’t have on my own. In fact, as my classroom moved toward more asking and less telling, I used to say that my most important work happened before the lesson, collaborating with other teachers and deciding what questions to ask. I’ve decided otherwise, though. My most important work happens in the moment, not just asking, but also listening. And then, if needed, adjusting what I planned to ask next based on the responses of the students in my care. And so the journey will always continue …

 
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Posted by on November 15, 2016 in Geometry, Rigid Motions

 

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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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Reflected Triangles + #ShowYourWork

I’ve used the Illustrative Mathematics task Reflected Triangles for several years now. This year students practiced “Show Your Work” (from Jill Gough) along with coming up with a correct solution.

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

Which of the directions can you follow exactly to construct the line of reflection? Are there any that need clarification?

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Students wrote first and then constructed their lines of reflection. I used the Class Capture feature of TI-Nspire Navigator to watch. We’ve spent longer on this task in the past, but this year, I only had about 5 minutes for a whole class discussion. Whose work would you select for the whole class to see?

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We started with Kaelon’s construction.

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How is Holly’s construction different? Can you tell what Holly did?

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How is Phillip’s different? Can you tell what he did?

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We are well on our way towards learning how to “describe or illustrate a solution in a way that the reader understands without talking to me”, as the journey continues …

 
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Posted by on September 8, 2015 in Angles & Triangles, Geometry, Rigid Motions

 

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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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The Center of Rotation

This is the first year we have tried Identifying Rotations from Illustrative Mathematics.

△ABC has been rotated about a point into the blue triangle. Construct the point about which the triangle was rotated. Justify your conclusion.

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This reminds me of the Reflected Triangles task, which we have used now for several years.

I got a glimpse of students working on the task using Class Capture. I watched them make sense of problems and persevere in solving them.

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We looked at all of the auxiliary lines that LJ made, trying to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Jarret’s work, who used technology to perform a rotation, going backwards to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Justin’s work, who rotated the given triangle about A to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Quinn’s work, who knew that if R is the center of rotation, then the measures of angles ARA’, BRB’, and CRC’ must be the same.

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Students took those conversations and continued their own work.

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The next day, Jared shared his diagram. What can you figure out about the relationship between the center of rotation, pre-image, and image looking at his diagram?

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In my last two posts, I’ve wondered what geometry looks like if we start our unit on Rigid Motions with tasks like these instead of ending the unit with tasks like these. Maybe we will see next year, as the #AskDontTell journey continues …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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The Line of Reflection

I’ve blogged about the Illustrative Mathematics task Reflected Triangles before. I really like that it asks students to determine the line of reflection given the pre-image and image instead of determining the image given the pre-image and line of reflection.

△ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

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How would you construct the line of reflection?

Work on your mathematical flexibility to come up with more than one way to construct the line of reflection.

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I used Class Capture to monitor student work and selected some students to share their approach with the whole class.

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Many students approached the task like Paris:

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What can we learn about the relationship between the pre-image, image, and line of reflection from the additional line in Max’s diagram?

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I’ve always used this task towards the end of our unit on Rigid Motions, but I am thinking about using it earlier in the unit next year. Maybe the task itself could be an #AskDontTell approach for students learning what we want them to learn about the relationship between the pre-image, image, and line of reflection. We will see next year, as the journey continues …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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