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The Center of Rotation

10 Mar

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.

0 03-10-2015 Image004

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.

1 Screen Shot 2014-08-28 at 9.30.07 AM

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.

2 Screen Shot 2014-08-28 at 9.34.22 AM

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.

3 Screen Shot 2014-08-28 at 9.34.59 AM

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.

4 Screen Shot 2014-09-02 at 9.53.36 AM

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.

5 Screen Shot 2014-08-28 at 9.37.34 AM

Students took those conversations and continued their own work.

6 Screen Shot 2014-08-28 at 9.58.49 AM

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?

7 03-10-2015 Image003

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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2 Comments

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

 

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2 responses to “The Center of Rotation

  1. howardat58

    March 10, 2015 at 9:32 pm

    I am enjoying following your journey through rigid motions a lot.
    After the previous one I thought “let’s try something different”. Take a parabola and find its axis. The damn thing is symmetric about its axis, reflective symmetry. Maybe some rigid motion stuff will help. I am seriously stuck !!!!
    The rotation thing does require some sideways thinking, or at least “What does a rotation do?”.
    Have they seen that two reflections produce a rotation?
    And two reflections can yield a translation….
    This really makes reflection the basic operation, and when I was reading this paper on mathematics for physics the author was explaining that the reason that students did not know about this was that the mathematics needed to describe reflections in a coordinate system is far too messy.
    There’s a lot more to this stuff than I first thought.

     
  2. Travis

    March 10, 2015 at 9:44 pm

    Had a thought….Somewhere along the timeline of this unit…have students make a triangle, rotate it, hide the point of rotation. Then swap calcs and try to find the center of rotation. They place their point/guess, then test it with a rotate about it. Then they can drag the point until coinciding occurs….or after a certain amount of time, Hide/Show to reveal the answer.
    If they know about two reflections across intersecting lines, then they can use that method.

     

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