Finding Centers of Rotation: Math Task

1. Draw a line segment [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]] on a piece of paper, then a point C not on [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]]. Imagine that [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]] is rotated around point C, the center of rotation.
a. What do you notice?


b. Draw [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]] at 5 different locations during its rotation around point C.


c. Describe the path of line segment [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]].


d. Describe the path of each endpoint of [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]].


e. Describe the path of the midpoint of [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]].

2. Take a look at the Geometer's Sketchpad applet "Rotations" available on your facilitator's Fostering Geometric Thinking Toolkit DVDl. Explore the path the line segment takes as it rotates around the center of rotation. Click on 'View several different locations ... " so you can see how the segment looks at different locations as it rotates.
a. What do you notice?


b. Describe the path that each endpoint of the segment takes.

c. How did viewing the Sketchpad file affect your thinking about the 5 positions for [[{"fid":"271","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]] from part I?

d. Laura is confused about centers of rotation and her computer just crashed so
she can't explore it with Geometer's Sketchpad. Keeping in mind what you
learned from exploring the applet, describe what a center of rotation is to Laura.
Make sure you tell her about the things that change and the things that stay
the same during a rotation.

3. Laura thinks this idea of centers of rotation is interesting and she wants to explore it some more. So far Laura has imagined herself watching from a distance as [[{"fid":"272","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]]rotates around C.


She starts to wonder if she might notice anything different about rotation if she imagined herself at the center of rotation (in the system). She imagines standing
on C and rotating with [[{"fid":"272","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]]so that she is always looking at the segment as it rotates.

a. Imagine standing on point C and rotating so that you're always looking at [[{"fid":"272","view_mode":"default","type":"media","attributes":{"height":22,"width":28,"class":"media-element file-default"}}]] while it rotates around you. What do you notice? Do you notice anything
that you hadn't noticed while looking at the computer? Explain.


b. Now that you've considered centers of rotation from a few different perspectives, would you add or change anything to the description you gave Laura in
part 2d?


4. Below are some pairs of congruent line segments. We know they are congruent because, for each pair, we were able to find a center of rotation and rotate the
segment on the right so it fit exactly on the segment to the left.

We tried several possible points for this first segment pair and we think
we've found a center of rotation. Check to see if our point is correct.

  • Place a piece of patty paper over the 2 segments.
  • Trace the segment on the right and our point onto the patty paper.
  • Use your pencil to anchor the patty paper at your center of rotation and rotate the patty
    paper to see if the traced segment lands on top of the segment on the left

Find the centers of rotation for the segment pairs below. Use the method outlined above to check your centers.

a. You may have tried several points, and many did not work. How did you find the center of rotation that succeeded in allowing you to rotate the right segment
onto the left segment?

b. How did you find the center of rotation?


c. Point A is one center of rotation. (Check with your patty paper.) A rotation of 90 degrees will do it. There is another point that also serves as a center of rotation. Where is that point, and how did you find it?

d. Find 2 centers of rotation. How did you find each center?

e. Find 2 centers of rotation. How did you find each center?


After finding the centers of rotation for the 5 segment-pairs in part 4, Laura is pretty
sure there is a general procedure that would allow her to find a center of rotation
for any 2 segments. Because you were so helpful to Laura earlier, she asks for your
advice again. 'What procedure, or method for finding a center of rotation, will help
me find a center of rotation for any 2 segments that I might draw?"
Before you respond to Laura, draw 2 line segments of the same length anywhere
on a piece of paper. How would you find the center of rotation P so that I of your
line segments can be rotated about P to get to the position of the other line segment,
no matter where you drew those 2 line segments? Describe your method to
Laura in a paragraph or two. Remember, your method has to work no matter where
Laura decides to put her 2 line segments.


© 2008 by Education Development Center, Inc. from The Fostering Geometric Thinking Toolkit. Portsmouth, NH: Heinemann. Reproduced with permission.