Day 157: It’s an Eclipse – NO, an Ellipse!
The patty paper activity we did today is so relaxing, yet kinestheticly engaging and intellectually satisfying for most students. Art Mabbott initially shared these activities informally during one of my TI User Groups, followed by using the TI Nspire documents to make sense of what had happened during the paper folding. I just loved the chance to have some hands-on activity within a higher level math that I seized the opportunity. And the students get a chance to talk to each other both about math and other things…we don’t get a chance to do that often.
After the experience, I spent some time rethinking/redoing the activity while writing guided explorations for the parabola, ellipse and hyperbolas. Here’s a snippet of the ellipse exploration:
- You will need a sheet of patty paper with a circle filling most of the paper. Does not have to be centered on the patty paper.
- Locate the center of the circle. Call it point C. Describe your method:
- Choose a point on the circle and call it point P.
- Mark a point inside the circle that is somewhere between the center of the circle and the circle itself (it doesn’t necessarily have to be in line with P and C)– this will be a focus. Call this point F.
- Fold point P (the point on the circle) onto point F (the focus point). Make a good crease. With your pencil, draw in the line on the fold. Also draw the line FP.
- What is that fold line you drew? Think back to geometry. There are two correct answers.
- Write on the fold line that this line is the …. (ask your teacher).
- Pick another point on the circle. Repeat the process from step 5 and make another good crease. You do not need to draw in the line on the fold, though.
As the kiddos were folding away, they were making conjectures about what they were seeing: is it another parabola? my lines aren’t making anything! Cool, it might be a circle. No wait, it’s too flat.
What I really like about using the patty paper is that the fold lines are easy to see and the kids can write on it too. Once the class finished their folding, we then explored some patterns and relationships with some guiding questions that eventually lead to the geometric definition of an ellipse.
Looking at the guiding questions, I need to think more about how “leading” they are or how “open-ended” they are. My sense in listening to the discussions was they weren’t sure what they were trying to do. I need to rethink these questions for next year:
- Now, go back to your original point P on the circle. Draw in the radius of the circle using point P and point C. The radius will intersect the original crease line at a point. Call this point E. Describe some properties of this point E:
- Using a ruler, measure segment CE, FE and CP.
- What relationship do you notice between these three segments?
- Is it true for other points that have the same properties as point E? Explain
- What has this to do with the Focal Radii definition of an ellipse?
Do any of you have some thoughts about these questions or what could be asked differently to get students sense-making, organizing, analyzing and reasoning to impose structure to this situation?