Wax Paper Folding - Discovering the Conic Sections
Initial Publication Date: December 31, 2016
Summary
Students fold "patty paper" to create each of the conic sections, one at a time. They create three "layers" for each conic section.
For the parabola, they fold one layer to create a parabola, a second layer that shows the relationship between the focus and the latus rectum, and a third layer that examines the locus of points.
For the ellipse, one layer is the ellipse, one shows the constancy of the distance from one focus to any point on the ellipse to the other focus, and one explores the relationship between a, b and c.
For the hyperbola, one layer is the hyperbola, one shows the constancy of the distance from one focus to any point on the hyperbola to the other focus, and one explores the relationship between a, b and c.
Learning Goals
Students should get from this activity that the focal diameter of a parabola is exactly four times the distance from the vertex to the focus, and that the distance from the focus to any point on the parabola is the same as the distance from that point to the directrix.
Students should also uncover that the sum of the distances from one focus of an ellipse to any point on the ellipse and then to the other focus is constant. If I do the "long" version, then students also learn the relationships between the values of a, b and c in an ellipse.
Students learn that the circle is a special case of an ellipse.
Students also learn the same types of properties about the hyperbola as they did about the ellipse.
Students only submit their portfolio once, so there is no opportunity to improve it by editing it, but this is something that I might entertain in future. I would have hoped that they would want to turn in quality work, since this has always been an exam grade, but alas, that has not been the case.
Students should also uncover that the sum of the distances from one focus of an ellipse to any point on the ellipse and then to the other focus is constant. If I do the "long" version, then students also learn the relationships between the values of a, b and c in an ellipse.
Students learn that the circle is a special case of an ellipse.
Students also learn the same types of properties about the hyperbola as they did about the ellipse.
Students only submit their portfolio once, so there is no opportunity to improve it by editing it, but this is something that I might entertain in future. I would have hoped that they would want to turn in quality work, since this has always been an exam grade, but alas, that has not been the case.
Context for Use
These activities would work well at the high school or college level. In fact, they are "borrowed" from a high school activity, but greatly extended to include discovery of the properties of each of the conic sections, ideas which were not present in the activity as I discovered it.
Class sizes have been 25-35, but I think this would work with anything from 10 to 40. More than that and the logistics of the activity would become untenable, and it would become difficult to answer questions in a timely manner. I have done this activity in classes where groups are in regular use, so students have each other to ask questions first. If you are not using groups, smaller class sizes would be preferable.
This set of activities takes longer than teaching the conic sections by lecture. In general, I generally allow one day to lecture each of the conic sections (two days for the hyperbola), but two days if I'm doing it by "folding".
I've had students do the folding in "layers" then present the layers, along with self-guiding worksheets, in portfolios for an exam grade at the end of the unit. This is time-consuming to grade, but rewarding, since the algebra they must perform is demanding for students at this level.
I've also had some classes do just the folding, and then I ask them to do some measuring to discover some properties, but they don't save their folded paper or present them. It depends on how much time I have at the end of the quarter where this unit usually falls.
There are no particular prerequisite skills, but the stronger their algebra is before doing this activity, the better, depending on whether you do just the folding or the folding with the worksheets.
Class sizes have been 25-35, but I think this would work with anything from 10 to 40. More than that and the logistics of the activity would become untenable, and it would become difficult to answer questions in a timely manner. I have done this activity in classes where groups are in regular use, so students have each other to ask questions first. If you are not using groups, smaller class sizes would be preferable.
This set of activities takes longer than teaching the conic sections by lecture. In general, I generally allow one day to lecture each of the conic sections (two days for the hyperbola), but two days if I'm doing it by "folding".
I've had students do the folding in "layers" then present the layers, along with self-guiding worksheets, in portfolios for an exam grade at the end of the unit. This is time-consuming to grade, but rewarding, since the algebra they must perform is demanding for students at this level.
I've also had some classes do just the folding, and then I ask them to do some measuring to discover some properties, but they don't save their folded paper or present them. It depends on how much time I have at the end of the quarter where this unit usually falls.
There are no particular prerequisite skills, but the stronger their algebra is before doing this activity, the better, depending on whether you do just the folding or the folding with the worksheets.
Description and Teaching Materials
The attached documents include instructions as well as a list of all required materials. I didn't do this the first time I did this activity, and when students were absent on the first day but present on the second day of the activity, I spent most of the hour trying to get them caught up, pretty much failing, and not answering other students' questions in the process. With the self-guiding instructions, absent students still require more help than those who were there on day one, but to a much smaller degree.
All the documents are my own creations. I still feel as though they need adjusted, but then I suspect that I will always feel that way about them. Instructions and worksheets for the parabola (Acrobat (PDF) 2.3MB Dec16 16)
Instructions and worksheets for the ellipse (Acrobat (PDF) 719kB Dec16 16)
Instructions and worksheets for the hyperbola (Acrobat (PDF) 643kB Dec16 16)
All the documents are my own creations. I still feel as though they need adjusted, but then I suspect that I will always feel that way about them. Instructions and worksheets for the parabola (Acrobat (PDF) 2.3MB Dec16 16)
Instructions and worksheets for the ellipse (Acrobat (PDF) 719kB Dec16 16)
Instructions and worksheets for the hyperbola (Acrobat (PDF) 643kB Dec16 16)
Teaching Notes and Tips
In the current version of the ellipse worksheet, I have students measuring the distance from a focus to a point and from that point to the other focus, and using "generic" labels for these distances and the distance formula to derive the standard form for the equation of an ellipse. This is a challenging and demanding piece of algebra for students who are used to doing 25 exercises for homework. I need to tell them, then to remind them regularly that this is not a five-minute job, and that there is a lot of work involved in what looks like one step. I hope they gain from this piece that not all "math" is something that you should be able to do in five minutes or less.
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Assessment
When I use these activities exclusively for the conic sections unit, students (now groups) turn in a portfolio that includes, for each section, the "base" layer with the folded shape, a second layer that illustrates some of the measurements and properties of the shape, and a third layer that demonstrates the property of tangency for that shape. The portfolio counts as an exam grade for the unit, and then a question that involves completing the square on a polynomial to determine whether it is the equation of a parabola, an ellipse or a hyperbola, and then graphing the curve is included on the final exam.
References and Resources
The original YouTube videos on which I based the activities:
Parabola: https://www.youtube.com/watch?v=vaLQawKuq8M
Ellipse: https://www.youtube.com/watch?v=psuTYtDfxPE
Hyperbola: https://www.youtube.com/watch?v=nEISCCjObPg
Parabola: https://www.youtube.com/watch?v=vaLQawKuq8M
Ellipse: https://www.youtube.com/watch?v=psuTYtDfxPE
Hyperbola: https://www.youtube.com/watch?v=nEISCCjObPg