# Wax Paper Folding - Discovering the Conic Sections

## 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.

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## Learning Goals

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

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

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

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## Assessment

## References and Resources

Parabola: https://www.youtube.com/watch?v=vaLQawKuq8M

Ellipse: https://www.youtube.com/watch?v=psuTYtDfxPE

Hyperbola: https://www.youtube.com/watch?v=nEISCCjObPg