Conic Sections are Slices of a Cone...Duh!
The conic sections...used but under-used! In my "olde" days, conic sections were given considerable attention in precaclculus classes. The memories: latus rectum, foci, eccentricities...even a proof that all parabolas are similar. But, today, students seem to equate parabolas with quadratic equations...and claim that they have heard of ellipses and hyperbolas.
Perhaps students need to experience more regarding conic sections....Apolonius of Perga would agree! The following are three resources that should help:
Other resources are available...when teaching conics in a precalculus classroom, my major support for innovative ideas was Martin Gardner's columns on the conics in Scientific American. If you can find them, that is a great start.
- J. Down's Practical Conic Sections (1993) is designed to support a teacher's lesson on conic sections. Content includes generation of the conics, reflective properties (e.g. echo locations in an ellipse), and their applications.
- K. Kendig's Conics (2005) extends the discussion far beyond Down's text, both mathematically and historically. Included are a complex numbers perspective, area, eccenticities, curvature and 8 different ways to look at a conic.
- A. Mazur's The Ellipse: A Historical and Mathematical Journey (2010) uses a historical perspective to develop known information specific to ellipses. In addition to being a fresh perspective on a single conic, it will introduce you to much of the history of mathematics along the way.