In my numerous blog posts about Exploring Conic Sections with The Geometer's Sketchpad, I've featured an assortment of hands-on and virtual models for drawing conic sections, ranging from pins and string to paper folding to 17th-century linkages. But in order for students to understand why any of these approaches work, they … Continue Reading ››
Category Archives: Conic Sections
Introducing the Elipso
Longtime readers of this blog will know that I get more than just a little excited by devices—both mechanical and virtual—that draw conic sections. I've written about the conic section-drawing devices of the 17th-century Dutch mathematician Frans van Schooten as well as other methods of generating conics in a … Continue Reading ››
The Swimming Pool Problem
In a prior blog post, I presented an uncommon method for solving the well-known Burning Tent problem. My solution, modeled on the approach in the Connected Geometry curriculum, used a dynamic ellipse to pinpoint the optimal solution. Now, I'd like to offer a related problem from Connected Geometry where … Continue Reading ››
The Cowgirl Problem
In a previous post, I described two different approaches to solving the Burning Tent optimization problem. Now I'd like to offer a related problem that I assigned many years ago to my pre-service mathematics teachers at New York University.
A cowgirl wants to give her horse some food and … Continue Reading ››
Race to the Burning Tent
How can you identify a lover of math? Casually mention a burning tent and notice if her first thought is how to minimize her path to a river and then to the tent to douse the flames. Here is a full statement of this classic geometry problem:
Ah, the great … Continue Reading ››
Pirate Treasure Awaits
In a 2018 blog post, I presented George Gamow's pirate treasure problem, which can neatly be solved by capitalizing on the geometry of complex numbers. There's more treasure to be had, however, so get ready for another adventure!
An island contains a giant boulder, a lighthouse, a cave, and a jail. Among … Continue Reading ››
A Plethora of Hyperbola Constructions
In my prior blog posts, I've presented methods for constructing ellipses and parabolas using both Web Sketchpad and paper folding. Now it's time for me to finally turn my attention to hyperbolas.
All of the Web Sketchpad models below (and here) are based on the distance definition of a hyperbola: the set of … Continue Reading ››
Tweaking the Expanding Circle Construction
In last month's blog post, I described a parabola construction technique dating back to the work of Persian polymath Ibn Sina (c. 970 – 1037). After I published the post, my colleague Scott noted that my construction could be more robust to allow for parabolas that are downward facing as well as upward facing. … Continue Reading ››
The Expanding Circle Construction
There can never be enough conic-section construction techniques—at least that's my philosophy, having grown up to think that conics existed purely in the realm of algebraic equations. So to continue my conic section construction series on Sine of the Times, I'll present a parabola construction attributed to Ibn Sina (Avicenna), a Persian polymath (c. 970 – … Continue Reading ››
A Trio of Parabola Constructions
In my prior blog posts, I've presented methods for constructing ellipses using Web Sketchpad and paper folding. The other conic sections are feeling a bit left out, so let's explore some techniques for constructing parabolas.
All three Web Sketchpad models below (and here) are based on the distance definition of a parabola: The … Continue Reading ››