The Web Sketchpad model below (and here) shows the function *f(θ)* = 1 – cos 2*θ* in both Cartesian and polar form. For each graph, the independent variable appears as a red bar that corresponds to a particular value of *x* (for Cartesian) or *θ* (for polar). The red bar has … Continue Reading ››

# Connected Geometry

It's that time of year when we start seeing "best of" lists for books, movies, music and the like. In that spirit, but stretching way beyond the past year, some of my favorite geometry textbooks include *Geometry: Seeing, Doing, Understanding* (Harold Jacobs), *Discovering Geometry* (Michael Serra), and *Geometry: A Transformation … Continue Reading ››*

*Symmetry Challenges*

*In his online article Simply Symmetric, Michael de Villiers observes that symmetry is a powerful but often overlooked tool for formulating proofs:*

Most primary geometry curricula around the world introduce the concept of line symmetry fairly early, and sometimes also that of rotational, translational and glide … 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 ››*

*Introducing Web Sketchpad at the 2021 NCTM Annual Meeting*

*NCTM’s Virtual 2021 Annual Meeting ran from April 21 through May 1, and in Session 299 Daniel Scher, Karen Hollebrands, and I presented an on-demand video workshop to introduce Web Sketchpad (WSP). Even if you weren't able to attend the conference, you can still take … Continue Reading ››*

*A Follow-Up to the Interior Angle Sum*

*This post is a follow-up to Sarah Stephens' guest post of a week ago, in which she described a lesson using embodied cognition to help students make sense of the interior angle sum theorem for triangles, not just as an abstract concept, but as a property grounded in their concrete physical experiences.*

*The Interior Angle Sum: An Embodied Investigation*

*[This guest post by Sarah Stephens, a senior at Pennsylvania State University, describes a lesson she created as part of her Senior Honors Thesis on leveraging embodied cognition to help students develop abstract mathematical concepts.]*

*As a soon-to-be classroom mathematics teacher, I have taken special interest in the field of … 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*

*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 ››*