Euclid Walks the Plank
Using Web Sketchpad, students construct a boardwalk path of equal-length planks to explore the key concepts behind Euclid’s Proposition 1.
Category: Geometry
Exploring Scaled Polygons
Below are some common methods that geometry curricula offer for constructing scaled polygons: To complement these standard approaches, I’d like to present a fourth option, one that allows students to form the original polygon and its dilated image simultaneously rather than starting with the preimage. In the Web Sketchpad model below (and here), begin by...
Injecting Surprise Into the Triangle Midline Theorem
Pi Day 2022 is now over, but I’m still thinking about a tweet from 10-K Diver: Take two random numbers X and Y between 0 and 1. What is the probability that the integer nearest to X/Y is even? The answer—spoiler ahead—is (5 – π)/4. (You can run my Web Sketchpad simulation of the problem...
A Paper Folding Investigation from Connected Geometry
In a prior post, I shared some good news: The Connected Geometry high-school curriculum authored by Education Development Center (EDC) is now available for free. I could easily devote every future blog post to a tasty Connected Geometry morsel, but I’ll restrict myself to just a few. The investigation below, taken straight from the curriculum,...
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 Approach (Coxford & Usiskin). There’s...
Symmetry Challenges
In his 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 reflective symmetry. However, in high school and even college geometry...
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 the pedagogical benefit of using an...
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 advantage of this workshop, and we will be glad to...
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 day before...
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 embodied cognition and integrating it...