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.
Sine of the Times Blog
A Triple Number Line Model for Visualizing Solutions to Equations
In Algebra 1, I was the king of solving for x. Algebraic manipulation was fun and satisfying, and I was good at it. But my confidence was shaken when I encountered a test question of the variety 4x + 5 = 4x – 3. After subtracting 4x from both sides, I was left with 5...
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,...
The Polar-Cartesian Connection
The Web Sketchpad model below (and here) shows the function f(θ) = 1 – cos 2θ 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 tick marks that show...
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...
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 water before returning to her tent....