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 … Continue Reading ››
All posts by Daniel Scher
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 … Continue Reading ››
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 … 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 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. … 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 ››
Protect the Sheep
A game of enclosing sheep and wolves in fences helps children to develop their conceptual understanding of polygons.