Using Web Sketchpad, students construct a boardwalk path of equal-length planks to explore the key concepts behind Euclid’s Proposition 1.

# 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 4*x* + 5 = 4*x* – 3. After subtracting 4*x* from both sides, I was … Continue Reading ››

# Exploring Scaled Polygons

Below are some common methods that geometry curricula offer for constructing scaled polygons:

- Place a polygon on the coordinate plane, pick the origin as the center of dilation, scale each vertex by some specified amount by using its coordinates, and then connect the scaled vertices.
- Draw rays from the … Continue Reading ››

# 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 … Continue Reading ››

# 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 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 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 ››*