# 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 … Continue Reading ››

# Exploring Scaled Polygons

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

1. 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.
2. 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 SketchpadContinue 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 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 ››

# 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 ››