# 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 ››

# 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 ››

# 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 … Continue Reading ››

# 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.

# Protect the Sheep

A game of enclosing sheep and wolves in fences helps children to develop their conceptual understanding of polygons.

# Adding and Subtracting Fractions on a Number Line

In my previous post, I presented a number line model for adding fractions. Now, I'd like to offer an updated Web Sketchpad fraction model below (and here) with two new pieces of functionality:

1. You can subtract as well as add fractions.
2. You can divide the number line into equal parts and choose the … Continue Reading ››