In geometry, we learn that if we erect squares on the legs of a right triangle, the sum of their areas is equal to the area of the square on the triangle's hypotenuse. This is visual way to conceptualize the Pythagorean Theorem. But now consider the image below that shows a bust of … Continue Reading ››
Virtual Cuisenaire Rods
I was happy to collaborate on this blog post with Dr. Stavroula Patsiomitou, a researcher at the Ministry of Education and Religious Affairs in Greece. Dr. Patsiomitou received her PhD from the University of Ioannina and has written extensively about the field of dynamic geometry environments, including Sketchpad and Web Sketchpad. … Continue Reading ››
A Bevy of Rhombus Constructions
In my January 2020 blog post, I presented a collection of Web Sketchpad construction challenges where the goal was to use each handpicked set of tools to build a rhombus. Could you, for example, construct a rhombus with just a Compass and Parallel tool? How about starting with merely the Reflect … Continue Reading ››
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.
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:
- 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 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 ››