*AB*and

*CD*of a circle intersect at point

*P,*then the product

*AP · PB*is equal to the product

*CP · PD*. You can see an illustration of this theorem in the Web Sketchpad model below. Drag points … Continue Reading ››

The power of a point theorem is one of the more surprising results in elementary geometry. The theorem says that if two chords *AB* and *CD* of a circle intersect at point *P,* then the product *AP · PB* is equal to the product *CP · PD*. You can see an illustration of this theorem in the Web Sketchpad model below. Drag points … Continue Reading ››

In years past, my colleagues and I at Key Curriculum conducted Sketchpad professional development workshops for teachers. One of our favorite activities challenged participants to construct a rhombus in as many different ways as possible. This assignment invariably led to great discussions because there are a multitude of ways to build a rhombus, and each method highlights its … Continue Reading ››

In a prior blog post, I introduced my new puzzle, Arranging Addends, that mixes arithmetic with logical thinking to create an engaging playground for mathematical discovery. Let’s revisit this puzzle and introduce some new variations.
Take a look at the puzzle below (and here), built with Web Sketchpad. Your goal is to arrange the … Continue Reading ››

Several weeks ago, my friend Martin shared the following probability puzzle with me: *Two points are chosen independently and at a random on a stick. The stick is broken at those points to form three smaller sticks. What is the probability these three sticks can form a triangle?*
This is a classic problem, dating back to … Continue Reading ››

Did you know that aside from being a source of news, *The New York Times* is also the place to get your weekly fix of mathematics? Their online Numberplay column features some very clever math puzzles. Last year, in fact, our blog featured a Numberplay puzzle about a flying squirrel-frog from former Key Curriculum … Continue Reading ››

The origins of this week's Web Sketchpad model date back to the Connected Geometry curriculum from the mid 1990s. I was one of the co-authors of the curriculum, working at Education Development Center with a wonderful team of math educators (Al Cuoco, Paul Goldenberg, and … Continue Reading ››

Simultaneous equations belong in elementary-school mathematics curricula. That's been my mantra for many years, and I want to examine it now in the context of an interactive Web Sketchpad activity.
When I say that elementary-age students should encounter simultaneous equations, I don't mean that they should be instructed in the standard algebraic procedure for solving pairs of equations … Continue Reading ››

For this year’s Pi Day post, I thought I’d continue our Web Sketchpad (WSP) construction theme. But rather than adapting the visualizations from last year’s Pi Day post to the new construction capabilities, I decided to take a different approach. Some time ago, I built a set of custom tools for … Continue Reading ››

Dan Meyer has posted a number of "What Can You Do With This?" activities on his blog. (Activities is probably too prescriptive a word; they're more in the nature of prompts for student thinking, noticing, and wondering.) One of the first was the image below, which he made by superimposing frames from a … Continue Reading ››