Below (and here) is a collection of four interactive number charts that we first introduced in our NSF-funded Dynamic Number project. Start by asking students to press the four directional arrows and to explore what they do. The right arrow, for example, moves the shaded square to the right, and wraps the square to the next row up … Continue Reading ››

Angles are a thorny concept to teach because of the fundamentally different ways in which they can be used and understood. In the article What's Your Angle on Angles?, the authors divide the concept of angle into three main groups: angle-as-figure, angle-as-wedge, and angle-as-turn. In the Web Sketchpad game below (and here), we focus on angle-as-turn. Given an … Continue Reading ››

Several years ago, I wrote a blog post about the value that students derive from writing mathematics with Sketchpad. The post included an example of a simple Logo iteration, easily implemented in Sketchpad, that produces some very complex and interesting shapes depending on the values of several input parameters. In the … Continue Reading ››

In my recent posts, I've introduced interactive models for comparing fractions and multiplying fractions. To continue the fraction theme, below (and here) is a Web Sketchpad model in which the need for equivalent fractions arises naturally through the rules of a game. The model displays two arrays. Dragging the four points changes the arrays' dimensions. The goal is to drag … Continue Reading ››

In my previous post, I presented an interactive Web Sketchpad model for visualizing and solving fraction multiplication problems. This week, I'd like to back up a step and focus on the more fundamental skill of visualizing and reasoning about the size of fractions. The fraction game below (and here) presents two random fractions at a time … Continue Reading ››

Last week, Scott and I attended a fraction symposium at NYU, and it made me realize how long it's been since I've written about our Sketchpad work with fractions. Below is a Web Sketchpad model for displaying and solving fraction multiplication problems. Representing fraction multiplication with an area model is a common approach, but it's challenging to … Continue Reading ››

For the past few years, Scott Steketee and I have collaborated with the author team of Everyday Mathematicsto integrate Web Sketchpad deeply into their curriculum. As part of that work, I just completed a websketch that nicely mixes practice with logical reasoning. Students are challenged to find a hidden treasure on … Continue Reading ››

Four years ago, my colleague Scott Steketee and I set out to develop an interactive game to help students develop strategies for thinking about and solving multiplication problems. As we examined the existing apps on the market, we discovered that most focused on the drill aspect of learning one's multiplication facts. We set our goals higher. We … 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 … Continue Reading ››

What does dilation feel like? I recently had the opportunity to work with a group of students who were testing activities that treat geometric transformations as functions (what I call geometric functions). I got lots of good ideas for improving the activities not only by watching the students, but also but also from their suggestions … Continue Reading ››