Tag Archives: Problem Solving

Stars, Polygons, and Multiples

I've always found my collaborations with teachers to be a great inspiration for curriculum development, and that was especially true of my work with Wendy Lovetro, an elementary-school teacher in Brooklyn, NY. Wendy coordinated an after-school math club at her school, and I used the setting as an opportunity to develop and field test Sketchpad activities for the … Continue Reading ››

Pythagoras Plugged In

If there were an award for 'Mathematical Theorem Most Amenable to a Visual Proof,'  the Pythagorean Theorem would surely win. The title of this post is a nod to the Sketchpad activity module Pythagoras Plugged In by Dan Bennett. Dan's book contains 18 visual, interactive proofs of the Pythagorean Theorem. And there are more:  The Pythagorean … Continue Reading ››

A Mathematical Mystery Story with Web Sketchpad

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 article* where … Continue Reading ››

An Equivalent Fractions Game

In my recent posts, I've introduced interactive models for comparing fractions and multiplying fractions. To continue the fraction theme, below 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 blue squares into the blue array … Continue Reading ››

Understanding Fraction Multiplication

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 sort … Continue Reading ››

A Coordinate Plane Logic Puzzle

For the past few years, Scott Steketee and I have collaborated with the author team of Everyday Mathematics to 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 the coordinate plane … Continue Reading ››

Putting the Power of a Point Theorem to Work

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 ››

Tailoring Tools to the Task

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 ››

Arranging Addends Puzzles Revisited

In a blog post last year, I introduced my Arranging Addends puzzle. Take a look at the websketch below, built with Web Sketchpad. The goal of the puzzle is to arrange the circles and the six numbers (1, 2, 4, 8, 16, and 32) so that three conditions are met simultaneously: The sum of the … Continue Reading ››

The Broken Stick Puzzle

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 ››