*f*mapping a compact convex set into itself, there is a point

*x*

_{0}such that

*f*(

*x*

_{0}) =

*x*

_{0}. This is a deep theorem, but one aspect of it is lovely, surprising, and entirely approachable by high-school geometry … Continue Reading ››

According to Wikipedia, the Brouwer Fixed Point Theorem, named after mathematician and philosopher Luitzen Brouwer, states that "for any continuous function *f* mapping a compact convex set into itself, there is a point *x*_{0} such that *f*(*x*_{0}) = *x*_{0}.
This is a deep theorem, but one aspect of it is lovely, surprising, and entirely approachable by high-school geometry … Continue Reading ››

In the early 1990s, Danny Vizcaino, a high school student at Monte Vista High School in California, wrote to Key Curriculum Press noting that Sketchpad did not come with a tool to draw an oval. Undaunted by this omission, Danny had built his own oval with the software and shared it with Key's editors.
As shown in the interactive … Continue Reading ››

When I was child, I loved to solve the brainteasers in logic puzzle magazines. You probably know the type:
*Ruth, Phyllis, and Joan each bought a different kind of fruit (orange, apple, pear) and a different vegetable (spinach, kale, carrots) at the supermarket. No one bought both an orange and carrots. Ruth didn't buy an apple or kale.* … Continue Reading ››

While I enjoy reading *The New York Times* for its news coverage, I especially look forward to each Monday when they post a new math puzzle online in their Numberplay column.
Several months ago, I shared a Numberplay puzzle from former Key Curriculum editor Dan Bennett. Now I'd like to recap the Numberplay puzzle … Continue Reading ››

There are certain topics in mathematics education not appropriate for polite discussion. Number bases other than 10 fit this category well, perhaps because of their association with the maligned "new math" of the 1960s. That's a shame because there is a lot to learn from them, especially when presented as interactive puzzles.
Below are eight dials, each with … Continue Reading ››

In my prior post, I presented an interactive Web Sketchpad odometer that is a great tool for introducing young learners to place value.
Well, technology moves fast these days, and the latest odometers are more powerful than ever. While our prior odometer featured '+' buttons above each digit, our newest innovation in number-tracking technology features '+' and … Continue Reading ››

Below is an interactive odometer built with Web Sketchpad. Press each of the '+' keys and observe their effect on the odometer's value. Also notice how your button presses are tracked in the table below the odometer.
I built this model as a way to support students' development of … Continue Reading ››

For the past eight months, my colleague Scott Steketee and I have collaborated with the authors of the elementary curriculum *Everyday Mathematics *to design interactive Web Sketchpad models for their next edition.
When it came time to create a Sketchpad representation of an isosceles triangle, I built the interactive triangle model below. Try dragging any of … Continue Reading ››

Every week, *The New York Times* challenges its readers to solve a mathematical puzzle in its online Numberplay column. This week's puzzle was proposed by none other than Dan Bennett, a former editor and author at Key Curriculum Press, and his colleague, Avery Pickford.
Here is their puzzle, as described in … Continue Reading ››

Last week, I read about a Kickstarter campaign for the math game Primo. The game assigns colors to the prime numbers 2, 3, 5, and 7 (2 = orange, 3 = green, 5 = blue, 7 = purple) and then represents composite numbers by displaying their prime factors in color-coded form. It's … Continue Reading ››