*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 ››

In my last post, I provided some dilation challenges and linked to a Dilation Function Family activity. In that activity students manipulate independent and dependent variables, observe their relative rate of change, restrict the domain, and use meaningful function notation.
This and similar activities involving “technologically embodied geometric functions” … 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 ››

For a while now, I’ve been intrigued by the ways in which the study of geometric transformations can provide students with a very effective introduction to function concepts. Daniel and I have written a couple of articles about this topic, and we created a number of activities to take advantage of what can arguably be … 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 ››

With the World Cup in our hemisphere, and the US squad having started out with a win over Ghana, my thoughts turned to the mathematics of soccer. My friend Henri Picciotto has a nice page about the shooting angle, the angle within which a shot is on goal, so I thought of using … Continue Reading ››