*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 ››

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 and the … 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 ››

Can mathematical curves be beautiful? Most certainly! Precalculus students glimpse the connection between mathematics and art when they graph roses, cardioids, limaçons, and lemniscates. But these curves give just a taste of the beauty that can be achieved when graphing equations.
In a recent article from the online science magazine Quanta, Pradeep Mutalik reviews a gorgeous new math book, Creating … 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 ››

In a prior blog post, I described the pins-and-string approach to drawing an ellipse: Press two pins into a corkboard, place a loop of string around the pins, pull the string tight with a pencil, and trace the pencil tip's path as you pull the pencil around the taut string. Guaranteeing that the traced … 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 ››

In my prior post, I presented a "zooming" number line model that allowed students to estimate the location of a point along a number line and then repeatedly magnify that portion of the number line to obtain ever-finer estimates, accurate to tenths, hundredths, thousandths, and beyond.
In a sense I got ahead of myself because I … Continue Reading ››