The Math Education Blog
Of all the conic section construction techniques, my favorite is undoubtedly the approach that requires nothing more than a paper circle. Here’s what to do: Draw or print a circle and its center, point A, on a sheet of paper. Cut out the circle. Mark a random point B anywhere on the circle. Then, fold...
Geometry tends not to receive much love in elementary curricula, and that’s a shame. In this post, I’ll describe some of my new ideas for using Web Sketchpad to introduce young learners to fundamental properties of circles. On page 1 of the websketch below (and here), begin by asking students to drag point P and...
David Henderson, the author of Experiencing Geometry, died this past December. I wrote about David in a prior post, and in particular, his approach of asking us to grapple with a small number of rich problems, allowing us to find our own, often non-traditional, ways of solving them over weeks at a time. In this...
For the past five years, Scott and I have featured interactive Web Sketchpad models in nearly all our Sine of the Times blog posts. As much fun as it’s been to build “websketches” to share with you, we’ve really wanted to put the creative power of Web Sketchpad into your hands. And now, finally, that’s...
In his 1947 book, One, Two, Three…Infinity, physicist George Gamow poses a pirate treasure problem that has since become a classic. Below is my reworded statement of the puzzle. Among a pirate’s belongings you find the following note: The island where I buried my treasure contains a single palm tree. Find the tree. From the...
Below are two maps of the United Sates, with the smaller map a 50 percent scaled copy of the original. The edges of the two maps are parallel. Imagine that the maps are printed out, with one resting on top of the other. Believe it or not, you can stick a pin straight through both...
A recent post on an NCTM discussion group asked about tools to help students visualize and understand addition and subtraction of integers. We wanted to make a visualization tool for students that they could use in multiple ways. The result is the Web Sketchpad model below (and here). Page one focuses on addition while page...
The picture below on the left shows a right triangle JML with altitude KM. This altitude is defined as the geometric mean of JK and KL. Using similar triangles, you can prove that JK × KL = KM². Another way to think about this construction is shown below on the right. Start with segments JK and...
Twitter is a great place to find geometry problems. The July 22, 2017 post of xylem presented the image below with two squares, ABCD and BFGE, sharing a vertex. Given that AE = 5, what is the length of DG? My first thought was that surely the problem was underspecified. Without knowing more about the...
In the interactive Web Sketchpad model below (and here), ABCD is an arbitrary quadrilateral whose midpoints form quadrilateral EFGH. Drag any vertex of ABCD. What do you notice about EFGH? The midpoint quadrilateral theorem, attributed to the French mathematician Pierre Varignon, is relatively new in the canon of geometry theorems, dating to 1731. Mathematics educator Chris Pritchard says the...