*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 … Continue Reading ››

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 … Continue Reading ››

As a longtime Sketchpad fan, one of the most interesting features of Web Sketchpad (WSP) for me is the way its behavior can be customized. WSP makes it possible to add JavaScript to a web page in order to interact directly with objects in the sketch. For instance, a JavaScript function could use the … Continue Reading ››

In the interactive Web Sketchpad model below (and here on its own page), *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 … Continue Reading ››

At the 2017 NCTM Annual Meeting I was invited to do a short Wednesday-afternoon presentation on Function Dances in the NCTM Networking Lounge. (Here's the handout from the presentation.)
The idea of function dances is to get students (or in this case teachers) moving around, acting as the independent and dependent … Continue Reading ››

In my last post, I presented a lovely geometry problem from Japan that was ideally suited to a dynamic geometry approach. Below is a new problem whose construction is nearly identical to the original one. The text says, "Five isosceles triangles have their bases on one line, and there are 10 rhombi. One length of the rhombus … Continue Reading ››

Here is a wonderful geometry problem from Japan: The five triangles below are all isosceles. The quadrilaterals are all rhombi. The shaded quadrilateral is a square. What is the area of the square?
I wondered at first whether the English translation of the problem was correct because with so many side … Continue Reading ››

Last year, I had the pleasure of co-organizing a geometry-focused coaching collaborative led by Metamorphosis, a New York-based organization that offers professional content coaching to transform the mindset and practices of teachers and administrators. I had so much fun that I decided to do it again! My workshop partners were Metamorphosis staffers Toni Cameron, Ariel Dlugasch, and … Continue Reading ››

Using dynamic geometry software, a student can draw what looks like a square by eyeballing the locations of the vertices. However, the quadrilateral will not stay a square when its vertices are dragged. Building a "real" square requires that it stay a square when any of its parts are dragged. This is only possible by baking … Continue Reading ››

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 ›› *

Angles are a thorny concept to teach because of the fundamentally different ways in which they can be used and understood. In the article What's Your Angle on Angles?, the authors divide the concept of angle into three main groups: angle-as-figure, angle-as-wedge, and angle-as-turn.
In the Web Sketchpad game below (and here), we focus on angle-as-turn. Given an … Continue Reading ››