In the previous post, I discussed virtual manipulatives in general, and a particular implementation for algebra, using a Google Drawings representation of the Lab Gear. In this post, I will explore GeoGebra as a platform for virtual manipulatives. Pattern Blocks Virtual pattern blocks are not hard to find on the Web. One good implementation is on the Math… Continue reading Virtual Manipulatives: Part 2

# Blog

## Virtual Manipulatives: Part 1

Learning Tools In 1981, after ten years in K-5, I switched to teaching high school math. In some ways, this felt like starting a whole new career: the math was more involved, the relationship with students less like parenting, and tradition weighed a lot more heavily on the profession. Still, in other ways, teaching is… Continue reading Virtual Manipulatives: Part 1

## Sum and Difference of Cubes

Teaching the factoring of the sum of cubes and difference of cubes was not a priority for me in my teaching, and the topic does not seem to be part of the Common Core Standards. However, some people do have to include it in their classes, and as a result the subject occasionally comes up… Continue reading Sum and Difference of Cubes

## Wallpaper Symmetry Part 2

This post is a continuation of Wallpaper Symmetry Part 1, which you should read first. Both posts are companions to the Wallpapers Catalog on my website. --------------------------------------------------------------------- A long time ago, in my twenties, I attended a lecture about the mathematics of wallpaper designs. The presenter gave an overview of the entire proof that there are only… Continue reading Wallpaper Symmetry Part 2

## Wallpaper Symmetry Part I

While sheltering in place, I decided to use some of my time to complete a project I had been contemplating for a long time: creating a catalog of the 17 wallpaper groups using pattern blocks. It is now live on my website: Wallpapers Catalog. The idea is to provide a bridge between schools, where pattern… Continue reading Wallpaper Symmetry Part I

## User-Controlled GeoGebra Animation

Here is an animation I created in GeoGebra: As far as I can tell, WordPress only shows you the animation once, when you open this post. You can find the actual .ggb file, and an interactive applet in your browser here. (Links to related transformations applets are on this page.) The idea was to illustrate reflection in… Continue reading User-Controlled GeoGebra Animation

## No One Way

My motto for this blog, and for my website, is “There is no one way”. It is a topic I have returned to many times, for example, in these posts: Catchphrases, where I mostly discuss the assorted slogans I have spouted over the years. How To, where I argue that there is no single “best way” to teach any… Continue reading No One Way

## Minimum Polyomino Cover

When I taught elementary school (1971-1981 or so) I had more time for “enrichment”, which for me meant excursions off the beaten curricular path, especially into the world of recreational math. As a result, my first publications were largely about geometric puzzles. (Those books were assembled a few years later. More info about my books.) Anyway, one… Continue reading Minimum Polyomino Cover

## Asilomar Report: Conic Sections

As my retirement starts to kick in, I no longer attend conferences — except for one: the annual meeting of the California Math Council (Northern Section.) Once again, I had a great day at Asilomar, a beautiful spot near Monterey, right on the Pacific Ocean. Here is my annual report. Conic Sections Figuring out an approach to… Continue reading Asilomar Report: Conic Sections

## Towards Inquiry

When I was much younger, I was under the impression that anything students “discover”, they will remember. Over time, I realized that this is not really true. First of all, what I hope they discovered may not be what they actually understand. But also, it’s not clear to them what is important about their discovery,… Continue reading Towards Inquiry