This is a sequel to my last post (Seeing is Believing?), addressing the same issue from a different point of view. Fractions, of course, are difficult. When teaching 4th and 5th grade in the 1970's I struggled with this, and came up with a powerful learning tool: fraction rectangles. The idea is that it is… Continue reading Fraction Rectangles

# Blog

## Seeing is Believing?

“Proofs Without Words” are proofs based on a visual representation of a theorem which provides a convincing argument about its validity without the need for any accompanying text. The genre has been much enriched by the increased availability of computer animation. This is of course relevant to math education: many of the concepts we teach can be illustrated visually, including with… Continue reading Seeing is Believing?

## Transformational Geometry for Teachers

I taught geometry for decades, starting in the 1980’s, and loved it. I’m reasonably good at manipulating algebraic symbols, but I don’t especially enjoy it. In contrast, I am happy to spend plenty of time on visual puzzles, and I am enthusiastic about sharing that passion with colleagues and with students. Early in my high… Continue reading Transformational Geometry for Teachers

## Tiling in GeoGebra

In my last two posts, I promoted the idea of using tiling (tessellation) as an interesting context in geometry class, especially for the introduction of some basic ideas of transformational geometry. One reason this works is the connection with art, including the abstract patterns in Islamic art and the mind-bending creations of M.C. Escher. John Golden is a connoisseur of… Continue reading Tiling in GeoGebra

## Tiling and Transformations

In my last post, I argued that tiling is a good topic to include in a geometry program. Students find it engaging: on the one hand, it connects with art and culture; on the other hand, it provides a context for student creativity. At the same time, it presents many connections with the geometry curriculum in middle school and high… Continue reading Tiling and Transformations

## Tiling

Covering the plane with an unlimited supply of identical tiles is called tiling the plane, or tessellation. Over the years, I’ve developed a number of classroom activities about tiling. You can find links to those on the Tiling home page on my website. In a conversation with a teacher a few months ago, I realized… Continue reading Tiling

## Virtual Manipulatives: Part 2

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

## 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