Educational technology ("edtech") in mathematics classrooms once meant "we use calculators" (or overhead projectors!). To use the words of the memorable Virginia Slims ad, "You've come a long way baby!" Indeed, educational technology has come a long way. It now encompasses a myriad of web-based tools (Web 2.0), computer technologies (tablets & iPads), mobile technology, apps, and other "smart" technologies incorporating both hardware & software. All this technology certainly affects HOW we teach and learn mathematics. Today, though, I also want to talk about how this technology affects WHAT math we teach and learn. Specifically, I want to focus on a geometry topic that was largely omitted from K-12 mathematics before technology made it possible (and even desirable) to include it: isometries!

An *isometry* is a type of geometric "transformation" or movement that occurs when an object is moved in a special way within the plane to create a new image. We refer to the object as the "original" and the resulting figure as the "image". In an isometry, the original and the image are *congruent* (have the exact same size and shape). Isometries are often referred to as "rigid transformations" (rigid, because the original is not distorted in any way). And yes, there are non-rigid transformations as well -- think of what happens when you enlarge or reduce something on a copy machine (or in PhotoShop) -- but today's blog is focusing only on the isometries.

There are three rigid transformations that are now regularly taught in the K-12 mathematics curriculum: translations (aka "slides"), reflections (aka "flips"), and rotations (aka "turns"). The rigid transformations form the basis for understanding of symmetry, tessellations (tilings), and many other geometric concepts.

Before technology allowed us to create and easily manipulative planar objects on a computer screen, we taught the rigid transformations using paper or patty paper. Now, we can utilize any of a great number of dynamic geometry software tools. My personal favorite is The Geometer's Sketchpad (GSP). I've been using this software ever since it was first created and really love it. The original creators, Key Curriculum Press, have done a great job in subsequent versions by adding wonderful (and cool) new features and they have done an even greater service by creating a whole community of educators who freely share their sketches. The one down-side is that this software is not free (though there are free trial versions and a low-cost app for the iPad). For that reason, I also want to tell you about an open-source dynamic geometry software: GeoGebra. I admit I have not used it as much as Sketchpad, but what I've seen I really like and it being free means that many more students and teachers will get the chance to use it!

These dynamic geometry technologies now allow us to teach the rigid transformations in a much more engaging and, er, dynamic way! Students can discover properties and characteristics on their own instead of just being told what they are. Plus, the technology means the results are much more accurate than we could be by hand with paper AND can be arrived at much more quickly than we could do before.

Here's an online activity you can give to your students (or home-schooling parents, to your children): (click the image below)

As you saw, the activity is designed so the learner creates the sketch, exploring and discovering properties about the rigid transformations along the way. For those of you who'd like the "key" so you can see an example of the kinds of answers you'd like to see from your students or children, here's my screencast showing how their sketch may turn out:

Finally, I will end this blog post with a catchy and fantastic video on "transformation style": ENJOY!

As always, I'd love to hear from you. Contact me.

**References & Resources:**

**(Note: The Geometer's Sketchpad is abbreviated GSP)**

- Dr.Perdue's List of GSP activities
- Dr.Perdue's Transformations Activity for GSP
- Dr.Perdue's Help Page for GSP
- Dr.Perdue's screencast for the transformations geometry lab
- KeyPress site for GSP (info & free download)
- Tutorial from Key on using GSP in transformations
- Webinar from Key on creating transformations using GSP
- Link for GeoGebra (open source)
- Worksheet for rigid transformations using GeoGebra
- Nice YouTube video overview on transformations

The Solver Blog

Author: Dr. Diana S. Perdue