It all started over an hour ago when I clicked on the latest Math Munch blog entry. I was immediately intrigued because of this image:
I’m a sucker for tilings. The Math Munch post was about Marjorie Rice, who was influenced by Martin Gardner, who wrote about hexaflexagons, on which Vi Hart created a series of videos. Whew! Some people may call this web surfing procrastination I prefer to call it research. If you check out all of the links in the above two lines, you can decide whether your are procrastinating or researching. Whichever you want to call it, I think you’ll have fun.
Martin Gardner, who died in 2010 at age 95, was famous for his math puzzles published in Scientific American in the mid-20th century and wrote over 70 books. His work is often described as recreational math. The word recreational has the connotation of “play”. I’ve always enjoyed looking at fractals, tesselations, tilings, origami, etc. However, my appreciation has always been more visceral and artistic than mathematical.
When I was a high school teacher, I used these recreational math activities as fillers, but never studied them in their own right. Perhaps this is because of all of the other non-recreational (read dry and lifeless) math concepts that I was required to teach. More and more, in my work now as a mathematical consultant, I am trying to figure out ways that this so-called recreational math can be used in a way that reinforces the important mathematical concepts while engaging students.
One activity that I developed based on the MARS Roman Mosaic Task. The tasks calls for students to describe the mosaic verbally without the audience having the ability to look at the original mosaic. Describing the picture requires students to use many of the CCSSM Practice Standards: #1 Make sense and persevere, # 2 Reason abstractly and quantitatively, # 3 Construct viable arguments, # 6 Attend to Precision, # 7 Look for and make use of structure, # 8 Look for and express regularity in repeated reasoning.
I took this task one step further and asked students to create the task using geometric software such as GeoGebra or Geometer’s Sketchpad. This provides great practice in understanding the effect of transformations on the coordinate plane. It also adds CCSSM Practice Standards # 4 Modeling with Mathematics and # 5 Use appropriate tools strategically.
Roman Mosaic steps on GeoGebra The previous link gives an example of ONE approach to making this shape. In subsequent work with students and teachers, I have seen many different approaches, some more efficient than others. However, the decisions that are made for any approach to solve the problem provide a rich learning experience for students.
As I was reading about Marjorie Rice, I came upon another idea that could be used with students to reinforce, among other things (1) their understanding of mathematical communication via labeling shapes, (2) their understanding of the angles necessary to create tilings, and (3) the idea that mathematics is a dynamic subject that is still under construction.
My idea was spurred by this organized list of the 14 tilings for pentagons that have so far been discovered. Here’s an example of two of them. Notice how the pentagons are described using A, B, C, D, E and a, b, c, d, e.
Students, working in groups, would be given the 14 pentagon tiling.
Idea for tasks, one with clear answers and the other open-ended:
- sketch the pentagon unit that is tiled. Label all sides and angles with the appropriate capital letters and lower case letters to match the definition of the tiling that is given below each image.
- Compare and contrast the tilings. Group them into groups involving 90 degree, 180 degrees, 360 degrees, 120 degrees in their definitions. How are they alike? How are they different?
Artistic extension: Go to Marjorie Rice Intriguing Tesselations to see the tesselations that Marjorie Rice created from the tilings.
- Have students discuss the difference between a tiling and a tesselation.
- Long term project: Have students create a tesselation of their own from one of the 14 pentagonal tilings.
In conclusion, there are so many ways to get caught up in spirals upon spirals of ideas that are available at a click on my computer. I think it is time well spent. Others may call it procrastination. I think I’ll go play a game of Free Cell, which is my go-to procrastination tool…short and sweet…and I usually win!