Weaving your way through mathematics

Today’s guest contributor to Moebius Noodles is Patrick Honner, an award-winning high school teacher and a passionate math enthusiast. My favorite part of Patrick’s beautiful blog is the Math Appreciation section, because you can adapt most ideas there for young kids. When I was four or five, I spent a lot of hours weaving with paper. I believe it helped me fall in love with mathematics.

Before starting on Patrick’s game, check out his TED talk on creativity and mathematics! – MariaD

Weaving is a fun and creative way to explore real mathematical ideas. Simple “mat” weaving offers a way to experience basic concepts in geometry and number theory, while encouraging the development of representation and modeling techniques– fundamental mathematical skills.

With some colored construction paper cut into long, thin strips, and some glue or tape, you can get weaving right away! Here are a few introductory activities. More examples and ideas can be found at my website: http://www.MrHonner.com/weaving/

A good place to start is the checkerboard pattern. It is simple, intuitive, and helpful in developing facility with the basic techniques of weaving. Start with two sets of strips of different color; align all of one color horizontally and all of the other color vertically. Now, a simple alternating over-under weave will create the checkerboard.

A more challenging activity with just two colors, each aligned horizontal and vertically, is to weave a tiling of the plane.

This activity definitely requires some planning. Once a type of “tile” is chosen, the weaver must figure out what kind of weaving pattern will produce the desired tilling of the plane. The orange-and-black weave above uses a “short L”-shaped tile and an alternating, 1-over / 2-under pattern. The orange-and-purple weave uses a “long L”-shaped tile and a similar 1-over / 3-under pattern.

Here’s where modeling and representation come into the process. With a blank grid, one can plan out the weave ahead of time, hopefully figuring out what kind of weaving pattern will produce the desired mat. A standard modeling approach can be used, or the weaver can develop their own representation—in both cases, the important mathematical skill of modeling is being developed. Here are some examples of different approaches to modeling various weaves.

Through trial (and error!), the weaver can refine their modeling process and their plans to produce the desired weave.

Once the basic techniques of simple two-color weaving are mastered, more interesting and challenging projects can be undertaken. Using more than one color in an alignment (horizontal or vertical) opens up new patterns, as does using more than two colors. More challenging patterns, tilings, and inversions can be attempted. Here are some examples.

A fun mathematical follow-up to introductory weaving is to consider the question “Which kinds of patterns are weavable?” For example, the following two mats weren’t really woven—some pieces were cut out and taped over other pieces. An interesting and highly mathematical question is, “Would it be possible to produce these mats through weaving alone?”

With some basic supplies and a few simple techniques, significant mathematical ideas can be explored through weaving. And once you’ve mastered the basics, you can start investigating circular weavings, hat and basket weaving, and even try your hands at mathematical knitting!

For more ideas, visit www.MrHonner.com