My Math Education Blog

"There is no one way"

Thursday, May 25, 2017

Taxicab geometry

A few weeks ago, I led a workshop on taxicab geometry at the San Jose and Palo Alto Math Teacher Circles. Taxicab geometry is based on redefining distance between two points, with the assumption you can only move horizontally and vertically. So the taxicab distance from the origin to (2, 3) is 5, as you have to move two units across, and three units up. This has all sorts of geometric consequences which are fun to explore on grid paper (and with GeoGebra.)

The workshop was largely based on Labs 9.1 and 9.6 from my (free) book Geometry Labs. I quite enjoyed leading those sessions, as I learned a lot both about the topic at hand, and about how to best present it. The approach in Geometry Labs was based on separating a very basic introduction from further enrichment questions. The reason is that the basics are helpful as a contrast to Euclidean distance on the coordinate plane, and help students better understand the latter, while the further explorations were unlikely to be pursued in very many classes.

However in a session for teachers, or for a math club or an 11th or 12th grade elective, there is no reason for that separation. Moreover, in preparing those sessions, I got interested in other questions, including taxicab parabolas. The result is that I created a stand-alone worksheet on Taxicab Geometry that incorporates much of the material from Geometry Labs, somewhat reorganized and edited, and to which I added some interesting extensions. Download it here, and learn a lot about taxicab geometry! (If you use activities from Geometry Labs, be sure to periodically check the Geometry Labs home page for any new connections, corrections, extensions and revisions.)

This bizarre figure is the answer to one of the problems on the worksheet. Can you figure out what was being asked?

Here's a hint:


What prompted this post is that last Saturday, I attended the San Francisco Math Teachers Circle's last meeting of the school year. The session was led by Dr. Cornelia Van Cott, a math prof at USF. After an exploration of taxicab circles and ellipses, we got to think about what circles would look like in other metrics. This was both entertaining and thought-provoking. Each metric had a fun name: the elevator metric, the post office metric, the teleportation metric. Unfortunately, the resulting circles were not as satisfying as the elegant circles of Euclidean or taxicab geometry. Cornelia explained that to be as mathematically productive as taxicab distance, a metric needs to be a norm, i.e. satisfy additional constraints. She wrote about this in an award-winning article for the February 2016 issue of Math Horizons (MAA's journal for undergraduates.) The title of the article is "A Pi Day of the Century Every Year", because different norms lead to different values for π, and thus, you could get a value like 3.1418, which would be perfect for next year.

What blew me away was Dr. Van Cott's explanation of how you can reverse the whole thing. Instead of defining the norm, and seeing what a unit circle would look like, you can go the other way: define your unit circle, and build the norm from it. So taxicab geometry would be derived from a square with vertices at (1, 0), (0, 1), (-1, 0) and (0, -1) (or in fact any square centered at the origin!) Other geometries could be derived by starting from any convex figure that is symmetric around the origin, e.g. a 2n-gon or an ellipse. Apparently, such geometries are called Minkowski geometries. Read the article!

If this sort of exploration is interesting to you, you should join or start a Math Teachers Circle in your area!


Sunday, May 7, 2017


My early forays as a curriculum developer date back to my days as a K-5 math specialist in the 1970's. A key insight of my young self was that activities intended for students were that much more worthwhile if they were also interesting to me. I learned to view with suspicion activities that were boring to me and other adults, but supposedly good for children to slog through. Thus started a career-long search for low threshold, high ceiling problems and puzzles. (That way of describing such activities originated in the 1980's Logo movement, but that is another story.)

Among my first creations were pentomino puzzles whose difficulty spanned the whole range from Kindergarten to adult. (If you don't know what pentominoes are, visit Geometric Puzzles in the Classroom before reading further.) Those puzzles were collected in books and cards that remained in print for an astounding 30 years or so. Alas, those are no longer available, as they have fallen victims to test prep obsession, to the gobbling up of smaller publishers by mega-corporations, and to the latter's lack of interest in actual children's education. If and when time allows, I will make those puzzles available online on my Web site, as I did for the not-unrelated SuperTangram puzzles*.

Anyway, what prompted this post was this message from Meghan, a teacher in Santa Cruz:
I've been using your website like crazy this year. I'm teaching Geometry and used your transformation materials, and mixed in a little bit from your Space class. I used many of your puzzle problems to create a presentation project: each group had a different spatial/area puzzle including the polyarcs. One particularly artistic student drew these, which I thought you'd appreciate.
Polyarcs were my own invention, which you can read about here. Here are Aviva's polyarc creations:

Fun stuff! I'd love to see other student polyarc art! Start them off with this worksheet. (Prerequisites: they need to know how to find the area and perimeter of a circle.)

Meghan's project assignment also included these options, among others: Tiling Rectangles with Polyominoes, and various SuperTangram puzzles*.

Back in the day, I used geometric puzzles in my elementary school classes as activities in a weekly "math lab" session, and as menu options for students who finished other work early. The logistical key was that each student had a folder which listed all the options, and where they checked off the puzzles they had successfully solved.


* Alas, I am the only source of plastic supertangrams. If you sell math manipulatives and are interested in selling them, get in touch!