Seen Elsewhere

Today, I share some materials not from my website — though not unrelated!

Robert Wirtz

As a young elementary school teacher and math specialist in the 1970’s I came across Robert Wirtz’s materials, and was inspired by them. Here is an example, the one I call “Wirtz’s flags”:

Wirtz flag

The idea is to draw vertical lines, count the intersections, and write the addition that yields the desired number. (The vertical lines, of course, cannot stop partway.) What I love about this is that it can be solved by trial and error, so any kid can do it, but it also leads to interesting discussions and explorations about multiples — in this case, multiples of 4 and 7. This led me, as a teacher, to wonder which numbers can be written as sums of multiples of 4 and 7, and the more general question: which numbers can be written as a sum of multiples of p and q?

When my son was in third grade, we worked out (and sort of invented) the “McNuggets problem” on a napkin at McDonald’s. Seven or eight years later, I included the problem and its generalization in Algebra: Themes, Tools, Concepts. Read more about this, including a full lesson plan, here (scroll down to “Sample Lessons”).

The flag problems are typical of Wirtz’s best ideas: simple setups which simultaneously provide arithmetic practice and interesting problems. You can find much of his work at the Internet Archive. Search for “Robert Wirtz math”.


I have a large amount of geometric puzzle materials on my website, intended for classroom use. Much of that involves pentominoes and polyominoes. In fact, my first publications were pentomino puzzle books decades ago, so I know a lot about this part of recreational math. Still, I recently came across a question I had never heard of with respect to polyominoes: animal, vegetable, or mineral? This is the subject of an excellent activity offered by the Australian Maths Trust. The basic idea is to start with a polyomino, and move one of its squares to another location so that the resulting figure is congruent to the original. If you can move the figure on the plane by doing this repeatedly, it’s an animal. If it can move, but stays anchored to one location, it’s a vegetable. If it cannot move at all, it’s a mineral. If you want to explore this, find an illustrated worksheet here.

Nonperiodic Tilings

I like to include tiling (tessellation) activities when teaching geometry. In fact, I have a Tiling launch page on my website, with links to activities and articles about that. When students discover a tiling of the plane, I ask them to tell me how they know it can keep going “forever”. In some cases, a promising start is thwarted, as in this example (from the answers section in Geometry Labs:


Usually, students justify their tiling by pointing out that the tiles line up in infinite rows of some sort, and the rows can be juxtaposed. In other words, their tiling is regular.

I was always baffled by the existence of nonperiodic tilings. If they’re not periodic, how do we know they won’t run into trouble somewhere? Well, I know nearly understand at least one kind of nonperiodic tiling, the pinwheel tiling, discovered by John Conway. It is based on a 1, 2, √5 triangle. Here is a little piece of it:


I came across this in Patterns of the Universe, the stunningly beautiful coloring book by Alex Bellos and Edmund Harriss, which is where this figure comes from. Apparently the tiles appear in infinitely many orientations, and the vertices have rational coordinates. See an explanation and animation on Wikipedia. This all works because that triangle is a rep-tile. (What is that? See my Rep-Tiles worksheet!)

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