My previous blog explored the eight elements of the Common Core math standards:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

In this blog, we’ll look at another great project that brings 3D printing into the math classroom.

Anyone who has played with pattern blocks knows that many tiling patterns can be made with all the shapes in the set. But, while pattern blocks have polygons with three, four and six sides, the don’t have any regular pentagons. One reason for this is that you can’t tile a flat surface with regular (equal sides and equal angles) pentagons by themselves. Any attempt will fail and, as one of my math students said decades ago, “you need a lot of grout.”

But this can be fixed by adding one piece – a parallelogram that exactly fits the gap left when you try to build a tiling pattern with pentagons. And this is where 3D printing comes in. Using this powerful device and some free software, you can design you own set of tiles with two shapes: regular pentagons and special parallelograms. The software I recommend is Inkscape – a free two dimensional drawing program with a special plug-in that allows your shapes to be extruded into a 3D tile. I used a thickness of 2.5 mm, but you can use anything you want.

One reason for having students build their own tiles is that the process of designing the tiles incorporates several of the standards mentioned at the start. In particular, standards 1, 5, 6 and 7 seem relevant. And you haven’t even made the tiles yet!

Once the tiles are made, the real fun begins. To make the patterns interesting, students may want to have different colors for the pentagons and the parallelograms. A good set of tiles might have 50 of each shape, although most patterns will likely use more pentagons than parallelograms.

To get started, students might see if they can build a pattern that had rotational symmetry. One such pattern looks like this:

pentagonal pattern

While this pattern is pretty to look at, it also contains a lot of math. For example, what (if any) is the relationship between the number of pentagons and parallelograms in the pattern? What different patterns with this symmetry can be built with the same blocks? What patterns emerge in the number of tiles needed to expand this design with additional “rings” of polygons? The list goes on, and these explorations easily incorporporate all eight of the CCSS math standards.

While this pattern has rotational symmetry, can you build a pattern that has translational symmetry? Consider the various symmetries associated with this next pattern, for example:

pent tiles 2This pattern is completely different from the first, yet it is made with the same two kinds of tiles. This raises some more questions: How many different tiling patterns can you find? Is it possible to build a tiling pattern that has neither rotational nor translational symmetry?

Hours can be spent tinkering with these tiles, and all the while students are developing their skills at mathematical thinking.

If you want to experiment with these tiles in your classroom or school, send me an email at dthornburg (at) and I’ll gladly send you a pdf file with several educational projects that use 3D printers, each of which comes with step-by-step directions.

I’d say more, but the tiles are calling…