This week we have an activity that Occupy Math has used for math activity nights at schools. There is often a shortage of activities for younger children at a school math night and today’s post hopes to help with that. An earlier post on kid’s math books includes the idea of a reading table that gets parents and young children involved. The activity *Construction by Instruction* appears in the post on cooperative games. This week is the activity *Tiling the Wall*

You will need a large supply of copies of one of these tiles, depending on the size of your event, scissors, crayons or washable markers, and Scotch tape. You will also need an open stretch of wall on which paper can be taped. Here are the steps.

- Participants color in a tile.
- They cut out the colored tile, with help if cutting on the lines is difficult.
- The tile is then taped on the wall along with other tiles the same shape, showing how the shapes interlock.

The picture at the top of the post shows a small example of how this might come out, with very basic colors.

**Where do the tiles come from?**

You can tile the wall with regular triangles, squares, and hexagons. A *regular* polygon has all its sides and angles the same length. Those are the only regular shapes that will do the job, though there are a lot of other options if you follow the link. The tiles Occupy Math generated for this activity are based on hexagons. You pick a doodle and use it instead of the sides of the hexagon. The example tile above replaces the second half of each side of a hexagon with three sides of a square. You have to draw the doodle so that it faces the same way on opposite sides of the hexagon. In the example above, the square points out of the top side of the hexagon and in on the bottom side. This is what makes the tiles able to interlock properly.

The example tile above is named *Running Man* because it looks a bit like someone trying to run really fast in a cartoon. It is remarkable how, after you apply your doodle, the tile takes on an identity of its own. A way to take ownership of this activity is to make your own tiles. Sketch lightly in pencil to get the original hexagon and then carefully copy the doodle six times, fill with dark lines, and you have got your own tile. Remember: you could also start with a square or equilateral triangle.

**Here are the tiles available in the linked PDF above.**

**Other tiles!**

Occupy Math designed his tiles with a computer, using his own code. There are many different pieces of software on the internet that let you design tiles, but Occupy Math shies away from giving suggestions: they are ephemeral and end up turning into computer support issues. With that caveat, this activity is one that helps prepare younger students for reasoning about geometry and introduces the discipline of tiling in a fun way.

Tiling is a very old idea and a good place to start introducing young children to geometry. The requirement that the shapes fit together conveys the sense of ordered rules that is the mainspring of geometric reasoning without being overly dry or formal. Coloring the tiles permits the activity to also be artistic, helping make one of the natural connections between art and math that we so often miss in our modern teaching of math.

Tilings are a huge deal in both art and math. Mosques are often decorated with intricate tile patterns, like the one shown above. Penrose tilings are a bizarre type of tiling that never repeats its patterns periodically (many finite patterns repeat in a Penrose tiling, but spaced out irregularly). Finding pentagons that can tile the plane is a hard math problem. It even turns out that the regular tilings of the plane with triangles, squares, and hexagons are the infinite versions of the Platonic solids, which are the only completely regular solids.

Tiling is both a simple-to-run activity for your math night and an entry into a complex and diverse set of math problems that are still being worked on today. If you use this activity, Occupy Math would love a picture of the tiled wall (he forgot to take pictures when he was running this activity himself). Remember, Occupy Math is happy to work up activities you suggest as a post. Please comment or tweet!

I hope to see you here again,

Daniel Ashlock,

University of Guelph,

Department of Mathematics and Statistics