It’s been a while since I’ve done an evening of math with my two older kids (6 years and 8 years), so this week I decided I would teach them about rep-tiles. (If you don’t know what rep-tiles are, don’t worry. I’ll be explaining that shortly, or check out this excellent article on rep-tiles)
First we pulled out our copy of the board game Blokus (a great game involving placing polyominoes). I laid out the four monominoes on the board and asked them if they could make a square of the four. “Of course!”
I had told them previously that we would be working with rep-tiles, not the animals, but something to do with mathematics. They had been dying to know what a mathematical rep-tile was. I asked them if now they understood. “It’s a shape that you can make a bigger one out of a bunch of them.” I was surprised that they caught on so quick, expecting something more along the lines of “a shape that you can make a square from several copies of it.”
I had told them previously that Utah was a rep-tile (The pentomino that most people call “P”, I call “Utah”, me being from Utah originally). So we did that one next. It took them several minutes to piece four copies of Utah together into a bigger one and they passed the pieces back and forth several times. Here is what they came up with:
I had them select several other polyominoes and decide whether they were rep-tiles. The domino, for instance, was easy. The tetromino Zstumped them. With the aid of paper and a pen I gave them an argument that Z is not a rep-tile. They were convinced and later gave me a similar argument that the pentomino Xis not a rep-tile (how could you fill in a corner with a smaller copy?).
I told them that of course not only polyominoes can be rep-tiles. For instance what about circles? “No,” they said. Packing circles together always leaves gaps, after all. I had previously cut out some polyiamonds (also see: another polyiamond article) printed on card stock. We proceeded to determine which of these were rep-tiles. We spent quite some time on this, but I’ll skip the details and give you the end result. Namely, we pasted our rep-tiles to construction paper. The kids and I were pretty pleased with the results.