Homeschool Math: Exploring Pentominoes with Middle Schoolers

Shape puzzles are super fun and excellent for mathematical and logical thinking. I've been trying to strew more puzzles for the kids this spring, along with the sensory materials that I'm already used to offering them, and out of everything that I've offered, I think that shape puzzles have been the most popular.

And, of course, it doesn't hurt when *I* become obsessed with what I've strewn!

I happened upon pentominoes in the book Engage the Brain: Math Games, Grades 6-8. It was the first time I've seen them, and I can't get enough of them! They're diabolically simple: five squares must all share at least one side. Twelve original shapes can be made from that rule.

You can simply fiddle with them, putting them together however you like and seeing what you can make, or you can solve puzzles with them, either trying to assemble them into rectangles or squares or fit outlines that others have made. There are some easy, perfect-for-beginners puzzles out there, but we started with these more difficult ones, specifically the 6x10 rectangle with 2,339 reported solutions.

"Over two thousand solutions!" you say. "Why, that must be simple!"

I'm afraid I must disagree:

The kids and I worked and worked and worked on this! The most frustrating thing is almost solving it but for one single piece. Grr!



The younger kid developed the strategy of drawing her possible solutions rather then putting them together. Took much longer to do, but it did look lovely!

Don't tell the children, but I cheated. We were all working together at the school table, and they were so focused and intent that I didn't want to disturb them by leaving, myself, without a solution. If you homeschool, you likely know that the surest way for a child to lose interest in her work is for you to lose interest first. Go take a five-minute phone call and you'll find that the school table has mysteriously absented itself of children when you return!

I figured that the only way that I could walk away from the table without discouraging the kids is if I'd solved the puzzle, but that darn 6x10 rectangle just would not solve itself! And so I cheated. The puzzle page that I linked to earlier has its solutions diagrammed, so I began sneaking peeks at the solution, giving myself more and more pieces that were correctly placed to start with. I think I'd cheated half the puzzle before I finally managed to solve it:

And then, about ten minutes later and completely on her own, so did my thirteen-year-old.

Grr, indeed!

You can make a simple set of pentominoes using just graph paper (I'd recommend the one-inch grids), but you'll notice that we have these handy-dandy, ready-made plastic pieces:



Super-awesome pro tip: they come from our Blockus games! We own both regular Blockus and travel Blockus--AND an almost complete extra set of travel Blockus pieces--all bought from Goodwill. Blockus and Scrabble are two games that I almost always buy when I see them selling for a song at a thrift store or garage sale. I had it in my head that I really wanted to make DIY versions of pentominoes, so the younger kid and I experimented with some unfinished one-centimeter cubes that I have, and we did manage to end up with a couple of sets that I like okay:

I like that these handmade pentominoes are more tactile than a paper model, and that they're three-dimensional, so they have more utility and scope for creativity than the 2D versions. However, they're impossible to make so that they fit together as snugly as store-bought, machined pieces, and that U piece, in particular, I had to remake about four times, and after painting it I realized that I'll have to remake the purple one a fifth time--it's REALLY difficult to keep that middle space open more than a centimeter!

So in this case, I've finally resigned myself to the fact that store-bought plastic is simply better:

I had additionally been thinking that I should make a DIY magnetic version, perhaps to fit in some sort of metal tin, maybe made of Perler beads and with magnets on the backs, but then I realized: duh. I can obviously just use our TRAVEL BLOCKUS set. So that's one more problem solved!

The greatest thing about pentominoes, especially if you have gifted learners and learners at different levels, as I have both of, is how many enrichment opportunities there are with such a simple toy. There are a million ways to play with pentominoes, a million ways to structure activities, a million research projects, a million projects to solve, a million ways to incorporate them creatively into play. Here are some extension ideas and resources, some of which we've used, and some of which I've put on our to-do list for later exploration:

• Chasing Vermeer. We're listening to this right now as our car audiobook. 
• online pentominoes game. The younger kid enjoyed playing through this online game.
• lesson plan. If you need to more formally introduce the concept pentominoes, here's a full lesson plan.
• pentomino alphabet. These solutions are demonically tricky, but I think it would be really cool to cheat the solutions, then use them as templates and simply draw them and decorate them on graph paper for fun.
• printable pentomino puzzles. These didn't work great for us, because the printout diagrams didn't match the sizes of the pentominoes we already have. If you needed a quiet activity that kids could do independently, though, you could print these and the included pentomino templates. Bonus points for printing the pentominoes on magnet paper, popping it all into a metal tin, and having the travel pentomino set of my dreams!
• 3D pentomino puzzles. Here are some templates especially for pentomino sets made from blocks.

See! Now you can be obsessed with pentominoes, too!

P.S. Interested in more hands-on homeschool projects? Check out my Craft Knife Facebook page!