The idea came to me just before class, and the preservice teachers in my geometry & data for elementary course were willing to try and playtest. (Thank you!)

The class before we had defined and catalogued all the pentominoes. (Shapes made of 5 squares that only meet adjacent squares by sharing a full edge. In general, polynomioes.) I introduce them by asking about dominoes, and how do- is for two here. There's only one domino; that's when I impose the edge matching rule. Then triominoes, of which there are two. That's where we introduce the rule that if you can turn them to match, they are the same. On the board I drew

We skip right over 4, and I ask them to find all the pentominoes. We skip tetrominoes for several reasons. The objectives for this lesson are SMP 3 (construct and critique arguments) and running a mathematical discussion, in addition to the math content. We've been talking about persevering in problem-solving, too, so I'm trying to get them to be explicit about how they're trying to solve problems. Finding all the tetrominoes is sometimes a strategy that comes up for our big question: how do we know we have them all? I also want them to make the connection to tetris.

They work in groups (as usual) and occasionally I just ask the tables to say how many they've got. The first round was between 7 and 15. Second round between 10 and 13. Third round between 11 and 14. Time to put them on the board. The argument that usually comes up here is whether two pentominoes are the same if they are flips of each other. This day was a particularly lively discussion. Unusually, most of the class decided that the flips were different, with one main hold out. At one point, the chief counsel for flips are different asks "are we thinking of these as two-dimensional or three-dimensional?" "Ooh, good question!" I say. People argue both ways, and the square tiles we're using are the main argument for three. Then the holdout says "but a flip is just a turn in three dimensions!" We sort that out with lots of hand-waving and reference to snap-cubes, even though we don't have those out this day. (Point for Papert and the importance of physical experience.) Finally, they decide. Flips are different. They iron out to 18 and think they have all of them, despite the lack of a convincing argument that they do. And the frustrating refusal of the teacher to settle it by proclamation.

Next day, we're going to use the pentominoes for area and perimeter. The HW was there choice of questions about puzzles or making rectangles. One student found a 6x15 rectangle, which settled a question. I ask them for the area and perimeter of the pentominoes, and quickly someone says it's always 5 and 12. Conjecture! Rapidly disproved conjecture! Then I give some combo challenges: 3 pentominoes for a perimeter of 30 or more, 4 for 20 or under, 8 for exactly 26, 8 for exactly 36. The first is easy for most, but everyone gets stuck on one of the other three. (So hard to get at the thinking here, though.) After a reflection, finally I ask if they're willing to try a new game. Here's the rules we finally decide:

Materials: Two teams and a set of pentominoes.

Players will add pentominoes to a figure and get points = to how much the perimeter increased.

LOW SCORE WINS.

First team picks a pentomino and plays it. Instead of 12 points (unfair) they get one point for starters.

Second team picks a pentomino and adds it to the figure following polyomino rules. (Shared square edges.)

Alternate until all pieces are played.

Sample game:

Wow, team one was on fire at the end! It was pretty fun, and surprisingly strategic. Students invented more and more efficient ways to find perimeter, moving from one by one counting, to side counting, to eventually getting to a covered this many, added this many strategy. They were surprised you could score 0, and astonished when someone shared they scored negative points. The interesting question of whether trapped empty spaces count towards perimeter came up.

In the long run, I think the game gets repetitive, but it has given students a lot of experience with perimeter by then. If students wanted to play more, I'd challenge them to make a game board with obstacles. You could play this with the Blockus pentominoes, if you have a set, but making the pentominoes is a really good activity, too.

We're not sure about the name. Pentris was suggested. Reduce the Perimeter. Perimeduce. For now the placeholder is: Pentiremeter. But we're open to suggestions

PS: finally made a GeoGebra pentomino set that I like.

I love this! I like Perimeduce best.

ReplyDeleteThere are pentominoes for sale. I found 6 sets for $10 just now. I bought a bunch for $2 a set, so this is a good price, if you want to buy them.

I especially enjoyed your description of the class discussion.