Pentiremeter

New game! But a story first.

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.

What's a Problem?

We had a fun class in the elementary math course today. Introducing SMP1 - problem solving, we got to an interesting question: what's a problem?

Here's the story as told by the residue on my whiteboards.

Schema Activation: jot down about a time you solved a real problem in your life. You won't have to share with anyone if you don't want to - this can be private.

After a few minutes to write, I shared how one of the big justifications for teaching and prioritizing math in school, other than the jobs to which it gives access, is that it teaches problem solving.

 Actually more yes than I expected!

People asked if I meant did math class help with the problem that they journaled about or in general. I said we want to know about problem solving in general, but they could use their instance as a specific.







Next question: is it possible that math class could help teach problem solving? Short time to discuss at table.








These were definite yesses, with a lot of confidence.




One of the reasons I like teaching teaching math is teaching is so much like math to begin with. So rephrasing: our problem is how to go from the current situation to get what we want.

So the next prompt was to quickly brainstorm. Ideas for making this happen.

















I shared that I liked how many of these were things that were up to the teacher. And I paraphrased Marzano, about how there is bad news: a small percentage of factors affecting student learning are under the teachers control; good news: that percent still makes a big difference.

What did they like?

• The emphasis on real life. This brought out uniform hatred for unlikely impractical story problems.
• Logic problems: one of the students shared how engaging and powerful these were for here. I asked about the contrast with real world, but people were comfortable with the crazy logic problems because the emphasis was on how did you do it.
• Teachers can ask for multiple ways and have students compare them.

Okay. So if we're going to teach problem solving, we need problems. I asked a question, then asked them to think about if that was a problem or not. There was a tub of square tiles on each set of two tables.

After they all had answers, I asked them to think about the second part. After a short time to discuss, we went to +cheesemonkeysf 's talking points structure for the statement.

This is a problem.

We're still working on the structure, so some of my feedback was about that. The no comments idea is hard.

Pretty strong agreement. People were willing to share their reasoning with the whole class ...  even the small minority. Maybe especially the small minority?

I was really happy to see the "depends on what the teacher does" idea come up. And I added the "depends on students" too. This is not a problem for me. For first graders, a serious problem - there aren't enough tiles in the tub to cover! For them... well, they talked about methods, chose how to do it, discussed results; these are problem indicators to me.

Then we went back to the question. What answers did they find?

 There was shock at the diversity.

One question I ask a lot is: is this a question where different answers could all be right?

They discussed and...

There was concern about the largest answer being too big, but that table and an adjacent table figured out the first group's table is actually larger.

I shared how measurement is one of my favorite content areas, exactly for this reason that a diversity of answers is to be expected. It can be culture setting.

So, with all this, definitely time for a problem. One of my favorites. How many pentominoes are there?


We played with domino patterns last class, so that's a natural starting point. We agreed that there was only one way to make a domino with two squares. If they touch, they have to share an edge.

With three squares, the issue came up about putting them "in the middle." That's solved by the edge rule. What about the elbows in a different direction? No, the class agrees, if you can turn it to match, it's the same shape.

So what I want to know: is how many pentominoes?



People got to work with tiles and graph paper. No group found all the tetrominoes as a step, which I was trying to suggest. A couple of times I had tables report on how many they had and were they expecting more. 8 more, 9 more, 12 probably, etc. Next time, 12, 14, 15 and a mix of have them all and think there are more. When our time was up, they sent emissaries to the board to draw what they had:

Now the best part: math fight! Are flips the same or not?

They divided up into groups based on their answer: 14 flips matter, 9 flips are the same. They shared reasoning. Flips matter because these are flat things and a flip requires another dimension. And if you try to match them up they do not line up. Flips don't matter because you can get them to line up, and what's the difference between a flip and a turn, really?

I refuse to settle the issue, and ask them to make a complete set before Monday.

If you care to comment or tweet a response, I'll share your answers with them:

• How do you recognize what is a problem for your students?
• Are flipped pentominoes the same or different? Why?