Math Monday::Pentominoes, part 2.

When I last left off our pentominoes investigation, the kids had made the pentomino shapes out of square pieces of paper and laminated them. The sheets of pentominoes looked like this, and then we cut them out:

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So now that they each had 12 pentomino shapes as their game pieces (review of what a pentomino shape is linked through here) it was time to make the game board. The board was for a solitaire game of pentominoes–the goal being to get every one of the 12 pieces to fit inside the game board without any piece overlapping or going outside the board. So, we brainstormed. The question: how can we determine the size of the game board using the pentominoes as a reference.

No one really had any ideas, so I rephrased. ‘What we want to know is, how many pentominoes will fit exactly into a game board. How much room does each pentomino take up? (5 squares, because they’re each 5 squares.) So how much room total will you need if you want to fit them all in a game board?’ This kicked some activity off…counting started…some kids counted each square individually, some kids picked up their pieces and counted by 5’s, and some kids just did the mental math 12 pentominoes times 5. They all came up with 60 squares.

I wrote 60 down. So our game board needed to fit 60 squares, no more or less. To use mathmatical speak, the area of our board shape needed to be 60. Now that we knew that, we could find the dimensions of the board and then make one out of contruction paper.

So the next question was, what size board would give us an area of 60.

Not a lot of ideas.

Rephrase: “to find the area of something, you multiply the length times the width. We want to find the length and the width…what two numbers multiplied together equal 60?”

At this point, they could use calculators, multiplication cards with the math facts, or their own mind. Numbers started trickling in.

“1 and 60!”
“4 and 15!”
“6 and 10!”
“2 and 30!”
“3 and 20!”
“5 and 12!”

With those factors, we could start to narrow down our game board shape. Some of the dimensions wouldn’t fit any pentaminoes in them, like the 1 by 60 board, so we crossed that out. We kept crossing out number combinations until we arrived at the 4 shapes that were possible.

Each game board has it’s own possible solutions, so I used the wikipedia article here to explain which board would have the most possible solutions and which board would have the least. They decided on what their game board would look like, then started measuring and cutting it out.

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and then, of course, we laminated them. Any excuse to pull out the laminator!

So then they played around with trying to fit their pentominoes onto the game board.

There is a section on pentominoes in the book Shape Patterns (Let’s Investigate) which I notice is for sale on Amazon, used, for .88! (In fact, Amazon has most of the ‘Let’s Investigate’ book series for great used prices…I’ve found these books to be the best math related investigative books anywhere!)

In the shape pattern book there are a handful of two player pentomino games. A few of our favorites were:

*Two people each play with all their 12 pentominoes. They take turns laying down their pentominoes (without a board). Points are scored for each square of another pentomino that you touch with your own pentomino. You cannot overlap, but you can leave gaps and flip the pentomino around. Highest score wins.

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*Two people take turns choosing pentominoes from one (12 pentomino) set. Each player has one minute to arrange all their (6) pieces into a 5 by 6 rectangle. No overlapping, but they can be flipped and turned. Every square outside the rectangle counts as one point…the winner is the one with the lowest score.

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And we just played with the pentominoes for the rest of the time.

Yay for math!

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