The Pentomino Proof

IMG_4991My students enjoy talking about their ideas about books, about their writing, about new content we’ve been studying in Social Studies, and about math concepts. They question one another and listen to each other’s thoughts and strategies.

So why do the words, “Show your thinking,” instill dread and angst?

A wise staff developer once told me, “When we have questions about practice, it always comes back to purpose. What’s your purpose?”

The pentomino question existed long before I stumbled upon it, but when I did, I knew it had strong ties to the purpose of asking kids to show their thinking. I don’t want to see their thinking to know that it matches mine or that it conforms with a checklist I’ve been given. I don’t even want to see their thinking to know that they didn’t get the ‘right’ answer by accident. I really want to know what route(s) they took to arrive at an answer. What methods or ideas, misconceptions or certainties did they explore? The pentomino question creates an arena for kids to work on mathematical proofs without some of the baggage they bring to math (fixed mindsets, rushing to be done first, avoidance, computing without understanding, etc.).

First we imagined the words, monomino (do DOO de doo doo) (this is a single square), domino (two squares attached along one whole side of each square), and triomino (not the triangular game pieces, but three squares attached). These aren’t real things, but we wanted to build from a single square and show that there’s only one way to make a domino, and there are two ways to make a triomino.

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The complete collection- no flipping or rotating.

Then we showed them tetrominos. Together we explored how we could put together four squares to make different tetrominoes. While we were doing this, the terms “flip” and “rotate” came up- these actions do not create new tetrominoes.

From here we introduced The Pentomino Question: How many different pentominoes are there? How do you know you have found them all?

We got to work! We gave kids some time to work independently in their math notebooks first. Then partners came together to share their findings and discuss a plan. The work of composing a proof started here as children talked, questioned, argued, and defended. As they came to an agreement about how to communicate their findings, they began to create posters.

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During the exploration we noticed a few different approaches. Some kids went the random route. They kept drawing pentominoes and checking them against the ones they had already drawn.

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Other kids started with a pentomino, usually the simplest one (five squares in a line) and methodically moved one square at a time until they could no longer make any new shapes.

Some kids started from the five known tetrominoes. They tried adding one square to a tetromino in as many ways as they could to create different pentominoes. They repeated this with each tetromino.

This is about when we realized that having a systematic approach or a foolproof method was just the beginning. Students also needed to convince an audience that their process led them to absolute certainty that they had found all possible pentominoes, using a combination of numbers, pictures and words. We did a gallery walk and read each other’s posters critically.

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A poster that showed all 12 pentominoes and said basically that we know we have them all because if we add another square to any of these, it will not be a new one, was only somewhat convincing.

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A poster that narrated the process, describing each step, was a bit more convincing. As viewers, we wanted to be led through the steps.

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Children agreed that the most convincing posters described each step and also included graphic, or visual, information about how that step was done and what it yielded.

This last poster cleverly shows the tetrominoes on flaps. Lift each flap to see how that tetromino can lead to new pentominoes.

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Words were also important. After all of this, we drafted some guidelines for posters:

  • They should include the question or problem
  • They should describe a process or method, showing each step in order
  • When pictures and numbers aren’t enough, there should be words to help viewers
  • The answer or solution should be clearIMG_4069

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These have also become our guidelines when it comes to “show your thinking” questions. And now we are more aware of WHY this is important.

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