Unplugged Project: Boxes Without Topses and Pentominoes

graphic from the Wikipedia Pentamino article

The Unplugged Project this week is boxes. I often do the following "boxes" activity at school.

Our text book, Math Thematics, has a problem in which it presents several nets constructed with five squares and has the students decide which net will make a cube. I follow this problem with Marilyn Burns's activity "Boxes Without Topses" that can be found on page sixty in The I Hate Mathematics Book. I make a template of squares for the students and copy them onto oak tag or card stock. The squares are the size of the squares on the checkerboards that are in the school. Each student has to carefully cut five of these square out. I find that sixth graders and older can cut well with scissors. For younger students, I would suggest that you cut the squares for them.

Challenge the students to arrange their five squares in different ways with two rules: the edges must be touching and their corners must be aligned. Show them what is acceptable and what is not. There are twelve ways to arrange the five squares (please use the correct vocabulary: these are squares and not boxes). They are shown at the top of this post. Do not help them find the twelve combinations. They may get frustrated, but they will feel prouder of their accomplishment if you encourage and not enable. For this reason, it is best to have at least two students doing this activity. Have them sketch their combinations so that they can keep track of them. Sometimes I tell younger students that there are twelve combinations. I do not tell older students. Flips of shapes do not count as another shape.

After they successfully find all twelve combinations, we talk about the shapes and what letters of the alphabet they represent. This gives us a common vocabulary for discussing the shapes. I also tell them that these shapes are called pentominoes. I often conclude this part of the activity with a discussion and table of the other shapes with squares: dominoes, tetrominoes, etc. and how many different combinations can be made with one square, two squares, etc. Is there a pattern in the number of shapes that can be made with different numbers of squares? (I leave that exploration to you and yours.) Please be sure to read the Wikipedia Pentominoes article and become familiar with all of the enrichment materials that are easily available.

In the next part of this sequence, I have them create the shapes. They trace their squares onto clean and colorful cardstock and carefully cut them out. Even though they may not cut well, I feel it is important that they complete this part of the activity themselves, even if they use up a pile of expensive cardstock. Each child has her own unique color for her pentominoes.

When they are complete, I ask them to make shapes out of them (like tangram pictures). The ultimate challenge is that they use all twelve shapes to make one square rectangle. That is a very difficult activity. I have had only one student able to make that square a rectangle (and I have never been able to).

The next activity is with the checkerboard. I pair students and tell them the rules of the game: taking turns, they place one pentomino on the checkerboard. The last person to be able to place a piece on the board wins. Let them play often enough to develop a winning strategy. Would it make a difference if they played with one pentomino set or two?

For older, higher level students, I finally introduce them to John Conway's Game of Life. A good java applet for the game is here. If you have progressed this far with these activities, can you discover why Life is related to pentominoes?

The Game of Life opens up whole new worlds of mathematical exploration into his other game, Sprouts, and triangular numbers. I will explore that in a future post. I hope that teachers and parents develop their students' imaginations with this and other mathematical explorations. Mathematical games, puzzles, and recreations are rich sources of topics.

Visit other participants of the Unplugged Project here.

Next week's project is open-ended and the theme is THANKFUL.

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Boxes without topses? Do you remember this song?

In a cavern, in a canyon,
Excavating for a mine,
Lived a miner, forty-niner,
And his daughter Clementine.

Oh my darling, oh my darling,
Oh my darling Clementine
You are lost and gone forever,
Oh my darling, Clementine.

Light she was, and like a fairy,
And her shoes were number nine,
Herring boxes without topses,
Sandals were for Clementine.

Oh my darling, oh my darling,
Oh my darling Clementine
You are lost and gone forever,
Oh my darling, Clementine.

Technorati tags: boxes+without+topses squares pentominoes John+Conway sprouts polyhedral+nets

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