Judith Montgomery, Director Monterey Bay Area Math Project Polyominoes Judith Montgomery, Director Monterey Bay Area Math Project

Agenda for Polyominoes We will: Explore Polyominoes Learn and play BLOKUS, (Polyominoes game)

A Geometry Lesson- Polyominoes Students will Work with the concept of congruence Develop spatial relationships, Apply logical thinking.

…and they will gain experience with geometric transformations TRANSLATIONS (slides) ROTATIONS (turns) REFLICTIONS (flips)

Beginning the Lesson Consider the different shapes that can be made for 1,2,3,4 or 5 tiles. The shapes made from 1 square are called MONOMINOS. The shapes made from 2 squares are called DOMINOES. The shapes made from 3 squares are called TRIOMINOES The shapes made from 4 squares are called TETROMINOES. The shapes made from 5 squares are called PENTOMINOES.

Starting with dominoes... they all know this one.

Developing all the names of the shapes

Work with your group and use the tiles Each team member has 5 tiles and a piece of different colored grid paper. Use the 5 tiles to build a shape, Build all the shapes using 1,2,3,4 or 5 tiles. Record that shape on the graph paper, Be sure each shape is unique. (Not congruent to one you already found)

How to create a “legal” shape

In your group Start working. When someone finds a shape look to see if it a new one. Then all team members record the shape in their color and cut it out. Continue until you think you have found all the pentominoes. (How many exist ??)

How Many? How many shapes are there? Any idea why you think that?

The work can be posted

Students cut shapes and made a class chart.

Then they record in their math journals the shapes

These shapes are also the 21 shapes of the game BLOKUS

Let’s Play BLOKUS Pass out the rules

 The game is played on a board divided into 20 rows and 20 columns. There are a total of 84 game pieces, organized into 21 shapes in each of four colors: blue, yellow, red, and green.

The standard rules of play for all variations of the game are as follows:

To begin The order of play is based on color, with blue going first, followed by yellow, red, and green.

The first piece played of each color is placed in one of the board's four corners. Each new piece played must be placed so that it touches at least one corner of the same color, with only corner-to-corner contact allowed—sides cannot touch.

only corner-to-corner contact allowed! THIS SHOULD BE 30 – 45 INTO THE SESSION

When a player cannot place a shape, he or she passes, and play continues as normal. The game ends when no one can place a shape.

Quote from an exit interview about BLOKUS of an afterschool math club group. A math game like Blokus is a particularly special treat for students because it looks and feels like a proper board game - all numbers and theories are hidden in the alluring, tangible game-pieces - in other words, it successfully evokes the aura of something that ought not be played in math class, and thereby captivates it's participants. To play is to noticeably stop the running of math problems and activities, to stop class-as-usual, open up this big colorful box, divvy up the pieces and begin a relatively simple game with few rules, and a clear, exciting trajectory.

I like to start by having each player get to know her pieces, which undoubtedly most young people will do at once without instruction – the curious diversity of shapes beckon fiddling. I ask them to categorize the pieces according to the number of units, which results in five piles, as the piece range from having one to five units each. I will ask them if they can think of any other shapes (with five or less units) and when they cannot, we conclude that we have every possible combination, which is just neat. Now, while demonstrating, I teach them the basic rules: starting from your respective corners, each piece laid down must touch a previous piece, but no more than vertex to vertex, corner to corner. I reassure them that they may and in fact must get all up in and around their opponents' pieces, just no mingling with your own pieces! I display a picture of a completed game in all its chaos and explain that though it may seem all-to-easy at first, it will in fact get quite crowded. Because they have already organized their five-unit pieces in a pile, and recognize that there are certainly the most of these, I usually take the opportunity to give them one piece of advice, one hint before we begin: use up all of these big ones piled up first, and if you can, try and let the single unit piece be your last.

Due to the natural evolution of Blokus game play, many students start out allured by the specialness of the box, the board, the colors, and yet nonetheless begin laying out their pieces with disinterest. However, the magic of Blokus is that the first-time player does not often see the demand for a plan at first, though as the game goes on, and the game board fills, the need for a strategy arises, and suddenly everyone is eager to conclude the first game and start anew, big plans in mind. Generally, most kids have fun with Blokus, and will repeatedly ask for its return in class – though they may feel the need to inform their teacher, “This isn't math!”

Judith Montgomery judithm@ucsc.edu Some Polyominoes links from the MBAMP (http://mbamp.ucsc.edu/ ) website Polyominoes,  Part 1 lesson narrative for discovering Pentominoes ,   Part 2 pentominoes shapes.pdf,  Part 3 Polyminoes lesson plan.pdf and  Part 4 lesson plan pentominoes info.pdf