After students become familiar with pentominoes, they are ready to use them to explore some algebra. Teachers may present this activity either using the white board, or preferably, with the overhead projector and some pre-made transparent pentominoes (colored for easy location) when projected. Using a transparency grid (half-inch or one centimeter grids work well) and pentominoes cut to the same size grid make a great visual aid that can be used over and over for demo purposes. Since defining ordered pairs always begins with the origin, it’s nice to also begin pentomino graphing with one shape located with a vertex at the origin and sides aligned with the axes. Using the simplest shape, the I, record the coordinates of the four vertex points. This reinforces the concepts that ordered pairs located on the x-axis are always of the form (x, 0) and points located on the y-axis are of the form (0, y). Some simple ideas to extend this first task involve asking students: 1. How would the coordinates change if the I shape were rotated 90 degrees clockwise about the origin (keeping one vertex fixed at the origin)? 180 degrees clockwise about the origin? 90 degrees counter-clockwise about the origin? 2. How would the coordinates change if the I shape were translated 5 units to the right? 3 units to the left? 4 units down? 6 units up? These questions/explorations are important to ask/do since they build up visualization and prediction skills. One of the most essential things we can do for students is to give them immediate feedback on their progress; this allows them to gain confidence they are moving along the right track. Next, select another pentomino and place it so that it lies entirely within the first quadrant. This is good pedagogy since kids will be working entirely with positive numbers. After gaining confidence they can move along to quadrants 2, 3, and 4. After working entirely within these quadrants, you may want to place a pentomino so that it lies partially in two quadrants.
Subtraction of integers is another basic skill that can be incorporated into this activity. Asking the length of a pentomino’s side is equivalent to subtracting the integer coordinates.
Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon Golomb