Pentominoes

Materials: Handouts, grid paper

Goals:

¨ Carefully record observations from experimentation.

¨ Make and use manipulatives in problem solving.

¨ Carefully record the results of each step of a solution.

¨ Move from experimentation (i.e., trial and error) to a more systematic approach to problem solving.

¨ Create and describe game strategies.

Step 1:

¨ There is only one way to join 2 squares along an edge and the resulting shape is the usual domino. How many different ways can 3 squares be joined together along their edges? Do not count shapes that are simply flips or spins of another shape. These shapes are called triominoes. List all triominoes.

Suggestion: To help you find all the triominoes, quadominoes, etc., I recommend that you make or find some squares and physically move them around. You can easily record your figures on grid paper.

¨ How many different ways can 4 squares be joined together along their edges? These shapes are called quadominoes. List all the quadominoes.

Suggestion: To find all the quadominoes you might start with a specific triomino and find all the different ways to add one square to it. Be on the lookout for shapes that are made by flipping or spinning another shape.

¨ How many ways can 5 squares be joined together along their edges? These shapes are called pentominoes. List all the pentominoes.

¨ Consider an checkerboard like the one below. Can you fill this checkerboard up with quadominoes? Are there different ways to do it? How many different types of quadominoes can you use to fill up one checkerboard? Record your observations.

Suggestion: Outline an grid on graph paper and use a different color for each quadomino.

¨ How many squares are on an checkerboard? How many squares make up all of the pentominoes? So if it is possible to put all the different pentominoes on one checkerboard, how many square of the checkerboard would be left over?

¨ See if it is possible to put all 12 different pentominoes on the checkerboard at the same time. Is there more than one way to do it? Record your observations.

Suggestion: In this case it might be easier to actually make all the pentominoes and a checkerboard the right size and move the pentominoes around like puzzle pieces. Then record your results on the graph paper.

Figure 1: An checkerboard

¨ Complete "Pentominoes" from Mathematics for Elementary Teachers: An Activity Approach. Keep a record of the observations you made. Carefully describe the strategies you developed in the game from this handout.