pentommoes for fun and learning

Pentommoes for fun and learningAuthor(s): RICHARD A. COWANSource: The Arithmetic Teacher, Vol. 24, No. 3 (MARCH 1977), pp. 188-190Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189222 .

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Pentommoes for fun and learning RICHARD A. COWAN

At the time this article was written, Richard Cowan was a mathematics

field agent for the Del- M od System at the University of Delaware, where

he did inservice work with teachers, taught mathematics education courses, and was a resource consultant for the Science and Mathematics Resource Center at Delaware State College and Delaware TechnicalCommunity College. He is now coordinator for a mathematics basic skills

project in Roanoke Rapids, North Carolina.

л ^^m pentomino is a plane figure that is formed by grouping five congruent, square shapes together so that every square has at least one of its sides in common with at least one other square. Two simple pentominoes are shown in figure 1 . Before read- ing further, how many different pentominoes can you construct? Two

Fig. 1

pentominoes are different if they are not congruent; that is, if one cannot be made to match the other by flipping or by rotating.

Fig. 2

In figure 2, for example, a, b, c, and d are the same pentomino.

A good classroom exercise is to have your students find and cut out of graph paper all of the pentominoes. This is a les- son in perception, originality, discovery, and double checking that the students will find enjoyable.

The author has found that for students in the lower grades, it is better if the students are provided with five individual square

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188 The Arithmetic Teacher



pieces of paper that are the same size as the squares on the graph paper. The children are then encouraged to construct the pentomino shape with the individual square pieces before they cut the pentomino out of the graph paper. In a one-hour period in one third-grade class in which children were doing this, two students found all of the pentominoes. There are twelve different pentominoes, as shown in figure 3.

Once all of the students have made a set of pentominoes, there are many things that can be done with them, depending on the level of the students and the desires of the teachers. Several examples of ways in which pentominoes can be used with classes are included here.

In grades 3, 4, and 5, as students find and cut out the pentominoes, the teacher can point out some of the geometric principles of congruency. For example, a teacher might ask the following questions:

Do all of the pentominoes have the same area? The same perimeter?

Some of the pentominoes can be folded up to make a cube without a lid. Which ones are they?

Using all twelve pentominoes, can you construct a 6-by-10 rectangle? (This is the puzzle used in the commercially produced game, Hexed. A solution is shown in figure 4.)

Fig. 4

Other problems that would be appropri- ate for older students include the following:

Can you find two different pentominoes that can be used to make a 2-by-5 rectangle?

Complete the following table.

Number of Size of Can it be

pentominoes rectangle done?

N NX5 Yes or no

2 2X5 ? 3 3X5 ?

12 12X5 ?

Can you find three different pentominoes that can be used to make a 3-by-5 rectangle?

Given any pentomino, use nine of the other pentominoes to construct a scale model three times as wide and three times as long as the given piece.

There are many activities on tessellations or tilings with the pentominoes that could be used if a teacher finds tessellations inter- esting. Figure 5 shows two tessellations. By using the pieces for coverings, the pento-

Fig. 5

March 1977 189



minoes also can be used to develop young students' concepts of area.

There is also a game that can be played with the pentominoes and a checkerboard made with squares the same size as the squares in the pentominoes. The purpose of the game is for students to see if they can place all of the pentominoes on the checkerboard without overlapping the pieces. The game can be played in several ways. It can be a game of solitaire; or, two players with one set of twelve pentominoes can play against each other on the check-

erboard. Each player draws six pieces from the set, and then, in turn, each player places a pentomino piece on the board until no more pieces can be played. The last person to place a piece on the board is the winner.

More than two also could play the game. In another variation, the person who in three games has placed the most pieces on the board is the winner.

These are only a few of the things that children can find to do with the pentominoes. You and your students may have fun finding others.

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190 The Arithmetic Teacher



pentommoes for fun and learning

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