PENTOMINOES & KATAMINO IN EDUCATION

INTRODUCTION

The puzzle Katamino is based on the 12 unique pentominoes that can be created by connecting five congruent squares edge to edge to form various combinations.

Their potential as a motivating teaching resource at Key Stage Two cannot be over-estimated and there are a series of uses to develop students’ mathematical concepts and thinking, their spatial awareness and their problem-solving strategies. Students of all abilities can use them. Operating at a similar level to more able students may enhance a weaker students's mathematical confidence.

Pentominoes can be used to develop understanding of the concepts of area and perimeter, transformational geometry (including enlargement, congruence and symmetry) nets, volume and classification. All these can be illustrated through various activities.

Pentomino problems provide a high level of challenge and encourage students to develop some of the necessary skills such as trial-and improvement strategies, perseverance and reasoning.

Through playing KATAMINO, individuals develop winning strategies based on their developing spatial awareness and learning to plan ahead in predicting the future moves of their opponents.

STARTING OUT WITH PENTOMINOES

Summary

The best thing about pentominoes is that there are only 12 of them, no more, no less. If tetrominoes were used (4 squares joined by the same rules) or hexominoes (6 joined squares), there would be only 5 or 35 different shapes respectively, either too few for much investigative work and or too great.

A starting point is to see how many pentominoes can be found once students have been told what at a pentomino is. It can be left open-ended to see how many the students can discover. Providing the students with squared paper and scissors for this activity means that they can draw them easily and then cut them out to ensure that they have not just found a rotation or reflection of another. This investigation requires a systematic approach. This could be demonstrated by finding all the tetrominoes together in the introductory activity. It is best tackled in pairs so that students can discuss the shapes and begin to classify them as they search for more.

Learning objectives:

* Solve mathematical problems by working in a systematic and co-operative way
* Recognise the congruence of shapes in different orientations
* Make and describe shapes and patterns

Activities

Having collected our set of 12 pentominoes, the best way for describing and identifying each shape is to liken it to a letter of the alphabet. This helps students picture the various pentominoes and to classify them.

Sorting our shapes by different criteria:
* Length of perimeter
* Whether or not they could be folded into an open top box
* Number of lines of reflective symmetry
* Order of rotational symmetry
* Number of sides- naming of irregular polygons

Activities along these lines could be used to develop the following learning objectives:
* Classify and describe 2-D shapes referring to their properties
* Visualise 3-D shapes from 2-D drawings and identify different nets for an open cube
* Measure and calculate the perimeters of simple shapes
* Identify lines of symmetry in simple shapes and recognise shapes with no lines of symmetry
* Recognise rotational symmetry in simple shapes

USING KATAMINO IN THE CLASSROOM

The puzzleboard containing separate challenges to fit pentomino pieces into different sized rectangular grids is naturally differentiated for all levels of ability, depending on the size of the rectangle you choose to fill. Reasoning is developed as these challenges are attempted, e.g. you cannot leave a gap that is less than 5 squares on the board. The motivation to succeed grows with each successive success and the level of challenge increases.

Through attempting these challenges, teachers can draw pupils’ attention to the links to area, areas of rectangles, and the multiples of 5. It should be sharpening pupils’ spatial awareness, mathematical thinking and problem-solving attitudes.

In playing the two-person game of Katamino, students will develop some of the skills of strategic game play and develop further spatial awareness.

Pentomino Rectangles

A link can be made to factors of 60 through challenging the students to make different rectangles using all 12 pentominoes. They attempt to make rectangles with sides of 6 x 10, 5 x 12, 4 x 15 and 3 x 20. The last one is the most difficult as there are only three solutions.

Perimeters and Pentominoes

By combining two pentominoes by at least one side, can the pupil make a shape that has a perimeter of 15? 19? More than 19?

Using any two pentomino pieces, the students can be challenged to combine them in such a way as to create the largest and the smallest possible perimeter.

Pentominoes and Area

By using all 12 pentominoes, and ensuring that they touch by at least one side, what is the largest area that can be enclosed? For this activity, the students would need squared paper the same size as the pentomino squares.

Enlarging Pentominoes

All the pentominoes, except V and X , can be enlarged by a factor of 2 by using 4 of the other pentomino pieces. The piece being enlarged will not need to be used in this. Each pentomino can also be enlarged by a factor of 3 by using 9 of the other pentomino pieces. The piece being enlarged will not need to be used in this.

Congruent Pentominoes

It is possible to fit pairs of pentominoes together in such a way that they form the same shape as another pair of pentominoes. It is possible to use all 12 pentominoes to create three identical and congruent shapes using 4 pentominoes for each - a real challenge!

Missing Pieces

Using the Katamino puzzleboard, which gives you a ready made 5 x 13 rectangle, fit all the pieces in, leaving a 5 square gap somewhere in it which itself is in the shape of one of the pentomino pieces. In other words, the 5 empty squares are together on the board and create a pentomino shape. All the pentomino shapes can be ‘created’.

Checkerboard Challenge

Using the 8 x 8 square chessboard in the Katamino set, fit all the pieces in. It is possible to do this in many ways so that the gaps you leave form symmetrical patterns, square tetrominoes etc. This square tetromino can be positioned anywhere in the checkerboard and a solution is still possible!

This article is extracted from "The Uses of Katamino and Pentominoes at Key Stage Two", by Ruth Tomsett from Brunel University. Please contact us if you are interested in writing an article on Katamino or Pentominoes, or would like any more information on the uses of Pentominoes in education.