# Pentominoes

### Task 66 ... Years 4 - 10

#### Summary

A story shell in a family context is used to introduce some classic spatial problems which use a set of tiles, each made from five joined squares. The difficulty of the challenges increases and those listed on the card are only a sample of the possible problems that can be built around these pentominoes.

#### Materials

• 12 pentomino shapes
• 2 investigation boards

#### Content

• area
• spatial perception
• mental arithmetic
• problem solving

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

A seemingly endless collection of puzzles has developed around these 12 pentomino pieces and those on the card are just a few. To assist in describing solutions the shapes can be assigned a capital letter of the alphabet which is suggested by their shape. What letters would the students suggest for each shape?

Solutions to the card are:

Question 1

Question 2
Question 3
Students may need to tackle Question 3 several times before they find the solution. It doesn't help them to give the whole solution, but perhaps revealing a starting piece is reasonable.

• Question 3 can become a game. Players take turns to select from the twelve pieces until they have six each. Then they take turns to place the pieces on Shape C one at a time. The first player unable to place a piece loses.
• Each of the twelve pentominoes is 5 square units. Their total area is 60 square units. Shape C presents 60 square units as a 10 x 6 board. Other factors of 60 are 4 x 15 and 5 x 12. Students could make these boards and try to place all the pieces on them.
• Can the pieces be placed on an 8 x 8 board so that there is either a 2 x 2 gap in the centre or an uncovered square in each corner?
• One piece is a + shape. Place nine of the other pieces together to make a + which has each length three times longer than the single piece +.
• Eight of the pentominoes can be thought of as the net of an open top cubic box. Imagine which ones, then check your prediction either by cutting and folding graph paper, or using 3d Geoshape squares.
(Solution: L, Y, T, F, S, X, W, J)
• Pentominoes are made from 5 unit squares joined only by their sides. Are these 12 the only possible pentominoes? How do you know?
• What happens if the basic unit is an equilateral triangle. How many pentiamonds are there? How do you know when you have found them all?
• Construct your own set of pentiamonds and try to invent some puzzles using them.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

A pentomino set for each pair is one way to turn this task into a whole class investigation. Multiple sets can be purchased, or you could use the drawing or table tools of a word processor to quickly create a master copy. Solution 3 above could be the template. Another approach is to use this task in a work station, probably with other spatial tasks, and rotate students through the station over a period of time. Keeping a class 'wall journal' near the work station helps each group of learners build on the work of previous groups.

At this stage, Pentominoes does not have a matching lesson on Maths300.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Pentominoes task is an integral part of:

• MWA Space & Logic Years 5 & 6
• MWA Space & Logic Years 9 & 10
This task is also included in the Task Centre Kit for Aboriginal Students.