This activity investigates how you might make squares and pentominoes from Polydron.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
Caroline and Bronya from Tattingstone School sent wonderfully explained solutions to this problem. Caroline drew some diagrams to help her:
She explains:
The pattern is every time you add on a new smaller spike you multiply by 4 and divide by 3.
Here, Bronya describes how she approached the problem:
First of all I looked at the perimeter of the equilateral triangle. It was 3 units. Then I looked at the perimeter of the star. If one side is 11/3 units then the perimeter is 4 units. Then the perimeter of the third shape. I looked at a section like this: Each section = 1/3 + 1/9 unit = 3/9 + 1/9 = 4/9 unit There are 12 sections so the total perimeter = 4/9 x 12/1 = 48/9 I looked at all the perimeters as ninths: Perimeter 1 was 27/9 Perimeter 2 was 36/9 Perimeter 3 was 48/9 Each time the perimeter increases by one third. I think this comes about because in each section a third of the section is added on. The perimeter of the next shape would be 64/9 because: 48 divided by 3 is 16 therefore increase 48/9 by 16/9 Total would be: 48/9 + 16/9 = 64/9
First of all I looked at the perimeter of the equilateral triangle. It was 3 units. Then I looked at the perimeter of the star. If one side is 11/3 units then the perimeter is 4 units. Then the perimeter of the third shape. I looked at a section like this:
Each section = 1/3 + 1/9 unit = 3/9 + 1/9 = 4/9 unit There are 12 sections so the total perimeter = 4/9 x 12/1 = 48/9 I looked at all the perimeters as ninths: Perimeter 1 was 27/9 Perimeter 2 was 36/9 Perimeter 3 was 48/9 Each time the perimeter increases by one third. I think this comes about because in each section a third of the section is added on.
The perimeter of the next shape would be 64/9 because: 48 divided by 3 is 16 therefore increase 48/9 by 16/9 Total would be: 48/9 + 16/9 = 64/9
Thank you to you both - these are very well reasoned solutions. I wonder whether you could generalise to any shape in this series?