Caroline and Bronya from Tattingstone School sent wonderfully explained solutions to this problem. Caroline drew some diagrams to help her:
||Side AB = 1
Perimeter = 3 x 1 = 3
||Side ADB = 4 x 1/3 =
Perimeter = 3 x 4/3 = 4
||Side ADB = 4 x 4/9 =
Perimeter = 3 x 16/9 = 48/9 = 16/3 = 5 1/3
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
Thank you to you both - these are very well reasoned solutions. I wonder whether you could generalise to any shape in this series?