Number (patterns): Fascinating Fractals
The Problem
Fascinated by the film clip, Ismael decided to construct a simple sketch of a tree using straight lines. The first line that he drew represented the trunk. The second and third lines represented the tree’s main branches. At this point, Ismail recorded the number of lines that he had drawn. (See sketch 1 opposite.) Ismael recorded the number of lines every time a new set of branches were added. (See sketches 2 and 3 opposite.)
Can you draw a fourth sketch?
Using Ismael’s fractal pattern of tree growth, can you work out how many lines would be needed to create a sixth tree sketch?
Is it possible to calculate the number of lines without making a tree sketch? How do you know?
Did you know?
A fern is another example of a fractal pattern that occurs in nature. Look closely at the photograph and you will notice how its repeating pattern gets smaller and smaller.
Visualising the Problem and Getting Started
Bronwyn said, “Six is double three. This means that the sixth tree sketch will have double the amount of lines of the third tree sketch.”
Do you agree with Bronwyn’s reasoning? How do you know?
