Teaching Fractals to Ninth Graders

[profhiggins]

(sorry if this is in the wrong section) I teach a ninth grade class at a public high school and had a student ask me a question about fractals. We were reviewing complex numbers in Algebra 2 when she asked me how to write the equation of a fractal. How would I explain this or go about teaching this to a high schooler?

 

[nonequilibrium]

The idea is actually pretty easy to explain once the student knows complex numbers, isn't it? But maybe for inspiration you can check how some authors do it for their readers, e.g. Penrose in The Emperor's New Mind (excellent book, fyi).

 

[martane]

Hi there! Fractals are fascinating mathematical objects that have a repeating pattern at different scales. They can be found in nature, such as in snowflakes, trees, and coastlines, and can also be created using mathematical equations.

To write the equation of a fractal, we first need to understand the concept of recursion, which is when a process repeats itself over and over again. In the case of fractals, this process involves taking a simple shape or pattern and repeating it at smaller and smaller scales.

One way to explain this to a high schooler is to use the example of the Koch snowflake, which is created by repeatedly adding smaller triangles to the sides of an equilateral triangle. The equation for this fractal can be written as:

x = (1/3)(x1 + x2) + (2/3)(x3 + x4)

y = (1/3)(y1 + y2) + (2/3)(y3 + y4)

Where x and y represent the coordinates of the new points being added, and x1-x4 and y1-y4 represent the coordinates of the original triangle.

Another way to approach teaching fractals is to use computer programs, such as Geogebra or Mandelbrot Set Explorer, to visually demonstrate how fractals are created and how their equations can be modified to create different types of fractals.

I hope this helps and feel free to ask any follow-up questions! Fractals are a fun and interesting topic to explore in math.