# Teaching Fractals to Ninth Graders

• profhiggins
In summary, fractals are repeating patterns at different scales that can be created using mathematical equations. To write the equation of a fractal, we need to understand recursion and can use examples like the Koch snowflake or computer programs to demonstrate this concept.
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?

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).

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.

## 1. What are fractals?

Fractals are geometric shapes or patterns that display self-similarity at different scales. This means that as you zoom in on a fractal, you will see smaller versions of the same shape or pattern repeating. Fractals are found in nature and can also be created using mathematical equations.

Fractals are a great way to introduce students to the concept of self-similarity and the beauty of mathematics. They also have practical applications in fields such as computer graphics, biology, and economics. Learning about fractals can also help students develop critical thinking and problem-solving skills.

## 3. How can I introduce fractals to ninth graders?

There are many ways to introduce fractals to ninth graders. One approach could be to show them images of fractals in nature, such as in snowflakes or trees. You could also have them create their own fractal patterns using paper and scissors or a computer program. Another option is to teach them about the mathematical equations that produce fractals.

## 4. Is prior knowledge of math needed to understand fractals?

While some understanding of basic math concepts may be helpful, it is not necessary for students to have prior knowledge of math to learn about fractals. The concept of self-similarity can be easily grasped by students of all levels and the mathematical equations used to create fractals can be taught in a way that is accessible to ninth graders.

## 5. How can I make learning about fractals engaging for ninth graders?

There are many ways to make learning about fractals engaging for ninth graders. Hands-on activities, such as creating fractal patterns or using interactive computer programs, can help keep students interested and involved. You could also incorporate real-world examples of fractals in nature or in technology. Additionally, allowing students to work in groups or present their own fractal creations can make the learning experience more interactive and fun.

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