Discovering mathematics in nature.
Most mathematics that we study in school is old knowledge. Around 300 B.C. a mathematician by the name of Euclid organized the geometry we have been studying this year in class. You can thank him for all the beautiful postulate and theorems that we now have in our math toolboxes.
Much of fractal geometry, however, is new knowledge. Fractal geometry and chaos theory are providing us with a new way to describe the world. Many objects in nature aren't formed of Euclidís squares or triangles, but of more complicated geometric figures. Many natural objects - ferns, clouds, seashells - are shaped like fractals.
Fractal geometry is a new language used to describe, model and analyze complex forms found in nature. Chaos science uses this new fractal geometry.
Your task is to become introduced to fractals, both the history and some current applications. After getting a basic understanding, you will be asked to build a Kochís Snowflake fractal of your own.
∑ View the Fractals in Nature Gallery
∑ Access the information in the Reference section
∑ Using these Reciprocal Teaching strategies,
determine what fractals are and how they relate to mathematics.
1) What is a simple answer to the question What are Fractals?
2) Give an example of this simple answer and explain what it means.
3) What is Fractal Art?
4) What is not Fractal Art?
5) How long is the coast-line of Great Britain?
Lesson Procedures: Students will study properties of fractals by utilizing the fractal gallery and online resources. The timing of the WebQuest is at the end of the school year when all standardized and district tests have been taken. My hope is that it will launch the students into continue to investigate fractals during the summer. The lesson is designed for students to work independently.
Additional Web Resources:
Connection to standards:
2M-P2 Use appropriate technology to display and analyze data.
3M-P1 Model real-world phenomena using functions and relations
3M-P3 Analyze the effects of parameter changes on functions using calculators and/or computers.
3M-P4 Interpret algebraic equations and inequalities geometrically and describe geometric relationships algebraically.
4M-P1 Interpret and draw three-dimensional objects
4M-P2 Represent problem situations with geometric models and apply properties of figures.
4M-P6 Recognize and analyze Euclidean transformations
Answer to the question: How long is the coast of Great Britain -
How long is the coast-line of Great Britain? At first sight this question may seem trivial. Given a map one can sit down with a ruler and soon come up with a value for the length. The problem is that repeating the operation with a larger scale map yields a greater estimate of the length (Fig. 1). If we actually went to the coast and measured them directly, then still greater estimates would result. It turns out that as the scale of measurement decreases the estimated length increases without limit. Thus, if the scale of the (hypothetical) measurements were to be infinitely small, then the estimated length would become infinitely large!