## Teacher-to-Teacher

Why study fractals?
fractals, anyway?

Making fractals
Sierpinski Triangle
Using Java
Math questions
Sierpinski Meets Pascal
Jurassic Park Fractal
Using JAVA

It grows complex
Real first iteration
Encoding the fractal
World's Largest
Koch Snowflake
Using Java
Infinite perimeter
Finite area
Anti-Snowflake
Using Java

Fractal Properties
Self-similarity
Fractional dimension
Formation by iteration

For Teachers
Teachers' Notes
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Fractals on the Web
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Other Math Lessons
by Cynthia Lanius

One of the most exciting aspects of publishing curriculum on the Web is communicating with people using your materials - sometimes halfway across the country (or even the world). The following exchange between John Gross, now a former sixth grade teacher at H. Austin Snyder Elementary School in Sayre, PA, and me illustrates how the Internet expedites teacher collaboration, and explains some of the basic math used in the unit. With me walking him through the math ahead of time by email, John presented the lessons to 100 sixth graders.

Topic: Math Questions on the Sierpinski Triangle

John: When I worked the problems, I figured the amount shaded in percents rather than in decimals or common fractions. But then I couldn't see the formula. Which works best to represent the unshaded area, fractions or percents?

Cynthia: Some people avoid fractions like the plague, thinking they are really going to be hard. But look at the values below and see which is easier, fractions or percentages. Which form makes the pattern more obvious?

 Step No. 1 2 3 4 Fraction 3/4 9/16 27/64 81/256 Percent 75 56 42 32

John: Is there an easy non-technical way to explain the formula to the 6th grade?

Cynthia: Look again at the chart and notice the pattern. The top number is tripling and the bottom number is quadrupling. Now, how do you express this as a formula?

Look at the first term in the table. I could say that's 31 divided by 41. Now look at the second term. That's 32, right? And the denominator is 42, and the third term is 33 divided by 43.

Do you see the pattern now? That the step number is always the exponent? So, if you wanted to calculate it for 20, and you didn't want to calculate it step by step, you could calculate 320 divided by 420. You'd certainly need a calculator for that! Now for that last leap into the formula: call the step number n instead of 3, 4, or 20. Then the formula would be 3n divided by 4n.

 After the lesson John: Today's lesson generated a lot of oh's and ah's. Every child solved the first four problems when asked for the fractions. Most had them correct. I saw only one student who had developed the same formula that you helped me with. He readily admitted that his Dad had helped him, to which I replied, "Great, was it difficult?" "Yes," he said, "but Dad and I enjoyed the challenge." I have to admit that your page was the first I'd ever heard of fractal geometry. I'm not sure I fully understand it, but am having a wonderful time learning something new right along with the kids.

Discussions of the Koch Snowflake

lanius@math.rice.edu

URL http://math.rice.edu/~lanius/frac/Tch_Notes/sier.html