Cynthia Lanius

Let's Talk about the Math


Table of Contents


  Why study fractals?
    What's so hot about
    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
            Using Java

  Fractal Properties
    Fractional dimension
    Formation by iteration

  For Teachers
    Teachers' Notes

    My fractals mail
    Send fractals mail

  Fractals on the Web
    The Math Forum

  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?

Fraction or Percent Not Shaded  
Step No.1234
Fraction 3/4 9/16 27/64 81/256
Percent 75564232

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

copyright 1997-2007 Cynthia Lanius