Cynthia Lanius

  Let's Talk about the Math

Teacher-to-Teacher (2)

 
Table of Contents

   Introduction

  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
         Anti-Snowflake
            Using Java

  Fractal Properties
    Self-similarity
    Fractional dimension
    Formation by iteration

  For Teachers
    Teachers' Notes
    Teacher-to-Teacher

  Comments
    My fractals mail
    Send fractals mail

  Fractals on the Web
    The Math Forum

  Other Math Lessons
    by Cynthia Lanius

   
Topic: Area of the Koch Snowflake

John: Will you explain more about how the area doesn't increase? The lesson states that the area of the figure doesn't increase. It means the area of the enclosing circle, doesn't it? The area of the figure must increase as you draw the flake. I can see that the flake will never go outside the area of the circle. But it seems that the area of the figure itself must increase inside the circle. I also went to the page called chopping broccoli. It really helped me understand the fractal. Thanks for the moral support.

Cynthia: About this area business: once again your understanding is very good. Just as you said, the area of the Snowflake is smaller than the circle circumscribing the figure. It does increase, but it doesn't increase without bound (another way of saying it doesn't go to infinity) the way the perimeter does. It actually illustrates two very important concepts of calculus. The perimeter diverges, getting bigger and bigger without bound, while the area converges, getting closer and closer to a number but never going beyond it. You could tell your students that this is something called limits, and they'll learn a lot more about them in calculus in high school. They love learning something about infinity.

Cynthia: How are you doing with the perimeter formula? I am so impressed with what you are doing and have congratulated your students on the page for their good work.

John: The kids will be excited to see their school on the web - particularly Michael, who was out sick today and sent word to me that he wanted the new project if I gave it today. Made me feel good.

Now the area thing. That's exactly what I meant. I could see that the area of the snowflake had to be getting larger and I think I can get the kids to see it by drawing a circle around their original triangle with a compass. I'm going to try it and see how many kids notice that their flake will never go outside the circle. Then I'll try some of the questions about infinity and perimeter with my more advanced students.They'll see it right off.

You'll never believe how much this whole experience has meant to me. It's taken me back to when I had trouble with math in high school and made me realize that I could have done fine in math if someone had kindled an interest for me like this fractal thing has done.

Topic Below: Perimeter of the Koch Snowflake

John: Let's tackle the perimeter problems. My wife and I have worked on them for several hours and we think we have them, but we're not sure.

Cynthia: Remember the process: You divide each side into three parts, then replace one part with two parts. That means instead of it equaling 3, it now equals 4. Okay, if the original triangle has p = 9 and s = 3, then since the new figure (step two) has s = 4, then p = 12.

Think of it this way: on every step, you replace 3 parts with 4, even if it's 3/3 being replaced with 4/3. Does that make sense? So that means you multiply each time by 4/3. In other words, you increase by 1/3 each time. Look at the table below.

Koch Snowflake - Perimeter
Step No.1234
Perimeter91216 64/3

Now for a formula: We said to get the next step you always multiply the preceding one by 4/3. Let's add a row to the preceding table, rewriting the perimeter in a different form.

Koch Snowflake - Perimeter
Step No.1234
Perimeter91216 64/3
Perimeter9     9(4/3)1 9(4/3)2 9(4/3)3 

See the connection between the exponents and the step number? The exponent is always 1 less than the step number. So if we wanted the perimeter of the 20th step, it would be 9(4/3)19, right? That means we have a formula. If n = the step number, the formula is 9(4/3)(n-1)

After the Lesson

John: The fractal lesson went beyond my wildest dreams. I even had one group, oddly enough my most ambitious one, ask about the area when I mentioned that mathmatically the flake would never leave the circle. It was unbelievable to me. They understood the flake and why it was a fractal!

They understood the perimeter after I introduced it. They answered the questions, which I reworded to fit my sixth grade. Some even asked to color the flake. Many even did the bonus which was to actually measure and draw step four.

I told you I was using these challenge lessons as a break in the routine. Usually after the test is finished and turned in it's hard to keep them "on task." Today you could have read a book in there. All the "noise" was good noise. They were talking about the drawing and the questions. I heard the phrase "it goes up by 4's" and "This is 4 times bigger than that." I bet 50 kids brought the drawing to me, to show me how well it came out. In short, it was one of my better lessons. What a way to teach perimeter, area, and exponents.

 

lanius@math.rice.edu

URL http://math.rice.edu/~lanius/frac/Tch_Notes/koch.html
copyright 1997-2007 Cynthia Lanius