Koch Curve Lesson Plan

Curriculum Topics: Iteration, Limits, Infinity, Perimeter, Area, Trigonometry, Logarithms
Fractal Topics: Iteration, Koch Curve, Koch Snowflake
Number of days: 1
Grades: 7 - 12
Harlan J. Brothers    The Country School
Madison, CT
Objective: To use the process of iteration along with basic geometric formulas to confirm the often counterintuitive implications of infinite fractal length.
Linked worksheets  (snowflake.pdf)  (area.pdf)
Optional Lab Resources:
Koch Explorer (Logo software)
(download from www.brotherstechnology.com/math/fractals-edu.html)
Description of Lesson:
1) Introduce the Koch initiator, generator, and rule for iteration.

2) Ask students to calculate how the length of the curve changes with each iteration.

3) Draw and label a line one meter in length. Using it as an initiator, have students calculate how many iterations are required to produce a curve with a length equal to the height of a 10 story building (~ 30 meters). Ask what would happen if we could continue to iterate forever. Is there a limiting value to this process?

4) Draw or project an equilateral triangle on the board. Ask students to describe the shape that is generated at the first level by using each side as an initiator. What happens at the second level?

5) Use projection, handouts (snowflake.pdf), or Koch Explorer software, ask them to compare the amount of detail they see in levels 4 and 5. What is happening to the perimeter of shape, both qualitatively and quantitatively?

6) Using the worksheet (area.pdf), ask students to calculate the area of the inscribed figures. Students who finish all four exercises can try to derive a general formula.

7) Ask for observations about how the area changes at each level of iteration. Will the area of the snowflake ever exceed the area of the circle? What is the reason for their answer?

8) Once they have arrived at an answer, ask how it is that an object can simultaneously have an infinite perimeter and a finite area.

Optional activity: For an initiator 1 meter in length, use logarithms to calculate how many iterations of the Koch curve are required to produce an object the length of a football field (~ 100 meters); 1 kilometer long; the distance to the moon (~ 384 km). How far apart are any two arbitrary points on the curve?

Optional activity: Beginning with the area of the circles on the worksheet (area.pdf), ask students to figure out how close they can come to a rough upper limit for the area of the snowflake. Is this consistent with the value they obtained by directly calculating the area?

Here are more lesson plans from the archive at Michael Frame's Yale University site on fractal geometry.