Purpose: To introduce the Koch Snowflake and investigate the area and perimeter (in synthesis) of this snowflake. To achieve this, sigma notation, finite sums, limits of finite sums (series), approximating exact limit with finite sums and error considerations are all introduced.
Classroom Procedure: It is very helpful if you can get through the first page of this activity on the same day that you finish the Part I activity and let students go home to work on and practice sigma notation. This gives them time to become familiar with it before having to use it in a more sophisticated way to represent the area of the snowflake. Students will need time to work on 1 and then discuss it as a class to make sure everyone is understanding the sigma notation and what it means in this case. Then students are given time to work on 2 and 3 followed by a class discussion of these questions and 4. Then students can take time to work on 5 and think about 6 followed by class discussion of both. Finally if there is time to at least start 7 in class that is good although students can be sent home with that along with the synthesis questions for this activity if there is no time. There should be class discussion of 7 after students have had time to work on it as some confusion is highlighted hear about how the area of the finite snowflakes are related to the area of the Koch Snowflake.
Ideas this Activity Builds On: Again as this is a continuation of the first activity there is not a lot that it is building on - however it does build on students having a good understanding of the finite snowflakes developed in the first activity.
Introduction/Motivation of the Activity: As this is a continuation activity there is not much motivation given here other than continuing our work from Part I.
Need to Establish by the End of Activity/Wrap-Up:
Sigma notation and how to use it
Finite sums and how to calculate them in Mathematica
Series and how they relate to finite sums
That a series (an infinite process) can yield a finite number.
Additional Notes: There are synthesis questions for this activity that deal with the perimeter of the Koch snowflake and two other geometric series (one finite, one infinite). It is important for students to work on synthesis questions outside of class for time reasons and to give them experience with working independently on deep problems without an instructor being at hand.