### CA I.2 Snowflake Exploration Part II

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.

#### Teacher Journal: CA I.2

• Snowflake Exploration, Part II  Students struggle with writing a sum in sigma notation once the sum has gotten slightly more complicated than the original examples that they have worked with. In addition for the area formula the first term of the sum does not follow the pattern so it must be kept outside the sigma form. Once they have figured this out though the main confusion comes from the very unnatural conclusion in their mind that an infinite process can yield a finite number. Although when they consider the size of the snowflake physically they can see that the area is not going to be infinite, they still have a hard time connecting that conclusion to the series form of the area. Then they are even more confused by the notion that the perimeter could indeed be infinite despite the fact that it is contained inside a finite space. I think these are great ideas for students to have to confront immediately in this class as there are a lot of concepts both in series and improper integrals that deal with the notion of infinity and it is an area that students have many misconceptions. I like to have a quick discussion after students have turned in the synthesis questions to highlight some of these findings to the whole class as sometimes it seems that the students can have a reasonably good understanding of the concepts worked on but still might not have put them together to think about these bigger ideas.
Posted Jun 15, 2012, 8:53 AM by Mairead Greene
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