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GEOMETRY EVERYWHERE; FRACTAL, MATH & FORM
by Judy Butts
University of Idaho

ITV SERIES LIFE BY THE NUMBERS: PATTERNS OF GEOMETRY

OVERVIEW

This lesson comes after the study of similarity in triangles. The students will be introduced to the idea of self-similarity. They will learn to recognize self-similarity and appreciate how geometry can help natural scientists in their studies. They will watch Life by the Numbers video, Patterns of Geometry, program segment four, New Geometry, in order to develop a deeper understanding of how we can duplicate nature's patterns by using math and iteration of self-similar mathematical elements.

After the video, the class will break into groups of two. They will do the exercise, "A Tree Grows in Brooklyn," from the Challenge of the Unknown Teaching guide to create their own L-system. This exercise will lead them to make connections between the iteration process and exponentially increasing data. Today and over the next two days, the students in their groups will also do two online exercises. The first is a simple square construction that is divided with diagonals. Iteration is performed on the square dividing it up into smaller and smaller self-similar triangles. The square can be opened demonstrating in a striking manner the birth of a fractal from iteration. The second construction will allow the students to construct a Koch snowflake and design their own fractals. They will turn in one design, with observations about it, and an analysis of what small changes in the initial conditions of their fractal can do to the end product. On day two of the lesson, the students will play the Chaos Game. This will lead them to recognition of the order present even in random processes.

The students will all write an analysis of what they have learned about fractals, and how they see an understanding of fractal geometry as tool for the study of natural sciences. Students will be encouraged to do online explorations of fractals on their own in order to discover more applications of fractal geometry to add to their reports.


LEARNING OBJECTIVES

Students will value the importance of math to the study of the natural sciences, recognizing in fractals a self-similar, iterative geometric process that is everywhere in their natural world.

Students will be able to describe and recognize fractal characteristics such as self-similarity and iteration.

Students will recognize the structure present in random processes such as the Chaos Game.

Students will understand that vast changes can result from small initial deviations.

Students will construct fractals.

MATERIALS

One set of self-similar objects: deciduous tree branch, juniper branch, fern frond, broccoli, pinecone, Christmas cactus, rocks crystals, feather, and pictures of Pokemon characters that demonstrate self-similarity. One per student of the following: two-sided piece of graph paper, ruler, transparency, and a game die.


PREVIEWING ACTIVITIES

Construct the introduction so that the students discover for themselves the meaning of self-similarity.

The teacher will show students set of self-similar objects and ask them to find some characteristic that is shared by all of the objects. Collect their answers on the board. Prompt: If they are not getting it, Draw a circle around a small part of one of the pictures and tell students to think about that part.

If this doesn't do it, tell students that you are looking for the term self-similar and ask them what they think it means. Tell students that the concept of self-similarity has recently attracted the attention of scientists and artists of many persuasions. List several sciences and the causes of their interests (for example meteorologists studying weather patterns and cloud shapes, and Pokemon artists using self-similarity in their characters and backgrounds).

Point out that nature is full of self-similarity and even our bodies show a form of self-similarity in our internal organs and circulatory system. Tell students that self-similar objects have many mathematical characteristics, opening themselves up to explorations and discoveries by scientists using mathematical tools. This exciting new mathematical science is called fractal geometry. Tell students: we will now view a video that will introduce us to a Canadian scientist who is using fractal geometry to explore nature. He is using a computer to repeat patterns over and over so that he can construct natural objects, such as this fern frond, mathematically. Over the next three days you will construct many fractals yourself, some on paper and some on the computer.

 

FOCUS FOR VIEWING

To give the students a specific responsibility while viewing the video tell students that we will be starting a paper exercise today in which we will use the system the scientist in the video is using.

Ask the students to pay particular attention to his system so that they can tell you how it works and what is called at the end of the video. Also ask students to note the Mandelbrot Set, as they will be doing a research project that simply will ask students to bring me 6 facts about this set and a one paragraph write up about what use this set may be to scientific studies.

VIEWING ACTIVITIES

Begin the video at the New Geometry frame. Pause on the triangle frame and explain that they will have an opportunity to construct this fractal as one of their computer exercises. Resume video. Pause the video repeatedly as the camera zooms in for different views of the Mandelbrot Set.

Mention how much excitement this set has caused in many scientific fields. Tell students, you hope that their research will lead students to understand why.

Resume video. Pause when the first computer generated flower comes into frame. Tell the students all the flowers they will see in the video from this point on are computer- generated. Resume video. Stop video on greenhouse frame as Dr. P. walks away from the camera.


POST VIEWING ACTIVITIES

Discussion: What was the name of the system the Canadian scientist was using. Can someone describe it to me? Lets look back at the portion of the video where Dr. P. used the L-system. (Review portion starting at the frame with the rose, and ending on the frame with the finished L-system branch as the music comes up, before keyboard frame.)

Go over the rules for creating the L-system, break into groups of two, and handout exercise, "A Tree grows in Brooklyn", that is designed to be used with the video segment and graph paper.

Tell the students about the computer station online exercises they will be doing over the next few days. Have students sign up for time on each station. There should be enough time for each group to go to each station during the next three days of class.

First computer station online exercise:
The computer will beset on the online site Fractal, by International Educational Software. This site will allow students to easily see how a simple math process is repeated to form a fractal. At this station iteration will be defined for students. They will clearly see a square divided by diagonals again and again used to create a fractal. There will be a card at this station explaining in detail how to use the animation and asking students to write in a few sentences including the phrases self-similar and iteration describing how they think math could be used explore the final shape of the fractal.

Second computer station online exercise: The computer will be set on the animation Snowflake Online by The Shodor Education Foundation
http://storm.shodor.org/snowflake/
archive/186.html

This site creates the Koch snowflake fractal and also allows student to create their own fractals by altering the initial conditions of a straight-line graph and then iterating their initial pattern again and again with a click of the mouse.

You want students to understand what a large effect small variations in initial conditions can cause. You want students to become more familiar with the iteration process. There will be a card at this station explaining how to use the animation and asking the students to move the points around and observe what happens, and look for examples of self similarity. Have students save their best patterns and print out one favorite. After finishing this station, students should write a short paper explaining self-similarity to someone who has never heard of it before. They may include their printout or sketches. They should read these papers to their partners getting feedback on where they could make students clearer.

ACTION PLAN

Students will do "A Tree Grows in Brooklyn" exercise immediately following the video. In this exercise they will build an L-system like they saw in the video. The first two stages will be given and they will draw in the next four stages and create a table showing their findings. This will help them see the exponential nature of their findings. They will be asked to predict how many branches will there be after the tenth stage. Students will be asked to research the Mandelbrot Set online or in the library over the next couple of days. They should find six facts about the Mandelbrot set by our third day. They will turn these in along with a paragraph about the implications of the Mandelbrot Set for at least one area of scientific study.

 

POST VIEWING ACTIVITIES

Day two classroom exercise: The Chaos Game

Introduction: Yesterday we learned about fractals, self-similarity and iteration. We saw that small changes in initial conditions when reiterated could cause big changes in outcomes.

In mathematics we find that orderly patterns and structure sometimes result from what seem like random or chaotic processes. Today we are going to play a game that demonstrates this idea. It's called the Chaos Game.

Strategy: You all have a transparency with an equilateral triangle, ABC, drawn on it and a die. I want you to pick any point in your triangle and make a dot. Next you will roll the dice, if you roll a 1 or a 4, measure from your point halfway to the A vertex and make another dot. If you roll a 2 or a 5, measure halfway to the B vertex, and a 3 or 6 will take you halfway toward the C vertex. Roll the dice again, moving halfway to the point indicated. Continue this process for 10 iterations.

Conclusion: When the students are done you will stack their transparencies on the overhead. They will have created a Sierpinski triangle. After telling students of Polish mathematician Waclaw Sierpinski, Point out to students how self-similar it is.

Assessment: Ask the students to write a paragraph describing how the past two days of lessons have impacted their view of the natural world.

Third Day Activities:
On the third day finish up all of the fractal projects and writings. Students will share the fractals that they generated online and their observations about students. Students will also be asked to share their writings over the past few days. Students will be encouraged to do further research on fractals in the library or online. They will be encouraged to look for examples of self-similarity in the school and in their world. Finally, show the picture montage from Merrill Geometry: Applications and Connections appendix to encourage a discussion of what the future could be for fractal geometry.

In closing, give students this brain teaser: If you take a look in a mirror while holding a mirror is the image in the mirror of your reflection a fractal?

ASSESSMENT

All of the student's work will be assessed. The following short quiz will also be given on the fourth day during attendance. Look at the pattern shown below. What things in nature or our school follow this growth pattern? Draw the next stage. Define self-similarity and iteration.


For additional lesson plans and ideas relating to this topic and many others try TeacherSource at PBS Online! You will find activities, lesson plans, teacher guides and links to other great educational web sites! Search the database by keyword, grade level or subject area! Mathline and Scienceline are also great resources for teachers seeking teaching tips, lesson plans, assessment methods, professional development, and much more!


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