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Volume III: Fractals and Chaos


Investigating Fractals
Lesley Britton
Taylor Allderdice High School

This document contains objectives, guidelines for lectures, and materials needed to instruct advanced high school mathematics students in the basic structure of fractals, the algorithms involved in creating fractal pictures, fractal dimension, and leading them in a research project where they will investigate the basic properties and algorithms involved with fractals.


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Fractals
Roseann Y. Casciato
Taylor Allderdice High School

Fractals are a “hot” topic recently. Throughout this seminar I was given the opportunity to examine how fractals and chaos are present all around us in nature. Some of the mathematics that I teach on a daily basis goes back as far as 300 B.C. with Euclid and his work with geometry. My students are always asking why are we learning a certain mathematical lesson or when will we ever use this again. They feel as though the mathematics they are learning is from the past and they are just going through the motions of solving a problem. It will be beneficial for me to share with my class a new and more recent mathematical topic.

This unit of study on fractals will be incorporated into my 10th grade Gifted (Center of Advanced Studies – CAS) Algebra 2 class. The purpose of this unit is to introduce students to the topic of fractals. In teaching a gifted class, it is expected that I add enrichment into the existing curriculum. It is my job to introduce students to topics other than ones that are present in their textbook. I should be able to expose them to mathematical topics that are present in the real-world so they can see the relevance in them.  Hopefully my students will enjoy doing the mathematics necessary to develop a fractal as well as seeing that fractals are beautiful geometric shapes.


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From Order to Chaos and Back Again
Adam P. Deutsch
Taylor Allderdice High School

Chaos is one of the most important scientific/mathematical discoveries in recent years and is truly changing the way experts think in such diverse areas as physics, business and economics, biology, sociology, meteorology and many others. And it is an exciting new branch of mathematics. The results are surprising and relevant and ultimately beautiful. And one of the best things about chaos theory, at least from the perspective of a high-school mathematics teacher is that the mathematics which lies at the foundation of chaos theory is very basic. Ultimately, it is arithmetic. Some algebraic notation makes it easier to work with and represent, but at the most basic level, it’s all about arithmetic.

The purpose of the unit described herein is to provide students with the opportunity to explore the foundations of the mathematics of chaos. This unit will introduce students to the mathematical process that was first discovered to lead to chaos – iteration. And from there, they will explore the underpinnings of this new branch of science.


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Fractals and Chaos in the English Classroom
J. Gary Dropcho
Carrick High School

An English teacher taking a seminar on fractals and chaos seems random, but a closer look reveals an intersection of patterns between science and literature. Studying the science of chaos and its recursive figures of self-similarity is an interesting way into some of the literature read, discussed and written about in an accelerated twelfth grade English class.


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Making the Connection with the Sierpinski Triangle
Jane H. Fraser
Pittsburgh Classical Academy

Fractals are geometric shapes that are self-similar at different scales. The Sierpinski Triangle is one of the most well-known and versatile of all fractal figures. The possibilities for rich mathematical explorations and connections are numerous. The middle school Connected Math Program, currently taught in Pittsburgh Public Schools, is devoted to developing student knowledge and an understanding of mathematics that is rich in connections.

I have designed a unit that explores many of the intricacies of the Sierpinski Triangle while allowing students to revisit and discover numerous math concepts, such as measurement, fractions, and area. This unit is particularly suited for the seventh grade curriculum, with its emphasis on similarity and scale factor.


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Fractal Phantoms–Kindergarten Inspectors
Stephanie Johnson
Helen S. Faison Arts Academy

This curriculum unit is quite unique in the fact that it explores four content areas mathematics, science, technology and art. These wonderful blends of subjects make the activities exciting. I have taken the Gestalt learning theory; best teaching practices and related to the reason my student’s development. I wonder why anyone has not noticed a fractal. Would it be because it is not in their life space? I will discuss this in detail. The whole concept of fractals and chaos are very new. This very new subject has all of the content areas I have included in it. You will find that most early childhood classrooms will be able to use this unit. I have taken some specific strategies to bring about an understanding of the recursive patterns of a fractal. I have also included an activity to understand chaos. This is through a kindergartener's eyes. I have included family involvement and an interesting field trip. I do believe that the intermediate elementary classes as well can use it. Take your time and discover as my students will do and you will find it to be fascinating.


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Fractals, Chaos and Brain Functioning
Eric Laurenson
Peabody High School

My first introduction to fractals, unbeknownst to me, involved my early fascination with drawing leafless trees. As a child, I would incessantly draw the intricate, repetitive, but not identical patterning of a tree’s trunk, boughs and branches. I recognized the necessity of making the branches curve, split and shrink in scale in a controlled but not entirely predictable fashion. In effect, this was an introduction to fractals with a sense of chaos. Fractals demonstrate a repetitive scaling that makes them identical on all scales, whereas chaos introduces a seemingly random aspect to the patterning. It wasn’t until I took this class on fractals and chaos that I realized how pervasive fractal patterns are in nature, that I was seeking those patterns at an early age and that in the midst of that attraction to pattern was an innate sense that chaos provides a divergence from that pattern.

This unit is intended to introduce the pervasive concepts of fractals and chaos to high school physics students. The unit will provide the mathematical and conceptual ideas for a teacher to be able to present the unit. The level of the unit can be tailored to suit any level of physics class from general to A.P. The unit is meant to be included as a part of the modern physics curriculum but also can be included to supplement the mathematical concepts of a high school physics course. The information on brain functioning is meant to add relevancy to the topic. Brain functioning is meant to stimulate interest in the challenging concepts of chaos and fractals. Although this unit is intended for high school physics it is my hope that teachers in other areas who wish to explore brain functioning, chaos or fractals may find this unit engaging.  This could potentially include Biology teachers who are introducing the issue of brain function. I can also imagine this unit being used for the mathematical content on fractals and chaos. The unit could also be utilized by any teacher who wants to introduce the concept of chaos and finds the connection to brain functioning engaging.


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Chaos Theory (Mathematical) in Teaching Ecology and General Science Unit
Peter A Mamula
Oliver H.S.

Chaos theory (Mathematical) is enrichment to teaching Ecology and General Science. The collaboration (science and mathematics) will show students the integral relationship between the two disciplines. The primary example, in ecology, is to use population studies to explain the use of Chaos theory. This unit maybe used separately or part of the study of population growth in the past or future. This unit will help students understand old explanations of population. Also, the unit will help students use logistic equations to analyze population growth. Furthermore, it will give students a way to use computer analysis of old or new data to show how Chaos theory can be applied.


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Fractals and Chaos an Introductory Unit
Joseph McGuire
Oliver High School

The following unit is intended for Biology students in grades 9 through 12. The unit deals with the subject of fractals and chaos. The unit covers mainly fractals that are found within nature. Students are required to construct fractals. Students are also required to identify fractals that can be found within the environment. Students will also get a brief history of the contributors that led to the study of fractality and chaotic systems. Students are required to use math within the unit that satisfies the district’s math standards as well as address math anchors required by the various disciplines. Students will be required to write on the subjects covered as well as do research that deals with the subject matter. Pattern recognition and creativity will be explored within this unit. Upon completion of the unit students will have a working vocabulary that reflects the subjects of chaos and fractals.


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Fractals and Nature – It’s Elementary
Kathleen McHugh
Pittsburgh Gifted Center

This unit on Fractals will be used within a Geometry unit, to expand and enhance the concepts that are learned in Basic Geometry. Fractals are the study of the roughness of the world, its transformations and dynamical changes. Fractal Geometry studies the geometry in nature and the changes that take place all around us.

So what is a fractal? What does one look like? Who first coined the term? How can elementary students be taught to understand their significance?

Through interactive, engaging, and sequential activities on fractals, the student will develop a basic understanding of how fractals explain mathematical connections to nature. For some students, the knowledge will plant a seed for future investigations.


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Entropy in Chemistry
Maria Orton
David B. Oliver High School

This unit is meant to be supplemented into a high school chemistry course in order to introduce chaos and fractals to students and show its relevance to real life. The unit may be helpful for students in grades 10 – 12 in a regular education or inclusion chemistry classroom. The chaos theory can be related to entropy in chemistry and everyday life. This unit can also be tied into the exchange of energy in a chemical reaction, and may help further define the properties associated with both endothermic and exothermic reactions. Mathematics and science both are more meaningful when they are rooted in real-life contexts. This unit will allow students the opportunity to become actively involved in learning, they will be given the chance to both visualize the motion of particles in chemical reactions and also see the how fractals form in a single replacement reaction. This will allow them to visualize how the particles rearrange themselves in a seemingly chaotic way, and make connections both to the chemistry and the math behind the reaction they will be seeing.


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Fractal Fun
Rosemary Schmitt
South Brook Middle School

This unit on Fractals was written for a Middle School Saturday Program but can be used in addition to any regular curriculum. It starts with the basics of what a fractal is and lets students explore different fractal patterns. As students are introduced to different simple fractals they begin to recognize the pattern and try to continue to expand it. They will explore similarity of the changing shape and perimeter in some cases. They will try to appreciate both the simplicity and complexity of the pattern leading them towards creating tables and equations represented by the fractals.


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Fractals for Fourth Graders
Amanda Sill
Madison Elementary School

As a teacher I hear complaints and groans when it is time for Math. I feel that many students dislike Math because they do not understand it. I chose to create a unit about fractals that is brought down to the fourth grade level. My main priority for this unit is to spark an interest in Math in students by integrating the teaching of fractals into other subjects. I also want students to get enthusiastic about learning about Math so that they are willing to learn more.

This unit includes an introduction about fractals. The following lessons integrate the teaching of fractals into Math, Science, Writing, Computers, and Art. This unit takes a “hands-on” approach to teaching about fractals.

My theory is that when students are able to see and touch objects related to what they are learning they are better able to grasp the concept. By getting students involved in the process of learning I am hoping they will be encouraged to want to learn more about the mathematical concepts of fractals. Learning the basics about fractals will lay the groundwork for other math ideas.


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Chatham University
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Jointly sponsored by Carnegie Mellon University, Chatham University and the Pittsburgh Public Schools.
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