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Title: Technology - Chaos Theory Lab

Author: Andrew Renaud

Grade Level: 9

Subject/Content: Integrated/Applied Mathematics

Summary of Lesson: Through the application of math concepts, the purpose of the investigation is to examine the application of Chaos Theory that exists in one's everyday environment, in order to develop an understanding of how Noneuclidean Geometry opens opportunity for new ways to view our mathematical world.

Focus Question: What is Chaos Theory and why is this important to all life forms?

Databases(s): Student Resource Center, Opposing Viewpoints Resource Center

Procedures:

  • Materials needed: "Jurassic Park" novel, Chaos Theory handouts, graph paper (5 division/inch) , TI 83 Plus Graphing calculators, ruler, clip board, loose leaf paper, manila folders, tape, scissors, pencils
  • After our general review of the prerequisites: Direct Variation, Standard Form and Slope-Intercept Form for lines, Parabolas, Similarity and Fractal Geometry, Chaos Theory, students are randomly placed into groups of four.
  • Each person in the group is given an article related to a specific characteristic regarding Chaos Theory – topics may be chosen from this Web site: http://www.mathjmendl.org/chaos/. Articles may be copied on different colored paper (color-coded.)
  • Students will participate in a Jigsaw activity.
    1. They will read the article to themselves silently.
    2. They read with a focus: in their own opinion, they will highlight and number the most important ideas mentioned in a hierarchy (1 = most important, 2 = next most important, ..., 5 = least important) – there are many ideas in each article to choose their top five from
    3. They will get together with other students with the same color article outside of their diverse group and discuss what they choose for their top five using "ordered sharing" – this means one person talks at a time with others simply listening, making no comments, or emotional responses
    4. They will reconvene with their original group of four and ordered share their assigned article
    5. As a group they will agree upon their top choice for each article
    6. A speaker will share out with the large group their top choices
    7. Each student will write a five-seven sentence paragraph explaining "how the important items in the articles relate to concepts read in the Jurassic Park novel"
  • Students will review data presented about an animal theme park – similar to dinosaur data from the novel Jurassic Park. The teacher should make-up data that will form both parabolic and linear graphs. This can be done easily by selected problems from the classroom text.
  • This data will be modeled with equations, graphed on graph paper and on thee TI 83 Plus graphing calculators.
  • Students will explain the difference between the parabolic and linear graphs displayed.
  • Students will explain the relevance between these differences and explain how this ties to what Chaoticians (like Ian Malcomb in the novel Jurassic Park) say about complex systems like the one we live in.
  • Students will create a 3-D fractal and explain the relevance of the similar shapes throughout the fractal as these pertain to Chaos Theory.
  • Students create their own theme park as it pertains to what the learned above.

Steps/Activities by student(s):

  1. Anticipatory Set: Students view video clip on New Geometry hosted by Danny Glover. Students view a PowerPoint presentation on the importance of fractal geometry.
  2. Key terms/skills: Students will review the prerequisites including these math ideas: Direct Variation, Standard Form and Slope-Intercept Form for lines, Parabolas, Similarity and Fractal Geometry, Chaos Theory.
  3. Modeling: Students listen to teacher describe the lab procedures. Students take notes when clarification is needed.
  4. Student Test for Understanding: Students will follow this list of procedures with teacher support.
    1. Students participate in the Jigsaw as described above in the Procedure section.
    2. Explain the differences and similarities between the animal data sets.
    3. Using the manila folders, tape and scissors, and appropriate concepts relative to similarity, students will create a 3-D fractal. This can be in a mobile format or students can use their own creative minds to construct something unique with teacher approval. Students must then write how this relates to the novel. Special attention should be given to the iterations during this explanation.
    4. Students create their own theme park. This will be displayed in a PowerPoint format. The elements of the project will include: an overview/description of the park, the purpose of the park, a description of the biotechnology used in the creation of the theme park life forms, the location of the park an a detailed explanation for this choice, an analysis of the long-term impact, security considerations, an account and graphs for all possible outcomes, a Billboard add for the park, an analysis of the strengths and weaknesses for the park from a Jurassic Park character perspective, and moral and ethical implications.
  5. Recap – What did we learn? Explain verbally the important math skills that you utilized. What are important group skills (communication, cooperation, collaboration) used?
  6. Home link – Students answer these basic questions: Why is a sounds understanding of Chaos Theory important for this lab? What is the significance of linear versus nonlinear graphs? What do fractals have to do with complex systems? What does Chaos Theory mean in the context of math? Why is this stuff important for you and your life?
  7. Follow-up – Class presentation of the results. Students type a three paragraph summary in proper (Type III format – proper grammar, punctuation, spelling, 5-7 sentences per paragraph, at least ten word per sentence, no sentence starts with the same word in a paragraph.) Paragraph One explains what the lab was about. Paragraph Two describes what the student learned. Paragraph Three explains how this applied to the individual's own current or future life. The highlights of paragraph three are verbally shared with the class by each student.

Outcome: Through the application of math concepts (listed above), the students gain at least an application level understanding of Chaos Theory that exists in one's everyday environment.

Related Activities: H.O.T.S. (Higher Order Thinking Skills) – Stresses Bloom's Syntheses and Evaluation levels. Students participate in "Chaos in My Home Life." Students can record data that models the concept as it relates to experiences within their own lives outside of the classroom. The data can be modeled graphically and presented to the class using the graphing calculator viewing screen. Students evaluate how a Non-Euclidean geometry system like Chaos Theory represents plays in the role of this presentation.

Standard Date: October, 1998

Content Standard(s):

  • 1.1 Understanding numbers, ways of representing numbers
  • 1.2 Understanding the meaning of operations and how they relate to each other
  • 1.3 Use computational tools and strategies fluently and estimate appropriately
  • 2.1 Understand various types of patterns and functional relationships
  • 2.2 Use symbolic forms to represent and analyze mathematical situations and structures
  • 2.3 Use mathematical models and analyze change in both real and abstract contexts
  • 3.1 Analyze characteristics and properties of two- and three- dimensional geometric objects
  • 3.2 Select and use different representational systems, including coordinate geometry and graph theory
  • 3.3 Recognize the usefulness of transformations and symmetry in analyzing mathematical situations
  • 3.4 Use visualization and spatial reasoning to solve problems both within and outside mathematics
  • 5.2 Interpret data using methods of exploratory data analysis
  • 5.3 Develop and evaluate inferences, predictions, and arguments that are based on data
  • 6.1 Build new mathematical knowledge through their work with problems
  • 6.2 Develop a disposition to formulate, represent, abstract, and generalize in situations within and outside mathematics
  • 6.3 Apply a wide variety of strategies to solve problems and adapt the strategies to new situations
  • 6.4 Monitor and reflect on their mathematical thinking in solving problems
  • 8.1 Organize and consolidate their mathematical thinking to communicate with others
  • 8.2 Express mathematical ideas coherently and clearly to peers, teachers and others
  • 8.3 Extend their mathematical knowledge by considering the thinking and strategies of others
  • 8.4 Use the language of mathematics as a precise means of mathematical expression
  • 9.1 recognize and use connections among different mathematical ideas
  • 9.2 Understand how mathematical ideas build on one another to build a coherent whole
  • 9.3 Recognize, use, and learn about mathematics in contexts outside mathematics
  • 10.1 Create and use representations to organize, record, and communicate mathematical ideas
  • 10.2 Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately,10.3 Use representations to model and interpret physical, social and mathematical phenomena

Learning Expectation: Students will apply understanding of concepts related to linear and parabolic graphing, Similarity and Fractal Geometry, and Chaos Theory within the context of Technology.

Performance Indicators:

  • At Level 1, the student is able to: Write (show basic knowledge for) about the meaning of these: Direct Variation, Standard Form and Slope-Intercept Form for lines, Parabolas, Similarity and Fractal Geometry, Chaos Theory.
  • At Level 2, the student is able to: Comprehend and apply these concepts: Direct Variation, Standard Form and Slope-Intercept Form for lines, Parabolas, Similarity and Fractal Geometry, Chaos Theory.
  • At Level 3, the student is able to: Synthesize new ideas related to and evaluate concepts: Direct Variation, Standard Form and Slope-Intercept Form for lines, Parabolas, Similarity and Fractal Geometry, Chaos Theory.

Computer Literacy and Usage Standards 9-12:

  • Students demonstrate a sound understanding of the nature and operation of technology systems
  • Students are proficient in the use of technology
  • Students understand the ethical, cultural, and societal issues related to technology
  • Students practice responsible use of technology systems, information, and software
  • Students develop positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits, and productivity
  • Students use technology tools to enhance learning, increase productivity, and promote creativity
  • Students use productivity tools to collaborate in constructing technology-enhanced models, prepare publications, and produce other creative works
  • Students use technology tools to process data and report results
  • Students use technology resources for solving problems and making informed decisions
  • Students employ technology in the development of strategies for solving problems in the real world

ISTE NETS for Students

  • Identify capabilities and limitations of contemporary and emerging technology resources and assess the potential of these systems and services to address personal lifelong learning, and workplace needs
  • Make informed choices among technology systems, resources, and services
  • Select and apply technology tools for research, information analysis, problem solving, and decision making in content learning
  • Collaborate with peers, experts, and others to contribute to a content-related knowledge base by using technology to compile, synthesize, produce, and disseminate information, models, and other creative works

Information Power; Information Literacy Standards 1-4:

  • The student who is information literate accesses information efficiently and effectively
  • The student who is information literate evaluates information critically and competently
  • The student who is information literate uses information accurately and creatively
  • The student who is an independent learner is information literate and pursues information related to personal interests
  • The student who is and independent learner is information literate and strives for excellence in information seeking and knowledge generation
  • The student who contributes positively to the learning community and to society is information literate and practices ethical behavior in regard to information and information technology
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