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Description of Unit

For this project, students will examine mathematical patterns found in nature, such as tessellations, the Fibonacci sequence, the golden ratio, and pi. For example, the students can create a tessellation jigsaw puzzle. Each student draws a tessellation pattern on a sheet of paper, cuts it out, jumbles up the pieces, and passes the pieces to another classmate for reassembly. In this way, the student looks at tessellations from two different perspectives: once, when constructing his or her own, and from a different perspective when putting together a classmate’s tessellation jigsaw puzzle. Similarly, Fibonacci puzzles can be constructed using hexagons as in a beehive, or bricks as in a wall.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpuzzles.html

A student could make the pieces, write the list of rules for solving the puzzle, and pass the pieces with their rules on to another student for reassembly.

This guide links the Mathematics in Nature unit to the Texas Essential Knowledge and Skills (TEKS) for third graders. Mathematics in Nature is a science and mathematics unit that allows students to explore and gain knowledge about mathematical patterns found in nature, such as tessellations and the Fibonacci sequence. The unit also has interdisciplinary connections to other subject areas. For example, students will communicate clearly by putting thoughts and feelings into spoken words, as covered in the English Language Arts and Reading TEKS. They will also understand the concepts of time and chronology, which the Social Studies TEKS cover. The following document includes the applicable TEKS and the details of the Mathematics in Nature unit. The asterisks indicate the TEKS that are testable on the State of Texas Assessments of Academic Readiness (STAAR). The final section of this document presents the applicable Texas College and Career Readiness Standards adopted by the Texas Higher Education Coordinating Board (THECB) on January 24, 2008.

Phase I. Learning Experiences

  1. Introduce the concepts of patterns found in nature, such as tessellations, the Fibonacci sequence, the golden ratio, and pi. Talk about where such patterns might be found. Show examples and have students bring other examples.
  2. As a class, choose a pattern in nature, such as a pine cone, pineapple, turtle shell or starfish, that you all find interesting. Discuss its origin, how it is formed and why, and any special characteristics unique to that pattern. Which mathematical concept does it demonstrate?
  3. Discuss the various types of symmetry.
  4. Divide the class into small groups. Each group chooses a pattern and breaks it down into its components. Each group will discuss whether their pattern is symmetrical, how it is formed in nature, examples of the pattern in nature, and which mathematical concept it demonstrates.
  5. Each group presents their pattern and their findings to the class.

Phase II. Independent Research

A. Research process

  1. Selecting a topic. Each student should identify a natural pattern for more in-depth study.
  2. Asking guiding questions. Once students have selected their pattern, each student should think of three to five guiding questions, such as:
    • Where is this pattern found in nature?
    • What causes it to form that way?
    • What are the pattern's units (i.e., what does each piece of the pattern look like and how does it function in nature?)
    • Is the pattern symmetrical?
    • Which mathematical concept does the pattern demonstrate?
    While these examples are general, the student’s questions should be specific to the chosen topic. The questions should lead him/her to form individual research-based opinions. The student should also develop a hypothesis or some possible answers to the questions.
  3. Designing a research proposal. The student should include numerous components in the research proposal:
    • The type of pattern he/she will study
    • Three to five guiding questions he/she will investigate, as well as hypothetical answers to those questions
    • Resources he/she will need to find answers to questions, such as primary and secondary sources, correspondence with experts on the subject, etc.
  4. Conducting the research. After you have approved student proposals, each student begins using the resources he/she has identified and others he/she may encounter. During this stage, the student will need to keep a log, note cards, or resource process sheets of all the sources and what he/she has learned from each one.
  5. Drawing conclusions. Based on the research, students should work together to plan a class fair for other students in the school or people in the community. Each student will create an activity or display for the fair, exhibiting a natural pattern that they have explored.

B. The product

The teacher may wish to arrange a class fair in which other students from the school come to learn about symmetry and nature. Each student creates an activity or a display with an interactive component to illustrate the natural pattern studied.

C. Communication

The student will present his/her activity or display at the fair. He/she should provide some background—why that activity or display was developed—and offer any solutions or findings. The audience should be given time for questions and answers. The Q&A session should be impromptu and unscripted in order to accurately reflect student learning.

D. A completed project consists of:

  1. A research proposal, including guiding questions and answers
  2. A research log, notes, or resource process sheets
  3. The product—the activity or display presented at the class fair
  4. A Works Cited Page
  5. A videotape or audiotape of the student’s talk, including the unscripted Q&A session
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