Proportions with manipulatives

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					    Fractals using
Virtual Manipulatives



     Mary Corkery
                         Definition
• Fractal:
     – Patterns can be found in many objects
       in nature, including broccolli. This
       pattern of repeated shapes at different
       scales is part of fractal geometry.




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                        Math Objectives
• Understand
     – Iteration
     – Self-Similarity
     – Fractal




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    Mass Curriculum Frameworks
• G.G.5
• G.M.3
• 10.P.1




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                    Language Goals
                   ELPBO Objectives
• Essential vocabulary R.1.14.a
     – Fractals, Iteration, Self-Similarities, means
• Use Essential vocab in Speaking W.1.8.b
• Use Essential vocab in Writing W.1.8.b
• Show understanding of Essential vocab by
  Listening to teacher and group members S.1.2




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             WWW.SelectMath.com




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                        WWW.SELECTMATH.COM




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              Essential questions
• What is a Iteration?
• What is Self-Similarity?
• What is a Fractal?




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                  What is a Iteration?

• Process of repeating the same procedure
  over and over again




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             What is Self-Similarity?
• When we look at the smaller and smaller
  details of a figure, and we find the same
  geometric characteristics as the original
  form.

• Self-Similarity is an important
  characteristic of Fractals


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                    What is a Fractal?
• Fractal is a geometric figure created by
  iterations. The pattern’s structure appears
  to go on infinitely.




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                         Fractals
                   Sierpinski Triangle
                       Iteration 0




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                       Fractals
                Sierpinski – Iteration -1




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                         Fractal
                 Sierpinksi – Iteration 2




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                         Fractal
                 Sierpinski – Iteration 3




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                             Fractals
                        The Koch Snowflake
                           The Algorithm
• Step One.
     – Start with a large equilateral triangle.
• Step Two.
     – Make a Star.
          • Divide one side of the triangle into three equal parts and
            remove the middle section.
          • Replace it with two lines the same length as the section
            you removed.
          • Do this to all three sides of the triangle.
• Do it again and again.
• Do it infinitely many times and you have a
  fractal.
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                  Fractals –
           Koch Snowflake Steps 1-4




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                        Dynamic Fractals




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                        Dynamic Fractals




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                        Dynamic Fractals




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                        Dynamic Fractals




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                        Dynamic Fractals




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                        Dynamic Fractals




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                        Dynamic Fractals




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                   Lesson Objectives
Content Objectives.
     – Understand iteration
     – Understand self-similarity
     – Understand Fractals

• Language Objectives
     – Discuss iteration in small groups
     – Discuss self-similarity in small groups
     – Describe how to create a Fractal
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                          Materials
• Suggested materials for a class of 15
  students
     – Overhead projector
     – White board
     – Ruler
     – Geometry book( Glencoe)
     – Markers
     – Graph paper

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                            Launch
• Vocabulary
     – Think, Pair, Share,Worksheet
• Connect to prior knowledge
     – Iteration
     – Similar triangles,polygons,…




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                            Explore
• Practice/Application
     – Students will work in their groups and use
       white boards to draw successive fractal
       iterations.




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                        Accommodations
• Teacher will circulate among student
  groups to answer questions and make
  sure that the groups are on the right track




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                    Summary /Review
• Students will summarize the goals of the
  lessons by orally presenting their solutions
  on the white board to the entire class




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                        Bibliography
• Geometry Glencoe mathematics
• Massachusetts mathematics curriculum
• English language Proficiency Benchmarks
  and Outcomes for ELLs (ELPBO )




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