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.
• 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
– 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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