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Author:
Tammy Long

Link repeating patterns with geometry through the study of fractals. Practice fraction, pattern recognition, perimeter, and area skills. Introduce the idea of infinity, self-similarity, recursion, and iterations. Show that alternative thinking like Mandelbrot’s is valid and useful.

Discover how the different thinking of Mandelbrot brought about great discoveries in geometry and technology through the book, "Mysterious Patterns." Show the students the value of perseverance and being different. Use the idea of fractals and perimeter to explore how the perimeter changes with each iteration.

Tutorial

Geometry: Use visualization, spatial reasoning, and geometric modeling to solve problems

Source: Tammy Long

Pencil and Graph Paper

Fractal ppt

Copies of main worksheets:

Triangle Worksheet

Sierpinski Triangle – Remaining Triangles Worksheet

Sierpinski Triangle – Finding Area

https://www.chatham.edu/pti/curriculum/units/2006/Schmitt.pdf

Copies of Supplemental Worksheets:

Koch Snowflake – Perimeter

The Pythagoras Tree – Total Area of the Squares

Tortoise and Hare Race Exploration

file:///C:/Users/Tammy/Downloads/Tortoise_and_Hare_Race_Exploration_Questions.pdf

3D Fractals

Book: Mysterious Patterns by Sarah Campbell

(extra books about fractals and Mandelbrot)

Movie: Clouds Are Not Spheres

(available on DVD at most libraries and on Amazon Prime)

Source: various sources at each link listed

1. Have the warm review area and perimeter of rectangles, squares, and triangles.

2. Read through Mysterious Patterns, stopping at page 30.

3. Pass out the Triangle Worksheet. Following the directions on page 30, have the students create a Sierpinski triangle. Model it with them.

4. As they finish drawing, finish the book including the biography information about Mandelbrot in the Afterword. Emphasize how he didn’t feel like he fit in, but he followed his intuition about how he saw the world around him as complex mathematical patterns. His work went from being criticized to praised in a few short decades. Because of computer simulation and his work with fractals, we have the technology of our compact cell phone antennae and ways to predict where cancer will form (because the blood vessels fail to follow a fractal formation)

5. Start the power point, talking about the patterns in nature. Stop at slide 33 and ask them to discuss at their tables and write in their journals a definition for fractal.

6. Show slide 34. Have them compare it to their definition. Discuss the differences and their validity.

7. Show slide 35. Ask, “What is this term ‘iteration’?” Discuss any previous knowledge of the term.

8. Have students turn to their vocabulary section of their journal. “We have some new terms we’ll want to know as we explore making our own fractals.” Show slide 36. Give them time to copy then discuss each term briefly.

9. Use slide 37 to show how the terms are used when talking about the Koch Curve.

10. Show them slide 38: Cantor Dust. See if they can find the initiator and generator. For those that finish quickly, point out the fraction application in the picture. Can they continue the pattern?

11. Have them get their triangle worksheet back out and shade it according to slide 39. Pass out the second worksheet: Remaining Triangles. Together fill out many triangles remain after 0-3 steps. Have them finish the table together and try to come up with a rule for the nth step. Circulate and offer encouragement for finding the pattern. Give direction, but not the answer. Ask them questions that prompt their own thinking. “What do you see in the previous lines that are the same? Different? Can we use that to predict the next step and 20 more after that? When will the math be helpful? How many steps will you be able to draw with accuracy?”

12. Pass out the Finding Area worksheet. Using the number of triangles table and the area of a triangle formula see how many steps they can fill out in groups (or individually if they like to work alone). Can they come up with a rule for the nth step?

13. While they finish exploring cue up the movie. Also have the other worksheets and extra books available for those that finish early.

14. Have them try to come up with their own artistic fractals as they watch the movie. The movie covers the process of defining what a fractal is, Mandelbrot’s part in it all (with interviews of Mandelbrot himself), and how it has advanced today’s technology.

15. Leave the students with the option to create more art for the classroom or research and report back important figures like Cantor, Sierpinski, Menger, and Koch.

16, Warm up for the following class should include the reflection of how are the students different and how is that of value.

Source: Tammy Long

Since there are mostly big concepts, all learners will be able to participate on their own level (self-differentiating). For the early finishers or those really interested in fractals, have the extra worksheets available including 3D fractals. Also have various books about fractals and those written by Mandelbrot.

Diversity Responsiveness: Show how art and math can co-exist. Show how even what we once saw as chaos, we now can describe with mathematics. Especially emphasize how different thinkers make huge contributions in the world. Strengthen the idea for them to believe in themselves, much like Mandelbrot did.

Source: Tammy Long