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Math Explorations 2005
Shodor > SUCCEED > Workshops > Archive > Math Explorations 2005

Bethany began by introducing the students to the term, fractal. First she drew a diagram of "heads" on the board. Each day 3 heads were added to each head on the previous day. Repeating this pattern again and again would make a fractal. She explained that fractals are also seen in nature. The students then gave examples of fractals: trees and snowflakes.

Bethany explained that Geometry is the mathematical version of shapes. In order to show the students an example of Fractals she introduced them to Sierpinski's Triangle. The Sierpinski triangle is formed when a triangular shape connecting the midpoints of the sides of the original triangle is removed, forming three smaller triangles. The students discovered that to continue the pattern, they had to multiply the previous number of triangles by three. They were able to find this pattern by creating a chart, which described the ratio between the level and number of triangles.

Next they were challenged with finding the rule for Sierpinski's Carpet. The rule in this pattern was to multiply by 8. To make the activity more exciting, Bethany asked the students to find the length of each side and compare it to the original. For the each stage there is a 1:3 ratio. The first stage was 1/3 the second 1/9, 1/27 and so on. To find the length of sides, the students outlined the information in a chart once again. This activity allowed the students to identify the relationship between functions and patterns.

In the Flake Maker activity the center 3rd is cut out on each of the sides of a triangle. Each segment was replaced with 4 smaller segments. The students used Excel to create a spreadsheet to replicate the table that was created with the Sierpinski's Carpet and Triangle. Bethany showed the students by entering in Excel the formula "=b2*4". As the number of segments increased, the length of the segments decreased.

The Mandelbrot Set is known as the world's favorite fractal. The students discussed why the Mandelbrot Set is a fractal and how its designed. The equation used in the Mandelbrot set is "the previous stage times the previous stages plus a given #". The Mandelbrot Set describes the change of the constant value.

They used EXCEL to model the Mandelbrot set by entering in the equation "=b1*b1+\$b\$1". The students predicted the results of changes to the equation (i.e. 0*0+0=0) and saw that the column of numbers remained the same when the initial constant was zero, and other patterns with other constants (-1: alternating b/t 0 and -1; -2: all 2; positive decimals: increase slowly). Then the students designed their own fractals; the fractals were drawn three stages deep with each stage in a different color. The next activity allowed the students to explore fractals through a computer program. The program, Fractured Pictures produced pictures when the students selected the number of sides of the shape, the length of the sides, the scale factor, and the depth of the fractal. A second program Flake Maker allowed the students to manipulate points on a grid to create fractals. The students were given the final challenge of creating a fractal to display on the website.