Mathematics prerequisites and discussion:
Students should be familiar with basic concepts from geometry
such as regular polygons (equilateral triangles and
squares), properties of similar triangles/squares, calculating
the perimeters and areas of triangular and square
regions, use of the Pythagorean theorem to find the altitude
of an equilateral triangle, and finding the midpoint of a line
segment. In addition, students should have some background
on geometric sequences and series. (However, you
could introduce that topic as students work through this
pullout.) For Activity 4, students should also be familiar
with logarithms so that they can solve the equation rd = n for
d: d = log(n)/log(r) and then use their calculators to obtain
an approximation for d.
This pullout consists of four activities.
Activity 1 introduces students to
fractals in nature, in mathematics, and
in art. Then students draw fractal
trees. They investigate the sequences
formed by the number of tree branches
and branch lengths drawn at each
stage, which turn out to be geometric
sequences.
In Activity 2 students
draw by hand Sierpinski triangles.
The perimeters and areas of each
stage of their drawings also form geometric
sequences. Students discover
that as the stage number increases, the
perimeter grows larger and larger
while the area shrinks toward 0.
In Activity 3, students use GeoGebra to
draw several stages of the Sierpinski
triangle and Sierpinski carpet. With
both of these fractals , perimeters
grow with each stage of construction
while areas shrink. Because of this
property, the Sierpinski triangle and
carpet have been used as antenna
designs in wireless communication
devices.
Activity 4 focuses on the
problem of measuring coast line
length. Using the Koch snow flake
curve as an example, students discover
that as more indentations are
added to a representation of a coastline,
its length grows. It turns out that
coastline length is highly variable
depending on the length of the measuring
device used. Next, students are
introduced to the concept of fractal
dimension. They adapt this concept to
estimating coastline dimension, which
is a measure of the smoothness or
ruggedness of a coastline. This activity
concludes with a final challenge for
students to apply what they have
learned to create some fractal art.
The activities in this pullout address
the following standards from the
Common Core State Standards for High
School Mathematics:
