Fractals: Measuring fractal dimensionality
Topic  Fractals 

Program  Brown Science Prep 
Developed by  Alex Loosley 
Developer Type  High school students 
Overview / Purpose / Essential Questions
This lesson plan is for an interactive an engaging lesson that teaches highschool students about fractals and dimensionality. I tried it in September of 2013 on a grade eleven precalc class at Beacon Hill highschool and it was very successful. The lesson raised essential questions such as, what are dimensions; what is a fractional dimension; how can the fractional (fractal) dimensionality of everyday real objects be measured? The first question is answered using humor and brainstorming as part of a chalkboard talk, whereas the latter two questions are discovered by the students through guided interactive experimentation.
Performance / Lesson Objective(s)
The objective is to give high school student a generalized sense of what a dimension is and a quantitative tool for measuring dimensionality.
Lesson Materials
 Blackboard or whiteboard
 Different color chalks or whiteboard markers
 Coastline Paradox Materials:
 ~18x18" foam board with complex coastlines inscribed (see picture attached)
 ~100 push pins per island
 2 1" rulers per island, 2 2" rulers per island, 2 5" rulers per island, 1 10" ruler per island.
 Computer with Excel connected to a projector
Lesson Motivation
Procedure
 Humorous blackboard introduction (involving FRACTAL anagrams)
 I started with some funny words I came up with and then used the anagram word FLATCARS to get the show rolling.
 I used cardboard bowl to represent a car claiming that the highschool would not allow me to drive my car into the building... I asked the class if this was a 1D, 2D, or 3D car. They answered 3D. Then I flattened it (as best i could) and took a pool (hands up to vote), is this now a:
 2D car
 3D car
 2.5D car
 What does 2.5D mean? We'll find out later
 Group brainstorm
 Draw a box: class lists the dimensions (they should figure out that the dim's are height, width, and depth  the parameters necessary to define the box)
 Draw a sphere: class learns that this is only 1D (radius)
 Draw a stick figure of somebody important (like the principle)  some dimensions included (after I started the students off)  arm length, leg length, girth, percentage of head covered in hair (this exercise was very humorous and engaging)
 Recognize from the group brainstorm that we're still only working in integer dimensions, though we now have a more general understanding of what an integer dimension is. This was a qualitative exercise, now we'll look at dimensionality quantitatively.
 Draw line, square, cube... show that when the lengthscale is doubled, the number of lines doubles, the number of squares increases by a factor of 4, and the number of cubes increases by a factor of 8... this mathematically demonstrates basic spatial dimensionality.
 Show class the Koch Curve (snow flake), lengthscale must be tripled to get a curve that is quadrupled. Therefore, d~1.26 (weird)
 Show class another fractal like Serpinski's triangle if time (I usually leave this out)
 Introduce the concept measurements of length or area or volume depend on the lengthscale of measurement. If one looks at either of the fractals introduced in 5 or 6 at a big lengthscale, they miss all the details. Leave the class a little confused, get them into groups of 34 and hand out the foam board island. ACTIVITY: each group should measure the perimeter of their island using a 1" ruler, a 2" ruler, a 5" ruler, and a 10" ruler (they can use two 1" and 2" rulers to speed this process up).
 Collect their results: weird finding!! Throw the data back at the student the coastline perimeter changes depending on the ruler used?!?!
 Using excel spreadsheet (this should be premade), input the class measurements in the spreadsheet and fit the Mandelbrot power law formula to measure the dimensionality of each island. Well done, now everybody in the class has successfully measured their first fractional dimensionality.
 Which coastline had the highest dimensionality, does this one look the most complex?
 I finished with a bit about my research, in which I use this same technique to measure the complexity of migration trajectories of human white blood cells as the migrate from the bloodstream to sites of infection and/or injury. There, now everybody in the class can do the "impressive" mathematics that I do :).
Reference(s)
Alex Loosley
Alignment Info
Audience(s)  High school students Undergraduate students 

STEM Area(s)  Applied Math 
Standard(s) 

Activity Type(s)  Handson 
Grade Level(s)  High School 
Version  1 
Created  09/17/2013 10:02 PM 
Updated  09/15/2017 10:03 AM 