All posts by Benjamin Kirk

I am a New York State Master Teacher in Mathematics at Ithaca High School. I have been teaching at Ithaca High School since 2007.

AP Statistics Assignment for 9/20

Read the What If? and What Can Go Wrong sections from Chapter 4 (pages 93-95).

Over the weekend, write your response for the Chapter 4 Investigative Task (Auto Safety). As before, you may use your textbook and notes to help you write your response, and you may use the Internet to access Google Docs and Stapplet or whatever other graph-making tools you want to use, but you may use no other resources other than those. Do not go online looking for more information, and definitely no working with other students on this assignment.

Your response should be completed in a Google Document and submitted to me ( by the start of class on Monday, the 23rd. Please name your file appropriately; it should have the format “LastName.FirstName.Ch4InvTask”.  For example, mine would be “Kirk.Benjamin.Ch4InvTask“.

If you would like to rewatch or explore the data of the documentary from in class, you can find it here: The Fallen of WWII. Try the interactive version to explore the data more thoroughly.

Fractals & Chaos Recap for 9/19

Today was an important day.

We continued a thought started yesterday, acknowledging the difference between intrinsic dimension—the dimension of an object itself—and extrinsic dimension—the dimension of the space the object occupies. For example, the edge of a circle is one dimensional, since there is only one axis of movement around the circle, but the circle exists in two dimensional space. We further observed that the extrinsic/containing dimension of an object must necessarily be greater than or equal to the intrinsic dimension of the space.

We also discussed how the dimension of an object affects the dimension with which we can measure it. We cannot find the volume of a square, and we can’t find the area of a cube. Put another way, if we tried to use a cubic centimeter measuring device to measure the area of a circle, we’d end with a measure of zero. And if we tried to use a square centimeter measuring device to measure the volume of a cube, we’d get a measure of infinity. In general, too small a dimension of measure results in a measure of infinity, and too large a dimension results in zero.

Finally, we considered the implications of this to the Sierpinski Triangle. When we attempted to measure the area of the Sierpinski triangle, we resulted in a measure of zero (an infinity of triangles each with area zero results in a total area of zero). When we attempted to measure its length, we resulted in a measure of infinity (an infinity of segments each with length zero results in a total length of infinity). Thus, 2 is too high a dimension to describe the Sierpinski Triangle and 1 is too low. It must therefore mean that the dimension of the Sierpinski Triangle is strictly between 1 and 2, and is therefore fractional.


Of course, this raises an immediate question: If the dimension of the Sierpinski Triangle is somewhere between 1 and 2, what is it?

Fractals & Chaos Lesson Recap for 9/18

We had a short conversation about the nature of dimension in class today, after yesterday‘s wild results. In a quest to identify “true” two-dimensional entities in our world (since the conventional example of a piece of paper still does have some thickness), we observed that there may be a difference between “intrinsic” dimension, that is a characteristic of an object versus the “extrinsic” dimension of the space it occupies. For example, a desk is clearly a three dimensional object, but the surface of the desk could be thought of as 2D and the desk’s height could be considered 1D.

We resolved to think on this some more for a further conversation tomorrow, and transitioned to working some more with FractaSketch. I demonstrated how to make a fern using the program and made available some basic templates that you could use as inspiration for the upcoming 2019 Fractal Art Show.

The exact date of the art show will be determined later, but I will expect from each of you one entry in three of the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

For tomorrow: continue to think about what the term “dimension” really means, per yesterday’s conversation.

Fractals & Chaos Lesson Recap for 9/17

Today something weird happened.

We’ve already seen the Sierpinski Triangle appear in two ways, by iterating the process of breaking a triangle into four equal pieces and removing the central one, and by coloring the even elements of Pascal’s triangle. Today we found a third way, by Iterating a Curve.

By starting with a solid, filled in triangle, and step-by-step removing stuff from that area, we reach the same exact figure as we get by starting with a simple line segment and lengthening/bending it. The area clearly starts as 2-dimensional, the line segment clearly starts as 1-dimensional. Yet both processes have the same end result. What does this mean for the dimension of Sierpinski’s Triangle? Is it 2D or 1D? It certainly can’t be both, so the only option is… neither?

When I first took this course (back in the days of Mr. Drix), this was the first moment where I realized all of this talk about fractional dimension may be more than nonsense. Maybe there’s something to it after all…

Your homework: think critically about what dimension really means. What does it mean to say something is 1-dimensional or 2-dimensional. We’ll dig into this in the next few days…

Fractals & Chaos Recap for 9/16

After a brief discussion of the weekend’s reading, we started today sharing the images we created by coloring in the even elements (i.e., multiples of 2) of Pascal’s triangle, finding that it produces the same Sierpinski’s Triangle pattern we observed only a few days ago. Wow!

This sparked our curiosity about what other patterns can be found by coloring in other multiples in Pascal’s triangle, and we got into groups to examine the same patterns for 3, 4, 5, 6, and 9. Your homework is to finish these pictures.

We spent the rest of the period playing with FractaSketch,

Fractals & Chaos Recap for 9/13

We looked briefly at Pascal’s Triangle today, and some of the neat patterns that can be found there. I hinted at some hidden fractals that could be found by removing numbers from the triangle, so your homework is to fill in circles in this smaller version that would represent removing every even number from the triangle (remember, we observed that two filled in circles create a filled in one, two empty circles create a filled in one, and an empty and filled circle create an empty one).

We wrapped up class by playing with FractaSketch some more (linked at left). Before everybody left, I also handed out the next assigned reading for the course: this Science News article from 1997 (Fractal past, Fractal future) and this supplementary article from a 1997 issue of Popular Science about the Heartsongs album mentioned in the first one.

AP Statistics Assignment for 9/13

Over the weekend, write your response for the Chapter 2 Investigative Task (Race and the Death Penalty). As noted in the document, you may use your textbook and notes to help you write your response, and you may use the Internet to access Google Docs and Stapplet or whatever other graph-making tools you want to use, but you may use no other resources other than those. Do not go online looking for more information, and definitely no working with other students on this assignment.

Your response should be completed in a Google Document and submitted to me ( by the start of class on Monday, the 16th. Please name your file appropriately; it should have the format “LastName.FirstName.Ch2InvTask”.  For example, mine would be “Kirk.Benjamin.Ch2InvTask“.

Because some folks have asked, you can find instructions on taking screenshots using a Chromebook here. I recommend the Ctrl+Shift+Window Switch shortcut to take a screenshot of only the particular items you want to use.

Fractals & Chaos Recap for 9/12

We finished our discussion of the Sierpinski triangle, noting that just like the “final” version of the Koch Curve is “nothing but angles,” this geometric oddity is “nothing but edges,” as the area of the triangle converges to zero as the iterations continue. This conclusion also presented an interesting contradiction. For the Koch curve, we argued that an infinite number of segments, each of length zero, resulted in an infinite perimeter (effectively, ∞ * 0 = ∞). Here, we have an infinity of triangles, each again with an area of zero, resulting in an area of zero (effectively, ∞ * 0 = 0).

What this reveals is that the expression “∞ * 0” is what is called an Indeterminate Form, an expression the defies definition. We can create a reasonable argument that defines it as infinity, and we can create a just-as-reasonable argument that defines it as zero. Therefore, it must be defintionless.

We finished the day by opening up the PC laptop mobile lab and downloading FractaSketch to each device. We’ll be using this software extensively over the next few weeks!


AP Statistics Assignment for 9/12

Your main priority tonight is to familiarize yourself with your first Investigative Task about Race and the Death PenaltyDo not start yet. Just read it over and come in with any questions you might have to understand the task.

See here for some additional reading about Simpson’s Paradox.  You should also take a look at questions 41 and 42 from page 42 of the text.


Fractals & Chaos Recap for 9/11

We finished our discussion about the Cantor Set, noting that infinite set of endpoints that are left over with each segment removal are countably infinite. If the claim is that the cantor set is actually uncountable, that requires there to be other elements of the set that are not segment endpoints. And there are such points, uncountably infinitely many of them. See this post for more information on how this works.

We went on to draw a picture of the Koch Snowflake, a figure devised by Swedish mathematician Helge von Koch as an example of a continuous curve with no tangents. As the fractalization process continues, the number of segments that make its perimeter increases without bound, but the length of each segment shrinks to zero. Mathematically, however, the total perimeter also increases, resulting in a figure with finite area and infinite perimeter!

We ended the period by starting to draw an image of another fractal, called the Sierpinski Triangle. Start with an equilateral triangle (connect the dots in the worksheet), then bisect and connect all three sides. Fill in that middle triangle, effectively removing it and leaving you with three triangles at the three corners. Then do that process again for each of the three triangles: bisect the sides, connect them, and fill in the middle triangle. Do that as many steps as you can fit. Don’t cheat and look up what the final result looks like!

Mathematical Musing: The Cantor Set is Uncountably Infinite: A Proof

This post is intended for my Fractals and Chaos students. Other students may find it interesting, but in order to understand its full context, you will have to take my class! For those looking for an explanation of this third property of the Cantor Set, read on.

Continue reading Mathematical Musing: The Cantor Set is Uncountably Infinite: A Proof

Fractals & Chaos Recap for 9/10

We continued our study of the Cantor set by spending some time thinking about its properties, in particular how it has a length of zero yet still has an (uncountable) infinity of points contained inside.

That the length is zero is fairly easy to see from the fact that we remove 1/3 of the set in the first step, 2/9 in the second, 4/27 in the third, 8/81 in the fourth, and so on. That sum 1/3 + 2/9 + 4/27 + 8/81 + … forms an infinite geometric series, the sum of which is 1. And since the length of the original segment is also 1, the length of the “final” version of the Cantor Set is 0.

Yet it clearly contains an infinity of points! With each stage, we create endpoints of segments that never get removed, and an infinite number of stages produces an infinite number of endpoints. But not only that, I claimed the Cantor set is uncountably infinite, which required some explanation of the realization that some infinities are bigger than other infinities.

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