All posts by Mr. Kirk

I have been Mathematics Teacher at Ithaca High School since 2007. In that time, I have taught Algebra 1, Intermediate Algebra, Honors Algebra 2, Honors Pre Calculus, Advanced Placement Statistics, Math AIS, and a one-semester elective course in Fractal Geometry and Chaos Theory.

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.

Additional viewing:

Intro to College Algebra Assignment for 9/9

Finish the Four 4’s puzzle you started in class: Using four 4’s and any combination of operations, write ten expressions that each evaluate to the numbers 1 through 10 (and 11 through 15 if you’re interested in a challenge!). You must use exactly four 4’s (no more and no fewer!) and you may not use any other numbers than fours (so no powers of two or three).

Fractals & Chaos Recap for 9/9

We added to our list of fractals and discussed some observations from Wilson’s excerpt. We observed that dimension is tied to measurement, with lengths being one-dimensional, areas being two-dimensional, and volumes being three-dimensional. This suggests that any notion of fractional dimension must correspond to measurements that don’t fit within those three categories.

Furthermore, the article illustrated an interesting point (one that we will come back to soon): shrinking our scale of measurement increases the quantity of measurement we can find, disproportionate to the shrinking scale. The amount of space found at progressively smaller scales increases rapidly, and with it biodiversity.

After a brief distraction with a mouse, we introduced our first mathematical fractal: the Cantor Ternary Set, developed in 1883 by German mathematician Georg Cantor to challenge contemporary ideas of infinity and length. In particular, the set has three seemingly contradictory properties:

  • Every member of the set is a limit point
  • It has an uncountably infinite number of points
  • Yet its Length is zero

We will explore these ideas in the next few days.

AP Statistics Assignment for 9/9

Visit this post from the beginning of the year for the Student Information Survey and Textbook Log Form.

Also, I need everyone to register for the online AP Classroom for messages and occasional assignments. Sign in to (use the same login you use to access prior AP scores or to register for the SAT) and click “Join a Course or Exam” and use the code 9XKQKJ (period 1) or KWKAJG (period 7) to register for the class.

Tonight, please read Chapter 1 (pages 1-9) of your textbook. Skim it if you don’t have time to read it closely, but work on questions 1, 3, 9, and 13 from pages 10-11.

Finally, look over the student survey results and identify 2-3 interesting statistics and/or observations you can glean from the data.

Breaking Math News!

Too often, students view math as a stationary field, one that was invented (or discovered, depending on your perspective) hundreds of years ago and has remained unchanged since, with everything knowable having been figured out long ago. It’s easy to think that the only questions that remain are the incredibly difficult, abstract, esoteric ones that would require decades of math study to even understand. But that’s simply not true!

Consider the equation x³ + y³ + z³ = k. Easily understood: take three integers {x, y, z}, cube them, and add them together. In 1955, mathematicians at the University of Cambridge asked if a set of {x, y, z} could be found to add to every positive integer k less than 100. Some were easy to find: (-5)³ + 7³ + (-6)³ = 2; 2³ + (-3)³ + 4³ = 45; 25³ + (-17)³ + (-22)³ = 64. But others proved surprisingly challenging, requiring cubes of much larger numbers in order to form (51 is the sum of the cubes of -796, 659, and 602, and the solution for 30, found only in 1999, required the cubes of 2,220,422,932, -2,218,888,517, and -283,059,965). Even more unfortunately, there appeared to be not much of a pattern in the trios of numbers that worked, and so finding new solutions mostly amounted to an enormous guess-and-check procedure. A pair of mathematicians proved in 1979 that any number that the expressions 9n – 4 or 9n + 4 evaluate to (4, 5, 13, 14, 22, 23, 31, 32, 40, 41, 49, 50, 58, 59, 67, 68, 76, 77, 85, 86, 94, and 95) could not be expressed as the sum of three cubes, which took out several elusive numbers, but the search wore on to finish the list. Until this year.

At the start of 2019, a sum of cubes had been found every possible positive integer k < 100 had been found except for two: 33 and 42. Computer programmers had been checking possible combinations for more than 50 years, and had progressed with their checks up to numbers beyond 100 trillion, but with no success! Finally, Andrew Booker, a mathematician at the University of Bristol, came up with a new way to search for likely trios much more efficiently and set a university supercomputer to the task. It took only three weeks to find a solution:

33 = (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³

When he found the solution, Booker said he literally jumped for joy. But his job wasn’t done! There was still one number to be solved, and he knew this task would be too large for even his university’s computer.

So he turned to MIT’s Andrew Sutherland and a worldwide computer group called Charity Engine, Members of the group from around the world run a program that uses their computers’ downtime to do data crunching, effectively donating their devices’s computing time to a variety of causes. Fans of Douglas Adam’s The Hitchiker’s Guide to the Galaxy will notice a similarity here to the story, where a computer the size of a planet is constructed to find the “Ultimate Question of Life, the Universe, and Everything,” which a previous supercomputer had identified the answer as 42.

After a combined computing time equivalent to almost 150 years, they found the answer this month:

42 = (-80,538,738,812,075,974)³ + (80,435,758,145,817,515)³ + (12,602,123,297,335,631)³

The next-lowest number to have an unknown sum of cubes is 114, and in fact there are only ten numbers less than 1000 for which such a solution is unknown.

If you’re interested in learning more about this mathematical puzzle, I’d suggest you start with this Numberphile video that Booker says was his inspiration to start on his hunt. You can see an interview with Booker after his cracking of 33 here, and a recent followup here.

Fractals & Chaos Recap for 9/6

We finished our discussion of the ideas inspired by the Jurassic Park excerpt, including looking at a few theories of using fractals to predict financial markets (see the silver and bitcoin articles here if you’d like to read them more closely).

From there, we discussed last night’s assigned reading and used it to form some properties about fractals. In particular:

  • They demonstrate self-symmetry or self-similarity (each part could be viewed as a scaled-down version of the whole thing)
  • They are Non-Euclidean (for a Euclidean curve, no matter how wiggly it is, zooming in far enough eventually makes it look linear, but for a fractal zooming in just reveals the same level of detail).
  • They have fractional dimension (unlike one-dimensional lines, two-dimensional squares, or three-dimensional cubes, fractals live in a space between and could have a non-integer dimension)

This last idea is, of course, pretty wild, and if you feel skeptical about it, you should. Hold on to that skepticism! Let me convince you.

We finished the day making a brief list of ideas of fractals, including snowflakes, ferns, feathers, trees, and river deltas. Your homework is twofold:

  1. Continue to think of examples of fractals in the world around you, and
  2. Read over the excerpt from Edward O Wilson’s book The Diversity of Life: Living Labyrinths.  As before, make a note of 2-3 passages that seems significant or questions you have.


Intro to College Algebra Assignment for 9/6

At the start of class today, you were asked to fill out this Student Information Survey. If you didn’t finish it, please do so this weekend.

Also, take some time to read over the Course Expectations for this class. There will be a brief Kahoot! quiz on Monday, with a prize for the winner!

Finally, we will be starting in on Unit 1 on Monday, so please obtain and bring the 3-ring binder and spiral notebook or bound journal as detailed in the Course Expectations. You will need both on Monday!

Fractals & Chaos Recap for 9/5

We started out today with a discussion about course expectations. This is a pass/fail course but I still expect you to take it seriously. I will often ask you to read an article or do some math work at home, and I expect that work will be done the next day and ready to be discussed. I expect everyone to actively participate in class discussions and engage in the work we do during class time. This is not a study hall, so please don’t bring other work to do during this class or I will ask you to leave.

After a discussion of some of the pre-existing notions on what fractals are, we read an excerpt from Michael Crichton’s Jurassic Park and discussed some of the big ideas to be found there (you can see a clip of this scene from Steven Spielberg’s 1993 film version here).

For your first reading assignment, I’m asking you to read the article distributed in class: Fractals: Magical Fun or Revolutionary Science, from the March 21, 1987 issue of Science News. Take some notes as you read, jotting down the 2 or 3 major points of the article.  Pay particular attention to how this article defines the term “fractal”.

AP Statistics Assignment for 9/5

Complete the Smelling Parkinson’s simulation that we started in class today. Go to the simulation page here and change the settings to match those in the screenshot below (though feel free to tell the simulator to stop at 10,000 trials or “never” to get even more if you’re curious). Record your result for 1000 trials on your paper, and bring it in tomorrow.


If you would like to read the interview with Target’s lead statistician, where I got the story about the father and his pregnant daughter, you can find it here.

Welcome Students and Parents!

Welcome to my website!  I designed this site to be a simple and useful means for me to communicate with you outside of class. How will this site be useful to you?


I will post the daily homework assignments (and daily notes whenever possible) here on this site by the end of the school day.  If you are absent from class, either from illness or some other reason, you are expected to come to this site to find out any homework assignments you may have missed.  You will receive extra time to complete them as I discussed in class, but this way you can stay caught up without even coming in!


The comments for every homework post are open and anonymous.  If you ever have questions about any aspect of the homework, come here and make a quick post.  I will check the site often and answer your question as quick as I can.  You don’t need to give your actual name when you post, so don’t feel embarassed about asking!  Besides, I’ve never seen the case where only one person had a particular question; if you’re wondering, somebody else is probably wondering too.

If you want to come get some extra help during the day, see the Class Information page at the top!


That’s right!  I will always announce an upcoming test or quiz here on this blog before I announce it in class!  Visit the blog every day so you can plan out your studying and preparation.


In my Statistics class in particular, but occasionally in Intermediate Algebra, we will do a few projects over the course of the year.  Any details about these projects that I give in class I will also post here.  If you forget when a part of a project is due, visit the site and check the post about it!


Introduction to College Algebra and AP Statistics both have a major end-of-year test (Fractals and Chaos does not).  Over the course of the year, I will post links, advice, and test taking strategies to help you prepare for these tests.


Finally, any time I come across something interesting about the field of mathematics, I’ll post it here.  It could be a news article, a nifty website, or just something I heard.  The world of math is constantly changing and evolving, despite what you may think.  New strategies of solving puzzles are found, old puzzles are being solved, and those solutions are leading to new puzzles every day!

I hope you will visit this site often, and I look forward to working with all of you this coming year.  Thanks for coming and I’ll see you in class!  Oh, and incidentally, my favorite color is green.