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.

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.

Mathematical Musing: What is r (Part 2)?

In Part 1 of this question, we explored how the correlation coefficient is calculated, and how that calculation relies heavily on the covariance between two quantitative variables. We left off with a few questions: why is r bound between -1 and 1, and why does a value of r near 0 indicate a weak association (and near an extreme indicate a strong one)? In this post, we will answer these questions!

Continue reading Mathematical Musing: What is r (Part 2)?

Mathematical Musing: What is r?

A student asked me a really interesting question recently; a pair of questions, really. We have just discussed the correlation coefficient as a measure of the direction/strength of a linear association between two quantitative variables, and I demonstrated in class that the calculation for this quantity, referred to by the letter r, can be found by the formula

Formula 1: In-class formula for r

In other words, for each point of a scatterplot, find the z-score for the x-coordinate and the y-coordinate of that point and multiply those together. Do this for all of the points in your scatterplot, add them together, and divide by n-1 to get your correlation coefficient.

We discussed various properties of this quantity, and my student asked me that question that teachers always hope for (if not without a bit of dread sometimes!): “Why?” Why does this formula produce a quantity that measures the strength of a linear association? Also, why must the value of r necessarily be bound between -1 and 1? In this post, I seek to start an answer to these questions.

Continue reading Mathematical Musing: What is r?

Mathematical Musing: Editing your Math Pessimism

Feeling down about your status in math class?  Your pessimism just needs some editing!  Ben Orlin over at Math with Bad Drawings has a new post about just that topic.  Go check it out!

The thing of it is, he makes some really good points with this post.  A lot of the things we learn in high school took mathematicians centuries to come to terms with.  The concept of a complex or imaginary number, i = sqrt(-1), wasn’t really accepted until the 18th century (why do you think they’re called “imaginary numbers,” after all?).  So don’t dispair if it takes you a little while to understand something in math class.  Mathematicians of old died before they could come to terms with it!

Mathematical Musing: Will I be Buying a Powerball Ticket?

Correction: The cost of purchasing every possible ticket combination was miscalculated in the previous version of this post. It has been changed to the correct value.

Every so often, the news media becomes all abuzz when a particular lottery jackpot starts to grow really large.  Right now is one of those times, with no winner on Saturday putting the jackpot for Wednesday’s drawing at around $1.3 Billion, the largest lottery jackpot in US History.

My students sometimes ask me, as a math teacher and a guy who “knows numbers,” whether I play the lottery. Usually I just smile and tell them I buy the occasional scratch ticket for the fun of it, but almost never anything beyond that. It would require a “special occasion” or a “huge jackpot” for me to consider buying one.

This certainly seems like one of those special occasions.

To understand how to approach this question from a math standpoint, we first need to understand the probability of winning.

Continue reading Mathematical Musing: Will I be Buying a Powerball Ticket?

Mathematical Musing: Simpson’s Paradox

No, nothing about Homer or OJ (is that too much of a nineties reference?), this paradox is about a statistical phenomenon where analysis of pooled data can lead a researcher to make a conclusion in direct contradiction to the one that unpooled data would lead.  There have been several prominent examples of Simpson’s Paradox arising in areas of college admissions, treatment of kidney stones, and baseball batting averages.

The gist is this: Say you need to have a major operation done and there are two hospitals in your town where you could have it.  You’re worried about post-surgery complications, so you do some research into the hospitals and find that in the past year, patients at the larger hospital suffered post-surgery complications in 130 out of 1000 cases, and patients at the smaller hospital suffered complications in only 30 out of 300.  Based on these results, it looks like the smaller hospital is the better bet: only 10% of patients had complications after surgery there versus 13% at the larger hospital.

However, not all surgeries have the same rate of complications.  Relatively minor surgeries are less invasive and would probably result in a lower complication rate.  With that in mind, you look further at the data and find that, at the large hospital, 120 out of the 800 major surgery patients experienced complications compared to 10 out of 200 minor surgery patients, and at the small hospital, 10 of the 50 major surgery patients suffered complications compared to 20 out of 250 minor surgery patients.  In other words, broken down by type of surgery, the complication rates at the large hospital were 15%/5% for major/minor surgeries while the small hospital saw a rates of 20%/8%.  We see now that the larger hospital has a lower rate of complication across the board, regardless of the type of procedure done.

So why the different conclusion?  It has to do with how many of both types of procedures the hospitals did.  The vast majority of the larger hospital’s 1000 surgeries in the last year were major surgeries, which have higher complication rates across the board.  The majority of the smaller hospital’s 300 surgeries were more minor procedures, which generally have lower rates of complication.  As a result of this imbalance, the overall, pooled complication rates for the two hospitals are biased: the larger hospital towards a higher rate and the smaller hospital towards a lower rate. So it only appears that the smaller hospital has a lower complication rate because most of the surgeries performed there are less likely to have complications.

Check out this website for another explanation of Simpson’s Paradox, as well as some clever interactive animations that demonstrate how and why it can arise.  It’s an important lesson as consumers of data and statistics: while the saying may go “Less is More,” when it comes to how much detail to include in your research, sometimes less is wrong.

Update: It appears that the above VUDLab link is dead, which is too bad. Instead, you could check out this Towards Data Science article or this MinutePhysics YouTube video for some more information.

Mathematical Musing: Happy Pi Day!

Happy Pi Day, dear readers!  Try not to party too hard, and take some time to check out the following links!

Mathematical Musing: Is Algebra Necessary?

Back in 2012, an opinion piece was written for the New York Times asking Is Algebra Necessary?  The piece, written by Andrew Hacker, emeritus professor of political science from Queens College in New York City, suggested that making math education mandatory for all high school students:

Prevents us from discovering and developing young talent.  In the interest of maintaining rigor, we’re actually depleting our pool of brainpower.

This post is not to explore the virtues or flaws with Professor Hacker’s arguments, but to point out what many bloggers have recently observed: that the New York Times answered their own question last week in an article about Sony’s controversial movie The Interview.

The movie in question, you may have heard, was pulled from theatrical release on the 25th after the studio’s computer network was hacked and threats were made against theaters showing the Seth Rogan/James Franco comedy that depicts the two actors as journalists asked by the CIA to use an upcoming interview with North Korean president Kim Jong Un as an opportunity to assassinate the dictatorial leader.  After public outcry from major Hollywood figures and even president Obama, Sony released the film in independent theaters and online.  The December 28th NYT article discusses the amount of money Sony earned off of online sales and rentals, but observes that Sony “did not say” how much of the $15 million revenue was from each source (sales vs. rentals).

The Interview - Is Algebra Necessary

It would appear that Algebra was not reporter Michael Cieply’s strongest subject either, as there is enough information in this article to set up and solve a simple system of equations to answer that exact question.

Let s = the number of $15 sales Sony made and r = the number of $6 digital rentals.  From the $15 million headline, we can write the equation 15s + 6r = 15 000 000.

The second paragraph also tells us that there were about two million transactions overall.  Therefore, we can make the second equation s + r = 2 000 000.

Solving this system of equations is a matter any 9th grader can do:

The Interview - Is Algebra Necessary - Eqn 1

Multiply the second equation by -6 and add vertically

The Interview - Is Algebra Necessary - Eqn 2

Substitute that value of s back into original equation, and you get:

The Interview - Is Algebra Necessary - Eqn 3

So there you have it, New York Times.  With about 2 minutes of high school-level algebra, we can see that The Interview saw about 300,000 downloads and 1.7 million rentals in its first four days.  Maybe you should employ more ninth-graders…

Mathematical Musing – People Love Random Digits!

I ran across this listing in Amazon for A Million Random Digits, a book of random digits used in statistics and other fields for simulations.  Clearly, not riveting reading.  What is very entertaining, however, are the almost 500 user reviews.  Some of the best reviews include one bemoaning the lack of an index, another suggesting the numbers be sorted in order “to better find the one I’m looking for,” and another suggestion that readers find the source in the original binary so as to not lose the most “significant digits” in the translation.

For another moment of bizarre, check out the “Also Viewed” section.