From your textbook, read pages 14-23

From page 11, do exercises 15 and 16

From pages 36-39, do exercises 11, 13, 17, 23

Finally, check out this article about the Inspection Paradox, a funny paradox we discussed in class.

From your textbook, read pages 14-23

From page 11, do exercises 15 and 16

From pages 36-39, do exercises 11, 13, 17, 23

Finally, check out this article about the Inspection Paradox, a funny paradox we discussed in class.

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:

- Hilbert’s Hotel (TED Ed)
- Cantor’s Diagonal Argument (Numberphile)

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).

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.

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 myap.collegeboard.org (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.

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 *x*³ + *y**³ + z**³ = 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 9*n* – 4 or 9*n* + 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

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.

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:

- Continue to think of examples of fractals in the world around you, and
- 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.

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!

Over the weekend, please take some time to complete the online Student Information Survey.

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”.

Finish the Paper Folding Task that we started in class. Step 5 in particular is a little tricky!

Also, please be sure to bring your Chromebook to class tomorrow. We’ll be reviewing course information and filling out a brief student survey.

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 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.

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!

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

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.

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!

**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?

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.

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

- Sweet Number Pi – Pi music video
- One Million Digits of Pi – can you memorize them all?
- Official Guinness World Record for Most Memorized Digits of Pi – the record is 67,890 places!
- Search for Your Birthday in Pi – mine starts at the 2,373,070th decimal place!
- The Tau Manifesto – Pi is probably not correctly defined and should be twice the value as it is. Many mathematicians call this number “tau” and there is a convincing argument to be made about their point!
- Other Pi Day websites – PiDay.org, PiZone.com