This is the finale of the math competition scene in Mean Girls. I can only guess that references to this scene are ubiquitous in calculus classes these days (either to the delight or exasperation of calculus teachers).
Here’s a graph of the actual function, referred to in the movie as an “equation”.
When I was exploring the tangent identity a couple weeks ago, I thought that one of the more peculiar aspects of it was that the sum of three numbers was also equal to the product of those three numbers. After all, among the integers this is only true of (0, 0, 0), (1, 2, 3), and (-1, -2, -3).
But as it turns out, this is so common among the real numbers that given any two real numbers x and y, there is a third number z such that x + y + z = xyz as long as xy ≠ 1. This is fairly easy to show with some algebra.
What does a plot of all these points (x, y, z) look like? Let’s turn to WolframAlpha to give us a 3d graph:
I expected the above to be a little more interesting until I realized the boundaries were too narrow. Here’s a better look:
Interesting!
How many balloons are these two men carrying? Questions like this just come to mind when I’m walking down the street.
If I could have a mundane superpower — a knack or skill that is extremely useful but much less impressive than flying, invisibility, or super strength — I would choose to be an amazing estimator. If I could accurately estimate any quantity or measurement, I think I could do a lot of good.
2012 is a leap year, meaning that an additional day is added to account for the fact that the actual time it takes for the Earth to make one revolution around the sun is slightly more than 365 days. A leap year occurs every four years with one exception: a year that is divisible by 100 is a leap year only if it is also divisible by 400.
I recently learned that the Chinese calendar has a different definition for a leap year. The Chinese calendar is primarily a lunar calendar. That means that months are determined by the phases of the moon. Specifically, the start of each month always occurs on a new moon. A typical year has twelve months and 353, 354, or 355 days. The correction to account for the difference between a Chinese calendar year and an astronomical year is a little more drastic: every so often, a new month is inserted because there will be thirteen new moons within a year.
Complicating matters even further is a rule that states the winter solstice must occur in the 11th month. Therefore, the point in the calendar where the new month is added will vary from leap year to leap year. How often do leap years occur in the Chinese calendar? The intervals are not regular, but it’s once every three years!
At the beginning of this school year, I had an idea. The idea was that what my students needed wasn’t a better explanation from me but the opportunity to confront their own misconceptions head on. So I came up with what I started to call thought exercises. Here’s an example of one of the first I used:
My goal changed from trying to be clear and concise to actively wanting to confuse my students. My hope was that their confusion would inspire curiosity, thoughtfulness, and eventually resolution. Of course, there was the danger it could produce hopelessness. Curiosity though tended to win out.
Here are some other thought exercises that generated particularly good discussions in class.
Confusion and uncertainty are powerful tools in teaching and learning. Here’s to 2012 being a particularly confusing year!
As each year ends, I often take the holiday time to reflect on things that went well and things that didn’t. One project of mine that I’m particularly proud of is my Math Minutes bulletin board at school.
Each week, I post between three and six problems outside my office. These problems come from MAA’s Minute Math blog (also where my bulletin board gets its name) and from math teachers I follow on Twitter, and some I even write myself. My goal was to offer clever, rewarding, and quick challenges to my students in the hopes of getting them to talk about math outside the classroom.
At the end of each week, I would post the names of students who submitted correct solutions along with the solution itself.
I had a lot of fun doing this, and it was amazing seeing so many students who didn’t even have me as a teacher participating. I just hope I have the energy to keep it up in 2012!
Inspired by last night’s #mathchat, I started exploring patterns. Specifically, I tried to create some patterns that could be “continued” in more than one way. For example, the sequence:
3, 5, 7, …
can have 9 as the next term (if we are listing odd numbers) or 11 (if we are listing odd prime numbers). Similarly, the sequence:
1, 11, 121, 1331, 14641 …
can have 161051 as the next term (if we are listing powers of 11) or 15101051 (if we are listing smashed together terms that form lines of Pascal’s Triangle).
But this only got me thinking: given any finite sequence of numbers, no matter how long, there are an infinite number of choices for the next number in the sequence, and all of them could fit into a plausible pattern! For example, you might think it’s obvious what the next number here is:
2, 4, 6, 8, 10, …
But I could just as easily claim the next number is 983, because the actual sequence I had in mind was:
2, 4, 6, 8, 10, 983, 985, 987, 989, 1001, 1974, 1976, 1978, 1980, 2953, …
Here’s a much more compelling question though, and it has close ties to some of the theory used in statistics and hypothesis testing. Consider this sequence again, but think about plausible choices for the next two terms:
2, 4, 6, 8, 10, …, …
You might believe, prior to all of this discussion, that the most likely choices are 12 and 14. If I reveal that the term right after 10 is not 12, then that completely undermines your belief. But what if I told you that it is 12?
2, 4, 6, 8, 10, 12, …
How much stronger is your belief now that 14 is the next number? Or put it another way, how does this new information affect your previously held belief?
Holiday weekends inspire this kind of activity, especially with the NFL heading into playoffs soon while the NBA begins its regular season.
I wondered if there should be more intersections between these three circles, but that may not be physically possible.
Happy holidays!
How do you know when a teacher is ready for winter break? He spends the morning making the following on the bulletin board outside of his office rather than preparing lessons for the day.
A lower school student walking by pointed at it and stated it looked like a Girl Scouts badge. I personally think it looks more like the Triforce from Zelda.
Here’s to the start of two very much anticipated weeks of rest!