Fermat’s Last Theorem Documentary

This is an engaging, accessible, and surprisingly moving documentary about Andrew Wiles and his lifelong pursuit of Fermat’s Last Theorem:

http://video.google.com/videoplay?docid=8269328330690408516

Although the mathematics of the proof could not possibly be explained to the layperson (there aren’t many people in the world who could really understand it in its entirety), this BBC documentary does a great job of narrating the struggles, setbacks, and triumphs of Wiles’ pursuit.

The story of the hero and the many peripheral characters (including John Conway) opens a wonderful window into the world of advanced mathematics.

3D Street Art

The painter Edgar Mueller uses dilation and perspective tricks to create absolutely mind-blowing three-dimensional pavement art.

http://www.metanamorph.com/

The website is a little clumsy, but there are some really cool projects here, including The Crevasse, The Waterfall, and Lava Burst.  There are also several videos showing how the artist puts these perplexing pavement paintings together.

For an interesting application of this idea, check this out:  a 3D pavement painting of a child playing in the middle of the street!  The idea was to get the attention of speeding drivers.  I bet it did just that!

NFL Playoff Overtime Rules: The Case for Deferring

I have enjoyed analyzing and writing about the strategy implications of the NFL’s Playoff Overtime rules, and we’ve seen those rules in action for the first time this year.

Here’s a short summary:  the first possession in overtime is determined by a coin flip.  The first-possessing team will be known as Team A, and the other team will be Team B.  If Team A scores a touchdown, they win immediately; if Team A scores a FG, then Team B will receive the ball and can win (with a TD), tie (with a FG), or lose (with a turnover).  Aside from the specific first- and second-possession situations governed by these rules, the rest of overtime plays out in classic sudden-death style.

I looked closely at whether it is useful for Team A to attempt a FG, but the question that occurred to me recently was “If you win the coin toss, is it obvious that you want the ball first?”

Consider the following chart.  Again, Team A is the first-possessing team, and Team B is the other team.  The chart shows all the possible sequences of outcomes on the first two possessions:  touchdown (TD), field goal (FG), or Turnover.  Here, a turnover could be a fumble, and interception, a punt, or a turnover on downs.

playoff-ot-possession-chart

The first-possessing team, Team A, certainly has an advantage in that they can end the game immediately by scoring a touchdown.  But if they don’t score a touchdown, Team B can always win with a TD and in some cases can win with only a FG!  Furthermore, as the original kickoff team, Team B may enjoy a strategic advantage in field position, which can further affect the above scenarios.

Now, in all situations where the game is not decided within the first two-possessions, Team A gets the ball first as sudden-death rules take over.  However, I recall analyses that suggest that the first possessing team doesn’t have a statistically significant advantage in sudden-death overtime.

This is an admittedly elementary look at the situation, but it seems to me that the answer to the question “Do we want the ball first in OT?” is not obvious.  Hopefully I’ve given the coaches something to think about this week.

Related Posts

 

Math Quiz: NYT Learning Network

Through Math for America, I am part of an on-going collaboration with the New York Times Learning Network.  My latest contribution, a Test Yourself quiz-question, can be found here:

https://learning.blogs.nytimes.com/2012/01/30/test-yourself-math-jan-30-2012/

This question is based on the late Joe Paterno’s record for major college football victories by a head coach.  How many years would it take for a contemporary to catch him?

 

Regents Exam Recap: January 2012

Having spent a great deal of time dissecting and analyzing the 2011 New York Math Regents Exams, I was quite interested to see the January 2012 tests.

The same kinds of issues are generally present.  There are instances of mathematical errors, poorly constructed questions, underrepresented topics, and 9th-grade questions on 11th-grade exams.  Here is a quick overview of the Algebra 2 / Trigonometry Regents, the highest-level state math exam in New York.

Mathematical Errors / Poorly Constructed Questions

The exam writers for the New York Regents exams continue to find new and innovative ways to construct erroneous questions.  Here is number 23 from the multiple choice section:

Which calculator output shows the strongest linear relationship between x and y?

This is a bad question to begin with, in that it really isn’t a math question.  The student isn’t being asked to ponder anything mathematical; instead, the student must recognize a step in an artificial procedure about which they have no real mathematical understanding.  (The mathematical technique for finding linear regression equations is not taught in this course; it is expected that the student will use the calculator to generate the equation).

What’s remarkable here lies in the “answer”.  The way you assess the relative strengths of regression equations is by comparing their correlation coefficients (the r values).  Generally speaking, the closer r is to 1 or -1, the stronger the correlation.  In answer choice (4), the value of r is closer to -1 than, say, the value of r is to 1 in answer choice (1).  Thus, (4) is the correct answer.  Pretty easy, right?

Amazingly, the situation represented in answer choice (4) is a logical impossibility.   Since the r value is negative, this means the correlation between the two sets of data is negative; but the regression line’s slope (the b value) is positive!  This cannot happen.

As a result, a scoring correction was issued (after the exams had most likely been graded) and all students were to be given credit for this problem regardless of what answer they put.  And so another flawed question makes it through the draft-revision-publish cycle and into the hands of thousands of students.

Underrepresented Topics

By my count, only one question on this exam (worth 2 out of 87 total points) required the use of either the Law of Sines or the Law of Cosines.  Now, I don’t know how many points should be allocated for these particular techniques, but they are fundamental ideas in trigonometry:  they should be a non-trivial part of the course.

Indeed, a quick look at the exam’s reference sheet is illuminating:  less than half the formulas that are traditionally provided for the student relate to questions on this particular test.

9th Grade Questions on 11th Grade Exams

The final question, and the highest-valued question (6 points) on this Algebra 2 / Trigonometry exam, asked the student to simplify the following expression:

Once again we see a problem from the 9th-grade Algebra curriculum playing a significant role in the 11th-grade Math Regents exam.  A quick look at the official Integrated Algebra Pacing Guide shows that dividing and simplifying rational expressions, the techniques that this problem requires, are part of the 9th-grade course.

This isn’t necessarily an easy problem, and it does require dealing with a cubic polynomial (although factoring by grouping is an optional part of the 9th-grade curriculum).  But this is yet another example in the evolution of this exam showing that, year after year, the hardest questions seem to get easier.

To be completely fair, it was nice to see the exam writers include asymptotes on their graphs this time (to avoid fake asymptotes like these), and they did demonstrate a little more understanding about 1-1 functions (perhaps they did a little studying after this disaster).  But overall, it seems to be business as usual.

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