Teachers As Scholars (Steven J Miller: sjm1
AT williams.edu)
"A-ha!" Moments in Mathematics
and the Sciences
Monday January 13 and
Monday February 3, 2014
Harvard Hillel, 52 Mount Auburn Street, Rabb Hall (1st Floor)
Videos and Links for
Topics:
Below are the videos of the main items I presented at the sessions, as well as
some links and supporting materials by myself and the class; discussions among
participants was not recorded. Feel free to share these resources with anyone
you wish. These are meant to be just quick summaries to remind you of what we
discussed; if there are items I can add to make these more useful, just let me
know. I'm happy to work with you on creating units for your students (and if
that means having me do new lectures and record and upload them from Williams,
I'm happy to do that).
-
My math riddles page.
Please feel free to share these riddles with your colleagues and your students,
and let me know if there is anything I can do to make the site more useful for
you and your classes. The goal is to continuing expanding the student /
teacher's corner to facilitate using these in classrooms. If you or any of
your students are interested in
helping (a great item for the CV!), or want updates on the progress, let me know.
- VCTAL (Value
of Computational Thinking across Grade Levels 9-12): A great group of
people, we work on computational modules for high schools. See the webpages or
talk to me about using these in your classes, or getting modules to peruse.
I've been fortunate enough to work with Jim Kupetz on two units (cryptography
and streaming video).
- Class 1: Pythagorean Formula and Dimensional
Analysis (http://youtu.be/Ih1qjaKahEA):
24 minutes: Discusses some of the standard proofs of Pythagoras, including
James Garfield's
(Williams, Class of 1856) proof, and then the dimensional analysis proof. We
then discussed other dimensional analysis problems, such as the period of a
pendulum. Starts with the need for proof by giving an example of something
that works for awhile, but eventually fails.
- Russell's Paradox (http://youtu.be/hOJ3VaIPs2o):
5minutes: Continues the theme begun earlier on why we need proofs: a
collection of objects that all have a given property is not always a set.
- Pascal's Triangle and Binomial Coefficients
(http://youtu.be/nrknXC8xmTU):
41minutes: goes through Pascal's triangle, what it is and why it works, why
the relations hold, and shows how to get from that to Chaos Theory and
Fractals (we can discuss more on this later). Includes a Mathematica video
generating the triangle modulo 2, discussions on related problems, and issues
in computing these triangles quickly. Gives story proofs of some binomial
identities.
- Tic-Tac-Toe (http://youtu.be/TAs271A4r78):
30minutes: Discusses the theory of games, how to count the number of
possibilities, how to analyze strategies, how to generalize and create new
games such as Tic-Tac and Tic-Tac-Toe-Toc.
- Fibonaccis and M&Ms (http://youtu.be/3TkuXuds8vU):
67minutes: We go through a proof of the Geometric Series Formula by
considering a game of basketball, and talk about the dangers of misuing math
formulas (i.e., not checking the conditions needed to use). We see
applications of our hoops proof (a memoryless process) when we look at the M&M
game. Our analysis of that problem, as well as Roulette, involves difference
equations. We talk about these connections, and see some of the issues.
- I gave a version of the M&M talk at a meeting of the MA Math Teachers
Association, and to my Calc III students at Williams College (and to
talented HS kids in a summer math program at Hampshire College).
- Here is a video (about 6.5 minutes:
http://www.youtube.com/watch?v=Esa2TYwDmwA) that I made on Roulette and
Difference Equations.
- Here is the Nature By Numbers video:
http://www.youtube.com/watch?v=Y8ivF5GqzKo
- Press Your Luck scandal: the lack of randomness:
- Birthday Problem, Marriage Problem and Ulam's
Spiral (http://youtu.be/74pLoajA3QM)
48 minutes:
- Cryptography and Binary (http://youtu.be/cQAs5YoDxtQ)
1 hour 32 minutes
- Continued Fractions and Irrationality
http://youtu.be/7FN-ufcKEvc : 20
minutes
- Benford's law
http://youtu.be/20eneZBJt64 (15
minutes)
- HOMEWORK PROBLEMS:
- Let A win \(p\%\) of its games, and B win \(q\%\) of their games (\(q\)
is not necessarily \(1-p\)). Imagine A and B play each other now. Which of
the following four formulas do you think does a good job of estimating the
probability A beats B: \((p \pm pq) / (p + q \pm 2pq)\)? Remember there are
four choices for the two signs: both positive, one positive and one negative
(two ways), both negative.
- Consider a \(5 \times 5\) chess board. It is possible to place \(5\)
queens on the board in such a way that there are three squares they cannot
attack, so that you can then safely place \(3\) pawns on those squares. How
can you do this?
- Watch the YouTube video (http://www.youtube.com/watch?v=Esa2TYwDmwA)
on Difference Equations and Roulette. Let me know if something like this
could be of use to you. What can I do to make videos like this more useful?
I can make several of these with OIT here, but input on what would be useful
to you would be greatly appreciated.
- The Euler totient function, \(varphi\), is a map from the positive
integers to the positive integers. It is defined as follows: \(\varphi(n)\)
equals the number of numbers in \(\{1, 2, 3, \dots, n\}\) that are
relatively prime with \(n\) (relatively prime means the only positive
integer dividing it and \(n\) is 1). For example, \(\varphi(12) = 4\), as
the only numbers in \(\{1, 2, \dots, 12\}\) which are relatively prime to 12
are 1, 5, 7 and 11. Compute \(\varphi(n)\) for a lot of values of \(n\). Try
to make some conjectures based on the structure of \(n\). Can you prove any
of them? We'll see this function when we talk about cryptography.
Description: There
are many stages and approaches to solving a problem. Before trying to find a
solution, it's often a good idea to determine if one exists, as if it doesn't
our search will be fruitless! If we're fortunate enough to know there is a
solution, the next step is to find it. Oftentimes we can solve a problem through
brute force approaches, but these are inelegant, give us no greater
understanding of the problem or why the solution works, and frequently cannot be
generalized to related problems. The goal of this seminar is to introduce you to
alternatives to brute force. There are many examples throughout mathematics and
science where, if you look at the problem the right way, what was originally a
long, involved calculation evaporates into a wonderful `A-ha!' moment! We'll
draw examples from problems that can be solved by tedious, long computation as
well as quick arguments if viewed the right way.
Probable topics, comments and reading:
- Combinatorics: Problems include:
- Formulas for the Fibonacci numbers (here's
a nice video introduction): brute force recurrence versus divine
inspiration to reach Binet's formula. Application includes double plus
one (see this video for
more on the problem).
- Products of binomial coefficients (see
the wikipedia page for the definition). Examples include the formula for
the sum of adjacent binomial coefficients, proof of the Binomial Theorem, as well as \( \sum_{k=0}^n \left({n
\atop k}\right)^2 \) (with \(n\) a positive integer). We can solve some of
these by brute force, but storytime is much better.
- Two nice applications of the binomial theorem: if know calculus, it
allows us to show \(e^x e^y = e^{x+y}\); it's also used to compute the
derivative of \(f(x) = x^n\) for \(n\) a positive integer.
- A nice application of the exponential relations is the proof of all trig
identities; this requires \(e^{i \theta} = \cos\theta + i \sin\theta\) (Euler's
formula).
- Partition problems: the cookie problem or the stars and bars problem:
how many ways are there to divide \(C\) identical cookies among \(P\)
distinct people? As a nice application can analyze the probability of
winning a lottery with and without repeated numbers allowed.
- Symmetry, Duality and Exhaustion:
Problems include:
- Tic-tac-toe: how many first moves are there? How many games?
- Chess pieces: Consider a 5 x 5 chessboard; place 5 queens on the board so
that 3 pawns may safely be put down.
- Legal 21: given the numbers 1, 5, 6 and 7, use each number once and only
once and create 21 by combining these with the four basic operations (plus,
minus, times, divide). Key is to enumerate all possibililties.
- Application: duality arises in linear programming (see
this video for more), in signal processing,
....
- Dimensional analysis: Problems
include:
- Prove the Pythagorean theorem: if \(a\) and \(b\) are the sides of a right
triangle with hypotenuse \(c\), then \(a^2 + b^2 = c^2\).
- Show the period of a pendulum near the surface of the Earth is
proportional to \(\sqrt{L/g}\), where \(g\) is the acceleration due to gravity
and \(L\) is the length of the pendulum.
- Morley's
theorem: we probably won't do, but it's worth knowing;
see this
site for some proofs.
- Fast Algorithms: Problems include:
- Solving polynomial equations: we can search for integer solutions and try
to factor by sight, but want a general formula. Quadratic equation is the
solution. Start with simple linear and keep generalizing.
- Babylonian multiplication: leads to the concept of a look-up table.
Sometimes some operations are cheaper than others, and want to exploit.
Another example is the
Strassen's algorithm
for matrix multiplication.
- Evaluating a polynomial efficiently: Horner's algorithm. Application to
fractal geometry and iterating polynomials. A great example is the Mandelbrot
set (see this video);
one of the first instances of fractals being used in movies was the Genesis
torpedo in Star Trek II: The Wrath of Khan (click
here for the video).
- Fast Exponentiation: one of the key ingredients in modern cryptography.
- The Euclidean Algorithm: allows us to find greatest common divisors, and
runs in a lot less time than we might expect (every two iterations is a
savings of a factor of 2).
- Let \(a\) be relatively prime to a prime \(p\). Can prove \(a^p - a\) is a
multiple of \(p\). Without loss of generality may assume \(a\) is in \(\{0, 1,
\dots, p-1\}\). We have \(a^p = (a-1 + 1)^p = (a-1)^p + p (\cdots) + 1^p\)
(this is because of the binomial theorem), so \(a^p = (a-1)^p + 1\) plus a
multiple of \(p\). We do this a total of \(a\) times and we get \(a^p = a\)
plus a multiple of p, proving the desired relation. After some obvious
algebra, we get \(a \cdot (a^{p-1} - 1)\) is a multiple of \(p\), and since
\(a\) and \(p\) are relatively prime, we find that \(a^{p-1} = 1\) plus a
multiple of \(p\) if \(p\) is a prime and \(a\) is relatively prime to \(p\).
This gives a primality test (called the
Fermat
Primality Test). If we choose an integer \(n\) and another integer \(a\)
relatively prime to \(n\), then if \(a^{n-1}-1\) is not a multiple of \(n\)
then \(n\) cannot be prime. Sadly the reverse direction need not be true (see
the Carmichael
numbers).
- Note the above primality test can tell us an integer is prime without
telling us a factor! This should seem strange: we can prove a factor exists
without being able to find it! In general, factorization is believed to be
hard, and is the foundation of many cryptographic schemes. We can talk about
\(N = pq\) for two large primes versus a large random number password.
- Key takeaway: there are faster ways of doing computations than you might
expect!
- Probably won't do, but another topic would be the different sorting
algorithms (BubbleSort, MergeSort, QuickSort, ...).
- Irrationality Proofs: Problems include:
- Standard proof that \(\sqrt{2}\) is irrational. Give Tennenbaum's
geometric proof and my generalization with David Montague (paper
is here).
- Proofs by Induction: Problems include:
- Maybe sum of odd numbers by induction versus the geometric proof. Could
also do triangular numbers by induction versus the geometric proof (use a
right triangle and not an equilateral triangle and remove the double
counting).
- Error Detecting / Correcting Codes:
Problems include:
- Sphere packing, strange probabilities....
Math Riddles
I also maintain a math riddles page.
Please feel free to share these riddles with your colleagues and your students,
and let me know if there is anything I can do to make the site more useful for
you and your classes. The goal is to continuing expanding the student /
teacher's corner to facilitate using these in classrooms. If you or any of your
students are interested in
helping (a great item for the CV!), or want updates on the progress, let me know.
