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IB Math HL — Mathematical Exploration

General info:

Topic selection:

  • Megan — mathematical modeling — S-I-R model for the spread of H7N9 in China
  • Tina — problem-solving technique — Vieta jumping (?)
  • Keshav — the mathematics of the perfect shot in basketball
  • Troy — The geometry of voting
  • Sang Sik — probability — methods for lotteries, raffles, picking prizes
  • Jeff — modeling the height of a bouncing ball
  • Angela S — spherical geometry — measuring distances on the globe
  • Issa — the mathematics of Linerider
  • Angela W– Fourier series and the mathematics of noise-canceling earphones
  • Baron — Rubik’s cube — algorithms, bounds on number of moves, different-sized cubes
  • Ye Jin — Crochet models for hyperbolic geometry
  • Joy — Friendship Paradox
  • Heidi — knot theory — classification of knots using the Jones polynomial
  • Helen — hyperbolic trig functions — catenaries
  • Kevin — odds calculations in televised poker
  • Jimmy — mathematics of square wheels
  • David — Mathematics of the Electoral College — mathematical partitions


Resources (These are divided into rough categories, with a lot of overlap):

New resources from Mr. Chao:

(These are new in the sense that I have only recently come across them; they are not necessarily new in the sense of newly posted.)

  • Mathematics of Planet Earth 2013
  • The Museum of Mathematics in New York City
  • MIT Sloan Sports Analytics Conference
  • The New York Times Numberplay blog is listed as a resource below. This is a link to a recent example that has an interesting problem, as well as an interesting video with links to other sources of potential topics.
  • The SAS library has recently received trial access to a library of e-books, with an academic/reference focus. While the selection of math books is not very large, if you are interested in a historically-based topic, you may find something here that will point you in the direction of other sources. The url is: http://ebooks.infobaselearning.com (See Mr. Chao or Mr. Boyer for login info.)

New resources from students in the class:

Authors

(These authors have all written on a variety of mathematical topics, at a level that should be pretty accessible to you):

  • Keith Devlin
  • Marcus du Sautoy
  • Martin Gardner
  • Douglas Hofstadter
  • Eli Maor
  • John Allen Paulos
  • Ian Stewart
  • Steven Strogatz

Books:

  • The Best Writing on Mathematics 2010/2011, edited by Mircea Pitici
  • e: The Story of a Number, by Eli Maor
  • Godel, Escher, Bach, by Douglas Hofstadter
  • How to Solve It, by George Polya
  • The Joy of x, by Steven Strogatz
  • Flatland

Online resources:

  • Wolfram Alpha — a computational search engine — you can type in almost any topic and you may be directed in an interesting direction
  • New York Times — Numberplay (weekly puzzle)
  • New York Times — Elements of Math (series of articles on a variety of mathematical topics by Steven Strogatz)
  • Math Masters website — www.qedcat.com
  • Mathematical Association of America (MAA) website — www.maa.org — look under Contests or Columns
  • ESPN Sports Science
  • ASMA or AMC contest sites
  • Mathematical Intelligencer — http://www.springer.com/mathematics/journal/283

Possible Topic Ideas:

A variety of possible ideas is listed below, but almost any topic is possible, as long as it leads you to use mathematics that is “commensurate with the level of the course”. A topic that you discover and that catches your interest is likely to be “fresher” than a topic from this list.

Applications:¬†Mathematics and …

(almost anything can go after this — some are very broad areas and would need to be focused considerably)

  • Architecture
  • Codes
  • Digital communication
  • Economics/markets
  • Elections
  • Games
  • Internet security
  • Medicine/health
  • Music
  • Environment
  • Sports

Mathematical topics:

  • Matrix transformations
  • Exponential vs logistic growth
  • Numerical integration, e.g. Simpson’s Rule
  • Hyperbolic functions
  • Theorems from advanced Euclidean geometry, e.g. Menelaus’s Theorem, Apollonius’s Theorem
  • Non-Euclidean geometry
  • Projective geometry
  • Perfect numbers/friendly numbers
  • Number bases
  • Quaternions (an extension of complex numbers)
  • Harmonic series
  • Hypercube and other higher dimensional objects

Historical topics (just be sure that the focus is mathematical, rather than historical)

Famous problems: (Some of these involve very high-level math, so you would need to find a way to approach the math in a way that is both accessible and sufficiently rigorous.)

  • 4-Color Theorem
  • Construction with a compass and straight-edge
  • Goldbach’s Conjecture
  • Fermat’s Last Theorem
  • Distribution of prime numbers
  • Poincare Conjecture
  • Traveling Salesman Problem

Other:

  • Mathematics of Numb3rs