The following are sample topics for the presentation at highschools. Student-pairs in the Outreach Program can also choose their own topic, with the help of their faculty mentor. [1.] Visualizing bifurcations using soapfilms. (Mentored by Monika Nitsche) A simple experiment can be performed in the classrooms in which students measure the size of two bubbles connected by a tube as a function of the volume of fluid within the bubble. From the size they deduce radius of curvature and pressure. The students can then obtain a plot of volume vs pressure which leads to a discussion of bifucations. Similarly, other experiments visualizing bifurcations can be performed. [2.] The golden ratio. (Mentored by Michael Nakamaye) The ancient Greeks as well as artists and thinkers from the renaissance were facinated by this number, believing for example that the most aesthetically appealing rectangles are those whose sides have ratio equal to the golden ratio. The golden ratio naturally links geometry, algebra, and thanks to ancient numerologists, art. [3.] The Fibonacci numbers. (Mentored by Michael Nakamaye) This very simply defined sequence of numbers turns up everywhere in art, nature, and mathematics. The Fibonacci numbers are a recursively defined sequence so some time and fun can be had discussing what this means and giving examples. The Fibonacci numbers have all kind of wonderful, suprising properties, and even the golden ratio appears as a limit of successive quotients of Fibonacci numbers. [4.] Voting theory. (Mentored by Michael Nakamaye) In large scale elections in this country, we only indicate our top choice of candidate on the ballot. It might seem reasonable, instead, to make a full list of candidates in order of preference as this would provide more information. Unfortunately, it is a well known theorem that as soon as there are more than two candidates, this type of preferential ballot leads to many unwanted and interesting consequences! [5.] Irrationality of the square root of two. (Mentored by Michael Nakamaye) In addition to the very well-known proof which uses the property of unique factorization of integers, there is a beautiful geometric proof which reveals the same problem. At a much higher level, it is possible to exhibit rational numbers which are ``too close'' to the square root of two, forcing it to be an irrational number. [6.] Fractals. (Mentored by Michael Nakamaye) Students can personally engage in building the beginnings of simple fractals such as the Koch snowflake or, collectively, the Sierpinski gasket. In addition to having some fun with geometric shapes and working together, this provides a nice opportunity to look at some geometric and arithmetic series, relating to the area/perimeter of these fractals. [7.] Regular tilings of the plane. (Mentored by Michael Nakamaye) There are only three regular polygons (triangle, square, and hexagon) which can be used to tile the plane. It is a nice combination of geometry and arithmetic to show this. Time permitting, one could talk about all of the marvelous pictures of Escher which are also tilings of a plane, this time the hyperbolic plane. [8.] The quadratic formula. (Mentored by Michael Nakamaye) The quadratic formula comes from a process known as ``completing the square.'' Most students do not realize that the algebraic process of completing the square corresponds to a geometric picture where you really are completing a square. This interplay of geometry and algebra would be very rewarding. [9.] Game theory. (Mentored by Michael Nakamaye) Here students play some simple games, both zero sum and not zero sum, and discuss their strategies. There are some beautiful theorems in game theory which can be discussed to accompany this. [10.] Probability theory. (Mentored by Michael Nakamaye) This is a rich source of hands--on problems, including simple questions about coins and/or cards. The important distinction between theoretical and experimental probability can be discussed and a link can be made with statistics (for example by plotting the class results for a simple experiment such as tossing a coin ten times over and over again). |
