Cluster 6
Mathematics
- Instructors: Lawrence Marx, Abigail Thompson, Alex Coward & Monica Vazirani
- Prerequisites: Algebra II
- Typical Field Trips: Exploratorium, California Academy of Science, & Electronic Arts
- This is a FIRST CHOICE option only.
Introduction
This cluster is designed to introduce students with a strong interest in mathematics to several different advanced topics. Many of these topics would ordinarily only be seen at the advanced undergraduate level, but all lend themselves to an introductory course at the high school level. No prior experience in any of these topics is expected, but enthusiasm for and interest in mathematics is essential.
Core Course (4 Weeks)
Combinatorics
Enumerative combinatorics provides a sophisticated way to count in complicated mathematical settings, and we will learn several important counting techniques in this course. Here are a couple of examples of combinatorics problems that we will encounter in our four-week journey through this interesting area of mathematics: 1. If a teacher returns a test to her class of 10 students at random, what is the probability that no student gets his or her own test? 2. Suppose there are six people in a room. Show that either there are three people who all know each other, or there are three people who all do not know each other.
Supplementary Courses
Knots, Links and the Topology of Space (3 Weeks)
From knotted strands of DNA, to tangled necklaces, to the basis of String Theory, knots and links arise in many different applications. We’ll look at the mathematical theory of knots, and learn what knot invariants can tell us about them. We’ll connect knot theory to the study of all 3-dimensional spaces. For example, we'll examine whether or not our universe is flat. We thought the earth was flat for a long time... what about the 3-dimensional universe we live in? If we send a rocket off into space programmed to go "straight", will it eventually come back to where it started, like what happens if you go “west” long enough starting at a point on the equator? We'll look at some of the possibilities by studying what is known about the geometry and topology of space. What possibilities are there for 3-dimensional spaces? Can you come to the “edge” of the universe? Can you fall off?
Symmetry (1 Week)
Symmetry is all around us: in nature, art, architecture, chemistry, music, and more. The petals of a flower, a bee's honeycomb, a Sudoku puzzle, a mandala, a water molecule, even a mathematical equation; all exhibit beautiful symmetries. Using mathematics we can describe those symmetries, and more importantly we can put them "to work" for us. What if you wanted to count how many different Sudoku there are? Listing them all would take a LONG time, but taking advantage of their symmetry, we can write down a formula that counts them. In this course, we will use symmetry to count these and other objects, such as "colorings" of regular polyhedra.
Modified 2010-01-13T22:37:51Z