Developing Mathematics: Probability Through Algebra: Focused on learning mathematics by working problems together, this course explores the fundamental mathematics on a topic that has its roots in secondary level, and is related to the mathematical theme of the Institute. Careful work on this topic allows teachers (and students) to understand exactly how elementary and more advanced procedures in the specific content area are derived and generalize. The course is structured so that each participant can work at his/her own level. Those who are more mathematically advanced may be asked to help those with less preparation. The course is conducted by teacher leaders from the PROMYS program at Boston University. The focus of this strand is entirely on mathematics, although opportunity is provided within the course for reflection on the approach used by the instructors and to consider the implications of such an approach for teaching in secondary classrooms. The topic for the summer is described below:
You're at deuce in a tennis game and are 60% likely to win each point. How likely are you to win the game? What is the probability that you roll a sum of 13 when 5 dice are thrown? What is the most likely sum when 5 dice are thrown? Take an expression like x^6+x^5+x^4+x^3+x^2+x and raise it to the fifth power.
What do you get? If you raise it to higher and higher powers, what is the distribution of the coefficients "in the long run?" How does the "random" button on your calculator work? What's the probability that two positive integers, chosen at random, have no common factor? And most importantly, what do all these questions have to do with each other?
In this three-week course, we'll investigate questions like these (and more). No background in probability or polynomial algebra is assumed, but by the end of three weeks, we promise beautiful mathematical ideas that will make your head spin.
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