Through various geometric constructions first discovered by Pappus, Desargues, Pascal, and Brianchon, students explore the ways in which mathematicians for 2000 years flirted with the ideas which we now recognize as projective constructions. The effort moved forward when artists in the 15th century began to wonder how to depict scenes on flat paper which appeared to be three dimensional. Projective Geometry only began to be developed in the 19th century. Students come to appreciate a completely unfamiliar space of reality which is just as valid, and in fact more generally true, than the one with which we are more used to dealing.
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Students work with the concept of expected value and develop a mathematical analysis based on an area model for probability. Probabilistic thinking is often counterintuitive; for this reason, the activities in this unit model concrete experiences. The gambler's fallacy that the next roll of the dice depends on previous rolls is held with conviction even by well-informed adults. One goal of this unit is for students to recognize this fallacy, both in dice games and in real-life situations. More broadly, students come to understand theoretical probability and see how and when it can be used to model and give insight into every day situations.