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81) Voting Paradoxes
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English
Description
Learn that determining the will of the voters can require a mathematician. Delve into paradoxical outcomes of elections at national, state, and even club levels. Study Kenneth Arrow's Nobel prize-winning impossibility theorem, and assess the U.S. Electoral College system, which is especially prone to counterintuitive results.
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"If there is something in life you want, then just make it happen!" Following this advice, learn to solve polynomial division problems that have negative terms. Use your new strategy to explore infinite series and Mersenne primes. Then compute infinite sums with the visual approach.
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English
Description
Professor Tanton reminisces about his childhood home, where the pattern on the ceiling tiles inspired his career in mathematics. He unlocks the mystery of those tiles, demonstrating the power of visual thinking. Then he shows how similar patterns hold the key to astounding feats of mental calculation.
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English
Description
When two lines intersect, they form a system of linear equations. Discover two methods for finding a solution to such a system: by graphing and by substitution. Then try out a real-world example, involving a farmer who wants to plant different crops in different proportions.
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English
Description
There's more than one way to be truly random! Delve deeper into probability by surveying several discrete probability distributions - those defined by discrete variables. Examples include Bernoulli, binomial, geometric, negative binomial, and Poisson distributions - each tailored to answer a specific question. Get your feet wet by analyzing several sets of data using these tools.
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English
Description
Delve into irrational numbers--those that can't be expressed as the ratio of two whole numbers (i.e., as fractions) and therefore don't repeat. But how can we be sure they don't repeat? Prove that a famous irrational number, the square root of two, can't possibly be a fraction.
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Description
The Banach-Tarski paradox shows that you can take a solid ball, split it into five pieces, reassemble three of them into a complete ball the same size as the original, and reassemble the other two into another complete ball, also the same size as the original. Professor Kung explains the mathematics behind this astonishing result.
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Description
Start with a hypothesized parameter for a population and determining whether we think a given sample could have come from that population. Practice this important technique, called hypothesis testing, with a single parameter, such as whether a lifestyle change reduces cholesterol. Discover the power of the p-value in gauging the significance of your result.
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English
Description
Discover that Pascal's triangle encodes the behavior of random walks, which are randomly taken steps characteristic of the particles in diffusing gases and other random phenomena. Focus on the inevitability of returning to the starting point. Also consider how random walks are linked to the "gambler's ruin" theorem.
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English
Description
After learning the definition of a function, investigate an additional approach to solving quadratic equations: completing the square. This technique is very useful when rewriting the equation of a quadratic function in such a way that the graph of the function is easily sketched.
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Description
Discover the fun of arithmetic and other simple mathematics. Begin with the game Krypto. Then try out the "four fours" puzzle. Next, see how perfect squares and perfect triangles reveal algebra and geometry working together. Finally, reason out why a negative number multiplied by itself a is a positive number.
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Description
Take your understanding of descriptive techniques to the next level, as you begin your study of statistical inference, learning how to extract information from sample data. Focus on the point estimate - a single number that provides a sensible value for a given parameter. Consider how to obtain an unbiased estimator, and discover how to calculate the standard error for this estimate.
100) Order of Operations
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English
Description
The order in which you do simple operations of arithmetic can make a big difference. Learn how to solve problems that combine adding, subtracting, multiplying, and dividing, as well as raising numbers to various powers. These same concepts also apply when you need to simplify algebraic expressions, making it critical to master them now.
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