Useful Links

I will add more links to this page as the semester progresses. If you find a site that you think should be included, let me know and (if I agree!) I will put it up.

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Area and Volume

• The Australian TIMES Project has excellent modules on Area, Volume, and Surface Area, and on Cones, Pyramids, and Spheres. (Their other modules are also good!)
• Ed Steltenpohl, a student in this class a few years ago, wrote some statistics code to implement the Monte Carlo method described in the book for approximating . He ran 2000 simulations, each consisting of 2000 trials. Here is the resulting distribution of the 2000 estimates for . The red line is the mean of the 2000 estimates, which turns out to be an excellent estimate. (In fact, the code also put a blue line at the exact value of , but it is not visible in the picture, because it has been completely obscured by the red line.)
• There are many videos on Youtube demonstrating Archimedes’ approximation of . Here are two of the best, by MathWithoutBorders: first, the basic approximation method; second, a follow-up showing the method can be extended to give both upper and lower bounds for . I have implemented up a version of the spreadsheet used in the videos if you want to explore the numerical results.
• Here is the Cut The Knot page with many, many proofs of the Pythagorean Theorem.

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Theory of (Polynomial) Equations

• In class, I mentioned that there is (as far as I know) no complete test for irreducibility of a polynomial over the rational numbers. However, Eisenstein’s criterion is a relatively simple test, which will detect some, but not all, irreducible polynomials.
• We discussed the Fundamental Theorem of Algebra, and used it to determine the irreducible polynomials over the complex (and then the real) numbers), but we did not give a proof of it in class. There is a reason for this: there is no particularly simple proof of the theorem, and no known purely algebraic proof (so you certainly couldn’t provide a proof for the students in your High School algebra class). If you are interested, however, the wonderful Cut The Knot website has a page on Proofs of the Fundamental Theorem of Arithmetic.

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Number Theory

• We know (how?) that there are infinitely many prime numbers, but proving that large numbers are prime turns out to be a hard problem. As of January 2018, the largest known prime is , a number with over 23 million digits! You can read more about it here.
• Our text does not provide a divisibility rule for 7. Googling “divisibility rule for 7” will result in several links, including this Wikipedia page on divisibility rules.
• You have seen that it is not true that n² + n + 41 is prime for every whole number n. In fact, it has been proven that there is no polynomial P (with complex coefficients) such that P(n) is prime for every whole number n. Remarkably, however, there are polynomials (with integer coefficients) such that if P(n) is positive and n is a whole number, then P(n) is prime. Moreover, P can be chosen so that all primes are produced in this way. Even more remarkably, people have even managed to find explicit examples of such polynomials. For one such example, and more information, see this page about Matijasevic’s polynomial.

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Common Core State Standards for Mathematics

• In June 2010, Wisconsin adopted the Common Core Standards for Mathematics and English Language Arts. These standards are therefore a description of what students in Wisconsin are expected to know and be able to do by the time they graduate from high school. As a high school mathematics teacher, you will need to become familiar with these standards, and we will refer to them throughout the course, but please do not feel you are expected to be intimately familiar with them at the start—or even the end—of this semester!
• You can download a PDF version of the standards at the Common Core website, but there is also a very useful hyperlinked version available at the Common Core Tools website.
• The State of Wisconsin Department of Public Instruction has developed a nice graphic to remind teachers (and others) of the importance of the Standards for Mathematical Practice.

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Professional Organizations

• The Wisconsin Mathematics Council is our state branch of the National Council of Teachers of Mathematics. You should consider attending the WMC annual conference at Green Lake in May; see the WMC website for conference and registration information.

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Miscellaneous Resources

• What is wrong with all the “surefire” shortcuts your math teacher taught you? Read Nix the Tricks and find out.
• The National Library of Virtual Manipulatives has some really cool stuff!