Teaching Blog Test Time!

Preparing for a test requires figuring out what you need to know. That involves two things: figuring out what's important, and figuring out what the teacher thinks is important. And what is that? In the broadest perspective, this class has two main themes: learn the skills of mathematical reasoning, and learning about the subject matter of mathematics.

I'll say just a word about the first part, mathematical reasoning. Math isn't numbers. It is the study of formal structures, which are systems of specific, clearly-defined objects along with rules by which the objects relate. Perfect examples include arithmetic in the integers, and the game of Chess. Imperfect examples include the system of traffic laws and the American legal system. Much of what we experience in daily life has nothing to do with formal systems, but in whatever you do in life, having the ability to formulate problems in formal ways, and to use formal reasoning to analyze problems, will be a benefit.

As for the second part, the subject matter itself, I present the following outline.

Proofs you need to know:

• There are infinitely many primes.
• square roots of 2, 3, etc are irrational
• The cardinality of the reals is larger than the cardinality of the natural numbers
• The Pythagorean Theorem
• There are only 5 regular polyhedra (aka Platonic solids)

Chapter 1
No real content. Just some puzzles meant to make a point, for instance that formal reasoning can be used to solve real-world problems.

Chapter 2 - Numbers (Sections: 2.1- 2.4, 2.6)

1. The pigeon hole principle
2. The Fibonacci sequence. Recursive formulas and direct formulae. Ratios of successive numbers in a recursive sequence. Limits. Infinite fractions.
3. Prime numbers. Divisibility (for instance, the exact  meaning of m|n). Unique factorization. The division algorithm. Quotient and remainder. Proof that there are infinitely many primes.
4. Modular arithmetic. Reduction mod n. UPC codes and check-digits.
5. Rational (numbers that are ratios) and irrational numbers (numbers that are not ratios). Proof that square roots of 2, 3, etc are not rational

Chapter 3 - Infinity (Sections: 3.1-3.3)

1. One-to-one correspondence. Something more primitive than counting: Cardinality.
2. The even numbers, the prime numbers, the integers, and the rationals all have the same cardinality as the natural numbers.
3. Countable and uncountable sets. Examples of countable and uncountable sets. The diagonal argument. Proof that the reals are uncountable.

Chapter 4 - Geometry (Sections 4.1-4.3, 4.5, 4.7)

1. The Pythagorean theorem.
2. Triangulation. The art gallery theorem.
3. Definition of the Golden Rectangle. Computation of the value of the Golden Ratio. The Golden Spiral
4. Polygons. Polyhedra. Duals of polyhedra. Stellation and truncation. Regular polygons. Regular polyhedra. Proof that there are only 5 regular polyhedra.
5. Definition of dimension (a degree of freedom, or a mode of measurement). Real-life examples of multi-dimensional problems and data sets. Imagining the 4th dimension. Analogy with 1, 2 and 3 dimensions.

Chapter 5 - Geometry and Topology (Sections 5.1-5.3)

1. Geometry is the study of shape. Topology is more primitive: it is the study of connectedness. Topological changes and geometric changes. How to prove two objects are topological equivalent, and how to prove two objects are topologically different.
2. Gluing patterns. The Torus. The Mobius strip. The Klein bottle.
3. Knots and links. Equivalent and inequivalent knots. The unknot.

Chapter 6 - Graphs (Sections 6.1-6.3)

1. Graphs. Euler Circuits. The Konigsberg bridge problem.
2. The Euler characteristic. Second proof that there are only 5 regular polyhedra.
3. Planar and non-planar graphs. E<3V-6 for planar graphs without multiple edges. The graphs K5 and K3,3. Application of graphs: maps and the 4 color problem.

Chapter 7 - Fractals (Sections 7.1-7.3)

1. Self-similarity. Infinite detail. Fractal-like objects and phenomena in nature.
2. Examples of fractals: simple instructions repeated indefinitely. Sierpinski triangle. Sierpinki carpet. Menger sponge. Koch curve and Koch snowflake.
3. Fractal dimension. Scale and content.

Chapter 8 - Chance and Probability (Sections 8.1-8.3, 8.5)

1. Definition of the probability of an event. Definition of relative frequency. P(not E) = 1-P(E). Yatzhee question. Birthday question. Law of independent events: P(E and F) = P(E)P(F) provided E and F are independent.
2. Streaky-ness and probability.
3. Nash equilibrium. Payoff matrix. Probability strategies (eg. pgs 630-631)

Chapter 9 - Statistics (Sections 9.1-9.3, 9.5)

1. Statistics is the organization of data. Ways statistics can be misused. Coin-flip strategies for poll-taking. Correlation.
2. Histograms. "Center" of a distribution: mean, median, and mode. Skewed and normal distributions. The desert island example. Distributions that are bimodal, trimodal, etc. 
3. The bell curve. Standard deviation. Coin flips.
4. Deeper analysis of data: the hospital example, and the university gender discrimination example.

Chapter 10 - Applications (Portions of Sections 10.1-10.3)

1. Expected value. Weighted standard deviation.
2. Testing for a disease.
3. Interest and compounding.