Let's wrap things up a little...

• Some of those games were dramatic foreshadowing... e.g. dodge ball
• Others merely taught us important life lessons: e.g., when things get complicated, try a simpler example; or just do it.
• Generalize interesting problems.

• You need useful referents. There's no substitute for a little knowledge socked away....
  • How many people in the world?
  • How many Americans?
• Approximation by mathematical objects
• Pidgeonhole Principle (remember musical chairs)
• Exaggeration may prove useful.
• Don't forget bounds (e.g. polls that are given with a "plus or minus 3%"). Estimates aren't always useful: bounds give limits on the possible.
• Pay attention to units.

• Numbers have personalities.
• There are many different ways to express a number:
  • Choose a given base in which to write it as a decimal (e.g. base 8, base 2 -- binary);
  • Use prime numbers and products (that is, prime factorization); or
  • Use Fibonacci numbers and sums.
• Nature is full of Fibonacci numbers and patterns
• Fibonacci Nim (how to play, and how to win!)

• We've seen some infinite sets: the natural numbers, primes, Fibonaccis, etc.
• One-to-one correspondence is essential for measuring the "sizes" of sets, and particularly infinite sets.
• Infinity is a subtle concept. The smallest one is the countable infinity, an example of which is the natural numbers.
• The power set (set of all subsets of a set) shows and diagonalization shows that there are small infinities and larger ones.
• There are, in fact, an infinite number of successively larger infinities (measured by cardinality).
• Hotel infinity examples; ping pong ball conundrum.

• 0, 1, 2, 3, 4-dimensional objects and beings
• Learning about complicated things (the 4th dimension) by analogy -- by considering something simpler first
• How do we mess with each others' dimensions?

• Golden Rectangles and golden ratio (Fibonacci again!)
• Logarithmic (golden) Spirals
• Regular polygons (e.g. hexagons)
• Platonic solids -- all five of them! Symmetry, duality
• Borromean rings (bringing it all together: icosahedra, golden rectangles, and interlocking rings)

• Measures of central tendency: Mean, median
• Inappropriate averages: "The average American has one testicle and one ovary."
• Measures of spread: range, deviations, standard deviation; spread in polls.
• Histograms -- visualizing mean, median, spread, skew (mean chases extreme values)
• Distribution of data (e.g. normal distribution, or bell-shaped curve)
• Using randomness to our advantage: answering sensitive questions (one- and two-toss methods -- lying to protect anonymity).

• Plurality versus majority
• Different voting schemes give different results
• All reasonable sounding voting methods are "bad" in some way: only the dictatorship satisfies very reasonable sounding properties, which means that the others fail to satisfy at least one!
• Don't let someone get the lay of the land and then propose the voting scheme.
• Our "democracy": How the Electoral college works

• Simple rules and Iterations may lead to complicated dynamics; as Dr. Cushing said, the "surprising consequences of simple rules"
• He also asserted that "tipping points" (the points -- bifurcations -- where a system changes dramatically and quickly) are more common than we think.
• Chaos: sensitive dependence on initial conditions.
• Deterministic versus stochastic: without noise versus with noise (randomness)
• Equilibria, steady growth, oscillation, chaos: lots of types of dynamics are possible.

• Tufte's rule: simple design, intense content. Alternatively, he says maximize your content to ink ratio.
• Watch the scales (several different things can go wrong)
• Watch for elements that are there to sway your emotions. Are you being manipulated?
• Using dimension to lie with graphs (if the graphic is of dollar bills, is it a one-dimensional length or the two-dimensional area that's the focus?)
• What is "chart junk"? While it may be hard to define, do you know it when you see it?
• Which type of graphic best suits the purpose?

• Simple rules lead to complicated pictures.
• "Worlds within worlds" (self-similarity).
• Nature must use the principles behind fractals, as evidenced by broccoli, coastlines, or ferns.
• Do it once; then do it again! We made fractals starting from a single straight line segment, or by dividing up geometric figures (e.g. squares or triangles), or even donuts!
• The collage-making process, and fractal dust.
• The chaos game.

• Compound interest problems
• How do Mortgages work?
• What is inflation? How does it impact your money?

I'll be testing this hope via your "show-n-tell" projects. By the way, those projects are potential final exam questions, so you need to be here for them. I'll ask a few short questions about selected ones.