List of fun things to do for the first 5-10 minutes of class
Explanation: I like to do something fun and totally unrelated to class for the first 5 (maybe even 10) minutes of class. This shows students the depth, richness, and beauty of math and after the semester's over they walk away with a change of perspective on math in general. This also makes them more engaged, and it seems sometimes they are at least 50% more engaged as a result; if you spend 10-20% of your class on this and as a result, the students are that much more engaged, then there is still an overall gain in how much they learn.
A few important pointers to keep in mind:
1. Students like puzzles, not lectures! So if you lecture, make it so the students discover the theory and it's not just you giving it to them.
2. Wrong attitude: "I want to stuff this beautiful math into my students' heads!" Right attitude: "I want my students to have fun discovering this beautiful theory themselves."
3. Make sure not to spend too much time on puzzles, or especially lecturing on extraneous material!
Here's my list of things you could do:
0. Dr. James Tanton's Exploding dots!
1. Hidato, Kakuro, Nurikabe, and/or any other NP-complete puzzles. The app "Logic Games" for smartphones contains a LOT of great NP-complete puzzles; I like to use Parks, Abc, and Skyscrapers. You can introduce NP-completeness through these puzzles, maybe a spend a whole lecture on NP-completeness around the time you talk about l'Hospital's rule or at the end of the semester. You can also introduce O- and o-notation here to help with talking about NP-completeness. The cool thing about this is that students can actually understand the statement of the P vs. NP Millennium Prize problem.
2. Complex numbers and Mandelbrot set series of minilectures. First class: play a file on YouTube that zooms in on the Mandelbrot set and have some nice music in the background. Then take a few 5-10 minute lectures in the beginnings of the next few classes to introduce basic properties of complex numbers, maybe give them extra credit problems to work on so they get the basics of complex numbers down (this might indeed be very useful to most math / science / engineering majors out there!). Finally, talk about the Mandelbrot set, and then maybe fractals in general.
3. Jumble! Take a word and rearrange its letters to form an anagram; have the students figure out what the original word was. It gets students excited about the class and gets their brains working (makes them smarter too!). You can just do jumbles in the beginning of class every day; that's what my calculus teacher in high school did.
4. Combinatorics series of minilectures. You can do jumbles and follow up with a series of mini-lectures about combinatorics, each taking about 10 minutes: first give them the formula for the number of permutations of a word. Then show how the number of ways to choose r objects out of n is the same as the number of permutations of AA...ABB...B (r A's, n-r B's) (assign numbers to the n objects and pick the A's to be in the locations of the r chosen objects). Then show how all this relates to Pascal's triangle via the following question: if a taxi starts driving at the top of a square grid turned 45 degrees, how many ways are there to get to the places below without ever going upwards? You get Pascal's triangle, and it's the same as the number of permutations of RR...RLL...L (r R's, n-r L's) if you are going to the r-th place in the n-th row. Also, you can easily derive (n choose r) + (n choose r+1) = (n+1 choose r+1) using this analogy. Then show how Pascal's triangle can be used to find the binomial expansion (x+y)^n. Also, to relate this all back to calculus, show how Pascal's triangle can be used to find the n-th derivative of the product of two functions. Do an example like the n-th derivative of xe^x. Later, you can use this to find the MacLaurin formula for xe^x.
5. Magic Squares! Together prove the sum of each row for a 3x3 magic square has to be 15, then ask them to find the 3x3 magic square. What's the formula for the sum of each row in an n x n magic square? Show them Durer's magic square and see how many patterns they can find within it. There's lots of really cool info on magic squares out there, even on Wikipedia, if you just want to show them that.
6. Mathematical Induction! I use mathematical induction to prove the formulas for the sums of natural numbers, squares, and cubes - it's good for them to see a proof. I also show some neat properties of Fibonacci numbers and assign extra credit problems to prove some Fibonacci identities. It's neat to prove DeMoivre's formula with induction. It's also neat to prove that, in order to break up a chocolate bar made up of n squares into n 1x1 squares, you need to break it n-1 times (strong induction); you can have students discover this for themselves by bringing in some Hershey's bars - fun stuff! Another fun strong induction problem: prove that for n > 5, a square can be decomposed into n smaller squares.
7. Polygonal numbers. It's good for students to see some recreational mathematics (most people don't know recreational math exists!). I use triangular numbers to show that 1 + 2 + ... + n = n(n+1)/2 and give them extra credit problems to find and prove the formula for the n-th pentagonal / hexagonal number. You can also do this with centered pentagonal / hexagonal / etc. numbers.
8. Modular Arithmetic! You can make a series of minilectures on modular arithmetic and/or cryptography. What is the last digit of 2^100? What are the last 2 digits? Possible extra credit problem: show that the integral of x^ne^(x^m) can be computed explicitly if m and n are positive integers and n = -1 (mod m).
9. Golden Ratio! Prove that 1/(1+1/(1+1/...)) = phi = sqrt(1 + sqrt(1 + sqrt(1 + ...))) and phi is the limit of ratios of the Fibonacci sequence. There is a great 3-part YouTube video series with really neat information on how the Golden Ratio connects botany, biochemistry, physics, and mathematics.
10. Irrational numbers. Have them prove (guiding them) that sqrt(2) is irrational. Is it true that an irrational power of an irrational number must be irrational? (Hint: sqrt(2)^2 = 2)
11. Mersenne primes and Fermat primes. Write out 2^n - 1 for the first few n and have the students guess when the numbers will be prime. Ask them to see if 2^11 - 1 is prime (it's 23*89). Tell them about GIMPS and the largest prime number ever discovered. Tell them Fermat guessed 2^(2^n) + 1 is always prime; note 2^(2^5) + 1 = 4294967297 = 641×6700417.
12. Zeno's Paradoxes, etc. Tell them about Zeno's paradoxes, supertasks, and Thomson's lamp.
13. The diagonal paradox. I use this to accompany the lecture on how to find the arc length of a curve.
14. Cantor's Set Theory series of mini-lectures. Tell them how Cantor proposed to measure the cardinality of sets, that the natural numbers, integers, and rationals have the same cardinality, and the real numbers have greater cardinality.
15. Numerical Invariants. Have them prove that (1) you can't 2-tile a chessboard with two opposite corners removed. (2) You can't make a Hamiltonian Path on a 3x3 square grid by only making vertical or horizontal jumps. (3) There are 6 trees lined up in a row, and a crow on each one; every hour two crows jump and land on an adjacent tree - prove they can't ever all end up on the same tree.
16. Draw a 3x3 grid of dots and have the students try and connect them all by drawing exactly 4 line segments without lifting their pencil (the line segments can extend outside the grid).
17. Students can discover (with some guidance) the approach for solving a linear differential equation - this can take a whole class, and it's much better than just telling them how to solve it.
18-oo. You come up with some and please tell me!
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