I was looking at a good motivator for proof by induction. A collegue warns the kids to “not follow the pattern off the cliff”, and I was looking for good examples of this. Sent this tweet out in the morning:
What's an example of a mathematical pattern that seems to hold together for a while, but then falls apart.
— Dan Anderson (@dandersod) February 6, 2014
Got all these responses in less than 30 minutes. The best.
@dandersod Odd numbers are prime?
— Patrick Honner (@MrHonner) February 6, 2014
— Dan Meyer (@ddmeyer) February 6, 2014
I ended up using the partitions of a circle
pattern, powers of 11 to create Pascals triangle (which actually doesn’t break), fifth roots, and the polynomial setup (find function values for x=1,2,3,4,5, … for f(x)=x-(x-1)(x-2)(x-3)(x-4) except give it to them in standard form).
I really liked this as an introduction to induction. By showing them some examples of intuition breaking down, it motivates the desire to prove something for sure. Induction is my second favorite type of proof (after contradiction), and I share this fact with the students. I let them know that these proofs are far closer to what they might do later in college (compared to two-column geometry proofs).