The hardest thing seemed to be getting the wff right! Only a few obtained the correct wff. "If something is slithy, then it is uffish." This is true for anything, and hence merits a universal quantifier.
| If | ||
| there is | ||
| a slithy something | ||
| then | ||
| there is | ||
| an uffish something |
The proof can be made much shorter than this person gives, but it's okay to be verbose: that shows me how you're thinking, and this person is thinking well! Of course we need to add the restriction that the rational number is not equal to 0 in order to prove the result....
The second person did an "induction proof", but then never used the P(k) case! That's odd, and what it means is that we could do the proof directly (which this student did).
The last two proofs are done in completely different ways: one using calculus, and the other using some facts about parabolas and a little calculus. Well done!
One minor note: notice that I had to change the summation variable to "i" -- the student had it listed as "n" (but in the formula it's given as "g(i)").
The first person seemed to understand well the counting task, and correctly deduced the number of operations from the "expand" part of "expand, guess, check" method. This student missed the n=2m assumption, however, and so wasted valuable time and energy on that.
The second person did what "I asked", writing the pow function as a recursive function, and guessing the number of multiplications required.
The third person literally wrote the pow function as a multiplication counting function, and got it right. I would have liked to see how the student leaped to log2(n), however.
