> They get 1/5 for leaving a question blank or saying something like "I
> cannot answer this question."
>
> They get 1 mark for a base case (verifying the basis in structural
> induction).
>
> They get 2 marks for setting up the induction hypothesis --- assuming
> that n is an element of the appropriate set, assuming the predicate
> for the appropriate elements.
>
> They get 2 marks for carrying through the induction step.  They should
> use the IH in the appropriate place, and they should derive the
> conclusion.

I was concerned most of all with how they structured their proofs, and
least of all with how clever they were at manipulating mathematical
objects.

It was important that they stated clearly what the induction hypothesis
was, and that it did not contain errors (e.g. if I.H. makes use of
predicate P(n), but P(n) is incorrectly defined or contains a universal
quantifier over n, then they would lose a mark).

For the induction step, I looked for use of the I.H. in the correct place
(i.e. they did not necessarily need to explicitly say they were using
I.H., but it should have been unambiguous that this is what they were
doing, e.g. they may have assumed "for all n, P(n)" at the beginning of
the I.S., and then at the critical place they could have said something
like "because P(n), therefore..."). However, if it was not clear and
unambiguous that they knew they were making use of the I.H. in the
appropriate place, then they lost a mark.

As part of the I.S. mark, I also looked for a statement of the conclusion
that made it clear that they understood why the conclusion followed (i.e.
if they said something like "by induction, for all n, P(n)", or "since,
for all n, P(1) and P(2) and ... and P(n-1) => P(n), therefore, for all n,
P(n)", then they got the mark; but if they omitted the conclusion, or
simply stated "for all n, P(n)" without making it clear that they
understood why it followed from base case/basis and induction step, then
they lost the mark).

I did not take off more marks than the maximum allotted for each part
(i.e. 1 for base case/basis, 2 for I.H., and 2 for I.S.).

Note that because of my focus on the structure of the proof, some students
received low marks despite having written down a lot of relevant stuff
about manipulating the mathematical objects. Since I did not have time to
write detailed comments, some of these students might not understand why
they did not get at least part marks unless you explain the marking scheme
as I described here.