Here are some tips for writing up homework solutions in Math 312.
Most of these were written by my colleague
Greg Martin. I am grateful to Greg for allowing me
to modify his tips for Math 312 and to share them with you.
Justify all of your answers, even numerical answers.
Simply stating the answer isn't enough; in other words, include
general facts that allow you to draw your specific conclusions.
Write enough to distinguish your solution from someone else
who doesn't understand the material but is a very good guesser.
Conversely, stating general facts isn't enough if you
don't connect them to the problem at hand. For example,
if the problem asks "Is 97 a prime number?", it is not
enough to write "97 is a prime number because
we cannot factor 97 as a product of
two integers, n and m such that n, m > 1."
This sentence correctly restates the definition of a prime number
but never actually indicates why the specific number
in the problem, 97, satisfies this definition.
One rule of thumb is the telephone test: if you read your
solution out loud over the phone to your friends (who are in
the class but haven't thought about that problem themselves),
would they be completely convinced, or would they have to ask
you to clarify parts of the solution? If you'd need to clarify
it to your friends on the phone, you should clarify your solution
Suppose a statement makes an assertion about every integer,
every residue, every prime, and you have to determine
whether it's true or false. If it's false, then you can simply give
one example where the statement is false. If it's true, though,
then examples won't suffice: you need to write a general proof.
It's very easy to make calculation mistakes (or even copying
mistakes) when solving linear Diophantine equations or linear congruences
doing the Euclidean algoithm. Take advantage of the fact that
you can always check your answer!
Make sure you write all the relevant details, rather than expecting
the reader to deduce them from the context. The grader can only grade
what's written, not what you were thinking when you wrote it. For example,
when you introduce an integer n does it represent
any possible integer, a specific integer from
somewhere else in the problem or a parameter that ranges over
all possible integers?