Calculus Homework Generic Outline

Steve Nuchia, University of Tulsa

You are writing to convince a fellow student of the correctness of your answer. Your (imaginary) audience knows all the math you know but has not studied the problem you are solving. There is an art to deciding when detail is needed to avoid the "then a miracle occurs" step and when detail just gets in the way of a clear presentation. Learning that art is an important objective in this course.

A good write-up begins *after* you have solved the problem. Nobody wants to read the history of how you discovered the solution. It may have been a lot of work but that is not an excuse for making your reader work hard! Once you really understand the problem, ask yourself what steps are really key and what mathematical and real-world ideas are behind those steps. Only then should you begin to write.

The following outline should serve as a starting point and perhaps a kind of checklist. Not every problem will fit exactly – you should adapt the outline to the problem and not the other way around!

- Restate The Problem In Your Own Words
- Your statement of the problem should be
*more clear*than that in the book. In particular, you should clearly state what is known and what is sought: a number, a function, etc. - Introduce symbols for the known and unknown quantities; see the writing style handout.
- Introduce Relevant Facts and Set Up Your Solution
- Name and summarize the mathematical facts (theorems and definitions) that are important in your solution.
- Many problems require "real world" information that may or may not be in the book. You need to give a concise and accurate mathematical statement of those facts here. Also, if you use an outside source (including informal conversations with peers) be sure to cite it or them.
- You should now be able to
*characterize*the solution; that is, state what mathematical properties a solution must have, as determined by the connections among the given and background facts. - The end result of this section is a purely mathematical problem that is equivalent to the original "messy" problem.
- Solve the Mathematical Problem
- Show enough steps that a reader will be able to say "Oh, yeah" if he has a question about a step, but not so many that the reader has trouble following the argument.
- Make sure there is an argument! All equations should function as clauses in a sentence – a sentence that makes a relevant point in the context of a paragraph.
- Think about how to organize the
*presentation*of the calculation. Often it will be helpful to show a lot of detail in one case, but then you should just give a table of results if there are several more identical calculations. - The end result of this section is a solution to the mathematical problem.
- Interpret Your Results In The Context Of The Original Problem
- You have just solved a subproblem that you
*claim*is related to the original problem (as you have stated it in the opening paragraph of your presentation). Now you need to use the mathematical result to state the solution of the original problem in the language and context of that problem. - Any method of checking your answer would be appropriately mentioned here. For instance, you might show that your solution agrees with a simple approximation.

Often the problem assigned will seem to be ambiguous or incomplete. Sometimes it actually is ambiguous! Your restatement of the problem plays a critical role here. If you have misinterpreted the problem but you correctly solve the problem *as you understand it* you will probably receive full credit. I have given students full credit for turning in good solutions to the wrong problem entirely.

Imagine that you have been asked by your boss or a client to solve this problem. If you give them the solution in the form "here is the answer to the question you asked" and you have in fact misunderstood the question, you will have given an *incorrect* answer. On the other hand if you say, "here is how I understand your question, and here is my solution" then your boss will have a fighting chance to notice that your understanding of the problem does not match her own.

Good writing does not happen in a vacuum. It results from practice informed by reading and constructive criticism. As you read mathematical arguments, not just in the Calculus textbook but in all your technical courses, pay attention to the way they are written. Some will be good, some bad; many will be hard for you to understand. You can learn a lot by thinking about why a written mathematical argument is hard for you to follow.