Guidance for doing homework problems, and laying out solutions:

When we talk to employers (companies, governments, universities, military, etc.) we hear over and over again that an ability to communicate clearly is the most critical factor in decision over hiring and promoting. It is absolutely essential to develop these skills. Tens of thousand of dollars of your future salary (cumulatively into the hundreds of thousands) will be contingent on whether you have these skills. In science, or in fact any quantitative endeavor, it is vital to be able to lay out a clear, concise, cogent, and correct analysis. This is one of your rapidly diminishing set of opportunities to develop these skills. Do please make the effort to do it right – your future depends on it.

Basically good scientific answers boil down to some very simple rules: a) be clear; b) include units; c) don’t write crazy numbers; and d) be only as precise as is needed.

In more detail, here are a few tips:- 

1.Hand in neat work. If you have to rewrite a messy solution take the two minutes needed to do that.

2.Always include units! Do the units make sense? If your answer is supposed to be a length, time shouldn’t be in there.

Doing this is called dimensional analysis and is a powerful tool. Sometimes, even without knowing the theory, you can guess at what the answer must be because there is only one way of combining the variables in a way that makes the dimensions correct for the answer you want.

A silly example: suppose you are told that heat of Q = 1000 J of energy is applied to a metal bar of mass of m = 10 kg, that a specific heat capacity of c = 1000 J kg-1 oC-1. What is the change in temperature? Even if you forget the formula you can deduce that it is given by ΔT = Q/mc:  that is the only way that Q,c, and m can be combined to give an answer that has oC as its units.

3.Check the magnitude of your answer. Does it make sense? Are your speeds, distances, whatever, at least plausible, given the problem and the physical situation?

If the answer does not make sense and you can’t figure out why not, make a note to that effect on your work. It is much better to comment that the answer seems wrong, than to have a crazy number down there. If the method is correct, and you only made a typo on the calculator, you’ll still get full credit, unless you could have reasonably seen that the number was crazy off.

Putting down a crazy answer with no comments implies you are not thinking about what you are doing.

4. 4.Never, ever give an answer with more significant figures than is in the information in the question.
   

      Round to the same order as the least number of significant figures in the problem. For example, if gravity is given as 9.80665 m/s², and the question states that ball is dropped from 12 meters, the answer for the time it takes to fall should only have 2 significant figures in it.

Logically, there is no extra information in the higher significant figures. Also keep in mind how accurate an answer makes sense. Is it a high precision kind of answer, or is something good to 10% good enough to make the point. A physical sense for what matters is important. Normally, no more than 2 or 3 significant figures (i.e., tenths of a percent) are worth calculating in a class like this.

5.Lay out your answers/arguments clearly in a logical/systematic order. Show all but the very simplest of steps.

There are many good reasons to do this. It is much easier to think about and find errors if all the steps are clearly laid out. Secondly it is much easier for someone else to follow your thinking.

6.Use symbols and variables for as long as possible when working through math questions.

It is much, much easier to work through a problem using symbols (e.g., h for height, A for area, etc.) rather than numbers. Tracking what may be long numbers through several lines of working is very likely to cause errors. Much better is to create your own symbols if need be (and clearly define them). Only put numbers in at the very end to get the quantitative answer.

7.Explain in words what your steps are. 

Unless it is obvious, indicate how you get to the next step. If you use new information, say so. For example ”conservation of mass means that…”, or “from eqn 1 above”, or “neglect the 2nd term”. This is all about sympathizing with your audience. Think about whether someone can easily follow your lines of working. Again it helps you to find errors also.

8.Sketch out the situation in the problem. Label the various aspects.

It can really help clear thinking by making a little sketch of what’s going on, so you can see the relationship between the different parts of the problem.

9.Begin before the night before! Give yourself plenty of time, ahead of time.

Kinda obvious, but if you wait until there is not chance to think/learn, its very hard to do problem sets under pressure.

10. 10.Carry around only as many significant figures as you need for the answer.
    

      Don’t barf out all 12 digits from your calculator every time you use it, if you are only going to cite 2 sig. fig. in your answer. It may not seem like it, but taking only 2 sig. fig. through out the working of the problem, if you round correctly, will get you the right answer in the end. Also, always convert into scientific notation if you have, say, more than 4 decimal places. i.e. always right 4.2 x 10-6 instead of 0.0000042, which can be prone to producing error.

11. 11.Give yourself lots of space when you do math.
    

       Crunched up working makes it hard to correct things and to see where errors might lie.

12. 12. Where possible, give your answer in units that make physical sense.
    

     Thus if it takes 19 x 107 s to cool down, make it into years (1 year = pi x 107 s (accurate to about 3 sig. fig., and just a coincidence). In other words it is much more intuitive to know that it takes about 6 years to cool, instead of 19 x 107 s.