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At the end of the school year, I gave my students a problem set related to the law of sines where many students engaged in approximation at various points. I noted two types of precision issues in their work.

1) Looking at the following pattern 19.1, 19.03, 18.95 and concluding that the rule is that sin(A)/a, sin(B)/b, and sin(C)/c are approximately equal. I think that in this instance students are paying excessive attention to precision, rather than trying to conclude an overall rule.

2) sin(60) = .9 For the purposes of this assignment and most trigonometry problems, it seems that a one digit approximation is not sufficiently precise. (Although the student who gives a 10-digit expansion also seems to be missing something)

What do you think is the actual issue represented by these two examples?

What knowledge/ skills are necessary for students to be appropriately precise when giving approximate answers? How do I build this capacity in my students?

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    $\begingroup$ Could you clarify (1)? For instance, do you mean that you gave students a triangle and had them calculate $\frac{\sin{A}}{a}$, $\frac{\sin{B}}{b}$, and $\frac{\sin{C}}{c}$, which came out to 19.1, 19.03, and 18.95, respectively? Perhaps, if this is the case, you could try specifying that students round to the nearest whole number in the directions for the activity? $\endgroup$ – Xi Yu Jul 4 '15 at 16:13
  • $\begingroup$ Students here in the US receive many years of mathematical training in which these ideas are never discussed, followed by their first course in a quantitative science (usually chemistry), in which the concept of significant figures is introduced. If you're teaching students who haven't taken chem yet, it's not surprising that they have no clue about this. $\endgroup$ – Ben Crowell Jul 4 '15 at 18:51
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Students have no basis for figuring out what "appropriately precise" means. How much precision is appropriate is a property of the purpose of a calculation, not the calculation itself, and students rarely think about why they're doing a calculation---they're doing it because they were told to.

If you actually want your students to make their own choices about how much and when to approximate (and make them well), that's a serious undertaking---it's really an entire additional topic to cover. You need to give problems which include both doing some calculations and then doing something with the conclusion (your first problem looks like the start of an example---calculate sin A/a, sin B/b, and sin C/c, and decide whether or not they're equal). More difficult, you need to have problems where the solution isn't always "more digits is better".

I've seen people (more often in the sciences than math, though it's certainly appropriate in math) work with numbers with errors (1.04$\pm$0.1) and expect students to report calculations correctly---that is, is accurately as possible given the uncertainty in the original data.

For your second example, I share your intuition that one digit is too little precision and ten digits is too much precision. But how would you explain that rigorously to a student, to a level of clarity that it would be fair to mark them wrong for not getting it?

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There are multiple issues here. When dealing with a series of equations, taking a power, multiplying and then entering into the next calculation, it can be pretty important to hang on to all the significant digits you can through the process. But then, it's important to know when to get back to a reasonal number of significant digits. Say you have an exponential population growth. Fractional rabbits or people make no sense, so do the math, then round to an integer. Yet, when taking the tangent of an angle, see the difference between the Tangent of 89 degrees and 89.5 degrees. In the end, you can help them understand that significant digits need to adjust to the situation. I wouldn't discuss it as an entire lecture but as a brief note in each sets of new problems.

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