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?