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made grammatical changes and simplified wording for clarity
1: \label{Formula_Response_Sampling}
2:
3: As you may know, it is extremely difficult to determine whether a given expression
4: is exactly equal to another expression in general. For example, is $\sin 2x=2\sin x\cos x$?
5: Symbolically proving it one way or another is impossible in general. Therefore,
6: LON-CAPA uses a sampling system. If your answer and the student's answer
7: agree at the sampling points within your given tolerance factor, the student's
8: answer will be accepted. If the student's answer does not agree at the sampling points within your given tolerance factor, it will be rejected.
9:
10: To specify where to sample the formulas for determining whether the student's
11: answer is correct, you need to put a sampling specification in the \textbf{Sample
12: Points}\index{Sample Points} field. The sampling specifications take the
13: following format:
14:
15: \begin{enumerate}
16: \item A comma-separated list of the variables you wish to interpret,
17: \item followed by {}``@'' (not in quotes),
18: \item followed by any number of the following two things, separated by semi-colons:
19:
20: \begin{enumerate}
21: \item a comma-separated list of as many numbers as there are variables, which specifies
22: one sampling point, OR
23: \item a comma-separated list of as many numbers as there are variables, followed
24: by a colon, followed by another list of as many numbers as there are variables,
25: followed by a \#, followed by an integer.
26: \end{enumerate}
27: \end{enumerate}
28: The first form specifies one point to sample. The second form specifies a
29: range for each variable, and the system will take as many random samples
30: from that range as the number after the \#.
31:
32: For $2x^{2}+4$, with one variable {}``x'', one could specify:
33:
34: \begin{itemize}
35: \item {}``x@2'', which will sample the answers only at 2. (This is generally
36: a bad idea, as the student could get lucky and match at that point)
37: \item {}``x@1:5\#4'' will takes 4 samples from somewhere between 1 and 5.
38: \item {}``x@1:5\#4;10'' will takes 4 samples from somewhere between 1 and 5,
39: and also sample at 10.
40: \end{itemize}
41: For $2x^{2}+3y^{3}$, which has two variables, one could specify:
42:
43: \begin{itemize}
44: \item {}``x,y@4,5:10,12\#4;0,0'', which take four samples from the box determined
45: by the points (4, 5) and (10, 12), and also sample the point (0, 0).
46: \end{itemize}
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