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* Templates for formula and mathresponse
* Adding documentation to the manual
* stringresponse, etc, are not a subsubsection of "Foils" in author manual

    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: LON-CAPA has two ways of finding out if it is:
    6: \begin{itemize}
    7: \item algebraically, using a symbolic algebra system
    8: \item numerically, using sampling points
    9: \end{itemize}
   10: 
   11: You need to determine which way is the safest in a given situation.
   12: 
   13: If you don't specify sampling points, the symbolic algebra system is used.
   14: 
   15: If you do specify sampling points, LON-CAPA uses them.
   16: If your answer and the student's answer
   17: agree at the sampling points within your given tolerance factor, the student's
   18: 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.
   19: 
   20: To specify where to sample the formulas for determining whether the student's
   21: answer is correct, you need to put a sampling specification in the \textbf{Sample
   22: Points}\index{Sample Points} field. The sampling specifications take the
   23: following format:
   24: 
   25: \begin{enumerate}
   26: \item A comma-separated list of the variables you wish to interpret,
   27: \item followed by {}``@'' (not in quotes),
   28: \item followed by any number of the following two things, separated by semi-colons:
   29: 
   30: \begin{enumerate}
   31: \item a comma-separated list of as many numbers as there are variables, which specifies
   32: one sampling point, OR
   33: \item a comma-separated list of as many numbers as there are variables, followed
   34: by a colon, followed by another list of as many numbers as there are variables,
   35: followed by a \#, followed by an integer.
   36: \end{enumerate}
   37: \end{enumerate}
   38: The first form specifies one point to sample. The second form specifies a
   39: range for each variable, and the system will take as many random samples
   40: from that range as the number after the \#.
   41: 
   42: For $2x^{2}+4$, with one variable {}``x'', one could specify:
   43: 
   44: \begin{itemize}
   45: \item {}``x@2'', which will sample the answers only at 2. (This is generally
   46: a bad idea, as the student could get lucky and match at that point)
   47: \item {}``x@1:5\#4'' will takes 4 samples from somewhere between 1 and 5.
   48: \item {}``x@1:5\#4;10'' will takes 4 samples from somewhere between 1 and 5,
   49: and also sample at 10.
   50: \end{itemize}
   51: For $2x^{2}+3y^{3}+z$, which has three variables, one could specify:
   52: 
   53: \begin{itemize}
   54: \item {}``x,y,z@4,5,3:10,12,8\#4;0,0,0'', which take four samples from the box determined
   55: by the points (4, 5, 3) and (10, 12, 8), and also sample the point (0, 0, 0).
   56: \end{itemize}