\documentclass[11pt]{book} \usepackage{../StyleSheets/Preamble} %\usepackage{microtype} \setcounter{secnumdepth}{-20} % sssssssssssssssssssssssssssssssss End PREAMBLE \begin{document} \chapter*{User Manual For Schremmer's Math 161} \ChapterToc The first thing you should do is to check \url{http://www.ratemyprofessors.com/ShowRatings.jsp?tid=633910&page=1}. There, you will see the two ways students react to the way I teach my classes: \begin{compactitem} \setlength{\itemsep}{0mm} \item%bullet \textbf{The students who want to learn mathematics.} These students try to understand how things build on each other \textbf{\emph{logically}}. Then they realize that it turns out to be easy. \item%bullet \textbf{The students who just want the credit.} These students try to memorize ``how to do it''. Then they realize that this is extremely difficult. \end{compactitem} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{What Is The Point?} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS The situation in mathematics is that you \emph{can} memorize how to do \emph{certain} things but then there are two problems with that. \begin{compactitem} \setlength{\itemsep}{0mm} \item%bullet How do you know you remember it right? \item%bullet What do you do when the problem does not look like any of the ones which you have memorized how to do. \item%bullet The more stuff there is, the more difficult it is to memorize but the easier it is to realize the sense they make. \end{compactitem} So, the point then is that memorization is essentially useless. It looks great only until you want to use it. What you need is to know \textbf{why} things are the way they are, what is the logic behind them because logic will always let you reconstruct things. So, the way to learn \emph{mathematics} is to realize how everything is related and what the logic of it is. \vspace{1mm} \noindent \begin{EEXAMPLE}% A problem in Precalculus might be to find the inputs for which a given quadratic function returns the output $0$. Now while there is a formula, $\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, on a exam you might suddenly not be quite sure you remember the formula correctly and then there is the fact that the formula does not always apply. But the formula has a logic to it and when you understand the logic, you know when you can use it and, in fact, how to recreate the formula. \end{EEXAMPLE}% And, if you want to read more, particularly about \emph{logic} versus \emph{memorization}, you might want to read the Preface. You might even want to explore \url{http://www.freemathtexts.org} a bit. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{What This Means To You} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS The fact that the focus in the course will be for you to learn how to decide whether something someone said is \textbf{true} or \textbf{false} requires in turn a couple of things in the way you will have to operate in this class: \begin{compactitem} \setlength{\itemsep}{0mm} \item%bullet Because you will want to know \emph{why} you should agree on the truth of what is being said, you should never accept anything for granted or as going without saying. You should accept the truth of a statement only after a convincing case has been made, just like in court where an attorney cannot just say ``My client is innocent'' but has to prove it. %And, of course, this goes both for you and for me. As a result, just as I will always ask you ``why'' you are saying whatever you are saying, you too will have to question the truth of whatever anybody---including myself---is saying. \item%bullet Because we will need to \emph{agree} on the truth of what is being said, we will need to be sure we know, and agree on, exactly what it is that is being said. As a result, you will have to get used to paying very close attention to the \textbf{meaning} of the words we will be using to describe to one another what we mean. Again just like in court, there can be no ``You know what I mean'' as the response will always have to be ``No, I can't read your mind''. \end{compactitem} \noindent The idea is not to leave any ``grey area'' in your mind because these are precisely where ``math anxiety'' originates. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{The TEXT} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS I wrote the \textbf{TEXT} in such a way as to focus on the \emph{sense} the stuff makes, on \emph{the reason why} things are the way they are\footnote{And surely not for you to memorize ``how to do the questions on the exam''.}. The book is called Reasonable Algebraic Functions and it is available at the CCP bookstore, from Lulu.com and freely downloadable from both Lulu.com and \url{http://www.freemathtexts.org/Standalones/RAF/Downloads.php}. The \textbf{TEXT} is divided in 18 chapters. Copies of individual chapters can be downloaded from \url{http://www.freemathtexts.org/Standalones/RAF/Downloads.php}. You will have to read each chapter \emph{before class}\footnote{See the Schedule at the end of this paper}. But, in mathematics, ``to read'' does not mean letting you eyes wander on the stuff. In mathematics. ``to read'' means to read each sentence (starting with the very first one) and ask yourself what sense the sentence makes to \emph{you}. Then: \begin{compactitem} \setlength{\itemsep}{0mm} \item%bullet If the sentence \emph{does} make sense to you, you should still check that sense by trying to rephrase it in your own words and \emph{decide} whether the two say the same thing. And/or you may submit your rephrasing to somebody else, another student and/or to the class. \vspace{1mm} %\noindent \begin{EEXAMPLE}% In Chapter I, a function is defined as being a \emph{relation} that meets the requirement that \vspace{-2mm} \begin{center} \begin{boxedminipage}[t]{80mm} \emph{No input shall be paired with more than one output.} \end{boxedminipage} \end{center} \vspace{1mm} Experience shows that, at first, students miss the point. So, you should explain to yourself what this means. Then check if this means the same to other students, Then you might bring the case in class for discussion. The good news is that, eventually, everybody says ``Of course''. But only after they have gone through the process. \end{EEXAMPLE}% \item%bullet If the sentence does \emph{not} make sense, you should narrow down what the problem is. What often happens is that one of the words does not make sense to you. But the rule in mathematics is that all the words used in any sentence must all have been \emph{previously} explained. So you should look up the word in the index which will tell you where the word was explained. Another thing that can happen is that even though you understand all the words, the sentence as a whole somehow does not make sense. Very often though, there will be an \textsf{\bfseries{E\smaller{XAMPLE}}} that illustrates the sentence. What can also happen of course is that I botched it. In that case you have to let me know. \end{compactitem} And then there is another, difficult issue: there is the fact that there is stuff that you can skip on the first reading but what stuff you can skip on first reading will depend on \emph{you} and it is going to take a judgement call on your part. What happens is that I tried to answer as many of the questions I know \emph{many} students may have. But of course, \emph{you} may not be one of them andor \emph{you} may have \emph{different} questions. So, whether``to skip or not to skip'' is going to be up to you and learning to decide whether or not to skip is a good thing but you should make sure that, when you do skip something, you keep in mind that you did skip it. After all, you never know, the stuff might turn out to be important after all, for instance it might explain something that you will use later. But then again maybe not. That's life. All of this is why mathematician talk of ``reading pencil in hand''. The good news is that, once you have tried it, you will get quickly used to it and taking notes while you go back and forth in a mathematics book with the help of the index will become second nature to you and will put you at a decided advantage in later courses. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{The HOMEWORKS} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS Reading the text, pencil in hand, even asking precise questions is all well and good but there remains the issue of your figuring out how well you are going to do on the exam. After all, you will \emph{think} you understand the text, and you may well be right, at least to an extent, but, on exam day, \emph{I} will be the one to ask the precise questions and so you need to make sure, before the exam, that you will be able to answer \emph{my} precise questions. So, the next thing to do is the \textbf{HOMEWORK} that comes with each one of the 18 chapters. I will hand out copies of the homeworks but they too can be downloaded from \url{http://www.freemathtexts.org/Standalones/RAF/Downloads.php}. Doing the homework is designed to be the first ``reality check'' on your understanding of the chapter. But for that check to be really ``real'', you need to use the spaces provided in the homework for you to explain how you arrived at whatever result you arrived at. The multiple-choices that are also there are for me to see what kind of mistake, if any, you made and to give you ``answers'' (That's the only place where I will do so) and to give you an idea of how things went as a whole. But your answers, right or wrong, will have no effect whatsoever on your final grade. \indent I will accept late homework but keep in mind that, since they are for you to check your understanding of what you read in the chapter, you should try to do both the reading and the homework in time. You can also download another copy of the homework, re-do it to learn from your previous mistakes and submit it. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{Specific Questions} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS OK, so did your thing, you read the chapter pencil in hand and you did the homework that goes with it. You did understand part of what is in the text but you have questions about other parts of it. Hopefully, you used your pencil to take notes on your understanding and, in particular, you wrote down the \emph{specific questions} you had with the text. You also did the homework and, sure enough, you probably had the same trouble with the homework as you had with the text. Now what? \indent The answer is: Just ask \emph{specific question(s)}. Don't worry about how this is going to make you look. It won't make you look anything and, of course, the chances are high that if you have a question, you are not the only one: \emph{You} just have a \emph{specific question} and it is up to \emph{me} to help \emph{you} answer it. Keep in mind, though, that I will respond always in kind: A precise question will get you a precise answer but too broad a question will get you a question in order to narrow things down a bit. \vspace{1mm} \noindent \begin{EEXAMPLE}% If you tell me ``I can't do quadratic functions'', what can I say? On the other hand, if you ask ``Why does the discriminant of $f$ have to be positive?'', we have the beginning of a good question that we can work on, but only the beginning, because, of course, the discriminant does not have to be positive unless there is a reason. So, my response is likely to be something like ``What do you want to know about the function $f$?'' \end{EEXAMPLE}% \vspace{1mm} Also, if I think that, in fact, you already know the answer, I will ask you a question to move your thinking in that direction. In short, I will do whatever it takes for \emph{you} to be the one who gets the answer to your own question. The reason you must ask questions whenever something is not entirely clear in your mind is because of the way the text is constructed, which is like a building in which each successive floor rests on the floors that have already been constructed. As a result, as opposed to what would be the case if we were just going to go over a number of separate topics, as happen in many other disciplines, if you miss something and don't catch it, as will surely happen more than once, it may not show up right away but, for sure, will trip you somewhere down the line. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{How To Get A (Hopefully Specific) Response} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS There a several ways you can proceed: \begin{asparaenum}[\bfseries a.]%a, b, c. \item%a One place where you can ask questions is on the Homework itself. The homework also gives you places where to write-in questions which I will try my best to respond to in writing\footnote{My handwriting is terrible but I will try my best.}. I will ``enforce'' your giving explanations as to how you got your result by \emph{not} checking the answers in the case of ``\textbf{No} e\textbf{X}planation''. One the other hand, I will be specific in the way I address the issue you raise. And I won't care if it takes half a page to do so. \item%b One problem with asking your question on the homework is that you are not going to get my response to your question until I return the homework to you the next class and that may create a problem. That is why \emph{class time} will be entirely devoted to the discussion of the specific questions \emph{you} have with the text and the homework. \item%c It will often happen that you won't even want to wait for the next class to get a response to a specific question you have. In that case, your just email me your question. \textbf{However, please DO NOT use my ccp,edu address and use instead my gmail address:} \begin{center} \url{aschremmerccp@gmail.com} \end{center} Here are a few pointers for keeping emails simple: \begin{compactitem} \setlength{\itemsep}{0mm} \item%bullet Don't \emph{copy} the whole text with which you are having trouble, instead just tell me where it is and get to the point. \vspace{1mm} %\noindent \begin{EEXAMPLE}% Say you have a question about something in \textsf{\bfseries{E\smaller{XAMPLE}}} 3 in \textbf{Chapter} 2. Once you have said so, as in ``In Chapter 2, Example 3,'' just say what the specific question is, for instance` ``it says that we can easily find the output for the input +2. But how do you do that?'' Note that if what you write is just ``How do you do that?''', I am not going to know what you are talking about so I will respond something like: ``How do you do what?'' On the other hand, once I understand your question, I will write however much I need to make the answer clear. \end{EEXAMPLE}% \item%bullet Writing mathematics in an email can be a bother. Here are a couple of standards that everybody in the trade uses: \vspace{3mm} \begin{center} \catcode`\¥=\active \def¥{} \begin{tabular}{@{} cc @{}} \toprule %Row0 To indicate & Type \\ \midrule %Row1 $x^{3}$ & $x \hspace{2mm} \widehat{}\; \;3$\vspace{1mm} \\ %Row2 $x^{+2\hspace{1mm} \oplus\hspace{1mm} -5}$ & $x \hspace{2mm} \widehat{}\; (+2\oplus-5)$\vspace{1mm} \\ %Row3 $\frac{2}{3}$ & $2 \diagup3$\vspace{2mm} \\ %Row3 $\frac{2x+5}{x-5}$ & $(2x+3) \diagup(x-5)$\vspace{2mm} \\ %Row3 $\xrightarrow{\hspace{5mm}FUN\hspace{5mm}}$ & ---$FUN$---> \\ \bottomrule \end{tabular} \end{center} \vspace{3mm} Whether or not to use parentheses is often a judgement call though. \item%bullet An alternative can be just to write your question on paper with black ink, then scan the paper and attach the scan to your email. \end{compactitem} \vspace{1mm} \noindent \begin{note} I am, in fact, pretty good at answering my mail promptly \dots and thoroughly. \end{note} \vspace{1mm} \item%d Should all else fails, we will make an appointment to deal with the difficulty you are having. My office hours, \emph{by appointment only} are: \vspace{3mm} \begin{center} \catcode`\¥=\active \def¥{} \begin{tabular}{@{} cc @{}} \toprule %Row0 Tuesday & Thursday \\ \midrule % %Row1 % 8:00 - 9:20 % & 8:00 - 9:20 % \\ %Row2 11:00-12:20 & 11:00-12:20 \\ %Row2 12:30-1:50 & 12:30-1:50 \\ % %Row3 % 2:00-3:20 % & 2:00-3:20 % \\ \bottomrule \end{tabular} \end{center} \vspace{3mm} \vspace{1mm} \noindent \begin{note} My reason for asking you to make an appointment is that for me to sit in my office just on the chance that someone might show up would be a waste of time. In return, though, \begin{asparaenum}[\bfseries i.]%i, ii, iii. \item%i I have six office hours instead of the three I am required to have, \item%ii I will spend with you whatever time is necessary to resolve the issue and, if necessary, will make an appointment at another, other than the above times. \end{asparaenum}%i, ii, iii. \end{note} \vspace{1mm} \end{asparaenum}%a, b, c. Finally, and by the way, I mentioned above that things would likely change as we go. One thing that usually changes is that, during the last third of the semester, as I walk-in I often find students at the board discussing specific questions they have written on the board, arguing and explaining matters to each other. \textsc{And that is really how it should be from the very start.} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{How I Will Help You} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS After all this about questions, though, I should say that I happen to remember the days when I was myself a student and when I often didn't understand what it was that I did not understand. So, as long as I think that your question is in good faith, no matter if it is not really a good question, I will do my best to deal with it. Since I already wrote in the text all I had to say, and since I cannot put understanding in a nutshell\footnote{If I could do it I would have done it.}, we will devote all our \emph{class time} to your understanding of whatever mathematical issue\footnote{I am unable to discuss the weather and, much to my regret am not allowed to discuss politics.} you raise. And if the class agrees that something is badly explained in the text, or that there is an outright mistake which is all too possible, then we will have to agree on a rewrite. But there is another issue: if you miss class time\footnote{See the calendar at the end.}, you will have missed your main opportunity to improve on your understanding of the chapter you have read by asking the questions on your mind and what will then happen is that you will have a lot of trouble with your understanding of what will come later on because chances are that will depend on stuff that was discussed while you were not in class. And, while I will spend whatever time is necessary, by email and/or on \emph{office hours}, to help a student resolve any specific difficulty s/he may have with a mathematical issue, no matter what, you are going to have to read pencil in hand and do the homework---which is one reason why I am accepting late homework---and formulate \emph{specific question(s)}. Of course, I will also help you when, for whatever reason, you had trouble with a question on the exam. But you are going to be the one who will have to word the question and you are going to have to be precise. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{The QUIZZES} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \indent So what my own ``role'' in the course will be is that of a \emph{resource} that you should use to help improve your understanding of what you have read in the text and done in the homework. But then, in order to help you, I have to have questions. And, if history tells me anything, it is that, at least during the first few weeks, until the students realize that I really mean it, the questions are few and far between. So, even though I hope that this paper will have convinced you, I will play it safe and, one way I will help you if I don't get any question is that I will hand out a \textbf{QUIZ} on the chapter in order to ``force'' questions. But the QUIZZES are only meant to generate \emph{precise questions} and will of course \emph{not} count towards your final grade in any way. That way, you will be entirely \emph{free} to ask ``stupid questions''. (Even though there ain't no such thing.) %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{How About The Grade?} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS Just because you want to learn the stuff doesn't mean you shouldn't worry about the grade. Your final grade will be computed on the sole basis of the scores you will have gotten on three multiple choice \textbf{EXAMS}. \vspace{3mm} \begin{center} \catcode`\¥=\active \def¥{} \begin{tabular}{@{} ccc @{}} \toprule %Row0 Final Total Score between & (Final Average between) & Final Grade \\ \midrule %Row1 0 and 179 & (0 and 60) & F \\ %Row2 180 and 209 & (60 and 70) & D \\ %Row3 210 and 239 & (70 and 80) & C \\ %Row2 180 and 209 & (80 and 90) & B \\ %Row3 210 and 239 & (90 and 100) & A \\ \bottomrule \end{tabular} \end{center} \vspace{3mm}The three EXAMS will be given in-class but, at the time allotted by the College at the end of the semester for the final, you will be able to \textbf{MAKEUP} any and all of the three in-class exams with the understanding that the scores on the makeups will automatically replace the scores on the in-class exams in the Final Total Score and this \emph{for the better or for the worse}. In other words, there will be no final exam and your grade will depend on your latest performance and only on that: \textsc{I will have no occasion to bend your grade in any way, up or down, no matter what.} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{Reviewing For The Exams} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS To help you get a better idea of what the EXAMS will look like and how you are likely to do on them, I will hand out \textbf{REVIEW Questions} for each exam along with \textbf{REVIEW Discussions}. These too are downloadable from \url{http://www.freemathtexts.org/Standalones/RAF/Downloads.php} \indent After we have dealt with the corresponding chapters and before you take the exam, I will give you a REVIEIW Test with exactly the same questions as in the \textbf{REVIEW questions} and the REVIEW Discussions and which I will score as soon as you hand it in. The only difference is that the REVIEW Test will be \emph{multiple-choice} and that the questions will be in \emph{random order}. Of course, the score is purely for your information and will of course not count in any way towards your final grade. %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS \section{What This All Means As A Whole} %%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS The fact that class time will be entirely devoted to our discussing the specific questions you ask has several consequences in terms of how I will have to run the class: \begin{asparaenum}%1, 2, 3. \item%1 As already mentioned, a question that is too broad will force me to ask questions to narrow things down. In terms of the class as a whole and since we will have only so much class time, this means that any time I take to narrow down your question will be time taken away from my responding to more specific questions asked by other students. So, please, if only in consideration of the other students, try to sharpen your question before asking them. \item%2 Moreover, if the question you ask is about something that was already discussed at a time when you happened not to be in class, whether because you were late or because you were absent, I will not stop the entire class to answer your question because I will want to answer the questions of those students who have run into difficulties with other stuff. \vspace{1mm} %\noindent \begin{EEXAMPLE}% Consider the following: Jane Doe says: ``I can see why \emph{xxx} on page X is true but I cannot see why it follows from that that \emph{yyy} is also true because how about if we had ... Couldn't then \emph{yyy} be false?'' This is of course a very specific question deserving of a thorough response. Now just after the dust has settled on that issue, John Smith comes in, half an hour late, and asks the very same, very good question while Mary Jones has another question. Obviously, I will not be able to spend any time on John Smith's question because that has now been already dealt with and that would take time away from Mary Jones' question. \end{EEXAMPLE}% \hspace{7mm} According to College Procedure \#5, students missing the equivalent of two weeks classes may be automatically dropped. So, to avoid any abuse, I will take roll and even keep track of lateness and I will enforce College Procedure \#5. In other words, you can miss up to the equivalent of two weeks classes without giving me any reason but you can't miss more even with a reason. \end{asparaenum}%i, ii, iii. \newpage \section{The Calendar} %\begin{center} \catcode`\¥=\active \def¥{} \hspace{-20mm} \begin{tabular}{@{} |c|lcc|c| @{}} \hline Week & Homework on Chapter: & \multicolumn{2}{c|}% {To be submitted on:} & \\ % \cline{3-4} & & Weekday & Date & Activity \\ \hline \hline %Row1 \textbf{1} & \hspace{38mm}\emph{Orientation} & T & Jan 18 & \\ %Row2 ¥ & 1 - Relations And Functions & R & Jan 20 & \emph{Discussion} \\ \hline %Row3 \textbf{2} & 2 - Towards Local Analysis & T & Jan 25 & \emph{Discussion} \\ %Row4 ¥ & & R & Jan 27 & Snow day \\ \hline %Row5 \textbf{3} & 3 - Graphic Local Analysis & T & Feb 1 & \emph{Discussion} \\ %Row6 ¥ & 4 - From Local Graphs to Global Graphs & R & Feb 3 & \emph{Discussion} \\ \hline %Row7 \textbf{4} & 5 - Regular Positive-Exponent Power Functions & T & Feb 8 & \emph{Discussion} \\ %Row8 ¥ & 6 - Negative-Exponent Power Functions & R & Feb 10 & \emph{Discussion} \\ \hline %Row9 \textbf{5} & 7 - Exceptional Power Functions & T & Feb 15 & \emph{Discussion} \\ %Row10 ¥ & ¥ & R & Feb 17 & \textbf{REVIEW Test I} \\ \hline %Row11 \textbf{6} & & T & Feb 22 & \textbf{EXAM I} \\ %Row12 ¥ & 8 - Operators & R & Feb 24 & \emph{Discussion} \\ \hline %Row13 \textbf{7} & 9 - Affine Functions: Local Analysis & T & Mar 1 & \emph{Discussion} \\ %Row14 ¥ & 10 - Affine Functions: Global Analysis & R & Mar 3 & \emph{Discussion} \\ \hline %Row15 \textbf{8} & & T & Mar 8 & Spring Vacation \\ %Row16 ¥ & & R & Mar 10 & Spring Vacation \\ \hline %Row17 \textbf{9} & 11 - Quadratic Functions: Local Analysis & T & Mar 15 & \emph{Discussion} \\ %Row18 ¥ & 12 - Quadratic Functions: Global Analysis & R & Mar 17 & \emph{Discussion} \\ \hline %Row19 \textbf{10} & 13 - Cubic Functions: Local Analysis & T & Mar 22 & \emph{Discussion} \\ %Row20 ¥ & 14 - Cubic Functions: Global Analysis & R & Mar 24 & \emph{Discussion} \\ \hline %Row21 \textbf{11} & & T & Mar 29 & \textbf{REVIEW Test II} \\ %Row22 ¥ & & R & Mar 31 & \textbf{EXAM II} \\ \hline %Row23 \textbf{12} & 15 - Rational Functions: Degree \& Algebra Reviews & T & Apr 5 & \emph{Discussion} \\ %Row24 ¥ & 16 - Rational Functions: Local Analysis Near $\infty$ & R & Apr 7 & \emph{Discussion} \\ \hline %Row25 \textbf{13} & 17 - Rational Functions: Local Analysis Near $x_{0}$ & T & \textbf{Apr 12}\footnotemark & \emph{Discussion} \\ %Row26 ¥ & 18 - Rational Functions: Global Analysis & R & Apr 14 & \emph{Discussion} \\ \hline %Row27 \textbf{14} & \hspace{40mm}\emph{Slack day}\footnotemark & T & Apr 19 & \\ %Row28 ¥ & \hspace{40mm}\emph{Slack day} & R & Apr 21 & \\ \hline %Row29 \textbf{15} & & T & Apr 26 & \textbf{REVIEW Test III} \\ %Row30 ¥ & ¥ & R & Apr 28 & \textbf{EXAM III} \\ %Row31 & \hspace{40mm}\textbf{\emph{Study Day}} & F & Apr 29 & \emph{Discussion} \\ \hline %Row32 ¥ & \hspace{65mm}8:00 to 12:00 & T & May 3 & \textbf{MAKEUPS} \\ \hline \end{tabular} \footnotetext{The last day to drop the course without penalty is the previous day: Monday, Apr 11} \footnotetext{The slack days are just that: in case, for whatever reason, we need, or just want, to push things back a day or two.} \end{document}