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\section*{Introduction}
This assignment is your first group project
for \lstinline{Computer Organization}. It is designed to teach you how
group work is to be done in this class. The method for the group work
may seem either silly or overly scripted. It is actually drawn from
research on brain function and learning theory. I ask that you do your
best to follow the plan but feel free to let me know if, for some
reason, this method interferes with \emph{your} learning style.

[Based in part on documents found on cspogil.org.]

\section*{Note}
\begin{itemize}
\item By \textbf{design}, this assignment includes questions about
  topics that you have not yet seen in lecture. This encourages
  \emph{creative} exploration and \emph{constructing} your own mental
  models.
\item Your group is expected to work together on \emph{each} question:
  the goal is \textbf{not} to fill in the ``right'' answers as quickly
  as possible but rather to take the time to read, think, and discuss
  the material \textbf{together}.
\item Constructing your own model is \textbf{short-circuited} by
  reading or parroting another thinker's answer. This means you
  should, as much as possible, use \emph{recall} of class material,
  looking up definitions in \emph{notes} or the \emph{textbook} when
  necessary, and \emph{avoid} just typing the question into a search
  engine on the Web. If you want more information, feel free to search
  \textbf{after} the group work is finished.
\item Each student in the group has more or less experience with the
  topics in the assignment. If you have greater familiarity
  with the topics, \emph{please} hold back a little bit so others can
  engage with building their own knowledge. You, too, must engage
  because there is always more to learn.
\end{itemize}

\section*{Assignment Goals}
After completing this group assignment, each student is expected to be
able to
\noindent
Learning Outcomes
  \begin{itemize}
  \item Name the members of your group.
  \item Describe the \emph{four} roles in the POGIL method.
  \item Manipulate decimal numbers using placeholder notation.
  \item Manipulate binary numbers using placeholder notation and
    translate binary to decimal values.
\end{itemize}

\section*{Procedure}

\textbf{Get out paper.} The group will turn in \emph{one}
document. Place the names of all present group members atop the
page. This page will also be used by the speaker when answering
questions at the end of the class.

\noindent
\textbf{Copy each question before the answer.} This makes the answer
sheet a worthwhile study guide w/o the worksheet. The answers written
down should be \emph{discussed} by the group and represent the
\emph{consensus} of the group. Sometimes a \emph{picture}, a
\emph{graph}, or something similar is the most appropriate way to
answer a question.

\noindent
\textbf{Assign Roles.} Students should take roles they
have not held recently (or, perhaps, ever):

\begin{description}
\item[Manager] Move discussion forward.
\item[Reflector] Monitor that everyone gets heard and is caught
  up. (This is a \textbf{group} obligation, really.)
\item[Speaker] Asks the facilitator questions and communicates what the team has done.
\item[Recorder] Writes the report.
\end{description}

\noindent
\textbf{Answer these questions.}

\begin{questions}
\question{} \emph{Introduce} yourself to your fellow group members,
  including completions to the following prompts. These are designed
  to help each of you be more comfortable with being \textbf{wrong} in
  front of one another. \textbf{Mistakes} are a major part of
  learning.
  \begin{parts}
  \part{} The \emph{longest} computer program I have ever written was
    \ldots{}.
  \part{} The most \emph{challenging} computer bug I have ever had to
    solve was \ldots{}.
  \end{parts}

\uplevel{Assume \emph{all} integers in this document are non-negative
  (also known as \emph{unsigned} integer values).}

\question{} What is the \emph{largest} decimal number you can store in
  exactly \textbf{two} digits? How do you know. What is the
  \emph{smallest} number.

\question{} What is the \emph{largest} decimal number you can store in
  exactly \textbf{three} digits? \textbf{Four} digits?

\question{} Consider the number 357$_{10}$. What does the 3 mean? That
  is, would 3 mean the same thing if it appeared where the 5 or the 7
  appears?

\question{} If we numbered the digits (with subscripts) in the number
  as $3_{2}5_{1}7_{0}$, write the value of the number in terms of a
  summation $\sum\limits_{i=0}^{2} \dots$.

\question{} Consider $D=\{0,1,2,3,4,5,6,7,8,9\}$, the set of
  \emph{digits} used in \textbf{base-10} (or \textbf{decimal})
  numbers. What is $|D|$? Explain \emph{why} the set has that
  cardinality.

\question{} How many \emph{different} decimal numbers are there that
  use exactly \textbf{five} digits.

\question{} What is the \emph{largest} value you can get when adding
  two \textbf{single-digit} decimal numbers? How many digits does it
  take to represent?

\question{} What is the \emph{largest} value you can get when adding
  two \textbf{two-digit} decimal numbers? How many digits does it
  take to represent? What are the digit values in the two two-digit
  numbers? Where do they appear in an ordered listing of $D$?

\question{} Imagine a decimal number $d_{n}d_{n-1}d_{n-2} \dots
  d_{2}d_{1}d_{0}$. Write a summation to calculate the value of the number.

\uplevel{A \textbf{binary} number is one written in
  \textbf{base-2}. In base-2, each higher digit position corresponds
  to a power of 2 (rather than a power of 10 as in decimal
  representations). For example, $101_2$ is $1 \times 2^2 + 0 \times 2^1
  + 1 \times 2^0$ or $5_{10}$.}

\question{} Consider $B = \{$the set of binary digits$\}$. $B$ is the
  set of \emph{bits} used to represent \textbf{base-2} numbers. What
  is $|B|$? What is $B$, the set of bits, itself?

\question{} What is the \emph{decimal} value of the \emph{binary}
  number $1_{3}1_{2}1_{1}1_{0}$? What about $1_{3}1_{2}1_{1}0_{0}$?
  $1_{3}0_{2}1_{1}0_{0}$? Explain how you are figuring this out.

\question{} What is the decimal value of the binary number
  $1_{5}0_{4}0_{3}0_{2}0_{1}0_{0}$?

\question{} Imagine a decimal number $b_{n}b_{n-1}b_{n-2} \dots
  b_{2}b_{1}b_{0}$. Write a summation to calculate the \emph{decimal}
  value of the number.

\question{} How many \textbf{two-bit} binary numbers are there? Write
  them all out. What are the \emph{decimal} values of each one?

\question{} What is the \emph{largest} binary number using exactly two
  bits? What about three bits? Four bits?

\question{} How many \emph{different} binary numbers are there with
  exactly \textbf{one bit}? \textbf{Two bits}? \textbf{Three bits}?
  \textbf{Twenty bits}?

\question{} How do the largest \emph{binary} representations with a
  fixed number of bits compare to the largest \emph{decimal}
  representations with a fixed number of digits?

\question{} What is the \emph{largest} value you can get when adding
  two \textbf{single-bit} binary numbers? How many bits does it take
  to represent?

\question{} What is the \emph{largest} value you can get when adding
  two \textbf{two-bit} binary numbers? How many bits does it take
  to represent? What are the bit values in the two two-bit numbers?
  Where are they in $B$?

\end{questions}
\end{document}

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