\documentclass[12pt]{article}

\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\usepackage{amsmath}
%\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\topmargin 0pt
\advance \topmargin by -\headheight
\advance \topmargin by -\headsep
\textheight 8.9in
\oddsidemargin 0pt
\evensidemargin \oddsidemargin
\marginparwidth 0.5in
\textwidth 6.5in

\newenvironment{packed_itemize}{
\begin{itemize}
  \setlength{\itemsep}{1pt}
  \setlength{\parskip}{0pt}
  \setlength{\parsep}{0pt}
}{\end{itemize}}

\begin{document}
\begin{center}
\begin{tabular*}{6in}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
Midterm  &\huge TMath 171  &\normalsize Autumn 2025\\
\end{tabular*}
\end{center}

\noindent As a reminder, you are welcome to use a non-internet accessing calculator (which includes Desmos Test Mode) and one 1-sided 8.5 in by 11 in sheet of notes. Show your work as you would for a \emph{colleague} unless otherwise indicated.

\begin{enumerate}



\item $[3]$ (MultActivity \#2) Let $a$, $b$, and $c$ be whole numbers. Determine if following statement is always true, sometimes true, or never true? Briefly justify your answer. 

$$a+(b\times c)=(a\times b) + (a \times c)$$
\vspace{0.9in}




\item (Quiz1 \#2) Suppose you own a square piece of land.  You trade your land for a rectangular lot.  The length of your new lot is 2 meters longer than the original lot and the width is 2 meters shorter than the original lot.  
\begin{enumerate}
\item $[2]$ If the original land is 50 meters by 50 meters, what are the dimensions of the new lot?
\vspace{1.0in}
\item $[2]$ Does the new lot have as much land as the old? Provide some justification.
\vspace{1.0in}
\item $[3]$ Do your conclusions above generalize for any square piece of land? Provide some justification.
\vspace{1.5in}
\end{enumerate}






\newpage



%\hspace{3.3in}\vspace{-.4in} \includegraphics[height=3.0in]{blank_10}

\item Consider the number written in base 4 with 2 flats, 1 long, and 3 units or $213_{\text{four}}$.

%\vspace{-.1in} \hspace{2.5in} \includegraphics[height=1.0in]{midtermc}

%\vspace{-1.1in}

\begin{enumerate}

\item $[2]$ (\S3.2\#8) Sketch the base pieces for $213_{\text{four}}$.
\vspace{1.2in}


\item $[2]$ (NumberSystemActivity \#4) Determine the total number of units in $213_{\text{four}}$, reporting in the Hindu-Arabic number system.
\vspace{1in}
 
\end{enumerate}



\item (ExtraPractice \S3.1 \#11) Consider the number of units shown below.

\vspace{-.1in} \hspace{2.5in} \includegraphics[height=1.0in]{midtermd}

\vspace{-1.1in}

\begin{enumerate}

\item $[2]$  Sketch the minimum number \\
of base pieces for base six to \\
represent the set of units shown.
\vspace{1.6in}

\item $[1]$ Write the number of 
units in positional notation for base six.
\vspace{.4in}

\end{enumerate}



\item $[3]$ Is Egyptian or Babylonian a positional number system? Provide justification for your answer.


\newpage
\item (ProblemSolvActivity \#1)  Consider the sequence of numbers illustrated below.  

\vspace{-0.1in} \hspace{2.0in}\includegraphics[height=1.3in]{midterma}

\vspace{-1.4in}
\begin{enumerate}
\item $[3]$ Find the next two \\
numbers in the sequence.
\vspace{1.5in}
\item $[2]$ Identify if the sequence is
arithmetic, geometric, or neither. Justify your answer.
\vspace{.8in}
\item $[2]$ Find the 60th number in the sequence.
\vspace{.5in}
\end{enumerate}



\item Show work \& compute (your audience is a colleague but still highlight any regrouping!):
\begin{enumerate}
\item $[2]$ (Quiz3 \#1)  $202_{\text{four}}+22_{\text{four}}$
\vspace{1.5in}
\item $[2]$ (\S3.4 \#8)  $62_{\text{eight}}\div 5_{\text{eight}}$
\vspace{1.5in}
\end{enumerate}





\newpage

\item $[3]$ Find a pattern for a sequence of numbers that
\begin{packed_itemize}
\item form a geometric sequence, and
\item begins with 3.
\end{packed_itemize}

\item Grade the work that follows.  The work may be correct or incorrect.  If correct, briefly justify why.  If incorrect, find the error(s) \& try to detect the reason for the error.

\begin{enumerate}
\item  $[3]$ (AddActivity \#1)  

\includegraphics[height=1.5in]{midtermb}   
\vspace{.7in}

\item $[3]$ (\S3.2 \#8)

\includegraphics[height=1.5in]{midterme}



\vspace{.3in}


\item $[3]$ (SeveralAlgorithms)

\includegraphics[height=1.5in]{midtermf}



\end{enumerate}


\newpage
\item Consider $45 - 3 \div 3 \times 4$

\begin{enumerate}
\item $[1]$ (Quiz3 \#2) Circle the operation above that should be performed first:

\item $[1]$ How would you modify the above expression to make it more clear the order of the operations?
\vspace{1in}
\end{enumerate}

\item $[5]$ (ExtraPractice \S3.4 \#33) Introduce how exponents and multiplication interact with each other as you would to an elementary school student who had forgotten.  Use the example $5^4\times 5^6$ in the explanation.

\vfill

\end{enumerate}

\hspace{-.8in}\includegraphics[height=1.0in]{EgyptNum} 


\vspace{-1.4in}\hspace{3in} \includegraphics[height=1.0in]{BabylonianNum} 

\hspace{4.5in} Babylonian




\end{document}
