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an assessment instrumentlist of questions for incoming to discrete math | |

\begin{enumerate} | |

\item Matching symbolic representations with pseudocode | |

%Stick breaking:\\ | |

%Start with a unit length.\\ % stick.\\ | |

%This unit serves as "what is in stock".\\ | |

%Every time a fractional part is required, a random fraction of the remaining %stock is taken.// %the stick is broken, one piece will be provided as output of the process, and the remainder will be held for future breaking.\\ | |

%When it is desired to have $p$ fractions, adding to length 1, $p-1$ iterations will satisfy this.\\ | |

\begin{figure}[ht] | |

\centering | |

\includegraphics[width=0.7\linewidth]{./pic8} | |

\label{fig:pic8} | |

\end{figure} | |

\[ | |

\sum_{i=0}^k \frac{1}{2^i} | |

\] | |

let num = rand()*10;\\ | |

let stock = 1;\\ | |

let amount = 0;\\ | |

let frac = 1/2;\\ | |

getRandomFraction()\{\\ | |

while (num $>0$)\{\\ | |

stock = stock*frac;\\ | |

amount = stock+amount;\\ | |

num = num-1;\\ | |

\}\\ | |

return amount;\\ | |

\} | |

let num = rand()*10;\\ | |

let stock =1;\\ | |

let amount = 0;\\ | |

let frac = 1/2;\\ | |

getRandomFraction()\{\\ | |

frac=rand(); \\ | |

amount = stock*frac;\\ | |

stock = stock - amount;\\ | |

return amount;\\ | |

\}\\ | |

\newpage | |

\item Matching figures/diagrams with symbolic representations\\ | |

Show which equation goes with which diagram, if they can be matched. | |

\begin{enumerate} | |

\item \begin{figure}[ht] | |

\centering | |

\includegraphics[width=0.7\linewidth]{./pic2GNUPaint} | |

\caption{} | |

\label{fig:pic2} | |

\end{figure} | |

\item \begin{figure}[ht] | |

\centering | |

\includegraphics[width=0.7\linewidth]{./pic4} | |

\caption{} | |

\label{fig:pic4} | |

\end{figure} | |

\end{enumerate} | |

\begin{enumerate} | |

\item $b^2 + ab = (b+a/2)^2 - (a/2)^2$ | |

\item $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$ | |

\end{enumerate} | |

\newpage | |

\item Matching figures/diagrams with pseudocode | |

\begin{enumerate} | |

\begin{figure} | |

\centering | |

\includegraphics[width=0.6\linewidth]{./pic9} | |

\caption{First} | |

\label{fig:pic9} | |

\end{figure} | |

\begin{figure} | |

\centering | |

\includegraphics[width=0.6\linewidth]{./pic10GNU} | |

\caption{Second} | |

\label{fig:pic10} | |

\end{figure} | |

\item let amount = 1;\\ | |

getAmount(n)\{\\ | |

amount = 0;\\ | |

for i = 1 to n \{\\ | |

for j = i to n\{\\ | |

amount = amount +j;\\ | |

end;\\ | |

end;\\ | |

return amount; | |

\item let amount = 1;\\ | |

getAmount(n)\{\\ | |

if (n==0)\{\\ | |

return(1) \}\\ | |

else\{\\ | |

return(2*getAmount(n-1)+1);\\ | |

\}\\ | |

\end{enumerate} | |

\newpage | |

\item Comprehending figures\\ | |

Explain why the figure matches the equation $\frac{a+b}{2} \geq \sqrt{ab}$ with equality if and only if $a=b$ | |

\begin{figure} | |

\centering | |

\includegraphics[width=0.7\linewidth]{./pic5} | |

\caption{} | |

\label{fig:pic5} | |

\end{figure} | |

\newpage | |

\item Comprehending symbolic representation\\ | |

What is the meaning of\\ | |

\[ | |

\sum_{ k \in \mathbb{N}}^{\infty} (2k+1) | |

\] | |

\newpage | |

\item Comprehending pseudocode | |

What is the meaning of | |

done = false;\\ | |

let a = 0;\\ | |

let b = 0;\\ | |

let n = 2;\\ | |

while(!done)\{\\ | |

while($a>0$)\{\\ | |

a=a+1;\\ | |

while ($b>0$)\{\\ | |

b=b+1;\\ | |

while($n>0$)\{\\ | |

n=n+1;\\ | |

if ($a^n +b^n = c^n$)\{\\ | |

done = true;\\ | |

\}\\ | |

\}\\ | |

\} \\ | |

\} \\ | |

\}\\ | |

\newpage | |

\item Applying symbolic representation to figure | |

Describe the significance of either or both of these figures in mathematical symbols: | |

\begin{figure}[ht] | |

\centering | |

\includegraphics[width=0.7\linewidth]{./pic1} | |

\caption{} | |

\label{fig:pic1} | |

\end{figure} | |

\newpage | |

\item Applying figure to symbolic representation | |

Draw a figure that expresses through any three points that are not collinear, two can be used to identify a line, and the third can be used, combined with that line, to identify a line parallel to the first line. | |

\newpage | |

\item Applying pseudocode to figure | |

Write pseudocode (e.g., as has been seen earlier in these questions) to correspond with this figure: %(p. 93) | |

\begin{figure}[ht] | |

\centering | |

\includegraphics[width=0.7\linewidth]{./p93} | |

\label{fig:p93} | |

\end{figure} | |

\newpage | |

\item Applying pseudocode to symbolic representation | |

Write pseudocode for this calculation: | |

${ n \choose k} = {n-1 \choose k-1}+ {n-1 \choose k}$ | |

\newpage | |

\item Synthesis: word problems to figures | |

Draw a figure for this word problem:\\ | |

Pick an angle between 0 and $\pi/2$ radians, call it $\theta$.\\ | |

A right triangle can be drawn, the height of which is 1 plus the tangent of $\theta$, the base of which is 1 plus the cotangent of $\theta$. The angle $\theta$ is adjacent to the base. It will be the case that the hypotenuse of this triangle is the cosecant of $\theta$ plus the secant of $\theta$. | |

\newpage | |

\item Synthesis: word problems to symbolic representation\\ | |

Express in symbols this word problem:\\ | |

The number of moves in a game of size n is given by twice the number of moves in a game of size (n-1), plus one more move. | |

\newpage | |

\item Synthesis: word problems to pseudocode | |

Write pseudocode for this word problem:\\ | |

Two trains, initially 40 miles apart on the same track, going opposite directions, are getting closer to each other at the rate of 40 miles per hour. An insect flies from one train to the other and back, repeatedly. How far does the insect fly, before the trains collide? | |

\end{enumerate} | |

operation procedure | |

implication | |

vacuous true | |

law of excluded middle | |

modus ponens | |

negation | |

multiple quantifiers | |

formative use it to tailor the course | |

McGowan Tall 2013 Jour Math Behav Formative works |