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\begin{document}

\noindent
{\Large 
Integers Worksheet \hfill Name:\underline{\hspace{2.8in}}\\ \\
MA~201}
\vspace{.25in}


\be 
\item Use repeated addition to explain the following multiplication facts.
	\be
	\item $5 \times 4 = 20$
	\item $5 \times -4 = -20$
	\ee

\item Let $m$ and $n$ be positive integers.  What can you say about $m
	\times (-n)$?

\item Can you use repeated addition to evaluate $-5 \times 4$?  $-5 \times
-4$  Let's try to discover what these products should be from another direction.


\item Examine the following products and discuss any patterns that you
find.
	\be
	\item $5 \times 5$
	\item $5 \times 4$
	\item $5 \times 3$
	\item $5 \times 2$
	\item $5 \times 1$
	\item $5 \times 0$
	\item $5 \times -1$
	\item $5 \times -2$
	\item $5 \times -3$
	\item $5 \times -4$
	\item $5 \times -5$
	\ee
Let $m$ and $n$ be positive integers.  Using the pattern above, what can you 
say about $m \times (-n)$?  Does this conclusion agree with your conclusion
in problem 2?

\item  Let's use patterns to examine other products of integers.  Complete
the following multiplication problems by following the pattern.
	\be
	\item 
		\be
		\item $5 \times 4 = 20$
		\item $4 \times 4 = 16$
		\item $3 \times 4 = 12$
		\item $2 \times 4 = 8$
		\item $1 \times 4 = 4$
		\item $0 \times 4 = 0$
		\item $-1 \times 4 = $
		\item $-2 \times 4 = $
		\item $-3 \times 4 = $
		\item $-4 \times 4 = $
		\item $-5 \times 4 = $
		\ee
		Let $m$ and $n$ be positive integers.  What can you say
		about $(-m) \times n$?  
	\item 
		\be
		\item $-5 \times 5 = -25$ 
		\item $-5 \times 4 = -20$ 
		\item $-5 \times 3 = -15$ 
		\item $-5 \times 2 = -10$ 
		\item $-5 \times 1 = -5$ 
		\item $-5 \times 0 = 0$ 
		\item $-5 \times -1 = $ 
		\item $-5 \times -2 = $ 
		\item $-5 \times -3 = $ 
		\item $-5 \times -4 = $ 
		\item $-5 \times -5 = $ 
		\ee
		Let $m$ and $n$ be positive integers.  What can you say
		about $(-m) \times (-n)$?  
	\ee
\item Based on your conclusions about $m \times n$, $(-m) \times n$, $m
\times (-n)$ and $(-m) \times (-n)$, what can you say about the properties
of multiplication on the integers?

\item What model for division makes sense when you are talking about
division with integers?  How would you define division for the integers
using this model?  

\item Using your definition for division, determine the rules of signs for
division of integers.

\ee
\end {document}
