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{\bf {\large Division Problems}}
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\noindent
{\bf PROBLEM:} Is $8 \div 2$ defined or undefined?  Use the missing 
factor model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:} If $8 \div 2$ is defined then there is a unique whole number
$c$ such that 
\begin{equation}
\label{eqn2}
2 \times c = 8 
\end{equation}
Since the only solution to equation (\ref{eqn2})
is $c=4$, it follows that $8 \div 2 =
4$.

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\noindent
{\bf PROBLEM:} Is $8 \div 2$ defined or undefined?  Use the repeated  
subtraction model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:} Let us examine $8 \div 2$ by using the repeated subtraction
model.  Suppose that you have a pile of $8$ coins.  We need to remove
groups of two coins from the pile of $8$ coins until all the coins are
gone.  If $8 \div 2$ is defined, then it is defined to be the total number
of groups we remove.
Removing one group of two coins will leave us with a pile of $6$
coins.  Removing another group of two coins will leave us with a pile of
$4$ coins.  Removing another group of two coins will leave us with a pile of
$2$ coins.  Removing yet another group of two coins will leave us wil a pile of
$0$ coins.  Since we had to remove $4$ groups of two coins
in order to remove all the coins from the original pile of $8$ coins, it
follows that $8 \div 2 = 4$.

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\noindent
{\bf PROBLEM:} Is $8 \div 2$ defined or undefined?  Use the partition 
model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:} Let's examine $8 \div 2$ according to the partition model.
Suppose that we have $8$ sandwiches and $2$ lunch boxes.  We need to 
place each sandwich in a lunch box,
and we need to make sure that each lunch box contains the same number of 
sandwiches.   If we can do this in a unique way, then $8 \div 2$ will be
defined to be the number of sandwiches in each lunch box 
The only way to evenly distribute all of the sandwiches into the lunch
boxes is to place $4$ sandwiches in each lunch box.  Therefore $8 \div 2 = 4$.

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\noindent
{\bf PROBLEM:} Is $56 \div 0$ defined or undefined?  Use the missing factor
model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:}  If $56 \div 0$ is defined, then there is a unique whole
number $c$ such that 
\begin{equation} 
\label{eqn1}
0 \times c = 56. 
\end{equation}  
Clearly equation (\ref{eqn1}) does not have any solutions.  Therefore $56
\div 0$ must be undefined.

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\noindent
{\bf PROBLEM:} Is $56 \div 0$ defined or undefined?  Use the repeated
subtraction model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:} Let's examine $56 \div 0$ according to the repeated
subtraction model.  Suppose we start with a group of $56$ coins.  We need
to remove groups of zero coins from the pile of $56$ coins until all the
coins are gone.  But removing groups of zero coins from the pile will
always leave $56$ coins in the pile.  That is to say, we will never be able
to remove all the coins from the pile in this way.  Therefore $56 \div 0$
is undefined.


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\noindent
{\bf PROBLEM:} Is $56 \div 0$ defined or undefined?  Use the partition  
model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:} Let's examine $56 \div 0$ according to the partition model.
Suppose we have $56$ sandwiches and $0$ lunch boxes.  To evaluate $56 \div
0$ we would need to put each sandwich in a lunch box and make sure that
each lunch box has the same number of sandwiches.  Then we would need to count 
the number of sandwiches in each box.  Since there are $0$ bags, we cannot put 
each sandwich in a lunch bag.  Therefore $56 \div 0$ is undefined. 

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\noindent {\bf Note:}
{\em $0 \div 0$ is a difficult concept, and rightly so
since it it undefined.  I think that the missing factor model provides the
simplest explanation as to why $0 \div 0$ is undefined.  I have used all
three models here, but I think the last two explanations are pretty
convoluted.}
\vspace*{.25in}

\noindent
{\bf PROBLEM:} Is $0 \div 0$ defined or undefined?  Use the missing factor 
model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:}  If $0 \div 0$ is defined, then there is a {\em unique} whole
number $c$ such that 
\begin{equation}
\label{eqn3}
0 \times c = 0.
\end{equation}
There are many solutions to equation \ref{eqn3}.  For example $c=0$ and
$c=5$ are both solutions to equation \ref{eqn3}.  Therefore equation
\ref{eqn3} does not have a {\em unique} solution and, hence, $0 \div 0$ is
undefined.


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\noindent
{\bf PROBLEM:} Is $0 \div 0$ defined or undefined?  Use the repeated
subtraction model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:}  Let us examine $0 \div 0$ using the repeated subtraction
model.  Suppose that you have a pile of $0$ coins.  We need to ask
ourselves how many groups of $0$ coins must be removed from the pile so
that no coins are left in the pile.  Since the initial pile does not
contain any coins, I could remove $0$ groups of $0$ coins  and leave a pile
with no coins.  So at first glance it might appear that $0 \div 0$ is equal
to zero, but let's not be hasty.  I could also, for example, remove $3$
groups of zero coins and leave a pile with no coins.  From this example, it
appears that $0 \div 0$ could also be equal to $3$.  Therefore, there is not
a unique solution for $0 \div 0$, and, hence, $0 \div 0$ is undefined.


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\bigskip
\noindent
{\bf PROBLEM:} Is $0 \div 0$ defined or undefined?  Use the partition 
model to explain your answer.  

\smallskip
\noindent
{\bf ANSWER:} Let us examine $0 \div 0$ using the partition model.  Suppose
you have $0$ sandwiches and $0$ lunch boxes.  We need to place each
sandwich in a lunch box so that each lunch box contains the same number of
sandwiches.  Then we need to count the number of sandwiches in each lunch
box.  The total number of sandwiches in each lunch box would be our answer.
However, we cannot count the number of sandwiches in each lunch box because
we do not have any lunch boxes.  Therefore $0 \div 0$ is undefined.

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