Chapter 4.3, Problem 2P

Here is Amy’s explanation for why the commutative property of multiplication is true for counting numbers:

Whenever I take two counting numbers and multiply them, I always get the same answer as when I multiply them in the reverse order. For example,

$\begin{array}{l}6\cdot 8=48\hfill \\ 8\cdot 6=48\hfill \\ 9\cdot 12=108\hfill \\ 12\cdot 9=108\hfill \\ 3\cdot 15=45\hfill \\ 15\cdot 3=45\hfill \\ \hfill \end{array}$

It always works that way; no matter which numbers I multiply, I will get the same answer either way I multiply them.

Discuss why Amy’s explanation is not complete. Then explain why the commutative property of multiplication is valid by viewing arrays in two ways.