Multiplication Not Distributive (MND)

In precalculus/algebra we become familiar with the distributive laws that address interactions between multiplication and addition/subtraction. Specifically, these laws say

\begin{displaymath}a(b+c) = ab + ac \qquad\mbox{and}\qquad (a + b)c = ac + bc. \end{displaymath}

We use these laws all the time, both in expanding
\begin{eqnarray*} 3x^2(x - 2y) & = & 3x^3 - 6x^2y \qquad\mbox{and} \\ (x - 3)... ...x-3)x + (x-3)(7) \;=\; x^2 - 3x + 7x - 21 \;=\; x^2 + 4x - 21, \end{eqnarray*}
and in factoring
\begin{displaymath}30x^2y - 12xy^2 + 3xy = 3xy(10x - 4y + 1). \end{displaymath}

We even use it (although we don't often think about it this way and usually don't include the middle step below) when combining like terms, as in
\begin{displaymath}4xy - 15xy \;=\; (4 - 15)xy \;=\; -11xy. \end{displaymath}

The problem is when students misinterpret these laws, thinking they also say something about interactions between more than one multiplication; that is, they ``invent" for themselves a law that looks something like:
\begin{displaymath}a(b\cdot c) = (ab)(ac). \mbox{\hspace{0.6in} \epsfig{0.5in}{mnd.eps}}\end{displaymath}

This clearly is false, as most would see if these were all numbers -- few (though I cannot go so far as to say no one) would assert, say, that
\begin{displaymath}7(3\cdot 10) = (21)(70) = 1470. \mbox{\hspace{0.6in} \epsfig{0.5in}{mnd.eps}}\end{displaymath}

But when the objects involved are expressions involving variables, the error is frequently made, such as in this case:
\begin{displaymath}5(2x^2 y^3) = (10x^2)(5y^3) = 50x^2 y^3, \mbox{\hspace{0.6in} \epsfig{0.5in}{mnd.eps}}\end{displaymath}

or
\begin{displaymath}3[(x-1)(7x)] = (3x-3)(21x). \mbox{\hspace{0.6in} \epsfig{0.5in}{mnd.eps}}\end{displaymath}




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Thomas L. Scofield 2003-09-04