## Undo Multiplication with Division (UMD);
also,
Undo Addition with Subtraction (UAS)

The properties of equality that were mentioned earlier are, by some
students, implemented incorrectly even when the situation calls for
their use. For instance, when solving an equation
like

two steps are called for:

and

Notice that, in the expression , the order of operations
dictates that the multiplication by 3 comes before the addition of
7, and the ``undoings" of these processes -- the subtraction of 7 and
the division by 3 -- were carried out in reverse order. That is not
to say that we could not have undone things in a different order, but
students who do so often make the following error. Dividing by
3, they often neglect the fact that *all* terms on both sides
are to be divided by 3. In other words, after dividing by 3 they
write

They are too set on the idea that they will be subtracting 7 from
both sides to realize that, having divided by 3 first, it is not
7, but rather , which must be subtracted, giving the same
answer as before. One other note is in order here. If
parentheses appear in an equation such as

then the order of operations are preempted (the subtraction within
the parentheses comes before the multiplication by 3). In solving
for , we may of course, distribute the 3, thereby eliminating
the parentheses and making the problem appear like the last one
discussed. Even fewer steps are required if one just ``undoes" the
multiplication and subtraction in their opposite order:

and then

Now let us return to the equation

and investigate the more telling errors that gave the titles
**UMD** and **UAS** to this section. Some
students recognize the need
for two steps (like those carried out when this equation was
being considered above) to isolate , but have little feel
for which operations will achieve this. For
instance, realizing that, like the on the right-hand side of
the equation, is a ``non-" term, a student may write

misunderstanding that she has subtracted 7 on the left
side, but divided by 7 on
the other side. *The original equation and the new one no longer
have the same solutions* as a result. The same student may then
recognize that she needs to move the over to the other side.
Since the 3 is multiplied by the , she should ``undo" this by
dividing both sides by 3. But she may (wrongly) write

having divided on the left but subtracted on the right. Again,
the solution is different from the one that solved the
original equation , namely .
Worse still is when a student thinks he can solve in one step
(that is, take care both of the multiplication by 3 and the addition
of 7 via one operation). Such a student may write something like

Again, the answer this student gets,
, is different
than the correct one .

Top Algebra Errors Made by Calculus Students
(full document)

Full List of Grading Codes

Thomas L. Scofield
2003-09-04