## Cancelling Everything in Sight (CES)

Seeing a complicated fraction become less ugly as
elements are cancelled from both the numerator and denominator
can be something of an enjoyable experience. One's first
exposure to this magical process usually comes in grade school
when reducing fractions, such as

High school algebra classes build upon this, showing us that
we may also cancel expressions involving variables, as in

What some students do not notice is that these cancellations
only are performed once the numerator and denominator are
*factored*. Factoring a numerator (or denominator) turns
it into an expression which is, at its top level, held together
by multiplication. For instance, in the expressions

To be sure that one performs valid cancellations only, it is
necessary to
- be patient, making sure to factor numerator and denominator
first, and cancelling only those factors common to both, and
- accept that many times no factorization is possible, at
least none that leads to a common, cancellable factor.

With this in mind, cancellations such as those below may only
be labelled instances of someone ``cancelling everything in
sight", with no attention given to the discussion above, and
having no validity whatsoever.

Any attempt to simplify the original fraction (rational
expression) should start with factoring:

at which stage we see that there is no matching factor between
those of the numerator -- namely, and --
and those of the denominator -- and . Factoring,
in this case, did not lead to any cancelling, as is often the
case.

Top Algebra Errors Made by Calculus Students
(full document)

Full List of Grading Codes

Thomas L. Scofield
2003-09-04