Explain how and why the cancellation of $ 6 $’s in $ \dfrac{16}{64} $ to get $ \dfrac{1}{4} $ is a fallacious statement. Based on what we know from elementary and middle school teachers, most of us know that 16/64 correctly equals 1/4 because 16/64 is simplified with a common divisibility of 16. However, there is another way to prove that 16/64 equals 1/4 without dividing the numerator and denominator by 16. Who can explain how and why that method leads to a fallacious statement?

They may well know that
$$\require {cancel}\frac {25}{50} = \frac{1}{2} $$
If their logic was always correct, this would also be, which is not:
$$\require {cancel}\frac {25}{50} = \frac{2\cancel {5}}{\cancel{5} {0}} = \frac{2}{0} $$
So their cancellation method is not bullet proof. Once it does not work in all cases, they learn to focus better in the underlying logic of the cancellation.
