Generally using "n" in the denominator gives smaller values than the population variance which is what we want to estimate. This especially happens if the small samples are taken. In the language of statistics, we say that the sample variance provides a “biased” estimate of the population variance and needs to be made "unbiased".
If you are looking for an intuitive explanation, you should let your students see the reason for yourselfthemselves by actually taking samples! Watch this, it precisely answers your question.