No, a Republic is a Republic. Depending on how it is chartered, it may use democratic mechanisms to do certain things. Mostly, our constitution leaves that up to the states. In virtually every regard, our constitution is a framework to prevent mob rule through what most people think of as big-D Democracy. The founders were very insightful that way: crowds make terrible decisions. Checks and balances that prevent democracy from burning witches, burning rights, burning property, burning the republic itself - that's what the constitution is all about. Pretending there's no difference between those two frameworks - philosophically, and practically - is what people do when they want the power returned to the mob, because they think the mob is on their side at the moment.Because a republic is a form of democracy. A representative democracy. As opposed to a direct democracy.
There's FAR more to our version of a republic than the fact that we use representatives in place of massive everyone-votes about everything all the time. The republic reins in the 51% mob, but it also reins in the 500 or so people they send to Washington. And it reins in the executive branch. And it reins in the judiciary. Leaving all of that out of your lecture while telling people they didn't pay attention in civics class is pretty funny.