extemporize

Extemporize means to say or do something on the spur of the moment, an appropriate meaning given the word's history. "Extemporize" was coined by adding the suffix -ize to Latin ex tempore, meaning "instantaneously" or "on the spur of the moment." "Ex tempore," in turn, was formed by combining "ex" and a form of the noun tempus, meaning "time." Incidentally, "ex tempore" was also borrowed wholesale into English (where it means "extemporaneously"). Other descendents of Latin ex tempore include the now rare "extemporal" and "extemporary" (both synonyms of "extemporaneous"), and, as you have no doubt guessed by now, "extemporaneous" itself.