Lindsey, although it sounds more natural and maybe is the current usage, the rule states that in either... or... / neither... nor... clauses, the verb matches with the nearest subject because the first verb is omitted and implied in the sentence (the second takes the place of the first):
Either the kids or Rodrigo is...
Either Rodrigo or the kids are...
Either Rodrigo or I am...
The first verb is omitted (that �s why we use this kind of construction: to avoid repetition)
Either the kids (are - omitted) or Rodrigo is...
Either Rodrigo (is - omitted) or the kids are...
Either Rodrigo (is - omitted) or I am...
As I previously said, I understand the problem because some constructions I have to teach in Portuguese as a teacher, I don �t use because they sound strange to me as for the listener.