The phrase "behave improperly" implies behavior that goes against accepted social standards or norms. There are several synonyms that can be used to describe this type of behavior. One such synonym is "misbehave," which entails not acting in accordance with the expected rules and regulations. Another alternative word is "act out," which suggests impulsive and disruptive behavior beyond what is considered normal. "Transgress" is another synonym which implies breaking or violating boundary and norms. Other synonyms of "behave improperly" also include "break the rules," "act irresponsibly," "act inappropriately," "deviate from the norm," and "violate ethics." All these synonyms carry similar meaning to "behave improperly" and can be used interchangeably depending on the context.