About the explanation of "on"

Bonjour Truc, Here is another locution that uses «avec» : Les gens ne devraient pas parler avec leur bouche pleine In fact, here is another translation to the phrase in your question: «On ne doit pas parler la bouche pleine» --> You do not have to talk with your mouth full . As can be seen from the different locutions, they say similar things in a different way, i.e. don't talk with your mouth full. After I reread the lesson, I better understand the syntax: «You can also use 'on' in a more general sense like this: Si on travaille dur, on gagne plus If you work hard you earn more Here "on" includes Men/people in general, or can be a theoretical statement such as"one ". This form is often used WHEN EXPRESSING RULES such as: On ne doit pas parler la bouche pleine. People shouldn't speak with their mouths full.» What I get from this as a takeaway is this: «On» is being used in a general sense in a politeness rule; sometimes in stating a rule, the very valid French grammar is not always adhered to and the lesson is on the use of «on» not about the use of «la bouche pleine» nor about «avec la bouche pleine». If we write this in l'imperatif this is the phrase: «Ne parle pas la bouche pleine» or «Ne parlez pas avec la bouche pleine». J'espère que ma réponse vous aiderait. Bonne chance et bonne continuation dans vos études en français, la langue de Molière et qui a été utilisé par le monde français depuis l’époque d’Hugues Capet Ron (un locuteur non natif )

The French grammar actually is correct. You don't need the word "avec" as in the corresponding English version "with your mouth full". -- Chris (not a native speaker).