The formal "you"
In the notes for this it states, "The person who is addressed with a "Sie" has to offer you a "du" before you can use it." What does this mean? That if the person addresses you with "du" you can address them with "du"?
This is not true. use Sie unless if they are family members or close friends. (All family, and if you are not in a business situation use du. If you just meet someone your age while hanging out with friends use du. If you are introduced to a teacher, police officer, etc. use Sie. As a sign of respect, refer to the elderly as Sie unless they are in your family. Oh, and they will use du even if you use Sie often. It's just the way of talking down, but respectfully. DuoLingo isn't exactly culturally accurate with German.
I think what they mean is that when you're on "Sie" terms with somebody, you can't just switch to the "du" without talking about it first. One of the two people involved has to offer the "du" to the other person, e.g. by saying something along the lines of "Why don't we call each other 'du' from now on? My name is Marc." It's supposed to be the older or higher-ranking person (e.g. your boss) who offers the "du" to the younger or lower-ranking person. There is also a ritual that traditionally marks the switch from "Sie" to "du": "brotherhood drinking" (Brüderschaft trinken), which involves drinking alcohol and sometimes also kissing each other. But this ritual is not much used anymore, I think. By the way, there's a cute scene in a 19th-century operetta called "Die Fledermaus" that shows a parody of the "Brotherhood drinking" ritual. During a wild party, the drunk guests suddenly decide to offer each other the "du", a decision which they will later come to regret: "Brüderlein und Schwesterlein, lasst das traute 'Du' uns schenken ... Erst ein Kuss, dann ein 'Du'. Du, du, du, immerzu." (Little brothers and little sisters, let's offer each other the intimate "du" ... First a kiss, then a "du". "Du", "du","du", forever. ) http://www.youtube.com/watch?v=_Swe0YFDnF4feature=player_embedded
See, this is something that has always baffled me. Over the years, the native German speakers that I know have all laughed when I broke out "Sie" on them. Consensus among them was that if someone is similar in age, perhaps social status, or whatnot, "du" is fine. "Sie" was considered over-the-top unless it was a formal situation/relationship, like in business, talking with an elder, etc. I explained to them that the way I have been taught through various different means was that if you dared used "du" with anyone, no matter how casual the setting, without some prior permission, you were darn near committing a crime. I guess it's better to start out with "Sie" and risk getting laughed at. shrug