If you go around saying "At what are you looking?" people will stare at you like you have a disease. Maybe the people you hang out with would react in this ignorant way.
"At what are you looking?" is the same construction as "To whom are you speaking?" and they are both correct.
-"That's an old, useless grammar rule." It is an old grammar rule. It is not useless. -"Hanging prepositions are a very common thing in English, no longer considered bad grammar" That is true, placing the preposition at the end of the sentence is officially allowed and for good reason. Churchill's quote effectively illustrates how strict adherence to this rule can be awkward and confusing. Language should facilitate communication, not hinder it.
The rule was amended, not abolished. Now speakers of English can choose.
Churchill was being facetious. He would have written: “I would not have put up with that”. This has the triple virtue of grammatical correctness, natural sound, and conciseness. If you read Churchill’s writing you will see that splitting infinitives was the exception to the rule. Since DuoLingo permits multiple translations, I recommend accepting the grammatically correct translation as an acceptable answer. It is hard to see the downside.
In a prior page the hover indicated that Na = of, about, for. Women this about/of kids. And in "Díváš se na to?" when you hover over 'na' it says Of, For and Per. So I wrote for this lesson, For what are you looking? but turned it around as What are you looking for? and it said that was incorrect. Is that not a translation of it and if so, why not? Diky.
You cannot simply translate word by word, particularly not concerning prepositions. You should learn complete phrases, because different languages use different prepositions in their expression.
"dívat se na" means "to look at" or "to watch", not "to look "for". The latter uses a completely different verb in Czech, namely "hledat".