Translation:Please don't look at the notebook.
The context says absolutely nothing about which you could use note or notebook. You can have both in class. You could look at both. In this case I believe they arbitrarily decided to pick one of the answers.
To help clear up some of the ambiguity I'm seeing here: Duolingo lacks contextual clues, and therefore refers to the most basic and simple meeting of the vocabulary, regardless of what can be said colloquially. To that end メモ is the word I'd expect them to use for a handwritten note (regardless of length), while ノート most directly refers to a notebook.
Because of Japanese linguistic magic, one can refer to the stuff inside a ノート as ノート as well, but it's a more indirect sort of reference. This same sort of indirect referencing happens a lot with colloquial Japanese, and works just fine because of context clues.
i don't think you can use this argument if they had all the other questions use note as the correct translation. it's inconsistent and only serves to frustrate.
Please don't look at the note.
You used the wrong word. "Please don't look at the notes."
I'm sorry, how am I supposed to tell if it's singular or plural?
Why was ノート "notes" in the other question ("You don't need to take notes") but it is "notebook" here?
Say it with me everyone: "because... context!" ノート in Japanese can refer to the notebook itself, the stuff written inside it (notes), or small pieces of paper for writing messages like sticky notes.
In this particular question, I assume the most common interpretation of the situation is that it is an instruction given during an test or the like, and the speaker is asking everyone to do it from memory. In this case, "Please don't look at your notes" is a perfectly valid translation. Presumably, looking at the notebook itself, i.e. its cover, is fine. At the same time, one can imagine an unruly student holding their notebook open during the test for everyone to see. This time, "Please don't look at the notebook" is a more correct translation. Note (pun intended) that in both scenarios, 「ノートを見ないでください」could be used, though arguably 「そのノート」is probably more likely in the latter case.
It's a set grammar structure, the negative request form. The general case is ～しないで which means "don't do ～".
If you wanted to specify that it was a diary, you can say 日記 (nikki).
Otherwise ノート should be translated as "notebook", regardless of what was written inside it.
"Please don't look into your notebook." Is meant here, right. "Please don't look at your/ the notes." Is also possible. "Please don't look at the notebook" doesn't make much sense.