It is clear that your daughter's teacher (or textbook) makes a distinction between adverb and adverbial.
Adverb is the traditional term for a class of words—those words which by themselves can 'modify' any syntactic entity besides nouns.
Adverbial designates syntactic entities—either single words or phrases—which perform the same function in a sentence as traditional 'adverbs'.
In this understanding, then:
MORE, BUT NOT DIRECTLY RESPONSIVE:
You (and possibly your daughter, too) should be aware that English pedagogy is currently in the middle of a very painful (and very slow) transition from a traditional grammar centrally based on word classes ('parts of speech') toward a 'modern' grammar based primarily on syntactical functions.
This is further complicated by the facts that a) the 'modern' grammar is still very much a moving target, and b) the 'current state' of grammatical theory trickles down at varying rates and with varying degrees of intelligibility to the teachers of English.
One consequence is that terminology is very much up in the air: what your daughter's teacher calls an 'adverbial' might be called an 'adverb phrase' by another teacher, or an 'adverbial noun phrase'/'adverbial preposition phrase' by a third. More sophisticated grammarians might speak of an 'ad-clausal' and a 'locative complement'—and quarrel enthusiastically about what terminology should be used.
All this is great fun for those of us who are intensely interested in figuring out what's actually going on in these sentences; but it can be hugely confusing to students. For homework purposes, you and your daughter will probably be better off putting your energy into figuring out exactly what her teacher means, and sticking as closely as possible to that. Avoid third-party analyses, unless you get personally fascinated with the nuts-and-bolts; and if you do get fascinated, think twice before roping in the teachers: they may be eager to enlarge their own understanding, but they may also be too overworked and too invested in their curricula to co-operate—and there are always some who will resent any challenge to their authority.