Joe is the student with the highest grade
With the highest grade is an adjective phrase modifying student, with "with" being a preposition and "highest" being the adjective.
But in the following sentence:
He played baseball with great care
With great care is the adjective phrase, but I cannot figure out the head adjective: a) is it great or care? b) great could be a degree adverb describing the intensity of the adjective care. One could say care was noun, but if I changed it to:
He played baseball with care
a) Care isn't a noun is it? The phrase "with care" is describing how the noun "baseball" is being played with.
Or if I were to make other sentences such as:
He played baseball with the greatest care
In the above, there is less ambiguity (greatest) is the superlative form - so this must be the head adjective while care could be construed as noun because it has a determiner, the.
Of course, if I removed greatest:
He played baseball with the care...
Then "with the care" wouldn't make sense unless I added more things in it, such as:
He played baseball with the care one would pay his grandmother