I have no problem with "approximate" as a verb. It's not exactly an everyday usage; most people would just use "guess", but it's perfectly valid. I've never known of any distinction between "approximate" and "estimate".

I approximated the number of requests I received. I use this word as a verb on a regular basis. I calculated the answer but it conformed within statistical accuracy.

You were thinking of "estimate" in your sentence. When you use "approximate" as a verb, you mean something is close in quality or type to something else, or you make something as close as possible to something else. Simulate is close in meaning. I moved the loop of string to approximate a hexagon. The new teacher tried her best to approximate the previous teacher's manner. The motions of the stars can be approximated in a planetarium. (TheFreeDictionary) Something approximating a just outcome will be ensured. (TheFreeDictionary)

In my experience, in everyday conversation, we do not often use 'approximate' as a verb. I would say, "It was hard to guess the number of the trees". OR "It was hard to guess the approximate number of trees". However, like your teacher said, you can use the word 'approximate' as a verb. I think in professions (medical, law, technical jobs) "approximate" is used as a verb more often.