[Maine-birds] Coopers/Sharpie questions

I just started working at a school in Bangor, in a classroom that has windows looking out on the front grass and a bush with bird feeders and suet near or in it.  The teacher told me that a hawk has come and snatched the occasional bird, which of course upsets the kids, but interests me. This morning I was distracted by a small flock of house finches, when a hawk flew in and landed in the bush.  I don't know if it caught anything, and it almost immediately flew away.  The squirrel which had been under the bird feeder immediately climbed a low branch in the bush and flattened itself against the branch. 

So I have two questions.  No doubt this was either a Sharp-shinned or a Cooper's hawk.  In the few seconds I had (while I was supposed to be answering a student's question about his poetry), I observed a fairly long, narrow tail with bands on it, and a gray-ish back.  That's it.  In size, it seemed significantly larger than, say, a robin or other largish songbird.  My field guides say the Sharpie is a 'tiny' hawk; this one did not look like a tiny anything to me.  It wasn't quite as large as a crow, however.

When I see these birds, and I have, several times, in Bangor but always while I'm driving or otherwise engaged, I see them for only a few seconds and never as yet when I have access to my binoculars.  Can someone give me some pointers on attempting to identify them?

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