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Everything you are saying sounds very doable. First you have to assign DID to a voice user for it to have a DID number so if there is no DID assigned then there is no issue with DID getting to teachers. You would setup multiple Auto Attendants one for 8am-3pm and on this auto attendant you do not allow dial by extension that keeps anyone from calling extensions. On the second auto attendant you can allow dial by extension that way you could call a teacher after 3pm if desired. Normally the way I do dial by extension is in the Auto-Attendant menu I set the option for the first extension of the extensions as Dial by Extension in the tree and then add the check mark to include leading digit. But for your scenario just set that menu option to transfer to the operator. As an example if all your extension begin with 5 on the day auto attendant you set option 5 to transfer to the operator.
Cris - Assuming you are using either a SIP trunk or PRI trunk in this case for the DID, you can also use a virtual user as the destination for those DIDs. In the coverage for that virtual user, you can use the system mode ( assuming you use system mode for compartmentalizing the time of the day or day of the week) to route the calls to those three SIP phone destinations you want to send the external calls to during all times, except during 8:00am to 3:00pm
This way the internal callers can call those SIP phone destinations direct at any time with no restrictions, and when outsiders call the DID, the DIDs first get route to the virtual users, which will then automatically route calls based on time of day.
thanks guys for the replies i ended up doing little of both suggestions. i really appreciate the replies.