Have you got your network set as work or private or public?
That IP seems like a router Ip address.
If you are worried run the scans in this doc
and if you are still worried call support and ask them. Sounds like the school network names gives iPs in the 10.*.*.* range
Welcome to the forums.
I'm not sure of your network (at home) configuration, so take this with that in mind.
Commonly, newer consumer grade routers and those routers designed with small businesses in mind, will create 2 separate networks.
One network is for private internal use, considered Home or Work, usually on the 192.168.1.x/255.255.255.0 segment. This is the traditional network most routers start with.
Newer routers may also include a Guest network, designed to give your visiting friend access to the Internet, but block access to your locally connected systems on the Home or Work segment. Often, the Guest network will exist on the 10.x.x.x/255.0.0.0 segment. Many times, the Guest segment have easy or no passwords assigned, whereas the Home or Work segment often use WPA or better security (for the wireless) set up to help prevent network breaches.
This sounds like what might be happening on you home router. For whatever reason, you failed to connect via the Work or Home segment (192.168.1.x) and instead connected to the Guest network on 10.x.x.x network.
Of course, I can't be sure of this, since I do not know the configuration and capabilities of your router.
Hopefully this helps.