Can the school see what I do on my chrome book at home while it is connected to my home wifi?

The Schools can see what I do on my Chromebook at home as usually they have a pre-installed software to track laptops and to keep them in sync with the school's network. With that software, they could potentially log your browsing history remotely (even if you are on a different network), but that is not likely. The most likely way that they could easily track your browsing history from that computer is if you were on the school's network. The school's network most likely has an advanced system so it can operate a firewall that can block certain websites the school does not find appropriate. That being said, they most likely the school network would have log internet traffic of students computers on that network. Besides off that network, the only way they would be able to check your history is if: They have software that DOES remotely send the students browsing history (as mentioned earlier, unlikely) They have physical access to your computer. (Keep in mind they will not need credentials if your computer is not encrypted).

Bottom line is, that I would not condone searching anything on a school laptop that you would not want your school to see. However, most likely they will not go poking around as long as they do not have a reason (like using to much bandwidth on the school network). Update: Also having seen you asked whether your school "can see your Facebook", the answer to that is no if I am interpreting your question right. Your school cannot (by using their network) read Facebook chat messages or really anything you do on Facebook because it is encrypted using SSL