The bath-trap split (/ˈbæθ ˈtræp ˈsplɪt/) is a phonological split that occurs in some accents of English, but particularly Received Pronunciation, Cockney and Estuary English, whereby the vowel in the bath lexical set is elongated to /ɑ:/ though those in the trap set stay as /æ/. Accents from the Midlands and Northern England do not have this split, including many who speak Midland or Northern versions of RP.
The boundaries of the bath and trap lexical sets are somewhat fluid, though the historic tendency has been for words to transfer from the trap set to the bath set.
This results in irregular spelling where the pronunciation does not match the spelling.
Students' knowledge of this split may be helpful for listening. However, considering speaking, if a student wants to adopt a model based on Received Pronunciation, Cockney or Estuary English, then this may be important, but otherwise it doesn't matter (probably). There is an argument that it is better to teach pronunciation that matches spelling where possible, and also to teach the pronunciation closest to the majority of English speakers, and both of these arguments go against concentrating too much on teaching this split.