交(jiāo) - to hand over, to submit, to make (friends) in Mandarin

交 (jiāo) is the verb for passing something into someone else's keeping. You 交 homework to a teacher, 交 the rent, 交 a report, 交 a form at a counter. The feel is official rather than casual: something changes hands because it is due.

The second big use is social. 交朋友 is to make friends, and 交 on its own carries that sense too, as in 我交了很多朋友. The link back to the crossing sense is that two lives cross.

The trap for English speakers is reaching for 给. 给 means give to a person and needs a recipient standing there; 交 means submit, and the recipient is often an institution or simply understood. 把作业交了 is fine with nobody named, while 把作业给了 leaves the listener waiting for the rest of the sentence. Note also that 交 is jiāo but 教 (to teach) is jiāo or jiào, and the two are different characters.