I have to apologise: I do not have favorite math hacks. I have never used hacks/tricks. The essence of mathematics is in universal methods for solving all problems in a particular class of problems. Use of “hacks”/”tricks” is a replacement of mathematics by cheap surrogates. Hacks/tricks can work at a certain level, but frequently obstruct students’ progress at the next level of learning of mathematics. Hacks/tricks are frequently favoured by teachers who always taught students at a certain level but never at higher levels. These teachers tend not to care much about their students’ progress beyond their class.
In my opinion, teachers have to be assessed not by their students’ marks in their classes, but by their students’ success at the next stages of education. In that environment, any desire to teach “hacks” somehow disappears.
I apologise to be so firm in my opinion on that matter, but for at least a decade I taught mathematics at a Foundation Studies programme in a very big university: up to 400 students in a lecture theatre who failed mathematics at secondary school and had to be brought up to the level where they could start studies in (relatively) mathematically intensive degree programmes at the university level. Hacks/tricks were the last thing they needed.