Personally, the perspective I recommend is kind of the reverse. Despite the linked thread, I find that most of the time, the mathematical term has some reason why it was picked for the mathematical concept in question. It wasn't picked in an arbitrary or malicious fashion. At some point when the practitioner first used it, it seemed like the best English word for the concept in question.
Of course, natural language has ambiguity and multiple meaning to most words: but if you dig into the English dictionary, usually one of the several senses for a word is directly applicable to the math concept. As someone who does a lot of reading/writing as well as math instruction (and usually wants to dig into the history of math concepts before teaching them), the English connotation is clear, and I get a bit frustrated that my students' English isn't strong enough for that to be immediately evident -- as well as that there's no space in normal books for that connection, nor time in class sessions.
So: On reading the OP's question my heart kind of leaped for joy a tiny a bit at having the opportunity to make these connections clear. I already spend time in some cases reflecting explicitly on the English meaning of some more difficult terms as I introduce them in my college courses -- e.g.: predicate, relation, equivalence -- but I often wish wistfully I had time to do that more comprehensively.
Let me pick 3 semi-random examples from the linked thread of "different meanings". For convenience I'm using dictionary.com as my reference.
Even: The primary definition in English is "level; flat; without surface irregularities; smooth". Now, this same term has been used since ancient Greece (written ζυγός αριθμός). The point is that the number is evenly divisible by 2, that is, when halved, there are no "irregularities" in the way of any awkward remainder or fraction in the result.
Series: Primary definition here is "a group or a number of related or similar things, events, etc., arranged or occurring in temporal, spatial, or other order or succession; sequence". Clearly that's appropriate for both a run of TV show and an ordered list of values being summed. (The only nuance being that in English series and sequence are totally synonymous, where in math one does need to remember they're a bit more specific.)
Volume: In this case the relevant sense is a bit further down the list: "a mass or quantity, especially a large quantity, of something: [e.g.] a volume of mail". So this sense of "filling an amount of space" is definitely present in English usage. Consider the English word voluminous, "forming, filling, or writing a large volume or many volumes" which has the same sense of filling-space, but is not (to my knowledge) used in mathematics.
A few other things that I find my students respond quite well to when I take a momentary historical detour: (1) the fact that the radical symbol comes from an over-time transmogrified "r", being an abbreviation for radix/root, (2) the fact that our summation symbol $\Sigma$ (Sigma) is the Greek equivalent to our capital "S", the obvious choice to abbreviate "summation" (and you can easily draw out the graphical evolution from one to the other; mostly, drop the lower bar), (3) likewise for capital $\Pi$ (Pi), which is the analog to our capital "P", the obvious choice to abbreviate "product".
So in short, you should look at this class as a huge opportunity -- not to highlight the differences, but to underscore the similarities of the meanings between the English and mathematical terms, that would in many cases by cryptic to people with weak English vocabulary. The more research and connections you can make like that, the stronger it should be in your students' minds, and you get to serve two goals at once with this project. In some sense I'm even a tiny bit envious of your opportunity. Good luck!
