There are a few things that helped me or someone I know to understand some concepts:
e: Hot/cold stuff passively reaching ambient temperature (why does it get colder quicker right at the beginning but so slowly at the end??) as well as the time constant for capacitors (which you could analogize with how long it would take for water to drain out of a broken dam: quickly at first, slowly as the water level decreases)
integrals: My brother designed a boat that he would build. He wanted to know how much volume it would displace until water would spill inside. The walls were not straight nor right-angled. Using the slicing method and estimates of "this is somewhere between an oval and rectangle, so the area of this slice will be between those shapes", he was able to figure out the ballpark volume.
derivatives, and this one takes a LOT of care to avoid confusion... position, velocity, acceleration. I would start with straight distance examples, but it's also cool to note: On a Merry Go Round, the rider's acceleration would only be toward the center if the rotating speed stays the same. It has to! If it were even a little backwards, you'd slow down! If the Merry Go Round were wider, the acceleration would be not so strong. If the Merry Go Round were so wide that it's impossible to tell you are spinning, there would be nearly no acceleration.
When you drop that ball from a window, gravity is its acceleration. Gravity never changes its direction or gets stronger. That is intuitive. The velocity will keep getting larger, but at the same rate for a while (until real life interferes aka drag). The position, you can plot that and see that every second it is a farther interval. That also makes sense: A ball thrown upward doesn't bounce off the air at the top and come down right away at the same speed. It hangs for a brief moment. But it doesn't hang anywhere else after its peak because the position is the second derivative of acceleration, and that second derivative only has one point where it stays nearly the same for a brief moment.
It might not stick for everyone, but you can explain that 6 dB is always "twice as loud". Thus, 12 dB is "2x2 times as loud" and 18 dB is "2x2x2, or 8 times, as loud". The dB amount is always addition, but the "as loud" part is multiplied.
Similarly, every 10 to 30 years (usually about 20) the value of money drops in half due to inflation. You're adding years but multiplying or dividing value. That lets you know that there's a power/logarithmic relationship.
One other concept is more useless but fun for me... There's product-over-sum. It shows up in "how long does it take N lawnmower pushers of varying efficiency to trim so-and-so acres of lawn?" (If Joe does a lawn in 20 and Jake in 30, they do it in 12 together.) but also in things like parallel resistance. In this case, I tend to think of the electrons as the blades of grass in the lawn and the resistors as lawnmower pushers. The electrons don't care who "trims" them, and whichever resistor becomes available to process an electron (at its own given efficiency e.g. conductance/anti-resistance) will do it.
I am a big fan of the Street Fighting Math series from MIT, but I feel like there were much better resources for explaining the concepts (e.g. videos, lectures) than I can find today. The book is not as approachable as the concepts themselves when explained at an audience-appropriate level. The only downside it its lack of general practicality, but it does move some advanced concepts into the napkin math domain.
In one part, Dr. Mahajan shows that because the derivative of the LN function is very close to the derivative of x at 1 (that is, 1), you can approximate powers close to 1 by subtracting 1 to get them into the log domain and multiplying by their exponent. This probably doesn't make sense, so I'll solve an example from the book...
"If 5% of bacteria are mutated during a radiation round, how many bacteria are unmutated after 140 rounds?"
You can eventually figure it will be (.95)^140 by starting with simpler examples (all heads in two coin flips, three coin flips). Once you have the equation, it means that you can approximate its answer by subtracting 1 and multiplying by 140: -0.05x140 = -7. The answer would be quite near e^-7, which is 0.09%. (Beyond that, if you recognize that e^-5 would be five time constants e.g. just below 1%, then e^-10 is 1% of 1%, so then e^-7.5 would be the halfway point; somewhere below but also close to 0.1%.) The right answer is approx. 0.08%.
An actual game that is engaging and shows one math concept (probability/statistics) in-depth? Dope Wars. The entire thing runs off random chance and expectation. You could probably even simulate the outcome of various strategies over many simulations (the same way that a decent investment recommendation would be a percentage of bonds/cash equal to your age with the remainder in stocks)
Even stuff like Pythagorean Theorem isn't as important to an everyman as its moral. "If you cut through the grass, it's less walking." This also extends to the value of social networks (Metcalfe's Law) or understanding why a small bump in speed limit takes a relatively large increase in gas consumption (Ignoring wind resistance, kinetic energy is proportional to v^2, so going 55 mph takes roughly the same energy as someone going 50 plus someone else going 23.)