"Negative times negative is positive" — everyone memorizes this rule, but ask them to explain why, and a lot of people hesitate. In fact, you don't need to memorize this rule at all — it falls right out once you follow multiplication's own pattern all the way through.
Start from 3×3=9. Decrease the multiplier one at a time: 3×2=6, 3×1=3, 3×0=0 — the result drops by a steady 3 each time. There's no reason for this pattern to just stop, so continuing it naturally means 3×(−1)=−3, 3×(−2)=−6. The moment the multiplier turns negative, the result turns negative too. That's exactly why "positive × negative = negative."
So how do we explain negative times negative? This time, look at the pattern for (−3)×B. (−3)×3=−9, (−3)×2=−6, (−3)×1=−3, (−3)×0=0 — the result climbs by a steady 3 each time (−9 to −6, −6 to −3, −3 to 0). Carry that same pattern forward and (−3)×(−1)=3, (−3)×(−2)=6 must follow, to keep the pattern connected smoothly. In other words, multiplying a negative by a negative to get a positive isn't some rule forced onto the math from outside — it's simply the natural extension of multiplication's own built-in regularity.
Division's sign rule can be explained even more simply — because division is the reverse of multiplication. "12 ÷ (−4) = C" means exactly the same thing as "(−4) × C = 12." What number can go in place of C so that multiplying it by (−4) gives the positive 12? From the multiplication sign rule, "negative × negative = positive," so C must be negative (and indeed, if C=−3, then (−4)×(−3)=12 checks out). This way, you can always verify a division's sign yourself just by recalling the related multiplication.
There's also a quick trick for figuring out the sign. Count the number of negative numbers among everything being multiplied or divided. If there's an even number of negatives (0, 2, 4…), the result is positive; if there's an odd number (1, 3…), the result is negative. For example, (−2)×(−3)×(−5) has 3 negatives (odd), so the result is negative, while (−2)×(−3)×5 has 2 negatives (even), so the result is positive. This trick is especially handy for complicated expressions where several numbers get multiplied together at once.
On our activity page, move two numbers with sliders yourself, and watch in real time how the result changes as you fix one number and decrease the other by 1 at a time. Switch between the multiplication and division tabs and move the sliders around a few times, and you'll see with your own eyes that this isn't "a rule you have to memorize" — it's simply "what naturally falls out when you follow the pattern all the way through."