Common Core Standards: Math
2b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
It's true that a house divided cannot stand. But how about a rational number? We can divide that thing till the cows come home.
Even if it's messy, students should understand that they can divide any two integers in the world and the result will always be rational (as long as they're not dividing by zero). That's, uh, the definition of a rational number. But as usual, they'll need to keep a careful eye on their signs.
Students should understand that dividing two negatives works exactly the same way as it did with multiplication—the quotient will be positive: . A pair of negative signs is like fire and ice; they'll always cancel each other out.
But if there's only one negative sign, it doesn't matter where we stick it, as long as it's there somewhere. Students should understand that having a negative sign in either the numerator or denominator will make the whole fraction negative: and are both the same thing as . It can bop around anywhere in the fraction, and the quotient will still be the same.
We hate to sound like a broken record here, but the best way to get these concepts across is with a boatload of real-life examples. For instance, if we've got 3700 broken records and 16 boats to store them in, that's 3700 ÷ 16 = 231.25 records per boat. Which is fine, since they're already broken anyway. (And, of course, don't forget to throw some negative numbers up in there, too!)