Common Core Standards: Math
2c. Apply properties of operations as strategies to multiply and divide rational numbers.
We're in the big leagues now. Rational numbers aren't always pretty, but multiplying and dividing a bunch of rational numbers in a row means your students have a choice in how they approach the problem. And the key to knocking these things out as efficiently as possible is having a solid grasp of the different properties of multiplication and division.
Memorizing a bunch of rules might sound impossibly boring to your students, so show them how much time they can save in the long run by busting out the commutative, associative, and distributive properties for different problems. Multiplying and dividing rational numbers can get messy (especially if there's a mix of fractions and decimals), but the right property might cut out three or four different steps, if they play their cards right. It's all about sizin' up the problem and strategizin' about how to tackle it best.
Their strategy will depend on the problem. If they're multiplying , for instance, it makes way more sense to rearrange everything using the commutative property first, since we can simplify those two fractions into a nice, even 4, like so: . That's waaaay easier than multiplying a fraction by a decimal twice in a row.
And if they give you any lip about it, just remind them that doing so means less time math-ing and more time playing video games, or reading Divergent, or doing whatever it is they'd rather be doing.