Common Core Standards: Math
2d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Like pop stars and fashion models, rational numbers love to wear different outfits. The trick is recognizing them when they've slipped out of their winter fractions into their more comfy summer decimal-wear.
Any fraction with an integer up top and another integer down below (with the exception of 0!) is rational, and we can turn any of 'em into decimals the old-fashioned way: long division. This shouldn't be big news to your students—they've known the fraction bar is interchangeable with the division sign for a while now. The new thing here is that we can use division to switch between rational fractions and rational decimals.
Make sure your students understand that the decimal version of a rational number might not be pretty; it might even be a decimal that's infinitely repeating (like reruns of Seinfeld). The fraction looks pretty chill in fraction form, but dividing it gives us 2 ÷ 3 = 0.66666666… (a.k.a. 0.6), with an endless parade of 6's marching off into infinity. Students should be able to swap between the two forms deftly, like a translator who's awesome at English and Korean.
They should also know that every rational decimal will either stop eventually or repeat itself infinitely. (In fact, they should know that even decimals that stop eventually actually have an infinite number of 0s at the end.) The one thing a rational decimal can't do is have an infinite number of non-repeating digits after the decimal point. That's a weird honor reserved for irrational numbers only, so make sure you emphasize the difference between "non-repeating, non-terminating" and "infinitely repeating." They're different kinds of infinity, ya know?