1c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

Remember those classic Scooby-Doo episodes in which the gang always unmasked the bad guy at the end? It was never really a ghost or zombie rattling and moaning through the rich dude's mansion; it always ended up being a regular old Joe in a zombie mask, usually with a serious grudge and some stylish '70s outerwear.

Well, it's time students knew that our old buddy subtraction has a secret, too. It might look and sound like a terrifying poltergeist, but subtraction is secretly just regular old addition in disguise. That's where the additive inverse comes in.

Students have been dealing with negative numbers for a while now, so it shouldn't be too much of a leap for them to see that 3 – 4 = -1 is actually the same problem as 3 + (-4) = -1. Subtracting a positive number is the same thing as adding a negative number! But this rule also applies to any rational number in the universe: 3.268 – 18.9993 is equivalent to 3.268 + (-18.9993).

This handy new trick means we can think of the distance between any two rational numbers as the absolute value of their difference, no matter which number we start with. How far apart are the numbers and on the number line? (We're talking the distance between 'em, not their sum!) Boom: their difference is , so these two guys are units apart. Students should be able to show and understand why this works.

Nobody likes a vague abstraction (they're seriously no fun at parties), so you'll also want to apply this whole distance trick to as many real-life scenarios as you can. For instance, if it's football season, you might ask your students to find the distance between a player standing on the home team's 11-yard line and another player on the opposing team's 30-yard line. Since the field's markings descend on either side of the 50-yard line, we can treat the 50 as 0 on a number line, then snag the absolute value of the difference between the two players.

If we think of the home-team side as the negative one, the first dude is -39 yards from the center, and the other dude is 20 yards to the right, so they're 59 yards away from each other. Can we get some nachos up in here or what?