1b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

When you're lost deep in the woods and night is coming on, a compass is probably the most handy tool you can have (with the possible exceptions of an entire portable McDonald's drive-thru). We can't help you with wilderness training, but we can point out that the ± sign on a rational number is basically a math-compass—it tells us which direction our number is heading on the number line. We'd hate for little -3 to get lost.

Students should realize that the points -5.5 and 5.5 are both the same distance away from 0, but in opposite directions. That's why they cancel each other out—if you walk 5.5 miles up the road and then turn around to walk 5.5 miles back the way you came, you'll be right back where you started (hopefully at a Starbucks—we're gonna need some refreshment after all that walking). The two distances "cancel" each other, putting you back at your starting point like you never left, latte in hand. In any case, it's a great excuse to get more caffeine into your system.

Obviously students will already know that 5.5 – 5.5 = 0, but this part of the standard requires students to show why this is true, visually and geometrically. Since -5.5 is negative, it's 5.5 units to the left of 0 because |-5.5| = 5.5. Similarly, the numbers 8.5 and -2.5 are both 5.5 units away from 3.

Students should already know that any rational number is made up of two parts: the number itself, which tells us how many units we're traveling along the number line, and the plus/minus sign, which tells us which direction we're headed. Only now, rather than just knowing about it, students have to show that it's true.

This standard also introduces the concept of the additive inverse, which sounds like some kind of robot wrestling move but is actually just the math version of what we saw in the previous substandard. A number's additive inverse is whatever number we need to add to get back to 0. Again, the idea itself is pretty basic, but emphasize to your students that we can prove this geometrically. We know that -6.453 + 6.453 = 0 because we're starting 6.453 units to the left of 0, then moving 6.435 units back to the right.

As usual, putting these concepts into real-life contexts is always super helpful. If a student got a crisp $20 bill for their birthday but they still owed a friend who spotted them $2.50 last week, they should be able to track this on a number line to see that they'd have $17.50 left over after paying their friend (20 units to the right, then back 2.5 units to the left). Enjoy that birthday cash while it lasts.