6b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

Mathematical tales of old tell that on a rainy, French morning, a bored René Descartes watched a fly crawl across his ceiling and thought up the coordinate plane in order to describe the fly's exact location. So when students complain about having to graph points with negative coordinates, tell them to blame the hairy pest that caused this whole mess. And that fly, too.

This isn't the first time students have come across ordered pairs, but it is the first time they're dealing with negative coordinates. The first thing they need to know? Quadrants are labeled as I through IV going counterclockwise.

Once they know the different quadrants, they'll be able to figure out how the signs on the <em>x</em>- and <em>y</em>-coordinates determine which quadrant the point falls in. Given a particular point, they should be able to tell you which quadrant it's in based on the positive and negative signs on the coordinates alone. If they have to whip out Google Maps or stop and ask for directions, go back to square one—or, more appropriately, quadrant I.

After mastering the quadrants and positivity/negativity of coordinates, students might start to notice a few reflections. (Unless they are vampires, it's bound to happen.) For instance, students might notice that (4, 2) and (-4, 2) look like the same point, just on opposite sides of the <em>y</em>-axis. And indeed, they're absolutely right.

Using paper folding, mirrors, or other visual aids (like…uh…graphs), help students identify the connection between signs on coordinates and their reflections across the <em>x</em>- and <em>y</em>-axes. Namely, that if we reflect a point across the <em>y</em>-axis, the <em>x</em>-coordinate of the point changes sign, and if we reflect a point across the <em>x</em>-axis, the <em>y</em>-coordinate of the point changes sign.