During our study of vectors, towards the end we stumble into 3D space. It's hard to teach for a few reasons. Paper is bad for 3D no matter how much you emphasize "into the paper" and "out of the paper." My solution a year ago was to build a 3D coordinate system in the room.

If you want to make one yourself, it's really easy. I had some yarn lying around (I have **everything** lying around at this point) and strung it up with push pins. Electrical tape helps alleviate some of the stress on the push pin. I looped the axes around each other at 0, 0, 0 so that the whole thing interlocked.

It worked well as a visual but last year that was it's only function. Over the summer I revisited the activity in a Textbook Makeover and thought about what could be done differently. The end result was a pretty simple addition but was useful.

In summary:

Kids walk in, see the yarn, and give me "what the heck?" looks. One set of students remarked "yeah, but it's Claydon, this is normal." That was cool. Anyway, we discuss the 3D coordinate system, the z-axis and how the positive and negative areas are oriented. We revisit the quadrants from 2D and I ask "how many divisions do you see?" Guesses range from 6 - 16. Most frequent guess is 8 which is a good sign. We locate quadrant I - IV in 3D space and introduce the newcomers V - VIII.

In three classes the discussion diverged at this point to talk about Roman numerals. Apparently teaching them has fallen out of practice. Fun moment of "ohhhhhhh" when a student realized the connection between the Roman numeral C and the Spanish word "cien." A few juniors were relieved that the "XV" on their class shirt was correct when I brought up this fail.

Back to the coordinate system. We locate the octants. We discuss why the word "octant" is appropriate for describing them. Then we discuss the sign conventions.

Finally, the task at hand. Given a sheet of labels with 3D points (in coordinate and i, j, k vector notation), locate the appropriate octant for the point. It'd be a little hard to locate the points exactly, so I settled for finding the appropriate region. White paper was on the walls labeled I - VIII for attaching the labels.

Discussion took 20 minutes or so. Wandering around to locate the points took about 10 minutes. I thought giving them a chance to wander 3D space was useful for the conceptual aspect of 3D space. Many strategies were at work. Several sat in the seat for a minute and located the coordinates based on sign convention, a few chose to navigate on the fly using the +/- x, y, z signs I posted.

Later when demonstrating the difference between a 2D and 3D vector, many were able to identify that a 3D vector traces a sphere. It made me wish I had a beach ball to add to the demo.

The next step is having them construct their own models of 3D space.