This continues my effort from two weeks ago where I took an Algebra II word problem and rearranged it. This week let's see what we can do with a Pre-Cal scenario.

Some Algebra II books will dive into this as well as a pre-cursor to 3x3 systems. 3D is strangely difficult. For one, paper is a bad medium for trying to work with it. Two, students have the strangest time visualizing math in three dimensions. Rather than fake an orthogonal grid on a piece of paper, I would probably do something like this.

We turn the room into a 3D coordinate system. If you grab enough string this is easy enough to set up. Last school year, I did this very thing to introduce the 3D coordinate system, however I need to do more. Activities:

- Have students determine the signs of the octants, hanging labels for + + +, - + +, etc in the appropriate spots. Hang additional string these could be attached to.
- Assign groups a set of 3D points, students can either clip the point in the octant where it belongs, or require that they demonstrate a little more accurately where the point is floating. Tick marks along your strings would help this.
- Create a race or a scavenger hunt of some sort. Have them identify objects in the room that are in a specific octant. Assign a point at random and have two teams see who can find it first. Have a bucket with 3D points in it and see which team can correctly identify the most.

Then if you desire, you could demonstrate how 3D points can be represented on paper. In Pre-Calculus, I would use this as an introduction to 3D Vectors. We would do one or more of the activities and then do some note taking on properties and operations. After discussing things like magnitude, addition/subtraction, cross products, you could set this grid system up again and have them model properly scaled arrows that should float in the plane. Or modify it. I had success building miniature versions.

Here a set of vectors was assigned that had to be mounted inside the coordinate system. You could extend this construction activity to include building scale models of two given vectors *a *and *b *representing the direction of their cross product, labeling of the angle in between them, or demonstrating where *a* + *b* points. That small scale construction could precede the making of this larger model.

3D is such a project rich topic that it deserves more than plotting on notebook paper.