This is a means of demonstrating how a plane appears when it intersects the three axes in coordinate space. This model can be easily constructed, then folded flat for storage in a notebook.

1. Copy the grid (or one like it) onto card stock. You will need another sheet (preferrably of a different color) to model the intersecting plane.

2. Cut the grid along the dotted line.

3. Fold the card inward along the axes.
 

4. Fold the card so that the two copies of the x-axis labels match up. The model should look like one corner of a box.

5. Calculate the x, y, and z, intercepts of a simple 3 variable equation. For example, these work well:
 

	x-intercept:	y-intercept:	z-intercept:
a. x + y + z = 2	(2, 0, 0)	(0, 2, 0)	(0, 0, 2)
b. x + 1.5y + 2z = 3 (pictured)	(3, 0, 0)	(0, 2, 0)	(0, 0, 1.5)
c. 2x + 1.5y + z = 4	(2, 0, 0)	(0, 2 2/3, 0)	(0, 0, 4)

 
 

6. Graph the three intercepts. Draw the lines connecting the intercepts. You may find it easier to do this while the model is flat. You should see a triangular outline of the line segments indicating the intersections of the plane of the equation you are graphing with the three planes of the coordinate axes.

7. Now, carefully cut along the line segments connecting the x & y intercepts (x y plane) and the y & z intercepts (y z plane). DON'T cut the line segment between the x and z axes, since that will cause the model to lose structural integrity.
 

8. On the second piece of card stock, cut a long slit about one-third of the way down the sheet.

9. "Hang" the new sheet of card stock onto the grid from the back by forcing the slit over the cuts you made between the intersections. You should be able to see the intersection of your equation as it passes through the first octant now, as well as some of the places it extends into the surrounding octants.

by Dean A. Medek and Caran Risciniti, January 8, 1997. Updated 3/29/99.