Visualizing 3D Shapes in Algebra : T³ Learns

17Feb 2015 by Steve Phelps

For the past few years I have taught an advanced Algebra 1 course. Some students in this course may have taken Algebra 1 in eighth grade and maybe did not do as well as hoped, while others may have taken Pre-algebra and did extremely well. In any case, these students typically had a good grasp of linear functions; they understood slopes and intercepts and constant rates of change. I was looking for ways to extend these topics and also integrate them with geometry.

It was through the 3D graphing features on the TI-Nspire, using the cross-sections of a cube, that we could integrate geometry and algebra and extend their understanding of key concepts.

If you have never used the 3D graphing features on the TI-Nspire, I have created a short screencast that walks you through creating a 3D graphing page that you can access at http://screencast-o-matic.com/watch/conbYjeCkG. Feel free to use this or any of the screencasts with your students.

Background on Using TI-Nspire

For those new to TI-Nspire I have included some basics to get started on 3D graphing.

The default view is a 10 x 10 x 10 unit cube or box. Anything graphed in the 3D view will be graphed inside this box. Anything outside the range of the box will be “chopped” off. This graphing box is a key element in this activity.

Some keyboard shortcuts:
Pressing R on the keypad allows you to Rotate the view by dragging your finger along the touchpad.
Pressing A on the keypad will rotate the view Automatically. Pressing the ESC (escape) button will stop the view from rotating.
Pressing X will change your view to one looking down the positive X axis. Pressing Y will change your view to one looking down the positive Y axis. Pressing Z will change your view to one looking down the positive Z axis.
You can restore Order by pressing O on the keypad and returning the axes to their original Orientation.
You can access a short screencast illustrating these keyboard shortcuts at http://screencast-o-matic.com/watch/conbYFeCpn

Classroom Exploration on Cross-Sections of a Cube

Now that you have been introduced to the 3D graphing features and have watched the screencasts, let’s discuss the cross-sections of a cube activity. Before I started with my students, I made sure they understood linear equation in the form ax + by = c and how changing the parameters in the equation changed the graph. I also wanted to make sure that kids understood how the graphs were created in 3D. If you have a tile floor in your room, it is not a problem getting students to visualize that the z-value in the function z = x + y represented the height above or below the floor. Treat a corner of your room near the floor as the origin, and the room itself as the first octant. The grout lines between the tiles represent the grid lines in the coordinate plane. If you stand at what would represent the coordinates (2,3) on the floor, there would be a point 5 units above this location.

The first function we graphed together was

z(x,y) = x + y

as shown below.

Press R on the keypad and rotate the axes to a more favorable view.

Students noticed that it appears that a regular hexagon was essentially the cross section of a 10 by 10 by 10 cube with the plane z = x + y.

Some of my Algebra students were unfamiliar with regular hexagons so they were unable to explain why this was one. Once we discussed how a regular hexagon has sides that have the same length and angles that have the same degree measure, my students could establish some facts about the shape. They understood how to compute distances in a coordinate plane and the connection to the Pythagorean theorem. Changing the orientation of the graphing box by looking down, say, the y-axis, we could count along the grid and compute lengths. For angles, on the other hand, I was satisfied with the “eyeball” test (if they look like the angles are the same size, they probably are).

As a class, I asked them how they might change the equation we had entered to move the object down 6 units, all of them knew how to accomplish that. The equation z = x + y – 6 produced the figure below. It looks like an equilateral triangle. Again, by changing the orientation of the graphing box and looking down an axis, we could count along the coordinate edges and see the sides must be the same length, thus confirming the shape was equilateral.

There are nine typical non-trivial cross sections (cross-sections that are not a point or a segment) of a cube in addition to the regular hexagon and equilateral triangle we constructed together as a class.

Many of these shapes will require multiplying the x or y variables by a constant. At first, my students didn’t realize this, so I built upon their previous knowledge of 2D graphing.

When asked how to change the steepness of a 2D line, they understood they would need to multiply one or both of the variables by a non-zero number. We then explored how this also applied to the steepness of a 3D graph. Loosely speaking, multiplying the x-variable by a number would change the steepness along the x-axis, and multiplying the y-variable by a number would change the steepness along the y-axis.

A challenge for you and your students is to try to create as many of these cross-sections as possible, in as many ways as possible.

The eleven cross sections are: