**Background and Challenge**

This past year was my first year teaching Pre Calculus. Last summer I spent a good month going through the book to refresh myself with all the formulas and equations. Since I taught Algebra and Geometry the last eight years it had been eight years since I had worked with higher level math concepts. Because it was my first year teaching I had no experience with which topics students would struggle with and which topics they would excel at. One topic that students had a problem with was visualizing graphing points in 3D. Pre Calculus is the first time students are introduced to 3D graphing and the “z” axis. By the time students get to Pre Calculus they are able to graph in 2D with no problems, but once they are introduced to the third coordinate (z) they are unable to really visualize what the outcome of the graph would be. This past year we graphed 3D points on 2D printed out graphs with all 3 axis on it like the picture below.

As you can see it’s pretty difficult to visualize what the graph will look like.

In my Makers lesson students will be creating a 3D graph to use while graphing points (coordinates), or planes in 3D. From my experiences teaching students do better on assignments and understand the material better they haare able to visualize what they are learning. I think creating a hands on model will help them be able to actually see where the point will be when it is graphed.

When creating lessons, educators should have the TPACK (the relationship between technology, pedagogy, and content knowledge) framework in the back of their head. Since students will be creating their own technology (a 3D graph), I have created a Voice Thread for students to watch and listen to while I walk around the classroom checking to make sure the graphs will be put together correctly. Combining these two steps with the knowledge of the content, in this case graphing in 3D, creates what is known as “The Sweet Spot,” or the intersection of technology, pedagogy, and content knowledge according to Matthew Koehoer and Punya Mishrs’s TPACK framework.

**Materials Needed**

Students will receive the following materials:

4 cardboard pieces (cereal boxes work great)

Several sheets of graph paper (enough to cover all of the cardboard pieces, I used 12)

Scissors

3D Scribble pen (if available)

Glue/Tape

**Directions**

Students should cut 4 pieces to look like the pictures below.

One piece, this will represent our z-axis, will have 3 cuts in it. One cut will be for the x-axis at the top to connect with and the other 2 will be made on the sides for the y-axis pieces to connect to. I made the top cut, to connect the x-axis, half way across and half way down the cardboard piece. The left and right cuts were cut half way across the length and the were cut as far down as needed so that the y-axis pieces will have a tight fit.

Two pieces should be cut half way across the length and then cut half way down. These pieces will represent the y-axis.

The last piece, the x-axis, will have only one cut again half way across and down half way.

Students will glue/tape the sheets of graph paper on each side of the cardboard pieces. The whole cardboard face should be covered (graph paper wasn’t available when the pictures were taken).

Then students will connect the z-axis and x-axis pieces so that they form a cross. Like the picture below. The z-axis is vertical and the x-axis is horizontal.

Next we will connect the y-axis cardboard pieces to create cubbies. Like the picture below

Students will be asked to write the signs of each of the coordinates at the top/bottom of each of the octants. Like the picture below.

Students will then be given a 3D Scribbler and asked to graph several coordinate pairs. Students do not need to have a 3D Scribbler to graph. If they do not have a Scribbler they can easily trace the axis (since they have graph paper glued on to the cardboard) with their pencils. The actual 3D graph will help them to move along the axis with success.

For teaching this lesson I was planing on using a Haiku Deck, but because of the limited the number of words per slide, I decided to make a video on voice thread. This way I students can see the pictures as well as hear the directions as I walk around to help.

When I collaborated with my classmates in my master’s program, they gave me the idea to make it a frame so it would be solidified and sturdy. They also gave me the suggestion to label each of the octants with the signs of the coordinates. I really liked the idea of building the frame, but I want students to be able to bring this to and from school and since cereal boxes are light weight and compact I decided to keep with the original idea. I put the idea of labeling the octants to use. It will make things easier for students to put the graph back together. It will also serve as a reminder of which octant the coordinate belongs in.

At first I wasn’t real excited/happy about my idea. I knew students would be able to visualize a 3D plane better but I thought my idea wasn’t “mind blowing”. After discussing my idea with other teachers I realized it was a good idea, not “mind blowing”, but a good idea that will really help students understand the material.

This 3D model will not only help students understand how to graph a 3D point (coordinate) but also how to see planes in 3D.

Click here for a link to my Voice Thread lesson.

Cited images

[3D graph paper image]. Retrieved from URL http://www.shmoop.com/systems-equations-inequalities/three-d-graphs.html