Intermath Workshop Support s1

InterMath

Title Nets of a Cube

Problem Statement The net of a solid represents all of the connected faces of the solid. When you fold the net at its creases, you should be able to recreate the solid. For instance, in the figures below, the net shown to the left will create the cube to the right if you fold the net at each of the lines.

There are a variety of nets that can make a cube. Devise as many possible nets for a cube as you can.

Problem Setup I am trying to determine and construct as many nets as possible that will make a cube.

Plans to Solve/Investigate the Problem I predict that there will be more ways than one to make a cube. I am not exactly sure how many nets can be created, but I do believe there is more than just the one shown above. I also believe that just by simple, rotations you can create multiple nets that look different but yet yield the same cube. I plan to solve this problem by using a paper model first where I will draw out different nets on paper and cut them out to actually create a cube. I will then think of possible ways that each net could be unfolded in a different way to create another net. I will then draw these nets in GSP using squares as faces for my cube.

Investigation/Exploration of the Problem In order to discuss how I went about solving this problem I will do it in a step by step process: 1. Since I cannot show my hand drawings, I will simply begin by recreating my nets in GSP. I know that a cube is a polyhedron where the cube is 3 dimensional and the faces are made up of polygons which in a cube are squares. So, to begin with I will construct a square for the first faces of my cube. a. For the square, I had to make sure all my angles were 90 degrees and that all my sides were equal in length. b. I will begin my square my drawing a segment.

c. Next, to get a 90 degree angle, I need to rotate this segment 90 degrees.

m AB C = 90.00

C d. Then, I constructed a perpendicular line through point A and point C and put a point where these lines intersected. I then created a segment from point A to point D and point C to point D.

m BCD = 90.00 m AB = 1.75 cm A B m CDA = 90.00 m BC = 1.75 cm m DA B = 90.00 m DC = 1.75 cm m AB C = 90.00 m AD = 1.75 cm D C I now have a square where each angle is 90 degrees and each side of the square is equal. 2. Now, since I had my square the next task for me was to begin creating the nets for the cube. I know that a cube has 6 faces, so therefore I will have to have a total number of 6 squares for each net I create. These faces will need to be the same size. So therefore, I will need to use the above square that I have created for each face of my cube when creating my nets.

This show one reflection across a segment. 3. To get the faces of the cube or net, the same size and shape, I will simply reflect the square I have created in GSP across a segment. I will continue this process until I have all 6 faces created. 4. Now, that I know how to create my faces, the next task is to play with the different designs of the nets to see which ones actually work and creates a cube. I explored this using my paper model. After playing around, I discovered that if you lined up 5 or 6 faces in a row the net would not work and a cube would not be formed. The reason is that one or two or the faces would overlap and the cube would be left without a top or bottom or perhaps both. However, I constructed many nets that had 2, 3, and 4 faces aligned in a row and these nets seemed to work fine and create a cube perfectly. 5. Now, I will create the nets I was able to come up with. Some of the nets are rotations and are just drawn out differently.

Extensions of the Problem Devise the nets for the following solids:

right square pyramid right cylinder A right square pyramid is a pyramid with a square base. This means that the bottom of the pyramid will be a square and the sides of the pyramid will be equilateral triangles. This means that the bottom face of the pyramid will be a square. The other four faces of the pyramid will be triangles. The following illustrations show the nets for the right square pyramid. I was able to find 6 possible nets for the pyramid. The following illustration is the net for a right cylinder.

Author & Contact Carla McNeely, Middle Grades Education student, concentrating in English/Language Arts and Math. I am currently a junior at Georgia College and State University. [email protected]

Link(s) to resources, references, lesson plans, and/or other materials http://intermath.coe.uga.edu/dictnary