Doing Math- Nets

In class on Wednesday, we created three nets that when put together made a cube. The three shapes you make are a triangular prism, a triangular pyramid, and a square pyramid. We discussed that the triangular prism makes up ½ of the volume of the cube. The square pyramid makes up 1/3 of the cube. Finally, the triangular pyramid makes up 1/6 of the cube.

During class all four members of our group tried to make our own nets. Towards the end of class we realized it would have been smarter for us to all work together because each of us were unsure what each of the nets should look like. I was able to complete the triangular prism and the square pyramid, but did not have time to complete the triangular pyramid.

Geometry is one of my favorite areas of math because I love visualizing different shapes in my head and it is fun translating that to paper. After class, I really wanted to make a new set of these nets so I could complete the cube. Also, I wanted to take a closer look at the measurements on the nets to get a better understanding of how they work and if each part actually is the right percentage of the volume.

First, I am going to make the triangular prism. To do this I visualized slicing a cube diagonally down the middle since is makes up 1/2 of the cube. I decided the dimensions of the cube would be (3cm)(3cm)(3cm). Thus, two sides of my triangular prism would be 3X3 cm. The other two sides would be these 3x3 sides cut in half to make two triangles. To find the hypotenuse of the triangles I used the Pythagorean Theorem as shown in the image below. Finally, I used the hypotenuse of the triangle to find the length of the rectangle. 

    Next I decided to make the square pyramid which makes up 1/3 of the cube. I knew I should start with another 3X3 square which makes up another face of the cube. From there, I realized I again had to cut this 3x3 square in half again to make two more triangles. Finally, I knew there were two more triangles that made up the square pyramid. I knew that one side would be the hypotenuse I found from the previous triangle- 4.2cm. Then, I used the Pythagorean Theorem again to find the hypotenuse for the next two triangles as shown below. 

     The last net I made was for the triangular pyramid which makes up 1/6 of the cube. This was the most challenging net for me to make. It should have been the easiest because I had the other two already made and I just had to fill in the blank, but for some reason I kept messing up when I was drawing it. 

      I noticed that this net contained two of the triangles made from cutting the 3x3 square in half. Then I used the previous nets I made to find the dimensions for the other two triangles. 

       I was very excited when I was able to put the three nets together and found that they did in fact form a cube. 

         Once I had formed all my nets and made the cube, I wanted to do the math to make sure that the triangular prism was infact 1/2 the volume of the cube, and the square pyramid was 1/3 of the cube, and the triangular pyramid was 1/6 of the cube. My math is shown below. 

          It turns out each of them gave the volume they were supposed to give. I was happy I was able to make my own set of these nets and gain a deeper understanding of how the shapes do in fact have the correct volumes. 

         For me, this activity was important because it made me wonder how and if I could incorporate creating geometric models into my classroom one day. I think this activity may be a little harder than sixth or seventh graders could handle, but I think they would learn a lot from creating other simpler nets in class. This would be a fun example to show them and challenge them once they have mastered simpler nets.

        Now, I want to see if I can take another shape, besides a cube, and cut it up into three new shapes. As many times as I have made nets, I have never thought about making nets that create shapes that can be put together.  

Nature of Mathematics- Euclid's Proofs

         Greek mathematicians, for example Euclid, were the first to use the concept of proof. They figured out that in order to show that a statement was true they must use logical arguments. Before Euclid, mathematicians such as the Egyptians and Babylonions “proved” statements from repeated observations and inductive reasoning. However, the Greek mathematicians realized that they needed to have logical explanations to prove that statements were true. These proofs needed to clearly show why the statement was true in a way that would last over time. In order for Euclid to use this new system, he had to create definitions and axioms that would be accepted by everyone. These were written and shared with others in “The Elements”. Some of the definitions in this book include point, line, and surface. “The Elements” also contain Euclid’s five postulates which he assumed to be true. Euclid used these definitions, axioms, and postulates to prove statements. When the statements had been proven, the statement became an accepted theorem. Most of Euclid’s proofs involved geometry, but ever since Euclid, all types of mathematics use his logical argument concept and create accepted definitions to prove statements.


                        

                                                              

            Euclid’s idea of proofs changed mathematics. He found a way to prove statements in a consistent way that is still accepted today. Euclid created the foundation for how mathematicians would solve proofs. Mathematicians start with a statement they would like to prove. In order to do this, they use the definitions and axioms to create a logical argument showing the statement is true. Once they have done this they have found a new theorem. Euclid changed the way mathematicians proved statements from a system of trial and error to forming logical explanations.

            No one has been able to disprove what Euclid found. He was a founder for Euclidean Geometry. However, after him, Non-Euclidean Geometry was discovered when you change one of Euclid’s main axioms or assumptions about parallel lines.

Citations

http://nrich.maths.org/5996

http://www.storyofmathematics.com/greek.html

http://www.biographyonline.net/scientists/euclid.html