What was hard and what was easy
The thing that was easiest this year was definatly getting to learn different things and understand the concepts of different math equations. one of which I kinda enjoyed was the logarithmic equations. The main reason being is that I was able to understand them quickly and efficiently. Now, the hard part of this year was definatly getting my assignments done and turned in. I'm not sure if Cathy knew the reason being but I was having a hard time being at home and I was always worried or upset and confused. I definatly do plan to do much better work next year and I feel like I have a grasp on how I can get my work done. Enough with the ups and downs, just allow me to tell you the one thing i was proud of this year that i did in my composition book. I felt I had an ecstatic time doing the chapter 4 because it's the only assignment i did and was able to understand without assistance from my teacher to get it done. Overall, this year has definatly been a roller coaster for myself and for my grades.
Platonic solids
Alicia Armstrong + CJ Harrison
Personal Connection: Platonic solids
I really liked this project because it was fun to learn about how platonic solids relate to the different elements of the world. I also liked how fun it was to make the platonic solids mainly because we didn't have to take a long time to get the materials, we just had to get pizza boxes and it was a simple cutting out the shapes. In my group I think we really work well together because were both neat freaks so it kinda evened out in the end so it was nice to get things finished neatly. One thing I learned was that platonic solids have to be less that 360°. I could go on about different things but I think overall we like math a lot because there are so many different things you can do to make it fun but learn at the same time.
Math connection: Platonic solids
Ancient Greeks studied platonic solids and came to the conclusion that only five platonic solids can be made. These are the tetrahedron,cube or regular hexahedron, octahedron, dodecahedron, and the icosahedron. Geometric proofThe following geometric argument is very similar to the one given by Euclid in the Elements:
1. Each vertex of the solid must coincide with one vertex each of at least three faces.
2. At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°.
3. The angles at all verticals of all faces of a Platonic solid are identical, so each vertex of each face must contribute less than 360°/3 = 120°.
4. Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for:
* Triangular faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
* Square faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
* Pentagonal faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron.
