Last year when one of our units worked on this part of trigonometry, I constantly had to look at the paper for the ratios. I didn’t quite understand how these formulas fit into what we were doing, and I was confused. This year, when we applied Soh Cah Toa into our bee problem, I understood why these things were important. First of all, when finding the side lengths and angles of a shape, such as a hexagon, I realized that cutting it up into smaller sections and using trigonometry was actually very useful. With this, I could figure out more complex things about shapes without having to guess, and it helped with our “bee dilemma” because we could compare and contrast different shapes more in depthly.
2. Visualization One thing that this semester has taught me is how to visualize math problems. This especially came in handy when we worked with lateral surface area, volume, and area. In order to correctly find the equations and formulas to get the shapes, I first had to count out the surface area and the amount of ‘squares’ that fit into the shapes. By making a table of these I was then able to find the patterns and make a formula that was applicable to all of the shapes. I couldn’t have done this without first understanding the basic way of finding volume and surface area, by visualizing the shapes in my head and pushing my brain to think of the shapes in a different way.
3. Strength of shapes An activity we did recently was understanding the strengths of different shapes. We connected marshmallows and toothpicks together to understand which angles were the strongest. Ultimately we learned that in all of the angles, the joint was the weakest point, and so the 180 degree angle was the strongest. After that we applied this to shapes. Again, the fault was the place where the two toothpicks connected, because they could be bent out of shape but the side lengths stayed the same. What we figured out for this was that triangles were the strongest shape (holding the most weight) because there was no room for deformation of the shape.
4. Pythagorean Theorem During this semester, another thing that I took away was understanding the pythagorean theorem more. I already knew how the formula worked, but I never understood why. When we started learning about the rug patterns, I had no idea how that related to anything. By the end of that short unit though, I realized that the pythagorean theorem has to do with squares, and that the area of the squares connected to the side lengths equal each other because of the square. This was kind of an ‘aha’ moment for me, especially when comparing right triangles to the other triangles and seeing that they were the only ones that worked for the theorem.
5. Corrals The last thing that we worked on this semester that really applies to the final bee problem is the corral workpages. In the book, there was a farmer who had a certain amount of fencing, but wanted to make different shaped corrals, rather than just rectangles. By examining this, we realized that, by finding the area and the side lengths using the pythagorean theorem, that triangles were less space efficient than hexagons or rectangles. By learning this, and applying this to the bee problem, I understood that hexagons are the best shape for bees because they tessellate, they are space efficient, and although they are not the strongest shape, putting several hexagons together strengthens the overall hive.