Measuring Your World Project
Unit Description
This unit that we'e had in math class for about a month now has covered various things relating to measurement. We began by learning (or being refreshed) about the Pythagorean Theorem - which states that a²+b²=c². We then learned how to measure simple distances using triangles and other shapes, then moved onto the distance formula - which helps us figure out the difference between two points. Using this formula allowed us to explore what shapes could look like if they had the same distance to any center point. In trying this, we created circles. Circles will always have the same radial distance to each of the points created, otherwise it wouldn't be a circle. We learned about points on any given axis and how they are similar, we learned how to do intersections using radial lines, and saw what angles were formed. We covered Trigonometry, learned what the Sine and Cosine of a shape are, and how they correlate - A Sine is the ratio between the hypotenuse of the triangle and the opposite sides. The Cosine's definition is the ratio between the adjacent sides of the triangle and the hypotenuse side. We learned how to apply these terms to the Pythagorean Theorem and the Distance Formula, which allowed us to prove tangents of any angle we desired. We began with simple degrees - 30,45, and 60 degree angles. We learned about radians as well, and in the final trigonometry portion, we were assigned the Mount Everest problem - How the British were able to figure out the height and position of Mount Everest without being allowed there by using a theodolite.
After doing our unit on Trigonometry, our class took a closer look at polygonal equations - which are equations used for various polygons. By using several habits taught to us in class, the 'Habits of a Mathematician', such as 'Starting Small', we were able to understand how trigonometry plays into all of which we were learning. We learned for example that a square can be split into triangles, which leads me to my project which i'll describe later. We also went into more exploration of using trigonometry to find areas of different triangles, and soon had the ability to make formulas for pentagons, hexagons, and so on for any -gon shape with any value. We also briefly explored volume in the third dimension and how to calculate areas within shapes.
After doing our unit on Trigonometry, our class took a closer look at polygonal equations - which are equations used for various polygons. By using several habits taught to us in class, the 'Habits of a Mathematician', such as 'Starting Small', we were able to understand how trigonometry plays into all of which we were learning. We learned for example that a square can be split into triangles, which leads me to my project which i'll describe later. We also went into more exploration of using trigonometry to find areas of different triangles, and soon had the ability to make formulas for pentagons, hexagons, and so on for any -gon shape with any value. We also briefly explored volume in the third dimension and how to calculate areas within shapes.
Designing Your Project
After finishing the above mentioned unit, we were assigned a task - we could create our own project. For the custom part of this, we were instructed to design our own project using one of the skills taught, and being able to demonstrate it. At the end we would be giving brief presentations on the topic we chose. I was very fascinated by the right triangles within a square, so I decided to take it to the next level - Pyramids within a cube. Now, a pyramid doesn't have to have a square base - it can really have any shape as a base. There exists a model of this - an ancient Chinese model called a Yangma. The name essentially denotes a rectangular-based pyramid whose vertex is vertically above one of the vertices of the base. I decided I would create my own Yangma out of a block of wood, and demonstrate it with my presentation. The Yangma is an interesting model - it is both a cube and three equal pyramids which can split away from the cube. You can see pictures from my process of creating it, some drawings, and my presentation below.
yangma.pdf |
Reflection
I never knew I could learn so much in a project. I learned a lot about trigonometry, something that sounded scary, but was actually kind of easy. The only challenge I really faced during this project/unit was finding out how to make the cuts for my yangma - the process took several tries and my work was stolen once. However, once I knew what to do, in no small part assisted by starting small, I was able to get the job done efficiently and nicely. Working both alone and with classmates on the unit was great - and the way I could choose who to work with (or none at all) for our custom project worked great too. I definitely had some great successes during this, such as becoming more attuned to geometry and other concepts that I haven't really touched base on in a long time.
I really liked the project and would gladly do it again.
I really liked the project and would gladly do it again.