This unit covered things that have to do with measurement. We started with the Pythagorean Theorem. We learn how to measure simple distances using triangles. We then learned the distance formula. Using the distance formula we started to explore what shapes could look like if they all had the same distance to a center point. In doing this, we created a circle. A circle will always have the same radial distance to each of its points. We learned about points of the x and y axis and how they're the same, intersections with radial lines, and what angles are formed. Then, we figured out that the Sine and Cosine are correlating as well. The Sine is the ratio between the hypotenuse and opposite sides of triangles. The Cosine is the ratio between the adjacent and hypotenuse sides of triangles. We learned how to apply these terms to the Pythagorean Theorem and the distance formula. By learning all this, we were able to start proving that a tangent of an angle is equivalent to sin/cos. To start, we used 30, 45, 60 degree angles. We incorporated learning about radians as well. In the final trigonometry portion, we were given a problem about how the British calculated the height and position of Mount Everest using a theodolite.
After exploring trigonometry, we looked closer at polygonal equations. By using the habit of a mathematician "starting small" we were able to learn and understand how trigonometry played into this. We learned that a square can be split into right triangles. We went into further exploration of using trigonometry to find areas of equilateral and isosceles triangles. We soon were able to make formulas for pentagons, hexagons, septagons, octagons, and so onto x-gons where x is any value. To finish up the content part of this project, we learned about volume in 3d shapes and learning how to calculate the area within shapes like rectangular prisms and cylinders.
Design Your Own Project
For the DYOP part of this unit, we were instructed to design our own project using one of the skills taught. We were told that we would be giving brief presentations on the topic we chose. I was most fascinated by the Mount Everest theodolite section. For my project, I wanted to see where two places in the world would have the exact same area using a triangle. You can see my work in the presentation below.
I did research on herons formula. I found that it would be easier to calculate the area in square miles of the two triangles using this formula. By inputting values for the variables I was able to successfully use the new formula that I hadn't been taught before. As you see in the presentation I had to solve of S first. I did this by adding A B and C together and then dividing it by two. I was then able to insert the S variable into the formula and get my solution.
Reflection
There were definitely new things that I learned in this project. I learned a lot about trigonometry and how to apply past knowledge into exploring further. The biggest challenge for me was learning about the volume of different shapes like a cylinder. I just didn't understand how to use things like a square and an angle with in a square to help calculate what the volume would be. Doing group work definitely helped because I was able to understand from a distance different perspective of my peers and how they work through the problem. I success I had in this project was the do it yourself portion. I was really comfortable talking about the material I had explored and help explaining it to my partner. For my project the habit of the mathematician that was used most frequently was starting small. The distances between periods were extremely difficult to calculate to be exact. I even switched destinations multiple times in order to have an accurate distance. I don't think I've ever used Google Maps more. By dragging the distance from the starting point to the final destination, I was able to create two triangles that had the same distance in lengths and had the same area of square miles. I feel like going into further topics having this background knowledge I will be comfortable because I can always refer back to my notes.