Measuring Your World
To start off the Measuring Your World unit, all students were given the challenge of proving the Pythagorean Theorem. The Pythagorean Theorem (a^2 + b^2 = c^2) is used to find a missing side length of a right triangle. To prove this theorem we used an area diagram that was dependent on similar triangles and squares, also known as “Proof By Rugs.” We then used the knowledge we gained from the Pythagorean Theorem and right triangles to derive the Distance Formula. The Distance Formula uses x and y coordinate points which also can give you a missing side length of a right triangle. Basically, it is used to find the distance between two points on a coordinate plane.
|
From the Distance Formula you can then derive the equation of a circle centered at the origin of a Cartesian coordinate plane (x^2 + y^2 = r^2). Know we can use the information supplied by the unit circle, a circle with the radius of one, and apply it to other circles due to the fact that all circle are similar. Proving this by translating two or more circles so their center points align and then dilating the circle(s) to compare their exact alignment. Since all circles are similar, we can conclude that by using a radius of one to create right triangles with angles such as 30, 45, or 60 degrees will be similar to any right triangle with these given angles and follow the same rules due to the fact that this triangle originated from the unit circle. We then used this conclusion to move into finding the points of the unit circle by using and adjusting the Pythagorean Theorem, plus to solve the points/lengths for the different angles listed above we used reflection. Once we got our first set of coordinates/point on the circle we use symmetry to find the rest. Since the circle is on a coordinate plane, each of the four quadrant holds one fourth of the circle. This allows for the first point on the circle found to be adjusted for the other coordinates. For example, (2,2) is in a point in quadrant one but if it is reflected into quadrant two it then becomes (-2,2).
|
Using what we know about the unit circle, similarity, and proportions, we were able to derive the sine and cosine equations. For example, by creating a right triangle using the radius of the unit circle as the hypotenuse and the theta angle you can develop the following analysis: The sine trigonometric equation (sin(theta) = O/H) can be used to find a y-coordinate on the unit circle, while the cosine trigonometric equation (cos(theta) = A/H) can be used to find the point of interception on the x-axis allowing for you to solve for a complete coordinate point. That leads me to the third trigonometric equation we learned, tangent (tan(theta) = O/A). Tangent is “a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point.” By using reflection you can determine that the coordinate plane makes ninety degree angles within a circle by using the tangent line which allows for proof of the tangent equation. Now for the inverse. Instead of finding points/lengths we wanted to find the angles. To do this you can use the arcSine, arcCosine and arcTangent functions (theta = sin^-1(O/H), theta = cos^-1(A/H), theta = tan^-1(O/A)). We also derived these equation from the unit circle, similarly to how we found our other equations just reversed.
|
From sine to the law of sines, right triangles to any triangles. We made this transition by finding solutions to The Mount Everest Problem. In the Mount Everest problem we were given a triangle that had no ninety degree angles. We then dropped a perpendicular to make the problem more familiar. Which eventually lead us to deriving the law of sines (sin(A)/a = sin(B)/b = sin(C)/c). To use this equation you need two angles and one side length which can be used to find the other missing side length. Next came the law of cosines (c^2 = a^2 + b^2 - 2abcos(theta)) which we derived from one of our previous equations, the distance formula. You use this formula when you have two side lengths and one angle. Both these formulas can be used to find missing side lengths.
|
Using what we learned and applying it to real life: LEGOs
During this mini project, we were able to choose an object(s) to measure and calculate using what we previously learned. My group and I chose to use legos as our measurable object. We chose legos because we thought they could easily be taken apart to help us better understand the material we were using. To see the legos we measured and our calculations please look below.
Reflection:
Presentation:
- I feel my group and I presented our project really well. We all understood the topics and could explain, in depth, the math process we used to solve for trigonometric functions, area, and volume. Also, we all spoke clearly and made sure to tie up all loose ends. I believe our presentation was clear and easy to understand.
- Overall I feel my group and I exceeded expectations. We were able to split up the work evenly, use our time wisely, and ultimately work well together. We were also able to overcome challenges and obstacles. For example, finding ways to calculate the math and articulate how we solved the problem. We overcame these challenges by using the habits of mathematicians take apart and put back together and collaborate and listen. By taking apart our shapes we were able to split up the calculation into more simple formulas. By collaborating and listening we were able to constantly stay in contact with our partners and follow them along the math process to keep from confusion. In conclusion, I feel my group and I fulfilled all the requirements and create good quality work.