Measuring Your World
Measuring Your World Project Intro
The Measuring Your World Project was one that was launched for the purposes of teaching our class about how measuring affects our day to day lives and the importance of the formulas that we are constantly being taught. We focused on the different formulas that are used to measure different values such as the area and volumes of different objects, and finding proof for as to why these actually work. Throughout the entirety of the project we covered various units of math and we continued to deepen our understanding of what we thought we understood. All of this within one project that lasted a few months and allowed us to begin to measure objects from our choosing.
Math wITH TRIANGLES ANd Trigonometry
The first concept that we really started with was, how triangles would be very important for the purposes of other concepts and how they exist everywhere. We began with the Pythagorean Theorem which, helps to calculate the lengths of a two dimensional triangle as long as the triangle has the interior angles 30, 60, and 90. This formula can be manipulated in various ways in order to find any of the sides of the triangle, as long as two values are provided.
With the two dimensional triangle we also worked on how to calculate the area, and finding proof as to why this formula really works. In order to calculate the area of the triangle, the formula is b*h*1/2. This is a very simple way to calculate the area of a right angle triangle, but what happens when the triangle is isosceles? In order to find the height of the triangle, one would have to go through certain steps to get the value of that length. This is where we were introduced to Trigonometry, which uses the interior angles of a right angle triangle to find the lengths of the triangle’s sides. The formulas for these are: Sine= Opposite/Hypotenuse Cosine= Adjacent/Hypotenuse Tangent= Opposite/Adjacent The sine calculates for the height of a triangle, using the length of the side opposite of the angle and the hypotenuse of the triangle. The cosine calculates the base length of a triangle using the length next to the angle and the hypotenuse of the triangle. The tangent calculates for the hypotenuse of a triangle using the length opposite to the angle being measured and the length next to the angle. The way we would use this in case we were measuring the area of an isosceles triangle, would be by splitting the triangle in two, to find the height of the triangle, which happens to be the sine. In unison, these simple formulas can help us to find some of the values that exist within triangles. In order to prove this work with trigonometry and triangles, we began to work with the unit circle which had a radius of 1. This circle would be placed on a coordinate plane with the center of origin on the (0,0), giving us the ability to fit triangles within the area. This new unit circle allowed us to not only prove our work with trigonometry, but it also helped to shows us that special triangles existed. This special triangle happened to have the interior angles: 30, 60, and 90. And this triangle had outside lengths of 1, 1/2, and the square root of 3/2. With these new findings on a smaller scaled triangle, we know that the lengths of any special right angle triangle will be proportional to this equation. |
Working With area and 2d shapes
As we advanced within the unit we began to focus on other areas such as area and how to calculate the value and the reason behind why it works. From an intuitive point of view, the formula for area works because you are saying how many of something there is in a column, and how many columns there are. Therefore multiplying both the length and width of an object should provide the area within. The formula that summarizes triangles is B*H*1/2 = A. The formula that summarizes squares is L*W = A. The formula that summarizes circles is πr2 = A. The reason as to why this equation works is because we have the radius being squared, which accounts for the distance between the edges of the circle and then Pi, which is the relationship between the circumference of said circle and the diameter. This is what allows for the area of the circle to be calculated Within this unit of the project, we also looked at GeoBoards, which have little pegs that one can wrap rubber bands around to create two dimensional shapes. With these we posed the question, of whether or not there was a relationship between the number of interior pegs, to the border pegs, and area of the shape itself. This was somewhat more complicated because different shapes and sizes had various values and so it was somewhat more difficult to find patterns and create a formula.
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Working with volume and 3d objects
After the area unit we began to look at volume and how it closely resembles the former, except that this has another direction in which it can go and creates a third dimension. Volume is roughly when you have the area of a certain shape and begin to stack it on top of each other until some kind of thickness is created that adds height. With height, another variable is added to the equations but it is very simple, as long as one can find the area of the shape. If we were to have a cube, we know that the base is a square and its height has been increased, therefore the equation would be L*W*H = V. With triangles if we were to make a triangular prism, the formula would be similar to the one for area, with height added once more. B*H*1/2*L = V. As for circles we would also be stacking the same circle on top of each other to create a cylinder, a circle with height. The formula for this object would be π*r2*H = V. In the same way that we added height to a shape to give it a third dimension, it is possible to find the volume of said shape, as long as one has the area of the base and the height of the object.
To follow along with the measuring of volume of different shapes, we learned about the Cavalieri Principle that proves that if an object with congruent bases, is slanted, the volume will remain the same. For example, if we were to have a cylinder, the dimensions would be straightforward and we would have the base times the height to provide the volume. If the object were to be oblique, the bases would still be parallel, but the sides would be slanted. And if this is the case, we know that the base stays the same, it’s just that the different “rings” are displaced by a few measurements. But since the “rings” continue to stay on top of one another, they have the same height and they have the same dimensions of the base, which in turn results with the same volume of a regular/right cylinder. And this is what Cavalieri’s principle proves, and can be applied to other shapes that have congruent bases that are parallel to one another. |
Design Your own project Introduction
During the course of the Measuring Your World Project, we also had the opportunity to find the distance/area/volume of an object that we wanted to do. For my project I decided to pair up with a peer and together we choose to find the area of a flag in Cuartel Morelos, Tijuana MX. This is where a huge flags stands on a hill and towers over the city, and it was something the two of us were very excited to measure since we liked Mexican flag very much. Therefore we had our object decided and all we had to do was find the measurements needed to solve the are.
information gathering and math
As we began the project, we knew that we would not be able to measure the flag in person since it was in another country, and was on a military base. Therefore we had to do some research that would give us the information necessary to solve this problem that we set up. One of the first things that we found, and were key to the project was that the flag mast had to have a height of 50 feet and have a ratio of 4:7 for the lengths, as decreed by the National Coat of Arms, Flag, and Anthem. Another bit of information that we would find later, but was equally important was that the flag width had to be 28.5 percent of the total flag mast height. This information put together would allow us to calculate the area of our flag. The first step that we took was to use find the length of the width so that we could find the next side. Since we knew that the width would be 28.5 percent of 50, we calculated this and found that the width of the flag would be 14.25 m. Using this length we could compare it to our 4:7 ration and see what the next side length would be. Since 4:14.25, then we know that 7 must have the same increment.
So: (14.25*7)/4=Length That means that our length for the flag is 24.93 m. And now that we have the two side values, we can find the total area of the flag by multiplying the sides. L*W= A 14.25*24.93= A 353.8275 m2= A |
Reflection
This was a quick project that helped to wrap up the this unit before the end of the this week. It got us to employ what we have learned in a quick and easy way that could be displayed and presented in front of our group. One success that we had during this project was finding the information that we would need in order to solve for the problem. Through this project we really didn’t encounter any challenges and so we didn’t have to worry much about whether or not we were doing the math correctly or if our measurements were wrong. If we were to have changed something it would be the object that we measured because, although we had a somewhat complicated way of finding simple measurements, we could have gotten a more complex object. This would have made our project much more enjoyable for us and to present to the group. But overall we had a successful piece of work that we were happy to present and fit the requirements of the unit.
Measuring your world project reflection
This entire project/unit has been a very long and educational journey that helped us as students to become more comfortable with equations that we knew and equations that we learned. The most important part about all this was that we weren’t just drilling the formulas into our heads, but understanding why they make sense and how they really came to be. This is because sometimes if you don’t know what something means in a formula or where it is derived from, it makes it even more complicated to use or even understand. And so it is necessary to have some kind of knowledge on the basics of formulas, and what they are specifically meant to help with. I would say that for me this was a project that covered different topics of measurement very thoroughly and was enjoyable all the way through. This was also a great opportunity to employ the Habits of a Mathematician because we were doing all sorts of work that provided for an opportunity for them. One that we used very often was starting small, and this was because in order to be able to generalize we would have to have somewhere to draw information from. And with small examples, this became much more manageable and allowed us to learn more easily. Another habit that we used and that I mentioned, is generalizing. And this is really where all formulas come from, when someone finds a way to combine all possible variables into an equation that will provide the results of said variables. And this was something we did quite a bit, because we were working with different cases, all of which could have different outcomes, and so we would find something that fit the needs of all the inputs and outputs. Another huge habit was looking for patterns, and this was very closely tied to generalizing because in order to create a formula, one will see patterns that help to lay the foundations for an equation. Not to mention, seeing patterns helps to visualize what’s happening and allows for a more complete grasp on what to expect from a certain measurement. Overall for me, this was a great opportunity to learn about certain concepts that I was eager to cover and wanted to understand better. For example, a goal that I had for this math school year, was to learn more about trigonometry and how it works. And luckily for, we were able to cover that topic within this unit, and I was very happy about that. We learned quite a bit of content, and I do believe that this prepared us for the following year and for upcoming tests that we will be taking soon.