We recently completed a project in math called "Measuring Your World", in which we went through a series of worksheets that all filed into one big portfolio. We started out with working with simple Pythagorean theorem: A^2 + B^2= D^2. This is stating that if you add up the squares of the two legs, (Shortest sides in a right triangle), it will equal the square of the hypotenuse, (Longest side in a right triangle). From there we worked our way into the distance formula, which is derived from the Pythagorean theorem. This is used to find a distance between 2 points on a graph that aren't on a vertical or horizontal line. This formula is Distance =√((x2−x1)^2+(y2−y1)^2). Next we worked with the equation of a circle, Which was mainly finding sides of triangles based off of the radius of a circle. What we did looked something like this:
After that we looked at the unit circle. This was kind of generalizing the whole circle equation for us. It showed us the standard triangles that you could make and how they could be altered. It looked like this:
From there we went into the definition of Sine, Cosine, and Tangent. These are ratios of one side of a triangle over another. Sine its opposite over hypotenuse, Cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. From these you can find side lengths. For example. if you have one 30 degree angle in a triangle, the side opposite of that would be the opposite, and the hypotenuse would be the hypotenuse. Say you know the length of the hypotenuse. Maybe it's 4. If you entered, (and make sure your calculator is in degrees mode), SIN(30)*4, then it would give you the length of your opposite side. Here is a diagram to simplify what I just said:
Next we went into to right triangle trigonometry, which is using what I just explained to ind side lengths of all different sorts of tight triangles. From there we began to work out way into area. The formula for that being Base*Height=Area. We looked into polygons that were tiled a bit, like a rhombus, but 2 dimensional. We were able to use right triangles to find the missing distance of height. We then looked into the area of a circle. Which is pi*r^2. We went through a big lesson explaining why it is that way. The thing is, with
area, we were only working with 2 dimensions. So why not add a 3rd? Which is what we did.
We started working with volume. The simplest volume formula is Base*Height*Length=Volume. We worked with that for a while, and learned that even a cube that is shifted, or tilted a bit. Oblique, I think it's called, has the same volume as a normal cube. This is what is known as the Cavalieri Principle.
area, we were only working with 2 dimensions. So why not add a 3rd? Which is what we did.
We started working with volume. The simplest volume formula is Base*Height*Length=Volume. We worked with that for a while, and learned that even a cube that is shifted, or tilted a bit. Oblique, I think it's called, has the same volume as a normal cube. This is what is known as the Cavalieri Principle.
We then looked into the area of different shapes, like cylinders. We learned that for many shapes, the volume is just the area of one face, multiplied but the height. If you had a special shape, like a cone or a pyramid, you would just follow the equation of Area*Height=Volume, then you could just divide by however much you needed to divide by. Most of the time it was 3. At least for a pyramid.
Design Your Own Project:
For this project, my group decided to find the volume of the moon. We chose this because we are all deeply passionate about space, and we are all part of the HTHNC Space Club.
The math was simple yet complex. We took a photo of the moon. The photo was 5472 pixels across. The moons diameter was 840 pixels. We knew that the camera lens had an angle of 3.4 degrees. Using this information, we made two right triangles that let us find the distance, in miles, of the diameter of the moon. From there we used the equation V=4/3 pi*r^3. That gave us the volume. Here is how we did it, starting with our picture:
We weren't very systematic when we started this project. We originally wanted to find distances to stars using a method called parallax. However, some time into doing that, we discovered that parallax was an extremely complex method. On top of that, the equipment that we had was like a toy, compared to the equipment that we needed. After we abandoned parallax, we brainstormed different ideas. We narrowed it down to a few ideas. Measuring the volume of my boat, measuring the volume of Kabir's body, measuring the volume of Carter's water bottle, measuring the volume of a slice of pizza, or measuring the volume of the moon. As you can see, we went with the route of measuring the volume of the moon. We kind of procrastinated in the end, but it turned out to be a pretty great presentation.
Overall measuring your world reflection:
This project was quite an interesting one for me. It definitely had its ups and downs. It started out smoothly, mainly because I already knew some of the topics we were learning about. When we moved into trigonometry, I started to struggle with my depth of understanding. Would just get handed papers, work on them, then hand them in. I never really got a full understanding. In order to fix this, I worked with some of my classmates, and they gave me some tips. That really helped me out. Later, when we were working with regular polygons, I found that taking apart and putting back together worked really well. Any regular polygon divides into triangles, which you can then divide by two, to get a right triangle. With that, you can use trigonometry to find the side lengths and eventually area of those triangles. Then you would put the shape back together and find the total area. I feel like the main habit that we all used during this project was generalization. Everything that we learned, built off of the last topic. Everything that we learned, still somehow connected to things that we learned in the beginning. We always used basic methods to find answers to more complex problems. Overall, I'd say that this was a pretty good project.
Overall measuring your world reflection:
This project was quite an interesting one for me. It definitely had its ups and downs. It started out smoothly, mainly because I already knew some of the topics we were learning about. When we moved into trigonometry, I started to struggle with my depth of understanding. Would just get handed papers, work on them, then hand them in. I never really got a full understanding. In order to fix this, I worked with some of my classmates, and they gave me some tips. That really helped me out. Later, when we were working with regular polygons, I found that taking apart and putting back together worked really well. Any regular polygon divides into triangles, which you can then divide by two, to get a right triangle. With that, you can use trigonometry to find the side lengths and eventually area of those triangles. Then you would put the shape back together and find the total area. I feel like the main habit that we all used during this project was generalization. Everything that we learned, built off of the last topic. Everything that we learned, still somehow connected to things that we learned in the beginning. We always used basic methods to find answers to more complex problems. Overall, I'd say that this was a pretty good project.