Measuring Your World Project
Description
After the Scaling your World project, it only felt natural to begin working with measurement in other ways. Much of the project involved reviewing things that we already knew about distance, area, and volume, but a lot of the project was spent going over new topics that work well alongside our previous knowledge. For example, everyone had already learned about the Pythagorean Theorem, but very few had knowledge of right angle trigonometry. We started the project with the aforementioned review topics. We discussed the Pythagorean Theorem, the distance formula and how it’s derived from the Pythagorean theorem, and the equations of a circle. We then began to discuss new topics, such as the unit circle, the definition of sine, cosine and tangent, right-triangle trigonometry, and how they all work together. A bit later we got into the topic of area and volume with the area of polygons and circles, and the volumes of many different three dimensional shapes, such as pyramids, prisms, and spheres.
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For my “Design your own project” I decided to measure the volume of a star decoration that hangs from a chandelier in my house. I wanted to do this originally because I had no clue how to find the volume of the shape. I knew that doing so would challenge my knowledge of finding volume. Every side length (besides the points) was 4 cm. I used a ruler to find the other measurements seen.
The Calculations
To start out, I decided to employ the habit of a mathematician taking apart and putting back together. I began by removing all the star’s 18 points. Each point was an identical square pyramid, so I knew that I would only have to find the volume of one. I very quickly did those calculations using the universal pyramid volume equation. (V = Abase* h * 1/3)
I ended up with one 26 sided shape with 18 square sides and 8 equilateral triangle sides. I was unsure of how to proceed for a while, but after taking the shape apart more, I eventually noticed that there was an octagonal prism within the whole. I knew that the equation for the volume of any prism is V = Abase* h. I knew that I had to find the area of the octagon face, so I divided it into triangles. Originally, I found the height of this triangles using angles and trigonometry. I found the area of the octagon that way, but the result I got looked very strange in the context of the other results. I chose to use the measurements I had taken myself instead. I ended up with a much more reasonable answer, and I moved forward.
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I had successfully found the volume of two major parts of the shape. I still had a strange shape to calculate before I would have everything I needed. After a lot of thinking, I ended up going deeper by taking this shape apart further. I decided on three basic shapes, one rectangular prism, four triangular prisms, and four triangular pyramids. I had some struggle with some of the measurements of the shapes. I couldn’t see any way of finding the height of each shape. I knew it wasn’t 4 like all the other measurements, but I didn’t know what it was, either. I eventually decided to use the approximated measurements that I had taken with my ruler. That led me to a height of 3 for each shape.
Finding the volumes of these shapes was simple once I knew the heights. I plugged the numbers into the pyramid and prism formulas and multiplied them by the amount of times they occur in the shape. After that, All I had to do was multiply the whole thing by two to account for the second, identical side.
With all of that done, I knew that the last step would be to add everything together. I ended up with the final total of 1,124 centimeters cubed. I am confident that this is the correct volume, but I’m willing to consider some areas where I may have made mistakes, such as the initial measurements.
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Early on in the mini project, I had gotten mentally stuck. I didn’t know how to move forward because I couldn’t visualize the shape. I got the idea to create a model of the shape I was measuring. I ended up with several paper representations of several different objects. One square unit on the graph paper I used was the equivalent of one square centimeter. This made approximating measurements easier.
Reflection
Throughout the course of this project, I felt very smart. Even before the “design your own project” I felt confident with the math I was learning. Particularly, I’m proud of my new knowledge of trigonometry. I knew nothing about sine, cosine, tangent, arcsine, arccosine, and arctangent before this project, but after lots of practice, I feel like a stronger math student.
As for the “design your own project” itself, I feel like there are things to celebrate and things to acknowledge. For example, I attribute the paper models to my visual thinking brain. I think that for such a complex shape, it’s important to have full comprehension of what you are working with. Logical brains may just need the measurements to be satisfied, but I knew that I would get confused without something physical to hold onto. Drawing out the 2D maps and folding everything together was a very gratifying learning experience. Not to mention, they helped me stay organized.
I do think that my measurements could’ve been better. Much of the preliminary measurements I did had to be approximated with a ruler. Therefore, they didn’t really feel exact. There are ways I could’ve found the same measurements while using purely mathematical methods. Next time I do a project like this, I may chose to challenge myself by approximating less information.
All that aside, I’m very proud of what I was able to accomplish with pencil, paper, tape, and the newly gained knowledge I had. I’m looking forward to diving into something new.
As for the “design your own project” itself, I feel like there are things to celebrate and things to acknowledge. For example, I attribute the paper models to my visual thinking brain. I think that for such a complex shape, it’s important to have full comprehension of what you are working with. Logical brains may just need the measurements to be satisfied, but I knew that I would get confused without something physical to hold onto. Drawing out the 2D maps and folding everything together was a very gratifying learning experience. Not to mention, they helped me stay organized.
I do think that my measurements could’ve been better. Much of the preliminary measurements I did had to be approximated with a ruler. Therefore, they didn’t really feel exact. There are ways I could’ve found the same measurements while using purely mathematical methods. Next time I do a project like this, I may chose to challenge myself by approximating less information.
All that aside, I’m very proud of what I was able to accomplish with pencil, paper, tape, and the newly gained knowledge I had. I’m looking forward to diving into something new.