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Measuring Your World
A Multi-Dimensional Exploration of Measurement
The Content
Through "Measuring your World", our class explored a wide range of concepts relating to distance, area, and volume.
Pythagorean TheoremWe started by looking at the Pythagorean Theorem, which acted as the base for the rest of the content we covered.
Unit CircleWe were introduced to trigonometry through the unit circle, where the radius is one. This was important because any right triangle is similar to one that is created on the unit circle with the same value of theta. Theta represents the angle centered on the origin of the coordinate plane.
Area Of a PolygonNow that we understood the relationships between sides on a right triangle, we were able to derive the equation for the area of a polygon. We did this by breaking the shape into right triangles and finding their area before adding them up.
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Distance FormulaWe used the Pythagorean Theorem to derive the formula for the distance between two points on a coordinate plane.
Sine, Cosine, TangentWe described the relationships between side lengths on a unit circle. Cosine refers to the value of the x coordinate given theta. Sine refers to the value of y given theta. The proportion of y over x is called Tangent. Each value of theta up to 360 degrees corresponds to only one value for each proportion.
Area of a CircleTo better understand the area of a circle, we imagined it as a polygon with an infinite number of sides. We were able to determine what the relationship between radius and area would be using the equation for the area of a polygon.
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Equation of a CircleApplying what we knew about the distance formula, we were able to find he radius of a circle centered on the origin.
Right-Angle TrigonometryDue to the proportionality between right triangles on the unit circle and all right triangles, the relationships between sides created on the unit circle can be applied to all right triangles. However, the unit circle is only useful for right triangles. This means that our ratios also only work for right triangles.
Volume of ShapesWe were able to extend our knowledge into the third dimension by exploring volume. Prisms were simply the base area multiplied by the height of the object. We used what we new about simpler shapes to break down complex objects.
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Cavalieri's Principle
After looking at prisms and cones, we explored what happened to the volume of a shape when it was not at a right angle with the base. These shapes are called "oblique". Cavalieri's Principle states that an oblique shape has the same volume as a regular shape with the same height and cross-sectional area at each point. We convinced ourselves of this by imagining a stack of CDs being pushed so that it is angled from the base. The volume of the CDs does not change.
Origamath - A Self-Designed Project
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My group and I were tasked with designing a mini project that would help us further explore concepts of distance, area, and volume. We chose origami because of it's two dimensional and three dimensional properties. We decided to look at an origami cube and use the known side length of the paper to calculate it's volume. At the end of the project, we gave a presentation on our work and what we learned. |
We unfolded the cube to analyze the shapes it had made in the paper. We noticed that the side length was broken into fourths, and used that to come up with an equation to describe the volume of an origami cube made from a paper length of any size. After we answered our original question, we wanted to explore further and look at surface area. As we looked at our fold map, it was clear that the net of the cube (the exposed faces) did not take up the majority of the paper. We wanted to find out what fraction of the paper was included in the surface area. We did this using the same measurement we used in the volume problem and confirmed our answer by counting the number of squares in the unfolded piece of paper, which was sixteen. |
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This project helped us to explore and practice both skills and content we have studied this semester. We explored the concept of nets, which take an object and unfold it into the lower dimension, surface area, and proportions. In trying to solve the problem, we were challenged to use many Habits of a Mathematician. Most notably, Take Apart and Put Back Together can be seen in our process. We folded the cube and took it apart before using the marked edges to visualize the connection between those two states. This process made it easier to Look for Patterns in the folds. Lastly, we had to use what we knew about the proportionality of all origami cubes to Generalize our findings and derive our equations. |
Reflection
This project was structured so that every new concept was derived from the last. This helped me to learn by Starting Small. If I ever forget a rule or equation, I will be able to go back to what I know and work my way up. In the beginning of this project, I was familiar with the concepts we were exploring. I had to work on Describe and Articulate so that my group members could understand. When we moved into concepts I struggled with, such as the equation for the area of a polygon, it was my turn to Stay Confident, Patient, and Persistent in asking those around me to explain their reasoning and help me. I worked to stay organized in individual problems as well as in keeping my project materials together. This is still an area for growth for me, and I plan to create a separate place in my binder and on Google Drive where I can keep track of the notes and assignments for that unit. This will help me to reference what we are learning now in future classes. This was a valuable unit for me because it demonstrated the connections between familiar concepts and equations. Because I learned how the equations were derived, I was able to extend them to more complex problems.