Final Exhibition
For our final product, we had a choice of making a kaleidoscope, a slice-form, a tessellation, or a circle pattern. I ended up making a slice-form of a raft. My accompanying description is posted below.
Raft Slice-Form
For one part of our project, we made our own creative math visuals. I had several choices, including kaleidoscopes, tessellations, or a slice form. In the end, I decided to make a slice form. Slice forms are three-dimensional solids made of slotted paper. In order to make my final product, I went through several steps. First, I brainstormed what I wanted to make. Once I had decided I wanted to do a raft, I drew the slices that I would put together. The horizontal pieces look like the top view of a raft, while the vertical pieces are circles with a radius of two. Then I cut slits in the pieces and fit them together. Once I finished the raft, I had some extra time, so I made the visual that you see now. One thing that I had kind of a hard time figuring out was how to relate math to my project. When I thought about it however, the raft was made up of several different solids that we have been working with for a long time: cylinders. By using the formula for the volume of a cylinder (∏r2H), I found that I could calculate the volume of my slice form. The main tubes are 20 squares tall (that’s the big H) with a two square radius. So, my formula looked like this: 4∏20. For the volume (V) of one main tube, I got 251.33 units3. To compensate for the other side of the boat, I multiplied this by two and got 502.65 units3. The ends were a little more difficult. As accurately as I could, I measured height and diameter, for a total volume (of each end) of about 175.94 units3. This multiplied by two plus the main tube sections gave me a total volume of 854.53 units3. After this project, I was able to understand how math could be used to find out things that were important to me. For example, if I wanted to buy a raft, I could use these skills to determine which model would have more overall flotation.
Raft Slice-Form
For one part of our project, we made our own creative math visuals. I had several choices, including kaleidoscopes, tessellations, or a slice form. In the end, I decided to make a slice form. Slice forms are three-dimensional solids made of slotted paper. In order to make my final product, I went through several steps. First, I brainstormed what I wanted to make. Once I had decided I wanted to do a raft, I drew the slices that I would put together. The horizontal pieces look like the top view of a raft, while the vertical pieces are circles with a radius of two. Then I cut slits in the pieces and fit them together. Once I finished the raft, I had some extra time, so I made the visual that you see now. One thing that I had kind of a hard time figuring out was how to relate math to my project. When I thought about it however, the raft was made up of several different solids that we have been working with for a long time: cylinders. By using the formula for the volume of a cylinder (∏r2H), I found that I could calculate the volume of my slice form. The main tubes are 20 squares tall (that’s the big H) with a two square radius. So, my formula looked like this: 4∏20. For the volume (V) of one main tube, I got 251.33 units3. To compensate for the other side of the boat, I multiplied this by two and got 502.65 units3. The ends were a little more difficult. As accurately as I could, I measured height and diameter, for a total volume (of each end) of about 175.94 units3. This multiplied by two plus the main tube sections gave me a total volume of 854.53 units3. After this project, I was able to understand how math could be used to find out things that were important to me. For example, if I wanted to buy a raft, I could use these skills to determine which model would have more overall flotation.
House Project
For this project, we renovated a house on Eastlawn drive (left). Working in groups of three, we found the most important things that needed fixing, then calculated how much it would cost us to make each remodel. When we finished, we had changed flooring, added a new bathtub, a matching stove and refrigerator, and a hot tub.
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House Project Introduction
Our first project has arrived! Last Thursday, we went to a house in Durango and took measurements of each room, counter, sink, and door. since then, we have been calculating areas, finding paint costs. calling realtors, and making blueprints. Next week we will present to a realtor our remodeled version of the house, complete with paint and flooring. We will also present how much our improvements have raised the value of the house, along with all of our calculations.
Summary of Geometry
In Geometry, we started with learning about inductive and deductive reasoning. Inductive reasoning is when you make a conclusion based on your observations of a pattern or repeating element in a problem or question. Deductive reasoning uses known methods like multiplication or division to solve a problem.
After that, we studied recognizing patterns and then putting them into equations. For this part of the year, we used the equation “f(x)=Dx+c” to find linear patterns, or patterns that added a given amount to the previous term to get the next term. An example of this would be 3,6,9,12… because it adds three to the previous term to get the next term. The equation for this pattern would look something like this: f(x)=3x.
Next, we studied special angle relationships. Some examples of these are vertical angles, complementary angles, supplementary angles, and consecutive angles. We used these to find the measures of angles in triangles and other shapes.
The next topic that we studied was the four points of concurrency in a triangle. They are the Orthocenter, the intersection point of the altitudes, the centroid, the intersection of the medians of a triangle, the incenter, the intersection of the angle bisectors, and the circumcenter, the intersection of the perpendicular bisectors. This also went along with studying constructions, or making a shape using only a compass and straightedge.
We completed working on discovering and proving triangle properties a few weeks ago. One such property is the angle relationships of a triangle. In any triangle, the sum of all three inside angles is 180 degrees. Knowing this allows us to find the third angle in a triangle if we are only given two of the angle measures.
We are now in the process of learning polygon properties. This past week, we discussed the exterior angle conjecture and formula (360/n, n being the number of sides) and the interior angle sum conjecture and formula (180{n-2}, n is also the number of sides.)
After that, we studied recognizing patterns and then putting them into equations. For this part of the year, we used the equation “f(x)=Dx+c” to find linear patterns, or patterns that added a given amount to the previous term to get the next term. An example of this would be 3,6,9,12… because it adds three to the previous term to get the next term. The equation for this pattern would look something like this: f(x)=3x.
Next, we studied special angle relationships. Some examples of these are vertical angles, complementary angles, supplementary angles, and consecutive angles. We used these to find the measures of angles in triangles and other shapes.
The next topic that we studied was the four points of concurrency in a triangle. They are the Orthocenter, the intersection point of the altitudes, the centroid, the intersection of the medians of a triangle, the incenter, the intersection of the angle bisectors, and the circumcenter, the intersection of the perpendicular bisectors. This also went along with studying constructions, or making a shape using only a compass and straightedge.
We completed working on discovering and proving triangle properties a few weeks ago. One such property is the angle relationships of a triangle. In any triangle, the sum of all three inside angles is 180 degrees. Knowing this allows us to find the third angle in a triangle if we are only given two of the angle measures.
We are now in the process of learning polygon properties. This past week, we discussed the exterior angle conjecture and formula (360/n, n being the number of sides) and the interior angle sum conjecture and formula (180{n-2}, n is also the number of sides.)
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
-J.H. Poincare (1854-1912) (Cited in H.E. Huntley, The Divine Proportion, Dover, 1970) |
Beautiful Math When I read this quote, my first reaction is that most people, or at least most High Schoolers would disagree with it. Throughout our school careers, we are always told that math is very useful, and that is why we learn it, not because it is "delightful." However, I can understand how someone who is very dedicated to math could find it delightful. One group of topics that we have studied that I found delightful, or at least very interesting, were the triangle points of concurrency. These were, as listed above, the Orthocenter, Circumcenter, incenter, and the centroid. These concepts had many applications as well as just the normal math ones. They also helped us find the exact center of a triangle and the center of balance of a triangle.
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