Desmos Drawing Project
Here is the link to my graph on Desmos: https://www.desmos.com/calculator/6ljzflwxgz
- How did you go about drawing this image? (Did you plan? Did you use patterns, reflections of curves? Did you experiment? Did you consult with peers/teacher?)
I went about drawing this image by picking something that I found simple and knew a lot about, therefore I picked a train. I found a picture on Facebook recently before I started this project that resembled an engine similar to this. and so I decided that I would go off of that picture and attempt to recreate it.
Q: How did using Desmos and creating this drawing help you understand function families and their transformations?
I now understand how many of these equations look on a graph. These will help me a great deal in Algebra 2.
Unit 3 Reflection: Area, Volume, Measurement
What content/skills have been then most interesting to you?: The process of matching minimal surface area to an equation for volume was the most interesting for me.
How have you grown mathematically?: This has helped me grow mathematically by allowing me to fit a minimal surface area to a set volume in order to create the ideal size for a 12 oz. soda can. This process can also be applied to many other things.
How have you grown mathematically?: This has helped me grow mathematically by allowing me to fit a minimal surface area to a set volume in order to create the ideal size for a 12 oz. soda can. This process can also be applied to many other things.
Problems of the Week (POWs)
How have problems of the week helped you grow mathematically? (Communicate ideas, breakdown problems, organize information, develop a growth mindset around mathematics):
POWs have helped me grow mathematically by increasing my ability to retain information and apply that information to fun yet challenging problems.
POWs have helped me grow mathematically by increasing my ability to retain information and apply that information to fun yet challenging problems.
Problem Statement
What is the ideal size of a 12-ounce soda can? From the perspective of the company, “ideal” means the soda can that will cost the least. We need to find the smallest amount of aluminum, smallest surface area, for the given volume. Find the radius and the height of a cylindrical can that holds 355cm cubed and uses the smallest amount of aluminum.
Process And Solution
I started by choosing radius’ 1-8 and solving what the height for each radius is. I made a table to keep organized. To do that, I used the equation below.
πr squared times height = 355cm squared
Then I made another table that had radius, height, and surface area. I inserted the radius and the height into the table then used another equation to solve for surface area (below).
2πr times height + 2πr squared = surface area
After I found the surface area of every radius I decided that a soda can with a radius of 4, and a height of 7.06 had the least amount of surface area. The surface area is 277.97cm squared.
Questioning The Result
My answer does not correspond to the current size of a soda can. I think that factories are not making the soda cans with the least amount of surface area because they want to have the most drink also. And you would have less drink if you used the least surface area soda can compared to the soda can they are using.
Evaluation
This pow made me think about volume and area, which I usually don’t think about. I also had to use a few different crazy equations, which messed with my brain a bit. I asked my peers for help when I got stuck and I also talked to them about what they had found to.
What is the ideal size of a 12-ounce soda can? From the perspective of the company, “ideal” means the soda can that will cost the least. We need to find the smallest amount of aluminum, smallest surface area, for the given volume. Find the radius and the height of a cylindrical can that holds 355cm cubed and uses the smallest amount of aluminum.
Process And Solution
I started by choosing radius’ 1-8 and solving what the height for each radius is. I made a table to keep organized. To do that, I used the equation below.
πr squared times height = 355cm squared
Then I made another table that had radius, height, and surface area. I inserted the radius and the height into the table then used another equation to solve for surface area (below).
2πr times height + 2πr squared = surface area
After I found the surface area of every radius I decided that a soda can with a radius of 4, and a height of 7.06 had the least amount of surface area. The surface area is 277.97cm squared.
Questioning The Result
My answer does not correspond to the current size of a soda can. I think that factories are not making the soda cans with the least amount of surface area because they want to have the most drink also. And you would have less drink if you used the least surface area soda can compared to the soda can they are using.
Evaluation
This pow made me think about volume and area, which I usually don’t think about. I also had to use a few different crazy equations, which messed with my brain a bit. I asked my peers for help when I got stuck and I also talked to them about what they had found to.
Unit 2 Reflections: Shadows, Similarity, and right Triangle Trigonometry
What has been the work you are most proud of in this unit?:
I am most proud of my trigonometry work in which we went outside and attempted to find the height of objects using Trigonometry. This really helped my understanding and it gave me a real world application that I could use trigonometry in. I can now, with the aid of a calculator, find the approximate height of any object, which will come in very handy when I need to know the height of a certain tree or object but do not have the appropriate means to do so. This will also come in very handy with Boy Scouts so I can get more badges.
What skills are you developing in math/geometry?:
I am developing my problem solving skills and my critical thinking skills. These skills assist my growth and understanding in geometry because they allow me to solve for problems quickly and efficiently. this can be proved by any paper involving trigonometry. these papers clearly demonstrate my ability to critically think and solve problems. This is because I can think through each step of the problems and complete them to the best of my ability.
Choose one topic: similarity or trigonometry. Explain what it is. Provide an example of how it is used in mathematics to solve problems. State an application of the topic in the adult world that interests you.
Trigonometry is the study or right triangles. It can be applied to find the height of certain objects. For example, If you wanted to find the approximate height of a tree, Trigonometry can be applied to find this out. First you have to measure you distance from you to the tree. After you have found out that distance, you need to obtain the angle of which you are looking at the top of the tree from. Knowing this, you can set up a problem in which you would use the tangent function to find the height of that tree. the final step is to add the height from the ground to your eyes, and then you have found the approximate height of that tree.
I am most proud of my trigonometry work in which we went outside and attempted to find the height of objects using Trigonometry. This really helped my understanding and it gave me a real world application that I could use trigonometry in. I can now, with the aid of a calculator, find the approximate height of any object, which will come in very handy when I need to know the height of a certain tree or object but do not have the appropriate means to do so. This will also come in very handy with Boy Scouts so I can get more badges.
What skills are you developing in math/geometry?:
I am developing my problem solving skills and my critical thinking skills. These skills assist my growth and understanding in geometry because they allow me to solve for problems quickly and efficiently. this can be proved by any paper involving trigonometry. these papers clearly demonstrate my ability to critically think and solve problems. This is because I can think through each step of the problems and complete them to the best of my ability.
Choose one topic: similarity or trigonometry. Explain what it is. Provide an example of how it is used in mathematics to solve problems. State an application of the topic in the adult world that interests you.
Trigonometry is the study or right triangles. It can be applied to find the height of certain objects. For example, If you wanted to find the approximate height of a tree, Trigonometry can be applied to find this out. First you have to measure you distance from you to the tree. After you have found out that distance, you need to obtain the angle of which you are looking at the top of the tree from. Knowing this, you can set up a problem in which you would use the tangent function to find the height of that tree. the final step is to add the height from the ground to your eyes, and then you have found the approximate height of that tree.
Tessellation project
What was the idea/theme with your project?
The idea behind my project builds off of my love for trains. I wanted to do something that would reflect what I love to do in my free time in my school work. The original tile was going to be a steam locomotive, but upon discovery that circles could not be used in tessellations, i changed my plan. The current tile and tessellation is a Durango and Silverton Narrow Gauge Railroad center-cab diesel locomotive.
What polygons did you start with and how did you alter it?
I simply started with a rectangle. I thought to myself about how I could make just a simple shape into a train. The easiest way was to just make the tile into a train was to turn it into a coach, which would involve simple tasks and alterations to make this. I figured that that was too simple. i took the rectangle, and made a cab out of it first. Then, obviously, I had to add wheels. Although a circle can not tessellate, semi-circles can. Therefore, i made the wheels seem almost hidden, as they are in real life, behind the wheel center base. By then, all I needed was to add color.
What transformations describe how your pre-image tile moved to create your two image tiles?
The tile either moved up, down. left, or right depending on where the tile was needed to fill space. the most common use of the tile was the move down.
In your opinion, are tessellations math or art?
It is hard to say one way or another. Tessellations require an understanding of geometry to create a product that can be admired as art. You start with mathematical concepts and end up with a product of math and art. In truth, tessellations are considered both, and that is how I view them.
The idea behind my project builds off of my love for trains. I wanted to do something that would reflect what I love to do in my free time in my school work. The original tile was going to be a steam locomotive, but upon discovery that circles could not be used in tessellations, i changed my plan. The current tile and tessellation is a Durango and Silverton Narrow Gauge Railroad center-cab diesel locomotive.
What polygons did you start with and how did you alter it?
I simply started with a rectangle. I thought to myself about how I could make just a simple shape into a train. The easiest way was to just make the tile into a train was to turn it into a coach, which would involve simple tasks and alterations to make this. I figured that that was too simple. i took the rectangle, and made a cab out of it first. Then, obviously, I had to add wheels. Although a circle can not tessellate, semi-circles can. Therefore, i made the wheels seem almost hidden, as they are in real life, behind the wheel center base. By then, all I needed was to add color.
What transformations describe how your pre-image tile moved to create your two image tiles?
The tile either moved up, down. left, or right depending on where the tile was needed to fill space. the most common use of the tile was the move down.
In your opinion, are tessellations math or art?
It is hard to say one way or another. Tessellations require an understanding of geometry to create a product that can be admired as art. You start with mathematical concepts and end up with a product of math and art. In truth, tessellations are considered both, and that is how I view them.
How I made my Tile:
1. First, I drew a rectangle.
2. Then, I cut off the section of the rectangle where the cab would go.
3. Then I cut quarter wheels off of the top and added them to the two smaller rectangles at the bottom as wheels.
4. The fourth and final step to creating my tile entails cutting little strips of paper between the wheels to create the smokestack for the diesel engines on each side of the cab.
1. First, I drew a rectangle.
2. Then, I cut off the section of the rectangle where the cab would go.
3. Then I cut quarter wheels off of the top and added them to the two smaller rectangles at the bottom as wheels.
4. The fourth and final step to creating my tile entails cutting little strips of paper between the wheels to create the smokestack for the diesel engines on each side of the cab.
Geogebra lab: Burning Tent
1.Once you have a minimal path, what appears to be true about the incoming angle and the outgoing angle?
The incoming angle is directly opposite of the outgoing angle.
2.Why is the path from the points Camper to Tentfire' the shortest path?
This is the shortest path because it offers the runner a lesser distance to run.
3. Where should the point River be located in relation to segment Camper to Tentfire' and line AB so that the sum of the distances is minimized?
The point River should be located close to the center of the two points, yet further towards the point Tentfire'.
The incoming angle is directly opposite of the outgoing angle.
2.Why is the path from the points Camper to Tentfire' the shortest path?
This is the shortest path because it offers the runner a lesser distance to run.
3. Where should the point River be located in relation to segment Camper to Tentfire' and line AB so that the sum of the distances is minimized?
The point River should be located close to the center of the two points, yet further towards the point Tentfire'.
Geogebra lab: Snail Trail
In the Snail Trail Graffiti Geogebra Lab we used line reflection symmetry to create symmetric designs. The first step to the lab was making a circle with on 60 degree angle, and putting a point on the perpendicular bisector of the angle on the circle. I then reflected the point over the angle lines creating 6 different colored points. Every time you move the original point, the 5 other points follow it creating a symmetric shape. The Snail moved together and there trails created symmetry with the following points. If you ignore the color in the trail you will notice that It makes a shape from all rotating sides.
Disclaimer: The picture above for the Soda Can POW with the table is not mine but rather a friends, as mine was lost in a heap of paper :)