Semester 1 ProjectsCookies Project |
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This was the second project of my first semester where we had to create an advanced word problem using data from a real business. The idea behind the project is that we would use these numbers to understand how real businesses calculate profit. For the real world problem I created, I used an example of a coffee shop where the owner (the person answering the problem) wants to add two bakery items to the menu. I then created constraints (the obstacles) of the problem and then I solved using a particular process which included tables, graphs, and inequalities, which this unit was about. Below is this project and my final product:
This project includes 3 main parts my problem statement, solution, and reflection.
Problem Statement:
You have just started a new coffee shop and you only have a few items to go with your coffee and drinks. You want to add muffins and pound cakes for your customers. The kitchen space in your coffee shop is limited. This means you only have two ovens to work with? How many pound cakes and muffins can you make within your budget?
Solution:
It is best to make 15 muffins and 9 pound cakes because you will end up with the highest profit of $115.38. Other combinations of muffins and pound cakes that I found using a graph resulted in less profit.
My Reflection:
Question: In what ways has the cookies project helped you gain perspective on doing real world mathematics?
I like how in this project we used data from a real coffee shop. Using this data helped me gain a greater understanding on what cost data and financials in a real business look like. I did not know that there were so many constraints when it comes to small things in businesses. I found it interesting when I created them because there is a lot more thought going on in the background of a business rather than just deciding to bake anything you want. When I made the profit table it turned out that if you only baked the muffins you would have almost no profit, so it would be too risky to limit the amount of items that the coffee shop bakes. It was also interesting how there was a lot of math involved in calculating the right profit like creating the feasible region graph, and how there were lots of tools you have to use to calculate your profit like a table, graph, and equations. Overall I learned a lot about how businesses calculate their numbers in bulk and the math process behind the decisions a business, or in this case a coffee shop, will make.
This project includes 3 main parts my problem statement, solution, and reflection.
Problem Statement:
You have just started a new coffee shop and you only have a few items to go with your coffee and drinks. You want to add muffins and pound cakes for your customers. The kitchen space in your coffee shop is limited. This means you only have two ovens to work with? How many pound cakes and muffins can you make within your budget?
Solution:
It is best to make 15 muffins and 9 pound cakes because you will end up with the highest profit of $115.38. Other combinations of muffins and pound cakes that I found using a graph resulted in less profit.
My Reflection:
Question: In what ways has the cookies project helped you gain perspective on doing real world mathematics?
I like how in this project we used data from a real coffee shop. Using this data helped me gain a greater understanding on what cost data and financials in a real business look like. I did not know that there were so many constraints when it comes to small things in businesses. I found it interesting when I created them because there is a lot more thought going on in the background of a business rather than just deciding to bake anything you want. When I made the profit table it turned out that if you only baked the muffins you would have almost no profit, so it would be too risky to limit the amount of items that the coffee shop bakes. It was also interesting how there was a lot of math involved in calculating the right profit like creating the feasible region graph, and how there were lots of tools you have to use to calculate your profit like a table, graph, and equations. Overall I learned a lot about how businesses calculate their numbers in bulk and the math process behind the decisions a business, or in this case a coffee shop, will make.
Tessellation Project
Cover Letter:
Topic 1
Area and volume are an important part of what it means to make an efficient shape. In math class we watched a video which describes what it means to make an efficient shape, and efficient shapes in nature. In this video bees and beehives were the main topic. People wondered why beehives had hexagons instead of another shape, and the answer to this question is efficiency. A hexagon maximizes the amount of space where bees can store honey as well as fit together so that there is no space wasted. I will explain how this works in my cover letter.
Now I want to explain this concept deeper, I started with the idea that the most efficient shape was a hexagon because of the beehive example we learned in class. I thought right off the bat that a hexagon was the most efficient shape only because it was able to fit together with other shapes. Let's imagine you are packing a box. What would be the most efficient way to pack it? Well if you fill a box with hexagon like objects then it will fill the plane in an efficient way. Here is why If you have a square which is inscribed in a circle then there is wasted space around the square within that circle. This makes the square less efficient. Now if you have a hexagon that is inscribed in that circle then it fills more of the circle. This means it is leaving less extra space than the square, which makes it efficient. This is why a hexagon is the most efficient packing shape.
Topic 1
Area and volume are an important part of what it means to make an efficient shape. In math class we watched a video which describes what it means to make an efficient shape, and efficient shapes in nature. In this video bees and beehives were the main topic. People wondered why beehives had hexagons instead of another shape, and the answer to this question is efficiency. A hexagon maximizes the amount of space where bees can store honey as well as fit together so that there is no space wasted. I will explain how this works in my cover letter.
Now I want to explain this concept deeper, I started with the idea that the most efficient shape was a hexagon because of the beehive example we learned in class. I thought right off the bat that a hexagon was the most efficient shape only because it was able to fit together with other shapes. Let's imagine you are packing a box. What would be the most efficient way to pack it? Well if you fill a box with hexagon like objects then it will fill the plane in an efficient way. Here is why If you have a square which is inscribed in a circle then there is wasted space around the square within that circle. This makes the square less efficient. Now if you have a hexagon that is inscribed in that circle then it fills more of the circle. This means it is leaving less extra space than the square, which makes it efficient. This is why a hexagon is the most efficient packing shape.
This diagram shows the wasted space of a hexagon vs a square inscribed in a circle.
This relates to the beehive example we learned in class. If a beehive were to be filled with circles then the bees face the problem that the circle leaves gap space between other circles making it not efficient because there is now gap space. But the bees don't have circles in their hive, they have hexagons, which is the most circular shape that tiles the plain meaning that the hexagons will not leave that unwanted gap space. This means that the bees now have the most amount of area they can to fit honey within the hexagon because it is closest to the circle. So the hive does not have gap space between the shapes this is why a hexagon is the most efficient packing shape.
This relates to the beehive example we learned in class. If a beehive were to be filled with circles then the bees face the problem that the circle leaves gap space between other circles making it not efficient because there is now gap space. But the bees don't have circles in their hive, they have hexagons, which is the most circular shape that tiles the plain meaning that the hexagons will not leave that unwanted gap space. This means that the bees now have the most amount of area they can to fit honey within the hexagon because it is closest to the circle. So the hive does not have gap space between the shapes this is why a hexagon is the most efficient packing shape.
This image shows the packing efficiency of a hexagon vs a circle.
However if we are talking only about the area a circle is the most efficient shape. Here is why because a circle provides you with the most area while having the least amount of perimeter. A circle is not the most efficient packing shape because it can not tile a plain because it can not fit together leaving no gap space like a hexagon, like I explained above.
Topic 2
In topic two I will explain how to find out how triangles are similar using postulates these are combinations which help us figure out if a triangle is similar.
However if we are talking only about the area a circle is the most efficient shape. Here is why because a circle provides you with the most area while having the least amount of perimeter. A circle is not the most efficient packing shape because it can not tile a plain because it can not fit together leaving no gap space like a hexagon, like I explained above.
Topic 2
In topic two I will explain how to find out how triangles are similar using postulates these are combinations which help us figure out if a triangle is similar.
These two triangles are similar because they are scaled versions of each other.
You can also use postulates to show triangles are similar. One similarity postulate is AA (stands for angle, angle) If two angles in one triangle are congruent to the corresponding angles in another triangle then the triangles are similar.
Another one is SSS (side, side, side). Also SAS (side, angle, side). Finally AAA (angle, angle ,angle). Using the same example above the corresponding sides of 3 and 12 and 4 and 16 are proportional. The included angle is congruent. Therefore by SAS (side, angle, side) the triangles are similar.
Reflection:
Making tessellations is hard. You have to create a very precise pattern using a straightedge and compass which needs to be able to repeat itself over and over again, as well as you need to figure out how to make a cool pattern. I had a whole new set of challenges doing my tessellation online. I used Desmos (the geometry tool) which is a math program which allows you to create shapes, polygons, tessellations ect… Using desmos I found it was hard to understand how the tool works. Because it was new to me it took multiple attempts to correctly use the tool in the way I wanted so it would create the tesselation I was intending to make. This led to frustration throughout the project as well as challenges I faced. It was also hard to take what I learned using a straightedge and compass and translate it to this online tool. It was basically like learning a whole new way to do it. This leads to a lot of extra time figuring things out. Using this tool was actually a lot like using the straightedge and compass, after a lot of trial and error I realized that doing it on desmos it was actually like a simpler way of doing it with the straightedge and compass. You are essentially doing the same thing you would on paper but instead you're simply drawing it on a computer which actually reduces the steps but you need to pay more attention to make sure there are no errors.
Which actually was a challenge in itself because making a tessellation with as many complicated parts like mine, it soon became hard to tell where in the process I was. Which I think is where the advantage of the straightedge and compass is, it makes it a little simpler to understand what you are doing when you have it right in front of you on paper and can follow basic knowledge of the tools you have while on desmos you have to make your process translate from your knowledge of the straightedge and compass to a new computer program. These were the most challenging things for me during this project. I actually learned a lot about geometry through these challenges, I feel as if I learned more than I would if I had made my tessellation on paper, this is because I had to translate the knowledge as I mentioned above.
I started off in this unit not knowing much about geometry. But by the end of the unit I could make a 6 fold complex tessellation which I was proud to present. This is how I grew my understanding of gromery. Through this unit I learned all about shapes though creating my tessellation.
Cookies Project:
Problem
You have just started a new coffee shop and you only have a few items to go with your coffee and drinks. You want to add muffins and pound cakes for your customers. The kitchen space in your coffee shop is limited. This means you only have two ovens to work with? How many pound cakes and muffins can you make within your budget?
Constraints:
Oven size:
The oven size is 24 by 24 inches. For the pound cake you need a 8 by 8 inch cake pan. In that oven you can fit 4 cakes. For the muffins the cupcake pan is 10 by 10 inches.
The total amount of pans you can fit in 2 ovens is 24. There are 3 racks in 1 oven You can get 12 trays in 1 oven for both muffins and pound cakes.
Cost:
You have a total of $150. To make your muffins you have $.32 cents for 1 muffin ($3.74 for a batch of 12). For the pound cake it costs $3.82 to make a batch.
Prep Time:
Pound cakes take 2 hours to make. 1 batch of muffins would take 35 minutes. 18 hours is the total bake and prep time.
Pricing:
Muffins Cost price: 3.82 300% markup: 3.82 times 4= 15.28 15.26 divided by 12= $1.28
Pound cakes Cost price: 3.74 300% markup: 4 timed 3.7= 14.96
Question:
What should you bake to maximize profit?
Semester 2 Projects
Rocket Project
Blueprint/Video-
Rocket Project Cover Letter-
This year in math 10 we worked on various topics. For our final project we learned how to apply these topics into real world situations through the rocket project. In which our objective was to create a rocket from plastic bottles, trash bags, string, super glue and whatever else we could use around the classroom. Our objective was to have a rocket that could successfully be launched at 100 PSI. To do this we learned how to use Quadratic Functions and how Quadratic Function relates to projectile motion. A projectile is an object that rises and falls under the influence of gravity and the quadratic function helps us understand its work and what path it takes. We use SOH CAH TOA to estimate the maximum height of our rocket. We use Quadratic Functions to estimate position (the place in which our rocket is located) ,velocity (the speed of our rocket in a given direction) and acceleration (the increase in the rate or speed of which our rocket moves).
This process was introduced to us at the beginning of the unit through a mini project where we created catapults. We started by creating a blueprint which was the beginning stage of the engineering design process (ask, research, and imagine) and eventually testing followed by improvement from our trial and error (plain, create, test, and improve). Then we took this into a bigger scale and when we added more depth when creating our rockets. Learning this process was our guide on this project and allowed us to create an end result we felt good about and was put together with care.
Rocket Project Calculations-
Time of Max Height
To find the time max height we first viewed a video that shows our rocket launching and its flight. We found that 12 frames on an iphone is equal to the max height of our rocket. Because on an iphone it moves 30 frames per second, we count how many frames it takes for our rocket to reach max height, this allows us to divide our frames by 30. In this case we had 12 frames so we would divide 12 by 30. So this means that our time of max height would be .4 seconds.
Max Height
When calculating max height our first step was to find the angle of the rocket. In our case the angle was 19. We calculated using SOH CAH TOA and applied the angles we know to the equation. Using SOH CAH TOA we took y/200 times 200 followed by taking our found angle of 19 and plugging that into the tangent which SOH CAH TOA shows us to use. So the equation would be y=200tan(19) and if we plug that into a calculator y= 68.86ft, so this means the max height is 73.61.
Initial velocity
The next step was to find the initial velocity of our rocket which is the velocity at which the motion starts.
To find the initial velocity we use the max height and time of max height to solve for the initial velocity which looks like this ]
h(t)=-12(g)(t)2+Vo(t)+yo
68.86=-12(32)(0.4)2+Vo(0.4)+1.5
68.86=-16(0.4)2+Vo(0.4)+1.5
68.86=-2.56+Vo(0.4)+1.5
68.86=-1.06+Vo(0.4)
+1.06 +1.06
96.92/0.4=Vo(0.4)/0.4
Vo=175.8ft/s
and this is our initial velocity
Theoretical flight time
our final step was finding the theoretical flight time and to do that we used the Standard Form of a Quadratic ax2+ bx+c and the Quadratic Formula
-bb2-4ac
X =
2a
As well as the formula for height at any given time: h(t)= yo+Vo(t)-12(g)(t2),
h(t)=-12(32)t2+175.8xt+1.5
a=-12(32)=-16
b=175.8
c=1.5
-175.8175.82-4(-16)(1.5)
X =
2(-16)
t=-0.01
t=11.0 total flight time =11sec
Reflection-
- What challenges did you encounter in this project and how did you navigate those challenges?
Our biggest challenge during this project was communication. We did not communicate with each other on what each of us needed us to do. This hurt our productivity because we would leave all the work on a particular thing to one person. At certain times I felt like I was doing everything and others felt like I was doing nothing, and I believe this is the result of not being able to communicate. I would say we really didn't navigate this issue and try to fix it, this is what caused a lot of stress for all of us in this project. An example of us not being able to communicate was our poster. I said that I would complete it but I ended up being out of school and I did not communicate this with my group so all the work fell on one of my group member's shoulders. This was an example of how I could have communicated better during this project.
- What were your successes in this project and how would you relay those successes to next year’s sophomores?
We were successful in the creation of the rocket and our planning (for example the blueprint was very good). The blueprint had such good quality because it was an example of good collaboration. When we made this it was the beginning of the project and we were all sitting together and were able to clearly communicate what we wanted to do with it. We were able to create something that was both neat and also very functional including everything we wanted it to be. This success is what contributed to the overall success of our final project. I would recommend to future sophomores that they should focus on creating an effective plan to help guide them towards the success of their project. I would also recommend to future sophomores that they communicate with their partners and make sure there is no lack of communication between each other.
- What was a turning point for you in this project and why?
- What lessons did you learn while doing this project, and how could you apply those lessons in your future projects and life?
I learned that it's hard to communicate with a group especially when there is a lot of stress involved. I learned that when there is that stress it is important to navigate that and figure out. Because the better the communication the more the stress we could reduce. So when I go into future projects I will make sure I am on the same page as the rest of my group. I would just do the work I need to do and get it done even if there is miscommunication about it. Also next time I do a group project I will make sure that we all know what we need to do before we start, and that we check in to make sure it is done and or going well.
- If you were to do this project again, what would you do differently and why?
If I could go back and do this project again I would have handled our group's miscommunication by better collaborating so everyone knows what they needed to work on at a given time. I would have taken initiative on what I knew I needed to do. During our project I would ask to help and be told I got it. Which made me think it was all fine but really my other group members were stressed, so I Would like to have just refused and done things anyway so hopefully there would be less stress.
In terms of the rocket I would have spliced in order to make the rocket more aerodynamic. Our rocket did not go as high as I would have liked. I think splicing would have helped achieve the height we were aiming for. Another thing was the nose cone. If we made a better nose cone design it would have helped the aerodynamics and how effective our rocket was.