During this project we studied projectile motion and the mathematics behind it. We then used these skill to build a functioning water rocket that we tested and then entered into a contest, where the highest flying rocket won.
Throughout this unit we learned about multiple aspects of statistics and probability, all coming together in a final research project where we came up with a question we wanted to answer. The question I went after was, "Is there a difference in belief in evolution in regards to the amount of sleep someone gets?" As a class we sent out a survey to look at these questions.
Cookies Problem
Problem Statement:Giovonni is a baker, he bakes peanut butter cookies and chocolate fudge. He has been hired to cater a wedding, because peanut butter cookies and fudge are what everyone wants at their wedding. He needs to make as much money as possible during this job, or his bakery will close. Question: What is the maximum profit you can make using the most material? Constraints:
We only have 12 hours to cook as many things as possible.
We only have 300 cups of chocolate chips
We only have 150 cups of butter
We only have 12 dozen eggs
Pricing: Given a 300% markup, outline the sell price and the cost price below.
To complete this we multiplied the cost to make by 3 which represents the 300% markup. Afterwards we added the cost to make it again. In the end I came out with 5.66 for 1.42 and 26.88 for 6.72. Solution
Steps labeled
Create a table with all your constraints and rules
Create inequalities for each of the constraints
Plot each inequality in desmos
Find feasible region and find the most efficient solution
Tables outlining all of your constraints
Screenshot of Feasibility region
A list of possible best solutions and profits for each.
20, 20 - This would come to 20 Fudges and 20 batches of peanut butter cookies. I would make $113.2 on the cookies and $537.60 on the Fudge. In total I would make $650.8.
0, 40 - I would only make 40 fudge - this would be $1,075.20
40, 0 - I would only make 40 cookies - this would make $226.4
The best solution for Giovanni would be to make 40 fudge and 0 cookies because he would make $1,075.20 compared to $650.8 and $226.4. Project Reflection
In what ways has the cookies project helped you gain perspective on doing real world mathematics?
The cookie problem has helped me build more knowledge and confidence in finances and business. We learned about organization of data and how to use that data. The first day we worked on the cookie problem. In the end I was not able to solve the problem, but I learned how to organize the data we had into a table, which I used later on when we really solved it. I never thought of how much a restaurant or bakery goes through to bake anything. Most might just wing it, but I have massive respect for anyone who goes through this process to solve their problems. It lets you take luck and unsurety out of the equation. Math and especially the math we learned in the cookie problem allow me to break the world apart into simple, easy pieces. An example of this is how we created inequalities to show real world constraints. I like to think of constraints as rules something has to follow, as we add more constraints, it becomes harder and harder to follow all the rules. Inequalities act like a code, if this, then that, but no more than this. Inequalities act as an algorithm for anything we need, we just plug the rules in and bam! In the cookie problem it is dough and icing, but we could use it for millions of other things (and we do, like actual computers). I can use this knowledge to simplify anything I do. I would like to create a business at some point and what we have learned will be invaluable. I have to say overall that I really enjoyed this unit and everything we learned. I think I really understand inequalities.
Tessellation Project
Describe the concept of area and volume in terms of efficiency. What is the most efficient shape and why? How can we measure efficiency?
We can measure efficiency in a few ways. Area vs volume is an easy way to measure efficiency. Area is the surface area of something and volume is everything on the inside. The difference between them equates to the efficiency of the shape. The ideal ratio is more volume to little surface area. This would be like building a box that can hold the most water using the least amount of wood or other materials. The higher the number of sides a polygon has, the greater its area to volume ratio. A circle is the most efficient shape because it has an infinite number of sides. However, a circle is only the most efficient shape if it is alone. Nature loves efficiency. It is constantly trying to create the most volume using the least amount of material. A great example of this would be how bees create honey combs. Bees go about creating circular honeycombs to store honey which is inefficient. A circle alone is the most efficient shape because it has an infinite number of sides. There is a problem though. A circle can’t connect in a large tiling without overlapping or creating gaps.
When bees start out creating their honeycombs they have little gaps that are a waste of space. Physics corrects this mistake. As bees move around the hive they generate friction and heat which melts the wax in the combs into the normal, hexagonal, honeycomb shape. The hexagon is actually seen in many other areas of nature. An experiment we can do to find this “most efficient shape of nature,” is the bubble experiment. In this experiment you start out with a single bubble, the bubbles materials stretch to form the smallest shape possible for the air inside. This ends up being a sphere. Now if you put two bubbles next to each other they connect to form a straight line between them, and if you place three bubbles next to each other… They form a 120˚ angle! This is the same angle as a hexagon. Nature proves that the sphere is the most efficient shape alone, but when there are other shapes involved, the hexagon is best. (The bubble tries to use the smallest amount of materials to the most space, so when it creates a hexagon it is using the least amount of bubble/soap to fit the most air inside)
Explain how we can prove two triangles are similar.
There are a few ways to prove two triangles are similar. Way 1.) AA - Two triangles are similar if two angles are the same. If two angles are the same on a triangle the side lengths don’t have to be the same. It still creates a scaled version of the triangle because the angles of a triangle add up to 180˚, if you are keeping two angles then the third angle must stay the same to equal 180˚.
Way 2.) SSS - We can tell if two triangles are similar by their ratio. If we take two side pairs that correlate to each other and divide them we should get the same answer for each triangle. In the example picture above we see side length 4.31 and 2.89 divide to equal 1.49 and on the other triangle the same sides, 1.92 and 2.87, divide to equal 1.49. This is a way to prove two triangles are similar.
Way 3.) Scaled - If two triangles are scaled versions of each other, then they are similar. Imagine having a triangle with the side lengths 2, 4, and 5, then having a scaled version of this triangle with the side lengths 4, 8, and 10. This would be a scaled triangle multiplied by 2.
Way 4.) SAS - If the ratio of the side lengths of two triangles are the same and at least one angle is the same, they are similar. The angle would stop the side lengths from going too far and it would stop the other angles from changing.You have the distance(the side lengths) and you have the a direction(the angle), it creates a copy of the first triangle.
Question: How do you think you have grown in your understanding of geometry?
Directions: Talk about challenges you faced in this unit, and things that helped deepen your understanding. Provide evidence and specific examples of how you have grown in your understanding of geometry. (1 full paragraph minimum)
I think I have grown massively in my understanding of geometry. I did not take 8th grade math, so I skipped that section of geometry(I haven’t taken it in 3 years). This caught me up again and taught me how to create geometric designs using construction. I had never done something like this and it has always been an interest of mine, so it was a lot of fun learning how to make crazy shapes with nothing but a compass and straightedge. Because of my lack of understanding, I think I struggled a little. Some others in the class have been doing this type of math for years, and even though lots knew less than me, it was still difficult to pick it all up quickly. The fact that we were creating Islamic designs kept me invested and interested in the project. I have been really into this type of history and I have been studying a lot of Islamic history, so all of this was really fun. This additional motivation kept me going when I struggled. Near the beginning of our project, when we had just started working on our tessellations, I decided to create a five fold tessellation. I worked with Joe to create a pentagon, but once I had it, I had no idea what to do. When I connected all the pentagons, it didn’t fit together with no gaps or overlaps. I talked to Ande and learned that I needed two more shapes to create the total tessellation. After this I was able to make a five fold tessellation. This taught me a lot about geometry and isometries.