For my growth mindset project, I decided to learn how to use Hawaiian poi balls over the two week period. I chose to learn how to use the Hawaiian poi balls because it was something that was sitting in my room for year and years and I never bothered to pick them up and learn. I'm also a part of a Hawaiian outrigger canoe club so I felt that learning how to use the poi balls would immerse myself more into the culture.
The growth mindset project impacted me because it showed me that if you actually put in the time and continuously do something even if you keep failing, eventually you'll learn what you're doing wrong and fix it. I hit myself or hit the poi balls together countless amount of times before I was able to do a run where everything went smoothly. But I eventually was able to get it right because I kept on going and set aside time to keep practicing. I didn't continue to try and improve on my topic after the two weeks over because my priorities were not that anymore. If I was in a dance where I needed to use them, then I probably would have kept it up but nothing was requiring me to continue to learn how to use them. |
The Cow ProblemProblem Statement: There is a 10' by 10' barn. A cow is attached to a corner of the barn with a 100' rope. We need to figure out what area of land the cow can graze. We used our learning of finding areas of different shapes to solve this problem.
I really enjoyed solving this problem because it was like breaking down a puzzle to figure out how the whole thing works. I liked how it pushed my thinking. I asked myself a lot through the unit, "What is the shape telling me? What am I missing?" It challenged me to think outside of the box and broadened the way I look at problems. |
Logarithms are an extension of exponents. In order to understand logarithms, you need to know the principle rules of adding, subtracting, multiplying, and dividing exponents. In this unit we learned how to manipulate exponents and turn them into logarithms as well as turn logarithms back into exponents. We also learned how to expand more complex exponents into logarithmic form.
I really enjoyed learning about logarithms because it challenged me. It was hard to adjust to the format of them and how you read one out. I feel like expanding the exponent gave me a better understanding of the exponent and all of its aspects, which in turn made me more fluent in exponents. |
Honors this year was a positive experience. I think, especially towards the beginning of the year, I enjoyed receiving packets that pushed my thinking of the original packet. Trying to figure out the elevated concept on my own helped me understand the core concept even better.
During the honors project this year I learned that I can be a very efficient worker. I learned how to multi-task. That was a very busy time, with three different projects I needed to work on to be exhibited at Festival del Sol. I found myself memorizing lines for our sketches while working on the drawing of my Fibonacci spiral. I learned that I am good at utilizing my time wisely. To the left is my drawing showcasing the Fibonacci Sequence. Something I suggest to improve honors next year is to provide more additional topics to learn. Sometimes I feel like the honors problems were just the same as the originals but with more numbers. I would like to see more learning of more complex units, maybe more pre-calculus to prepare for senior year. |
My initial attempt at solving this problem was to create a table that consisted of the number of knights, the winning seat, and the number of times the winner was met with King Arthur. When my group shared our different approaches, I found that keeping data on the number of times the winner was met with King Arthur was not going to be useful in the problem. From this knowledge, we were able to condense our different tables together to create a master table of the number of knights to the winning seats. |
From this master table, we found many patterns.
These patterns contribute to our solution because you know that the winner can never be an even number and the pattern is growing exponentially. However, it stops growing exponentially and resets to the number one after every base two exponent. These are aspects that you have to incorporate or think about in your final solution. |
My initial attempt at this problem was to draw out a graph with all four quadrants (shown to the right). I plotted a point at the origin and graphed y = 16 - x^2. Once I drew the line, I chose a random spot on the line and plotted a point there. My last two points met up the point on the line to the positive x and y axis. However, after discussing as a class, I realized I made the mistake of drawing a line instead of a parabola. If something is squared in a function that means it is a parabola.
For the final diagram (shown below), I plotted a point at the origin again. This time I drew out a parabola, using the function y = 16 - x^2. Like my initial attempt, my last two points met up the point on the line to the positive x and y axis. |
Initially when I was trying to solve for the maximum area, I went through all of the integers on the x-axis from 1-4, where the parabola ended. I found that the highest areas were 24 and 21. I got these areas from using 2 and 3 as the width of the rectangle or x-value on the graph. When I chose 2 as a value of x, I followed it up on the y-axis of the parabola to see the highest value of y I could get. The highest y-value ended up being 12. This means 12 was our height. I multiplied 2 and 12 and that is how I got the area of 24. I did the same for the x-value of 3 after to see which gave me a bigger area. |
My final, accurate diagram maximized the cow's rope to give it a bigger area to graze. Instead of wrapping around the barn to get to the other side, once the cow went one way and reached a certain point it would turn around and meet the other point on the other side. The little dimple is the point that I am talking about. |
2. The Pythagorean Theorem: a^2 + b^2 = c^2
In order to find the area of the triangle indicated below, we first need to figure out the diagonal length going across the barn. We can find this using the Pythagorean theorem. The diagonal length is c in the equation because it is opposite the right angle in the triangle, making it the hypotenuse. We know that a is equal to 10 and b is equal to 10. We learned about radicals to help us with this step. When we solve the equation out we get c=10√2. |
a) Now that we have the base of the triangle's length, we need to find the height. If we split the triangle straight down the middle from it's highest point to the base we get two symmetrical right triangles. Now that we have a right angle we can use the Pythagorean Theorem again!
b) If we are looking at just the left right triangle, we know that the hypotenuse in this triangle is 90 ft. because from the bottom left corner of the barn anywhere the rope goes is equal to 90 ft. Once the cow goes past the upper left corner of the barn the rope becomes 80 ft. c) We know the base of the right triangle is 5√2 because the whole base was equal to 10√2. When we spilt the triangle in half, we split the base in half making it 5√2. d) We have two lengths in the equation so we can solve for the third one. This time we have to be careful with what numbers we have and what number we're trying to solve for. We have the hypotenuse and the base. That means we have c and b of the equation. When we solve it out, we get the height of the triangle equals 89.72 ft. |
3. Area of a triangle = 1/2bh
a) Now we have all the numbers we can find the area of the triangle. The height is 89.72 and the base is 10√2. When we solve it out we get the area of the triangle equals 634.42 ft. b) There is one more catch to this. We have been solving for this triangle but a part of the triangle is in the barn. The cow is attached to the outside of the barn and the barn has no doors. The actual shape we are trying to calculate looks like the one in the picture shown to the right. This means we have to subtract the part of the barn that we have been including in our calculations to find the real area of the weird shape. c) We have been including exactly half of the barn in our calculations which is the shape of a triangle. We can use the formula for the area of a triangle again to find the area we need to subtract. The height of the barn is 10 and the base of the barn is 10. After we plug it into the formula we get the area of the triangle equals 50 ft. |
6. Add everything together!
Going back to the formula we wrote before, (area of the 3/4 of the circle) + (area of triangle in between two sectors) + 2(area of sector) - (area of 1/2 of the barn) = total area the cow can graze, now we have all the values we need to solve it out! Area of the 3/4 of the circle = 23,561.94 ft. Area of triangle in between two sectors = 634.42 ft. Area of sector = 3,501.07 ft. Area of 1/2 of the barn = 50 ft. 23,561.94 + 634.42 + 2(3,501.07) - 50 = 31,148.5 ft. The cow can graze an area of 31,148.5 ft. |