## Semester 2 Reflection

## POW

The Ideal Position

Problem statement:

If three points require a 4th that correlates directly to the other 3 then with the use of perpendicular bisectors you can identify a good corolating point. It is designed to find a place where each part is the same distance from each other point. This can be altered in order to find points for 4 and higher. By placing a circle that hits every point you can create a point in the middle that equally hits all points as seen in the visual representations.

Story time: One day Coral was watering his jellyfish, when all of a sudden he realized that the jellyfish was drying out. He had two sprinklers that were placed at 145 degrees and 25 degrees. Where does Coral need to place the sprinklers in order to keep the jellyfish from drying out.

Process: To begin this problem I looked at how pinpointing a certain point can be manipulated and then used the placement of where the flowers were to then see where to place the sprinklers. At first when this POW was assigned I saw it as any other POW, however the building and solving of this problem ended up being my favorite this year. The process of using item placement and geometry is one of my favorite uses of math and therefore helped me to keep going and try and find a realistic solution.

Solution:

The diagram on the left shows the ideal position for the three flowers when they are positioned in this way. By using perpendicular bisectors, it is easy to find the ideal position for the sprinkler to be placed. Simply draw a line from one flower to the next then draw another line directly through the middle. For four or higher however you will need to draw a circle that encompesense each flower then by placeing the sprinkler in the center you will have your ideal positions. There is a number of arrangements of flowers that may have ideal positions a straight line however is not one of them. The best place to place the sprinkler would be on the center flower however that would give the center flower to much water and the other flowers not enough.

Evaluation:

This problem was not that challenging. It was more of a puzzle than a math problem and did not need that much time of effort to complete. I think I could be more challenging if it was based on length rather than simply bisectors. It could also be adjusted to different situations such as paths of cars and where to place a stop light to prevent an accident.

Self Assessment:

I think I deserve an A because I had a solid answer showed my work and give reasons as to why it was true. In this POW I demonstrated my ability to problem solve using geometry and tests. I put a couple of hours into experimenting with other ways to solve this however this was the best answer I came up with.

Problem statement:

If three points require a 4th that correlates directly to the other 3 then with the use of perpendicular bisectors you can identify a good corolating point. It is designed to find a place where each part is the same distance from each other point. This can be altered in order to find points for 4 and higher. By placing a circle that hits every point you can create a point in the middle that equally hits all points as seen in the visual representations.

Story time: One day Coral was watering his jellyfish, when all of a sudden he realized that the jellyfish was drying out. He had two sprinklers that were placed at 145 degrees and 25 degrees. Where does Coral need to place the sprinklers in order to keep the jellyfish from drying out.

Process: To begin this problem I looked at how pinpointing a certain point can be manipulated and then used the placement of where the flowers were to then see where to place the sprinklers. At first when this POW was assigned I saw it as any other POW, however the building and solving of this problem ended up being my favorite this year. The process of using item placement and geometry is one of my favorite uses of math and therefore helped me to keep going and try and find a realistic solution.

Solution:

The diagram on the left shows the ideal position for the three flowers when they are positioned in this way. By using perpendicular bisectors, it is easy to find the ideal position for the sprinkler to be placed. Simply draw a line from one flower to the next then draw another line directly through the middle. For four or higher however you will need to draw a circle that encompesense each flower then by placeing the sprinkler in the center you will have your ideal positions. There is a number of arrangements of flowers that may have ideal positions a straight line however is not one of them. The best place to place the sprinkler would be on the center flower however that would give the center flower to much water and the other flowers not enough.

Evaluation:

This problem was not that challenging. It was more of a puzzle than a math problem and did not need that much time of effort to complete. I think I could be more challenging if it was based on length rather than simply bisectors. It could also be adjusted to different situations such as paths of cars and where to place a stop light to prevent an accident.

Self Assessment:

I think I deserve an A because I had a solid answer showed my work and give reasons as to why it was true. In this POW I demonstrated my ability to problem solve using geometry and tests. I put a couple of hours into experimenting with other ways to solve this however this was the best answer I came up with.

## Independent math function project reflection

The independent research project was very interesting and allowed all the students to experiment with math functions that interested them. I personally enjoyed researching properties of infinity and learning of ways to change it in order to maybe turn it into a normal number. Even though I never found the exact thing I was looking for it was super interesting to learn that anything can be infinite just increasing or decreasing at different rates. I would keep the project about the same but give a broader range of topics. I enjoyed the open end of this project and didn't enjoy that I had no idea where to start. I would keep the project exactly the same, just give a bit more direction.

## A Junior Year Reflection that is Math Focused

At the start of the year I chose not to set a goal because I am ore a go with the flow kind of guy and I much prefer to just do math and learn from it as I go. I feel more confident because I have learned new math forms and functions such as e, i and the usability of pi. Math is much broader sense junior year and I hope to use what I've learned here in Cyles class. I felt as if I had plenty of time and plenty of information to work on this class as well as the other classes that may have been challenging but it was doable. I hope that even though Hannah is leaving Animas to move onto greater things she won't forget her final junior class at Animas. Peace junior year.

## Semester 1 Reflection

Rats, rats, rats all I got is rats

Two rats arrive on an island. The first day they have a litter of 6 rats. Every 40 days after they will have another litter. The first litter will take 120 days to mature and have a litter of their own. Owen job was to find how many their will be after 1 year (365 days).

Restraints:

First I saw how many there were day one. Then added six till 120 days. Then I added another 15 for every day until 160 then added another 30, this continued. I used these numbers because it uses the previous amount plus the other litters I continued this until I found an answer.

This image represents how liters work, a new one of these starts every 40 days.

Results:

14 -325

20-285

44-245

86-205

196-165

278-125

536-85

974-45

1808-5

Evaluation:

I think that this POW wasn’t exactly hard, however I think it was a good addition to this last part of the semester. It wasn’t stressful or hard, however provoked thought and that seemed good for our workload.

Self assessment:

I spent a few hours outside of school but did most of it with Ryan, Saige, Cole and Jake. It was helpful to work with others. I think I could have done more to work with them, rather than counter their arguments.

Two rats arrive on an island. The first day they have a litter of 6 rats. Every 40 days after they will have another litter. The first litter will take 120 days to mature and have a litter of their own. Owen job was to find how many their will be after 1 year (365 days).

Restraints:

- 1 year = 365 days
- 1 month = 30 days
- 12 month = 1 year
- 1 liter 3M/3F
- 1 liter every 40 days

- Takes 120 days to mature to have own liter.

- 1 liter = 6 rats
- 9 liters in 1 year
- 51 total liters

First I saw how many there were day one. Then added six till 120 days. Then I added another 15 for every day until 160 then added another 30, this continued. I used these numbers because it uses the previous amount plus the other litters I continued this until I found an answer.

This image represents how liters work, a new one of these starts every 40 days.

Results:

14 -325

20-285

44-245

86-205

196-165

278-125

536-85

974-45

1808-5

Evaluation:

I think that this POW wasn’t exactly hard, however I think it was a good addition to this last part of the semester. It wasn’t stressful or hard, however provoked thought and that seemed good for our workload.

Self assessment:

I spent a few hours outside of school but did most of it with Ryan, Saige, Cole and Jake. It was helpful to work with others. I think I could have done more to work with them, rather than counter their arguments.

Arthur's peeps:

King Arthur has a circular table that is open to any and every knight. Every day Arthur plays a game, he goes around the circle telling every other that they are out. The first seat is in then he continues around. He does this until there is only seat left. What seat should you sit in to win?

The best way to solve this problem is to do trials starting with 1 and continuously going up. The optimal seat varies due to the number of knights are at the table. The equation that I found that applies to every situation is very complex in words and has variables. If you start at one you want to sit in the first seat (well I mean duh) then if there are two seats you still want to sit in the first seat. However once you hit 3 seats you want to sit in the third seat. Then if there are 4 seats you want to sit in the first. Then for 5 seats you want to sit in the third seat. Then if there are 6 you want to sit in the fifth seat. This sounds complex but once put in numbers and you have a couple of rules that you need to follow.

Rules:

Justification:

1=1

2=1

3=3

4=1

5=3

6=5

7=7

8=1

9=3

10=5

11=7

12=9

13=11

14=13

15=15

16=1

Equation:

Individual seats: X-(2^Y) * 2 +1

Reset seats: X^h

King Arthur has a circular table that is open to any and every knight. Every day Arthur plays a game, he goes around the circle telling every other that they are out. The first seat is in then he continues around. He does this until there is only seat left. What seat should you sit in to win?

The best way to solve this problem is to do trials starting with 1 and continuously going up. The optimal seat varies due to the number of knights are at the table. The equation that I found that applies to every situation is very complex in words and has variables. If you start at one you want to sit in the first seat (well I mean duh) then if there are two seats you still want to sit in the first seat. However once you hit 3 seats you want to sit in the third seat. Then if there are 4 seats you want to sit in the first. Then for 5 seats you want to sit in the third seat. Then if there are 6 you want to sit in the fifth seat. This sounds complex but once put in numbers and you have a couple of rules that you need to follow.

Rules:

- If you always start on one then you can automatically assume that all seats with an even number are out.
- For every one seat added to the table you want to sit two seats higher than the last.
- Sense it doubles each time that one seat is added then every eventually you should reach the point where the last seat is best seat to sit in.
- After you sit in the last seat in the sequence then it resets to 1.
- The separation in resets are the same distance as the number that comes after the reset. (I.E.- 3=3 reset is the first one, the second reset is 7=7, if you add 4 to 3 then you have seven. Then if you add 8 to 7 you have reset three, 15=15.)

Justification:

1=1

2=1

3=3

4=1

5=3

6=5

7=7

8=1

9=3

10=5

11=7

12=9

13=11

14=13

15=15

16=1

Equation:

Individual seats: X-(2^Y) * 2 +1

Reset seats: X^h

I am proud of these POWS because I used the needed skills well and I practiced new skills that we had learned in class. I felt that they were beautiful work and demonstrated my understanding. I enjoyed each of these because they gave me challenge that didn't boggle my mind of cause me stress but were just enjoyable to solve.

Over all I entered this semester with very little goals and was mostly focusing on just paying attention and passing the class. I feel that this was a good idea for me because I actually do worse with goals. I did well working with my peers and with Hannah, I think i worked well because I was joking yet productive and could have humor in my work. I am still the same person but somehow more sarcastic. Next semester I want to just do it. I want to pass the class and learn about something interesting.