**How did you like doing your independent research project?**

I enjoyed the independent project work, as it was nice to have a chance to study any mathematical topic that interests us, and since math is in everything; everyone had their own creative and individual projects. I loved doing this type of mathematical work as well throughout the semester, because sometimes it's nice to get a break from book work/working with only equations and math pages.

**What did you like and what didn’t you like?**

The only thing I didn’t like about this project was the fact that I was super stressed for all my classes, and although I am proud of the product I produced, I do wish that I had more time to work on this project because I feel with more time and less stress from other classes I would have made an amazing card counting project, and have multiple demonstrations ready for the exhibition.

**What would you change about this project, if anything?**

I would assign this project with more time provided, or assign it when there is not all school exhibition or a chemistry lab due shortly, the added stress made it really hard for me to focus on this project.

**What did you learn about your research topic?**

I learned how many mathematical, and statistical skills it takes to learn to count cards. Counting Cards is actually all about probability, and I learned counting cards takes a lot of research, practice, and mathematical skill.

If you set goals for yourself this year in Math class, were they achieved? Why or why not?

If you set goals for yourself this year in Math class, were they achieved? Why or why not?

My goal for this year in math was to get a good understanding for the content we were working on, and also to earn an A+ with my work. I fell short, as I received two A-’s but I am very happy with all the new knowledge I got this year, and it was a very fun semester. I loved the way Hannah’s math class combined math, real world applications, and fun, it really was a great environment.

**Do you feel more confident in your math abilities? Why or why not?**

I definitely feel like I gained a bunch of new knowledge during this year, but the major concussion I had this year made my memory a little fuzzy so I didn’t learn everything that I possibly could have. I am still pleased with the large amount of new math content I learned throughout the year though.

**Did you feel you were able to balance this class with other classes you were taking?**

Compared to other teachers, you were very understanding of us when we have a large workload, or when we are taking longer to finish a project than you had anticipated you have always been so supportive and generous with your deadlines. You truly were one of the teacher that helped me perceiver through this year, as you were always able to accommodate for me when I needed help, one on one time, or to retake assignments. I really appreciate you Hannah.

**Do you feel prepared for Math 4 as a senior and any other math electives you may choose to take?**

I feel well prepared for my Math 4 class next year, I am not yet sure whether or not I am prepared for pre-calculus, as it is a very difficult class, and the content I have seen from it looks insanely hard; but I am sure that I will be able to perceive and learn the content.

On a personal note hannah, thank you so much for helping me with everything I have needed throughout the semester, you are a great super supportive person, and an amazing teacher.

## Pow 4: The growth of the rat population.

Problem Statement:

Two rats climbed aboard a ship while it was anchored at a dock, one male and one female, and then in late December, they climbed off the ship and onto a desert island. Each time the rats produce a new litter, every litter is six, and three of those six rats are females. The original female gives birth to six young on January 1 and produces another litter of six every 40 days thereafter as long as she lives. Each female born on the island will produce her first litter 120 days after her birth and then produce a new litter every 40 days thereafter. The rats are on an island with no natural enemies and plenty of food, so no rats will die in this first year. The goal of this POW is to answer the following question; What will be the total number of rats by the following January 1, including the original pair?

Process:

My first approach in solving this problem, was simple. Just to try and figure out the most efficient way to solve it, I began working on the problem by simply drawing small dots, representing rats, and attempting to track their reproduction in intervals of 40 days, while making sure to apply all of the given information. This worked for the first few intervals, and then it became far too complicated to continue any further. When this didn’t work I decided on a modification of this process that would be better to attempt to solve this problem with. I drew a table that had a few sections, a section for total number of rats, a section that went from 120-0 in intervals of 40; to keep track of the amount of time until each litter would reproduce. This made it much more simple, because the rats reproduce every 40 days, and their offspring will reproduce 120 days latter, and after that every 40 days, so this made it much easier to figure out which litters would be reproducing, and when. Using this method alone, I probably would have been able to solve the problem, but eventually, even this table became very visually overwhelming, which made it very confusing, and difficult to check my answers on. Finally, I decided to use Google Spreadsheets, to make a digital version of this table that would be easier for me to work with visually/aesthetically. I started creating a similar table, that would be easier to use and track. I struggled with this a bit, and started to get confused. My confusion came from the fact that in every section I had to keep track of the date until birth, and number of litters born that day, while still remembering to move each litter down one 40 day interval until I got to zero, and when the rats reproduced, and I had to remember to move the rats that gave birth that day back into the 40 days until birth interval. I realized that this method could make this very easy for my to solve this problem concisely and accurately, I just needed to find a way to improve my table. I decided that my table could be improved by adding two additions, that could help me with organization, and make it easier to keep track of their population growth. I added a new section that was counting down from 365-0, in intervals of 40, that would help me keep track of how many 40 day intervals there could be within one year. After working with this for a while I found it was much easier to follow, but still kind of difficult and could be further refined to make my solution more accurate. I decided to double up on these numbers, so each date would be repeating, and there were 2 sections for every reproduction date. This way, I could use the first section as a placeholder, to keep track of how many litters/rats were already born, and how many 40 day intervals until they would give birth, allowing me to move them down 40 days until birth every 40 days. In the second section (a repeat of the same # of days) I would figure out how many new litters were born that day, and then move the rats that gave birth on that day back into the 40 days until birth section. This made it very simple for me to solve the POW using this method, as all I needed to do was some basic math and make sure I filled in the correct numbers in the corresponding categories/sections.

Solution:

The solution that I ended up with is as follows, after one year on the island the rats will have produced 89 new litters, and there will be a total of 536 rats.

Evaluation:

I really liked the way that the mathematical concepts of this POW were easy to grasp, yet it was still tricky/difficult enough to be fun to work on. I really enjoyed working on this problem it felt kind of like a fun little puzzle. I especially enjoyed finding my own method in which to solve this pow, and changing it and improving it until it worked.

Two rats climbed aboard a ship while it was anchored at a dock, one male and one female, and then in late December, they climbed off the ship and onto a desert island. Each time the rats produce a new litter, every litter is six, and three of those six rats are females. The original female gives birth to six young on January 1 and produces another litter of six every 40 days thereafter as long as she lives. Each female born on the island will produce her first litter 120 days after her birth and then produce a new litter every 40 days thereafter. The rats are on an island with no natural enemies and plenty of food, so no rats will die in this first year. The goal of this POW is to answer the following question; What will be the total number of rats by the following January 1, including the original pair?

Process:

My first approach in solving this problem, was simple. Just to try and figure out the most efficient way to solve it, I began working on the problem by simply drawing small dots, representing rats, and attempting to track their reproduction in intervals of 40 days, while making sure to apply all of the given information. This worked for the first few intervals, and then it became far too complicated to continue any further. When this didn’t work I decided on a modification of this process that would be better to attempt to solve this problem with. I drew a table that had a few sections, a section for total number of rats, a section that went from 120-0 in intervals of 40; to keep track of the amount of time until each litter would reproduce. This made it much more simple, because the rats reproduce every 40 days, and their offspring will reproduce 120 days latter, and after that every 40 days, so this made it much easier to figure out which litters would be reproducing, and when. Using this method alone, I probably would have been able to solve the problem, but eventually, even this table became very visually overwhelming, which made it very confusing, and difficult to check my answers on. Finally, I decided to use Google Spreadsheets, to make a digital version of this table that would be easier for me to work with visually/aesthetically. I started creating a similar table, that would be easier to use and track. I struggled with this a bit, and started to get confused. My confusion came from the fact that in every section I had to keep track of the date until birth, and number of litters born that day, while still remembering to move each litter down one 40 day interval until I got to zero, and when the rats reproduced, and I had to remember to move the rats that gave birth that day back into the 40 days until birth interval. I realized that this method could make this very easy for my to solve this problem concisely and accurately, I just needed to find a way to improve my table. I decided that my table could be improved by adding two additions, that could help me with organization, and make it easier to keep track of their population growth. I added a new section that was counting down from 365-0, in intervals of 40, that would help me keep track of how many 40 day intervals there could be within one year. After working with this for a while I found it was much easier to follow, but still kind of difficult and could be further refined to make my solution more accurate. I decided to double up on these numbers, so each date would be repeating, and there were 2 sections for every reproduction date. This way, I could use the first section as a placeholder, to keep track of how many litters/rats were already born, and how many 40 day intervals until they would give birth, allowing me to move them down 40 days until birth every 40 days. In the second section (a repeat of the same # of days) I would figure out how many new litters were born that day, and then move the rats that gave birth on that day back into the 40 days until birth section. This made it very simple for me to solve the POW using this method, as all I needed to do was some basic math and make sure I filled in the correct numbers in the corresponding categories/sections.

Solution:

The solution that I ended up with is as follows, after one year on the island the rats will have produced 89 new litters, and there will be a total of 536 rats.

Evaluation:

I really liked the way that the mathematical concepts of this POW were easy to grasp, yet it was still tricky/difficult enough to be fun to work on. I really enjoyed working on this problem it felt kind of like a fun little puzzle. I especially enjoyed finding my own method in which to solve this pow, and changing it and improving it until it worked.

## SIlverton Write up

Task statement:

On October 12th, the entire Junior class went up to Silverton. Where we tested the turbidity, pH, conductivity, streamflow, and temperature of three creeks; Cement Creek, Mineral Creek, and the Upper Animas. My pod tested Cement Creek, I worked with Claire, Janey & Saige. Ari worked with Sammy, Larrea, & Eli, and lastly, Megan worked with Lydia and Chloe. Our goal was to gather enough data to where we could be able to predict what the pH, turbidity, conductivity, streamflow, and temperature would be when all three creeks merged to form the Animas river. Once we made our predictions we would then compare them to the data that the USGS collected on that day, October 12th.

Introduction:

In this investigation we went to Silverton, Colorado, in order to test three creeks, Cement, Mineral, and the Upper Animas, all tributaries to the Animas River. We tested the water quality of each by taking precise measurements in order to find each of the following; conductivity (or the amount of positive and negative ions) , pH (potential hydrogen/the number of free hydrogen molecules), turbidity (the cloudiness of a fluid), temperature (the degree of the intensity of heat), and streamflow (cubic feet per second of water). After collecting these results we compiled our data as a grade, making sure every groups data was incorporated into the larger dataset. From there we used Standard Deviation, Mean, Median, Mode, Sum, Minimum, Maximum, to look for trends in the data and to compare our data we collected to other data collected by others. The end goal of this investigation, was to refine our dataset to the point in which it can be used to accurately predict trends/make conclusions about the animas confluence. We hoped to achieve the following during this investigation; Experience field data collection including pH, conductivity, temperature, flow rate, dissolved oxygen and turbidity, describe how people interact with their environment, particularly their water, and how these interactions impact water quality, treat hard-rock mining in Silverton as a case study, describe how geology and ecology can impact water quality

treat sulfide minerals and the equations resulting in AMD as a case study describe the current water quality in the Animas River headwaters and action steps which are being taken to improve water quality after the Gold King Mine spill.

Visual Representation:

Once we had all of the results from our investigation, we then made a table that contained all of our information in one place.

Methods/Process:

When we went up to silverton we had to test 5 things, pH, Turbidity, conductivity, streamflow and temperature. We used a pH rod to test the pH, a turbidity sensor for turbidity, conductivity rod for conductivity, a rod that worked as a temperature gage, and we used a ruler and measuring tape for figuring out the streamflow of the creeks.

The point of the project was to predict what the 5 things would be when the three creeks merge we had to make the predictions based on the data that was collected in the three creeks and put into a google spreadsheet. Megan, Chloe, Claire and I all worked together to try and figure out how we would make the predictions. We ended up finding the average of all the temperatures from the data set to predict what the temperature would be when the three creeks merged. To predict what the Streamflow was we added all the max’s in the data set and found the average. To find the conductivity we got rid of all the outliers and we found the average. Finally to predict the pH we did the same thing we did with conductivity, we got rid of all the outliers and we found the average.

My group only got help from Claire and Chloe when we had to figure out the predictions, we also got a few tips from Steve on the problems we encountered when comparing our data to the USGS data website. The problem that we came across was called the ice conundrum. Other than the ice conundrum we did not have any problems.

Importance:

It is important to find the pH, temperature, streamflow, conductivity and turbidity when working with water quality because if a body of water that people frequently use, such as the Animas river, is contaminated or toxic, and gets consumed, can be very dangerous and make people very sick. Testing the pH in water tells you how acidic or basic the water is. If the water you test is lower than the pH of 7, then the water is acidic, thus meaning it is toxic and can cause harm. Turbidity causes cloudiness or haziness to the water, which makes testing turbidity a major factor in testing water quality because it is a good indicator of the effectiveness in the filtration system. If turbidity is high it can make it harder on aquatic life because the water is unsuitable for their living, due to the particles being suspended. Testing streamflow is also an important factor because aquatic life depends on the flow of the stream. Rivers that flow fast can receive pollution discharges and be little affected, while smaller streams have less capacity to decrease and help get rid of wastes. Conductivity measures the waters capability to pass electricity, which is related to the absorption of the ions in the water. Temperature effects the dissolved oxygen levels within the water, which impacts the chemical and the biological components of the water.

Solutions/predictions:

Once we had all of our classes data collected, we all made predicted values for the water quality parameters in the Animas River. Our group predicted that the temperature would be 7.02, the conductivity would be 1,999.3, the stream flow would be 45.78 and the pH would be 5.55. We predicted the temperature by finding the averages, for streamflow we added all the max’s to find the average and for conductivity and the pH we used the x outliers average. Observing the student data, a trend that we noticed was that the standard deviation was always less than the average in each creek. Comparing our data to the USGS data, we predicted that the temperature was seven, which it was, but for the streamflow, the USGS data said the streamflow was 103, while we had predicted it to 45.78. Since there was no data for pH, conductivity, or turbidity in the USGS data, we had to look at the data over the weeks before we went, to find what the parameters would be, so we could compare it to our predictions. We then had to make calculations that helped find our predictions. We started off by finding the mean and used each mean for the parameters which then would find the weighted averages. Our predictions weren’t very accurate compared to the USGS data, because we found that when our data was being measured, it wasn’t as precise as theirs.

Evaluation:

Doing this experiment in Silverton, I think was worthwhile, because it provided us with important information that is crucial to testing water quality. We learned how to use the instruments that you need to test water quality, and also what they are used for. We were also able to learn why we need to use those particular instruments to test the water, and the importance each test has. Conducting this experiment was interesting in the fact that it is an actual body of water that can affect us, so we were able to learn what it means if the pH is low and the turbidity is high, and how it can affect us.

Self-Assessment: I think my group should get at least an A- because we worked the best we could and we communicated to the best of our abilities. In the beginning we all came to a consensus on how we would split the work, and we helped each other out when we needed it.

On October 12th, the entire Junior class went up to Silverton. Where we tested the turbidity, pH, conductivity, streamflow, and temperature of three creeks; Cement Creek, Mineral Creek, and the Upper Animas. My pod tested Cement Creek, I worked with Claire, Janey & Saige. Ari worked with Sammy, Larrea, & Eli, and lastly, Megan worked with Lydia and Chloe. Our goal was to gather enough data to where we could be able to predict what the pH, turbidity, conductivity, streamflow, and temperature would be when all three creeks merged to form the Animas river. Once we made our predictions we would then compare them to the data that the USGS collected on that day, October 12th.

Introduction:

In this investigation we went to Silverton, Colorado, in order to test three creeks, Cement, Mineral, and the Upper Animas, all tributaries to the Animas River. We tested the water quality of each by taking precise measurements in order to find each of the following; conductivity (or the amount of positive and negative ions) , pH (potential hydrogen/the number of free hydrogen molecules), turbidity (the cloudiness of a fluid), temperature (the degree of the intensity of heat), and streamflow (cubic feet per second of water). After collecting these results we compiled our data as a grade, making sure every groups data was incorporated into the larger dataset. From there we used Standard Deviation, Mean, Median, Mode, Sum, Minimum, Maximum, to look for trends in the data and to compare our data we collected to other data collected by others. The end goal of this investigation, was to refine our dataset to the point in which it can be used to accurately predict trends/make conclusions about the animas confluence. We hoped to achieve the following during this investigation; Experience field data collection including pH, conductivity, temperature, flow rate, dissolved oxygen and turbidity, describe how people interact with their environment, particularly their water, and how these interactions impact water quality, treat hard-rock mining in Silverton as a case study, describe how geology and ecology can impact water quality

treat sulfide minerals and the equations resulting in AMD as a case study describe the current water quality in the Animas River headwaters and action steps which are being taken to improve water quality after the Gold King Mine spill.

Visual Representation:

Once we had all of the results from our investigation, we then made a table that contained all of our information in one place.

Methods/Process:

When we went up to silverton we had to test 5 things, pH, Turbidity, conductivity, streamflow and temperature. We used a pH rod to test the pH, a turbidity sensor for turbidity, conductivity rod for conductivity, a rod that worked as a temperature gage, and we used a ruler and measuring tape for figuring out the streamflow of the creeks.

The point of the project was to predict what the 5 things would be when the three creeks merge we had to make the predictions based on the data that was collected in the three creeks and put into a google spreadsheet. Megan, Chloe, Claire and I all worked together to try and figure out how we would make the predictions. We ended up finding the average of all the temperatures from the data set to predict what the temperature would be when the three creeks merged. To predict what the Streamflow was we added all the max’s in the data set and found the average. To find the conductivity we got rid of all the outliers and we found the average. Finally to predict the pH we did the same thing we did with conductivity, we got rid of all the outliers and we found the average.

My group only got help from Claire and Chloe when we had to figure out the predictions, we also got a few tips from Steve on the problems we encountered when comparing our data to the USGS data website. The problem that we came across was called the ice conundrum. Other than the ice conundrum we did not have any problems.

Importance:

It is important to find the pH, temperature, streamflow, conductivity and turbidity when working with water quality because if a body of water that people frequently use, such as the Animas river, is contaminated or toxic, and gets consumed, can be very dangerous and make people very sick. Testing the pH in water tells you how acidic or basic the water is. If the water you test is lower than the pH of 7, then the water is acidic, thus meaning it is toxic and can cause harm. Turbidity causes cloudiness or haziness to the water, which makes testing turbidity a major factor in testing water quality because it is a good indicator of the effectiveness in the filtration system. If turbidity is high it can make it harder on aquatic life because the water is unsuitable for their living, due to the particles being suspended. Testing streamflow is also an important factor because aquatic life depends on the flow of the stream. Rivers that flow fast can receive pollution discharges and be little affected, while smaller streams have less capacity to decrease and help get rid of wastes. Conductivity measures the waters capability to pass electricity, which is related to the absorption of the ions in the water. Temperature effects the dissolved oxygen levels within the water, which impacts the chemical and the biological components of the water.

Solutions/predictions:

Once we had all of our classes data collected, we all made predicted values for the water quality parameters in the Animas River. Our group predicted that the temperature would be 7.02, the conductivity would be 1,999.3, the stream flow would be 45.78 and the pH would be 5.55. We predicted the temperature by finding the averages, for streamflow we added all the max’s to find the average and for conductivity and the pH we used the x outliers average. Observing the student data, a trend that we noticed was that the standard deviation was always less than the average in each creek. Comparing our data to the USGS data, we predicted that the temperature was seven, which it was, but for the streamflow, the USGS data said the streamflow was 103, while we had predicted it to 45.78. Since there was no data for pH, conductivity, or turbidity in the USGS data, we had to look at the data over the weeks before we went, to find what the parameters would be, so we could compare it to our predictions. We then had to make calculations that helped find our predictions. We started off by finding the mean and used each mean for the parameters which then would find the weighted averages. Our predictions weren’t very accurate compared to the USGS data, because we found that when our data was being measured, it wasn’t as precise as theirs.

Evaluation:

Doing this experiment in Silverton, I think was worthwhile, because it provided us with important information that is crucial to testing water quality. We learned how to use the instruments that you need to test water quality, and also what they are used for. We were also able to learn why we need to use those particular instruments to test the water, and the importance each test has. Conducting this experiment was interesting in the fact that it is an actual body of water that can affect us, so we were able to learn what it means if the pH is low and the turbidity is high, and how it can affect us.

Self-Assessment: I think my group should get at least an A- because we worked the best we could and we communicated to the best of our abilities. In the beginning we all came to a consensus on how we would split the work, and we helped each other out when we needed it.