__Semester 1 Reflection__

Content Skill (Learning):

The topic/content skill which I have mastered this semester is that of converting logarithmic equations into exponential equations and vice versa. The activities which have helped me learn this skill are Exploration #15 and Exploration #16. Both are shown below, in the order in which they are listed, first represented by their given questions, then by my corresponding work in exploring and answering these questions, and finally by an analysis which describes how these assignments have benefited my understanding of this topic;

The topic/content skill which I have mastered this semester is that of converting logarithmic equations into exponential equations and vice versa. The activities which have helped me learn this skill are Exploration #15 and Exploration #16. Both are shown below, in the order in which they are listed, first represented by their given questions, then by my corresponding work in exploring and answering these questions, and finally by an analysis which describes how these assignments have benefited my understanding of this topic;

The above Exploration #15 (an exploration whose basis was that of logarithmic to exponential equational conversion) was given to me at a time when I had not yet mastered the skill of converting logarithmic equations into exponential equations and exponential equations into logarithmic equations. Even though the problems on my answer sheet should be fairly correct as we reviewed them in a peer seminar, I recall that the exploration in itself was quite difficult, as I didn't have the formulas for this type of equational conversation memorized. My developing skills on this topic are represented well in problem #1, in which I was able to successfully convert a few simple logarithmic and exponential equations (not without several glances at the notes in my binder). My need for growth in the study of this is represented in problem # 4, in which I failed to properly analyze a logarithmic equation utilizing a variable, an error born out of a naivety to how to properly utilize my knowledge of these formulas in response to solving for a variable.

The above Exploration #16 (once again, an exploration heavy on logarithmic to exponential equational conversion) represents a slight progression in my use of logarithmic to exponential and vice versa formulas. Not only was I again able to confidently convert simple equations (as represented in problem #1), but I was now able to analyze how these conversion formulas can be seen in terms of their variables, (as represented in problem #4) and how a conversion of these equations into a variable format can actually prove more helpful in terms of discussing their properties and solving for unknowns in varying situations. However, I was still not quite ready to apply this knowledge in a real-world situation or one with varying factors.

Content Skill (Mastery):

The topic/content skill which I have mastered this semester is that of converting logarithmic equations into exponential equations and vice versa. The activities which demonstrate my mastery on this subject are POW #7 and Exploration #17. Both are shown below, in the order in which they are listed, represented by their given questions and by my corresponding work in answering these questions, and finally by an analysis which describes how these assignments exemplify my understanding of this topic;

The topic/content skill which I have mastered this semester is that of converting logarithmic equations into exponential equations and vice versa. The activities which demonstrate my mastery on this subject are POW #7 and Exploration #17. Both are shown below, in the order in which they are listed, represented by their given questions and by my corresponding work in answering these questions, and finally by an analysis which describes how these assignments exemplify my understanding of this topic;

This exploration (Exploration #17), as my final exploration in which I was tasked with analyzing and converting exponential and logarithmic equations, represents my full and proficient knowledge of this area in equational conversion. Throughout all the problems, both simplistic and containing variables, I demonstrated a knowledge of how to go about converting exponential equations into logarithmic equations and vice versa with memorized formulas and fully valid processes. Compared with Explorations #15 and #16, my skills in this arena of equational conversion are certainly demonstrated as more proficient, consistent in their proficiency, and based off of a well-learned foundation of formulas.

As I received a 100% correctness grade on this POW which focused completely on exponential and logarithmic equational analysis and conversion, I believe that it represents my full understanding of exponential to logarithmic and vice versa equational conversion. This POW handled varying assortments of variables interspersed into both logarithmic and exponential equations, and I was able to successfully solve and comprehensively display my work for each task concerning each of these subjects.

Problem-Solving Skills:

The Habit of a Mathematician which I have demonstrated this semester is that of Developing General Rules. To me, this skill means that in situations involving values which are able to be incorporated into another format such as an equation or expression whose derivation is that of an equation incorporating the initial values, one can confidently utilize their knowledge of certain equational formats and how variables interact with one another in order to create a rule of conversion for these variables into their expression and vice versa. I have demonstrated this skill in my creation of a general rule which converts a binomial into a perfect square trinomial. The activities which have helped me demonstrate this particular facet of the skill are Exploration #19 and Exploration #20. In Exploration #20, I was also able to demonstrate this skill in that I was able to analyze how the variables of a quadratic equation affect its graph. Both are shown below, in the order in which they are listed, represented by their given questions and by my corresponding work in answering these questions, and finally by an analysis which describes how these assignments exemplify my understanding of this topic;

The above Exploration #19 (an exploration whose basis was that of finding an equation converting binomials into perfect square trinomials) was handed out to the class when we had no knowledge of what binomials or perfect square trinomials were. This meant that we were required to grapple with converting a binomial into a perfect square trinomial on our own, yet utilizing our perviously-learned expansion property of FOIL, which essentially dictates the order in which the values within an expression are to be multiplied with each other in order to produce an expanded expression. My developing skills on this topic are represented in problem #1, in which I was able to successfully convert the given binomials into perfect square trinomials through the development of an equation which had the ability to convert these values. I found this equation through the application of the already-known FOIL method, in which I expanded Question 1a. I then analyzed this expansion to determine what its values were and how they could be rephrased in terms of the squares of the initial values. After many trial runs concerning this same method, as well as a bit of research concerning what binomials were generally used in within mathematics in order to provide me with clues as to the base construct of a binomials perfect square trinomial counterpart, I was able to successfully derive a formula out of my logical analysis followed by guess and check cumulative strategy. My understanding and application of these previously-derived equations is exemplified throughout the rest of the exploration.

The above Exploration #20 (once again, an exploration mainly focusing on the conversion of binomials into perfect square trinomials and vice versa, but also including a section on quadratic equations) represents the application of my solid understanding of this first-mentioned region in expressional conversion, as derived from my initial development of the rule in Exploration #19. Not only was I again able to confidently convert simple equations (as represented my process of problem #1), but I was able to take my knowledge a step further and simplify my perfect square into an even simpler expression, (as represented in my solutions to problem #1), thus effectively creating yet another rule for this strain of expression. My skill of developing a rule was thus proven in that my knowledge of this rule and its derivatives permitted me to more deeply and accurately analyze and apply its properties in varying situations. Another application of this my abilities int he creation of a general rule was in Question #2 of this exploration, in which it was asked that I analyze a quadratic equation in terms of how its equational variables affect its graph. After a simple entering of this equation into the Desmos program, I was able to clearly analyze the relationship of these variables and their physical application and once again demonstrate my ability to create a general rule (in this case for each variable in relation to its singular graphical application).

__Semester 2 Refletion (End of Year Reflection)__

__Habits of a Mathematician Analysis:__

__Domain #1- Generating Ideas__

This year I feel I have represented a considerable amount of strength in the Habits of a Mathematician Domain #1 through my abilities in identifying multiple mathematical tools which have the potential to aid me in presenting and analyzing my mathematical work as a critical thinker. I feel that I have the ability to apply many different varieties of presentational formats as well incorporate creative ways of utilizing mathematical tools in my work in order to solve problems. This is demonstrated in my most recent Algebra 2 project; my independent stock market brokering simulation. Within this project, I monitored the fluctuations of several selected singular company shares or put-together funds which I then analyzed to determine the most successful and profitable methods for stock brokering. I recorded and received information on stock fluctuation with the use of tables present in the application of Google Sheets, and also used these tables as well as their corresponding graphs to determine mathematical patterns present in my subject of study. Upon presenting this project, I utilized my constructed graphs and tables as well as several developed equations to exemplify the concepts which I explored within this project.

Below are pictured examples of the graphs, equations, and tables which I utilized in the above listed areas of mathematical processes:

Aside from my strengths in this area, an aspect of Domain #1 within which I could demonstrate some improvement is the section delineating a mathematical habit which describes the formulation of a plan in relation to commencing solving a math problem or completing a project. I believe that my lacking in this area is exhibited by my Spreadsheet Project, which I completed along with the help of a group. In this project, our team wanted to develop a series of editable equations on Google Sheets which determined the overall cost of college for a student with the consideration of several different factors. When we began this project, we did not explore the possible hazards which could be presented in accordance with our minimal experience in manipulating Google Sheets, and we were thus forced to a standstill when our faulty expertise hindered our ability to create complex and interconnected equations. It was necessary for us to ask for professorial assistance, and although this solved our initial problem, it is quite possible that we could have had a richer learning experience had we been able to dissect our mishaps and overcome these issues on our own. Furthermore, it is also possible that if we were prepared for an issue like this to arise, it could have been prevented completely with measures such as tutorials before the project began, and our group would have thus been able to focus on the content of the project rather than its mechanics. Although I do not doubt the importance of this experience with regard to our base of knowledge in operating a useful mathematical tool, a better analysis of what elements may have proved liabilities prior to beginning our work would have allowed us to grapple with the issue ourselves and would most likely have imprinted upon us a better-retained base of operative knowledge.

__Domain #2- Communicating Thinking in a Clear and Accessible Way__

I believe that this year I have demonstrated success in Domain #2 through my ability to respond to the ideas of others. When working in groups, I can understand others well and do my best to listen to, question, and build on their ideas. This is demonstrated by my continuous work in our Exploration Seminar groups. Within these groups, I find strong motivation to efficiently solve the problems presented, and I find that this is most easily completed with the help of others. Thus, I highly encourage my peers to comprehend and try to solve for themselves the problems which we are confronted with, I spur myself to listen to new perspectives. I also try to understand the concepts which are presented to me. As the Explorations deal with complex and challenging questions, our seminars witness a large amount of revision and trial and error. This is exhibited well in my Exploration 33, upon whose messy surface is the evidence of profound thought and its accompanying concepts of refinement and editing.

Below is pictured Exploration 33 after its analysis and refinement by my seminar group:

Despite my successes in the area of Domain #2, I have also demonstrated weakness in this region of personal habits, represented by my lack of effort in attempting to solicit contributions from quieter members of groups. I felt that this was evidenced by our class discussions, in which every student within our class period was required to present their own opinions of how the class is run, how challenging the subjects are, etc. Within each of the few discussions which took place this year, although I myself contributed sufficiently to the conversation, during moments of silence or those lacking in enthusiasm I failed to solicit the opinions of those who had spoken only briefly, apathetically, or not at all. I now realize that including all members of the group and encouraging them to contribute more wholly could have enhanced this experience for everyone involved, and perhaps even led the classroom environment to adapt new ideas which would have otherwise been passed over due to the shyness of their owners. As a result of this failure and its following reflection, I can see quite clearly what potential losses may have occurred in part due to my own apathy in indirectly contributing to the quality of the classroom environment. The valuation of everyone's opinion can truly help to determine how well a classroom environment reflects the needs of its individual students, and it is certainly an unfortunate occurrence if this is not achieved and student are left disinterested and unhappy. In the coming year, I will attempt to apply this realization and feeling of regret to a more positive outcome; that of aiding those in my new classes who are perhaps more reluctant to contribute in expressing their views and being an active part of a healthy classroom atmosphere.

__Domain #3- Recognizing and Solving Errors__

This year, I have excelled in the area of Domain #3 by being able to correct logical flaws in my work. I feel that in my work I am exceedingly thorough and if I do encounter a problem, I enjoy troubleshooting to find an answer. Recognizing one's own mistakes in solving logical problems leads to a definite state of mathematical mindfulness, confidence in the given skill, and overall success within a given assignment or environment. This was specifically evidenced this year in my completion of P.O.W. #11, in which I was tasked with solving a logical problem regarding the strength of a beam of wood and its corresponding equation which required analysis and manipulation if the problem was to be solved correctly. Early in this assignment, I made a mistake in my reasoning which I did not catch, however when Dan pointed this out to me after I had already turned the P.O.W. in, I still felt that in order to glean everything that I could from this lesson it was necessary that I go back and not only discuss with Dan what went wrong, but attempt to understand and remedy my mistake using my own reasoning. After a process which consisted of consulting resources as well as analyzing the initial problem, I was able to understand the problem to completion and gain a valuable mathematical skill in the process.

Below is pictured my P.O.W. #11, complete with teacher comments which highlight my original errors and successes:

My work in Domain #3 this year also witnessed my struggles with the application of the concept of precise computations. In our Taxes Workshop, the specific assignment which I will focus upon required that we create a paycheck based on made-up factors of income, and I made several computational errors concerning the calculation of tax withholdings. Due to the interconnected nature of the boxes which needed to be filled in on the paycheck, it ultimately resulted in every computation being inaccurate to some degree. I didn't realize this until a third party pointed it out. Even after this, I still experienced quite a bit of confusion in re-calculating these values due to the fact that I was still trying to work through them quickly and get the project over with. Needless to say, the amount of precision which I was placing into my calculations was minimal, and I only arrived at an accurate answer to the question of what these values should have resulted in in terms of withholdings when I finally consented to sit down and complete this process with attention to detail. I still find that if I do not completely enjoy a project it is very difficult for me to place 100% of my effort into the corresponding work required, however I do believe that as I continue to develop this skill it will sponsor in me a newfound appreciation for the subjects which I am exploring and lead to a more meaningful experience in each and every one of my projects.

Below is pictured the page of my taxes workshop upon which I placed my finalized paycheck. Although my mistakes are not readily apparent here, the interconnectedness of many of the boxes can give one an idea of how great an effect my simple computational errors created:

Below is pictured the page of my taxes workshop upon which I placed my finalized paycheck. Although my mistakes are not readily apparent here, the interconnectedness of many of the boxes can give one an idea of how great an effect my simple computational errors created:

__Domain #4- Reflecting and Synthesizing__

Within this listed category of mathematical habits, I feel that I have demonstrated strength in justifying my ideas and processes within assignments and projects. When I am confident that I have found a satisfactory answer and corresponding process for a mathematical problem, I find that I can clearly and eloquently state what I did to arrive at my conclusions. This ability has helped me find success this year in that I have utilized it in verifying the stability of my knowledge of a given subject, and upon confirming this knowledge, I have built up a base of skills which I have not only referenced time and time again throughout this school year, but will continue to do so as I move through high school. I have explicitly demonstrated this concept in the reflection section of P.O.W. #7, which focused on exploring the rules and mannerisms of exponential/logarithmic equations through tracking earthquake magnitude in relation to intensity. In this reflection, I not only referenced specific points in my problem solving process to back up my claims of realism in my answers and their unseating of my initial predictions, but utilized specific values and logical processes which were present in my reasoning for this same purpose. This supplements my claim of strength in the area of the justification of conclusions within mathematical problems as represented by my use of diverse and explicit evidence as it supports a logical conclusion.

Below is pictured the entirety of my P.O.W. #7 (the above-detailed reflection can be seen at its base):

The area of Domain #4 in which I still encounter challenges is that of connecting abstract ideas to real-world examples. This connection involves the application of theoretical or generalized mathematical concepts to tangible situations, and was present in this year's Taxes Workshop, during which I exhibited a definite weakness in applying what we learned concerning mathematical concepts (such as multiplication and division techniques, certain aspects of addition and subtraction, etc.) which were later to be applied to our own tax calculations. Although these were simple actions and I am exceedingly well-versed in them as they are, the added complexity of the many steps which go into determining tax amounts confused me to a large degree and I had trouble actually applying these ideas to my tax workshop. This was specifically exemplified by my fantasy pay stub which I created for this unit, in which I had to not only calculate my income tax, but several other facets of tax deductions and tax breaks. The concepts which I applied were simple, but when they were combined in several very complex steps riddled with common and possible errors, I had to re-calculate my pay-stub three times before I achieved an accurate result. Working on seeing mathematical generalizations in the light of real-world application would most likely have aided me during this project in that I would have been able to apply my reasonings for my final tax calculation with more clarity and understanding, and I intend to practice this skill so that mishaps such as this may be avoided in the future and I may produce work with more quality and efficiency.

Below is pictured my finalized and accurate pay stub, arrived at after many revisions:

Below is pictured my finalized and accurate pay stub, arrived at after many revisions:

Final Project Reflection

Final Project Reflection

__Final Project Questions and Responses:__

Question #1- Were you able to come up with original ideas for your project, either for the end goal or for smaller steps within your project?

During this project, I was able to come up with original ideas for my project, mostly at this project's commencement. In deciding what to do for this project, my partner and I considered the fact that when analyzing the stock market through the lens of virtual investment and experimentation concerning how differing types and arrangements of holdings would prove profitable for different types of investors, a focus on time versus profit was essential, both in tracking general trends as well as those of individual companies and sectors. In order to address this issue, we came up with the idea to focus on the problem of experimenting with profit, and what arrangements of shares in what quantity would most benefit investors in varying situations. However, this idea required that we diligently monitor and record the trends of the stock market, so instead of doing a simple analysis and recording of these values, we decided to transfer everything contained in our project to Google Sheets, which served as a base from which we could include monitoring tables, arrangements of stocks which we were tracking, and graphs visually demonstrating our findings. On Google Sheets, we could also directly compare and contrast all of our findings, evidencing an innovative new way in which to base and complete a large project.

Question #2- Did you research/apply techniques used by experts?

Throughout this process I believe that I did apply techniques utilized by those in the field of stock brokering. Although my partner and I's information came to us very slowly (in comparison with actual brokering stock monitors), when we began to prune some of our funds we applied principles which we had researched concerning what were the more effective methods of altering funds for profit. The fact that we adhered to professional principles was proven in that the funds to which we applied these strategies were wildly successful when we altered them in accordance with the rises and falls of both market and individual funds, as well as in accordance with what we deemed to be logical estimates of what share-holdings would benefit us in the future versus which ones were less likely to do so.

Question #3- In what ways did you shape your project without input from your teacher?

The ways in which my partner and I shaped our project without input from our teacher were in the areas of using multiple mediums to analyze and present our findings. Although Dan encouraged our research of professional techniques in stock brokering, we completely constructed our own ways of demonstrating what our use of these techniques yielded. My partner and I decided that in order to get the most accurate results from both our tracking and alteration of funds, we needed several layers of analysis, and varying forms of analysis at that. This was accomplished not only through log recordings of our actions and impressions, but tables and graphs which represented our own funds and shares. On top of this, we often referenced internet resources when checking the status of the market in general. The thoroughness granted us by our original idea to use varying forms of analytical media to our advantage drastically improved the accuracy of our results as well as the way in which we were able to apply these results in our experimentation.

Question #4- What is a challenge you tackled on your own? What did you need help with?

A challenge which I tackled on my own was my final analysis of this stock brokering experiment. Prior to the all-school exhibition in which I was to present this project, my partner was no longer attending Algebra 2, and I thus had to engineer a way in which to tangibly present my experimental findings to a broad audience. I understood initially that in order to clearly present what I did, I would have to make my terminology and processes as explicit as possible in order to induce audience comprehension, so I began creating a series of comparable equations calculating the general trends of the holdings which they accompanied which corresponded with the graphs which I had created to represent my differing funds and stock holdings. I also did this for the S&P 500 Index over the course of the time for which we had been working on this project. I then printed out my graphical representations of my varying profits as well as a graphical representation of the S&P 500 over this given time period, and directly connected these graphs with my equations to create a clear overview and analysis of my project for audience members at exhibition.

The elements of this project with which I needed help were the setting up of our Google Sheets stock market monitor. The application which my partner and I utilized required an installation, and in all honesty, my partner did the majority of the work in this installation. In connecting our differing graphs to this singular adherence to the real world of brokering, we also enlisted the help of Dan who had more knowledge in operating Google Sheets than we. However, from both of these situations, I picked up on skills which are essential in using Google Sheets, despite the fact that I did not initially possess them. I am now more confident in designing equations and connecting different elements in this spreadsheet application, and I owe it to the help which both my partner and Dan offered in improving my own skills.

Question #5- From the outset, was your project too ambitious, too easy, or a perfect balance? How did you adapt?

From the outset of my project, I believe that said project was a perfect balance. This project incorporated several mathematical skills, as well as spanned enough time to constantly fill our working agenda. The challenge of not only operating complex graphs and tables in Google Sheets but of harkening back to our class's earlier units dealing with equations and equational trends made my experience in this project stimulating and explorational at the same time. This being said, the main goal of this project was not adapted over time to better suit our needs, and I feel that my partner and I were wise in selecting a project which included these aspects of mathematics.

Question #6- What principles of mathematics did you apply to your project?

The principles of mathematics which I applied to my project were those of representing a given problem using several mediums as well as generalizing information represented graphically through the use of equational representations of trend lines. The first listed of these principles was included in our project throughout its duration, and consisted of representing our varying stock fluctuation and profit information with mediums that made it easier to view and track, such as graphs and tables which corresponded with the profits which we were trying to track in relation to time. I believe that this skill is important in that it has improved my ability to comprehend a math problem or set of mathematical information in all its forms, a comprehension which indicates a deeper understanding of the problem itself.

The latter listed principle was present at the conclusion of this project, when I wanted to sum up all of my information using simple measures that would appeal to an audience. In order to do this, I created linear equations which summarized the trends of each set of graphically-represented data, and for this had to incorporate a knowledge of the relation of an equation to data. I exercised skills which we had utilized at the beginning of the year and thus had not only to review earlier units, but to re-acquaint myself with terminology, thus stabilizing my grasp on these simple but necessary concepts.