Letter to Math Analysis Honors Students (2013-2014)

Dear Future Math Analysis Honors Students,

          For many much of my classmates this year has been filled with struggles, lots of due dates, but most importantly, PROGRESS. While some may feel that math is easy and others might find it on the challenging side, this math class is like unlike any other math class you will ever have -- and I mean that in a good way. Your year in math will be functioning in a way that will make you responsible for your own learning. You will be tested on how well you manage your time, how seriously you take every piece of work (including the WSQs), and how determined you are to learn the math. Will certain units may be easy, you will be thrown harder units as the year progresses and you can not simply rely on your prior basic algebra skills to solve. In this classroom, you must carry yourself with integrity. Yes, cheating on your PQs and PTs will get the job done, but when test day come, you will not have mastered the Unit as well as you thought. Some may think that they can cram and do quick review and some may even get away with it, HOWEVER, on the final you will not have it this easy. You will suffer from your lack of comprehension of the unit and that will definitely hurt your grade. There will be times when you feel extremely overwhelmed and frustrated with the work load, but its best to throw those feelings aside and just get through the work rather than waste X amount of time complaining. Sometimes you may feel like that one WPP is a waste of time and is meaningless, but it really will help you better understand that concept. As I said before, this all depends on how motivated and focused you are. Moreover, do not expect something you do not deserve. For example, if you cheated on your PQs or PTs or did not try hard on the other work that was assigned to you, don't expect to get an A on the test-- be reasonable. This is one of those classes where hard work counts. Its not just memorizing, it is UNDERSTANDING. The flipped classroom may take some time to get used to, but with time, it will become very easy. Mrs. Kirch has given you all of the tools (videos, SSS Packets, WSQ Charts, WSQs, etc.) it is up to you to use them, and use them wisely. With the flipped classroom you can learn at home and bring your questions and do your work at school. Of course, as I have mentioned before, you MUST use your time wisely. If you spend half the class period pretending to do your work, than for that night not only do you have to watch the videos for the new concept, but you also have to do the PQs for the previous concept. DON'T GIVE YOURSELF MORE WORK. In class you actually get to have one on one time with the teacher and ask your classmates for help. You can also get ahead when you know that you have a huge project coming up or an essay due soon. Personally, because I came into this class with an open mind I easily adjusted to the flipped classroom and my work ethic drastically increased. Lastly, in a regular classroom you probably did a few math problems for homework, the teacher lectured in class, then assigned homework on the lesson. With the flipped class room, you move past the normal homework practice problems and you literally interact with math. You make your own problems, your own videos on how to work your own problems. You get to make tests and you experience math in new ways. You finally move past knowing a concept, you actually realize its importance, its derivation, and how that comes into play with the real world. In the end, it is all up to you. Mrs. Kirch can not force you to do the work-- it is up to you to decide to do it. It is your responsibility. That does not mean you are alone. You still have support at school. That is nice to know, isn't it? Good luck, everyone!

- Yours Truly, Arlene S. (Period 6 - Princess table D with Tien N.,  

 Tiffany T., Grecia G., Miriam L., & Jessica R.)

Unit V. BIG Question

• What is this video about?   This video goes over how to derive the difference quotient. It also describes the resemblances of small secant lines to tangent lines.

• What does the viewer need to pay special attention to in order to understand the concept? Understand that a secant line hits a graph at TWO points whereas a tangent line hits the graph at only ONE point. Moreover, the smaller the h (delta x, dx) the more the secant and tangent lines resemble one another. Lastly, solving the difference quotient is like evaluating the derivative except for the derivative we take it a few steps further by using the "Limit Definition of the Derivative." In the video I refer to this in mathematical terms: the limit as 'h' approaches 0.

Unit U. BIG Questions

Unit U. BIG Question #1

 

Unit U. BIG Question #2

 

Unit U. BIG Question #3

Unit T. BIG Question #4

• What is this video about?   In this video I explicitly focus on why sine and cosine do NOT have asymptotes and why the other four trig graphs do.  I also go over the basics of the trig functions when referencing the Unit Circle.

• What does the viewer need to pay special attention to in order to understand the concept? Although I don't quite center the attention on the visuals of the graphs, they are a big part on why I made this video. Asymptotes are what are drawn on the graphs and I simply made this video as a rationale to demonstrate the purpose of why not all of the trig graphs have them (redundant, I know! But that is how important this is if I am emphasizing it twice!!!)

Unit T. BIG Question #3

• What is this video about?   This video  goes over why a "normal" tangent graph is uphill where as a "normal" cotangent graph proceeds downhill. Besides their differences, I also detail some of the similarities between tangent and cotangent. 

• What does the viewer need to pay special attention to in order to understand the concept? Pay attention to the use of the unit circle ratios of sin(x)/cos(x) for tangent and cos(x)/sin(x) for cotangent. By understanding this, you will see why we have asymptotes where we do. Also understand how asymptotes play into the why "normal" tangent graphs are uphill and "normal" cotangent graphs are downhill.

Unit T. BIG Question #2

• What is this video about?    In this video I give an in-depth clarification on the relationship between sine and cosine parent graphs with tangent, cotangent, secant, and cosecant. I also highlight asymptotes and their role in the parent graphs of these trig function. 

• What does the viewer need to pay special attention to in order to understand the concept? For best results, be sure to pause and rewind as needed throughout the video. Don't be afraid to give your brain time to process all of the information I am throwing at you!!! 

Unit T. BIG Question #1

Unit T: BIG Question Part 1a.

 

• What is this video about? 

    This video gives a thorough explanation for the reason that sine and cosine have a period of 2pi, whereas tangent has a period of simply pi. With this information and knowledge, one will be able to understand why the graphs (whether they are parent graphs or shifted graphs) of sine, cosine, and tangent look as they do.

• What does the viewer need to pay special attention to in order to understand the concept?

     Pay attention to the importance of the Unit Circle and how that relates to why the pattern of the periods are the way they are. 

Unit T: BIG Question Part 1b.

• What is this video about? 

    This video goes over how sine and cosine have amplitudes of 1 and why the other trig function do not have amplitudes. I do cover asymptotes as well so that is basic review from previous units. The Unit Circle will be very prominent as far as amplitudes and asymptotes go.

• What does the viewer need to pay special attention to in order to understand the concept?

      When I say "You'd think their graphs would look the same..." I am comparing the cotangent parent graph to the tangent parent graph. [7:38 of the video]. Moreover, when describing cotangent I say "amplitudes" but I really meant to say "asymptotes."  Lastly, please do pay special attention to how the Unit Circle relates to the parent graphs of each trig function - I did show it many times and mentioned it countless times for a reason!!!

Student Video #5: Unit S. Concept 7. #5

• What is this video about? 

    This video goes over how to solve equations with half angles. With it, we incorporate the use of the half angle formulas. Moreover, this trigonometric concept allows us to further explore the utilization of the Unit Circle.

• What does the viewer need to pay special attention to in order to understand the concept?

      You must be able to incorporate your knowledge from past units as well, not just simply Unit S. For example, in this particular problem I was able to identify that I would need to use the Pythagorean Identity for sin^2(x). For these problems as well as this unit, you need to able to focus on what you need to do in order to solve. If that means digging through your brain for formula or proofs from past units, than so be it.     

(Unit S. Concept 4.) Half Angles Formula and it's similarity with (Unit R. Concept 1) Sum Formula

Solving the Sine, Cosine, & Tangent of 75* with Half Angle Formula

• What does this picture show?

          This picture demonstrates how to solve for a half angle using the Half Angle Formula. Although this looks difficult and challenging, it really isn't--it is just plugging in and using simple algebraic skills to solve. Please understand that this is only one of the methods used to solve for the sin(75), cos(75), and tan(75); one of the other method is demonstrated in the picture below. I have done this problem step-by-step so that it is evident how to use the Half Angle Formula in relation to half angles of the Unit Circle.

• What does the views need to pay attention to in order to understand the problem?

             The quadrant in which we determine whether sine and cosine will be a positive or negative value is dependent upon the 'u/2' NOT the 'u'. The 'u' value is twice as much as the 'u/2' value. In this case twice of 75* is 150*. Although 75* is not apart of the major angle values of the Unit Circle, 150* is. The cosine and sine used in this problem come from the ordered pair for 150*. Knowing these key parts of information along with the Half Angle Formula allow us to solve for the trig function values of half angles like 75* as shown in this example.

Solving the Sine, Cosine, & Tangent of 75* with Sum Formula

• After having solved for the sin(75), cos(75), and tan(75) using these methods, how do you know that the answers are the same?

         Without even plugging anything into your calculator, you automatically know that although the answers look different between the sin(75), cos(75), and tan(75) because different methods were used, they are same. This is because the sin(75) can only be the sin(75), cos(75) can only be equal to cos(75), and tan(75) can only be the same as tan(75). This sounds very obvious, but many people assume that because the way the answers appear are different from one another that the they are different values, but they are not. Once plugged into your calculator, you will see for yourself that the approximate answers for all of them are the same. This is what the picture below shows. 

Student Video #4: Unit S. Concept 3 - #5

• What is this video about? 

     This trigonometric video lesson goes over how to solve problems by using the     power-reducing formulas.

• What does the viewer need to pay special attention to in order to understand the concept?    

      Please understand that we substitute the power-reducing formula to the entire given trig function, not just the (x). Moreover, note that for your final answer, you want the highest power to be 1.

***DISCLAIMER: These problems look a lot scarier than they are. With practice, they become super fast, easy, and even fun! :D

Student Problem #10: Unit R. Concept 3

*What is this problem about?

This problem in particular depicts how to use the sum formula in solving the trig of an inverse trig functions. Although this may sound big and scary, it really is NOT! As you can see from the above problem, the inverse trigonometric function is inside the brackets (thats the inverse trig part if that wasn't obvious enough.) The regular trig part is what is outside the problem. Now that you can distinguish each part, it doesn't sound so scary now, right? 

*What does the viewer need to pay close attention to in order to do the problem correctly?

Please understand that the way I got my coordinate pairs was through my referencing the Unit Circle. Yes, the Unit Circle is being used again! Moreover, for tangent, I realize that I could have used the ordered pair of (-1,0) as well because 0/-1 is also 0, however, I don't need to. I am picking the values closest to quadrant I. This has absolutely nothing to do with some sort of mathematical process, its just the way we solve these kinds of problems for my Math Analysis class, however, feel free to solve using that as well. :)

Student Problem #9: Unit R. Concept 2

What is this problem about?

This photograph shows how to the sum and difference formulas (look at Unit R. Concept 1 for reference) are integrated in the use of solving for right triangles.

What does the viewer need to pay attention to in order to do the problem correctly? 

Be aware that these triangles do lie in different quadrants! Because of this, the problem is still solved the same way as you normally would if both triangles were in the same quadrants, however, note whether sin, cosecant, cosine, secant, tangent, or cotangent are positive or negative [this depends on which quadrant we refer to.]  Please note that triangle drawn on the left is the 'u' and the triangle drawn on the right is the 'v' [my penmanship wasn't all that great here]. Be sure that you chose the correct formula for what is being asked. I can't stress that enough. If you use the wrong formula, evidently you will get the wrong answer!

Student Problem #8 :Unit R. Concept 1

Sum Formula:

*What is the purpose of this picture?

This picture demonstrates the use of the sum formulas for sine, cosine, and tangent being put to use to find exact values for these trigonometric functions.

*What should you pay close attention to in order to understand the concept?

Understand that 450* is not a normal angle from the unit circle. Here, I have added two values FROM the Unit Circle [of the normal reference angles to 30*, 45*, 60*, 90*, etc.,] that amount to 450*, those being 300* and 150*. (* = degrees symbol)

Difference Formula:

*What is the purpose of this picture?

This picture depicts the use of the difference formulas for sine, cosine, and tangent being put to use to find exact values for these trigonometric functions.

*What should you pay close attention to in order to understand the concept?

Due to the fact the 450* is going on to a second revolution of the Unit Circle, I have to use 720* because its a large value and can subtract another value (270*) to make 450*.  >>>NOTE: 720* is exactly two entire revolutions of the Unit Circle and its reference angle is 360* thus the ordered pair is the same. 

***CHECK: check if you are correct by plugging in the sin(450), cos(450), and tan(450) into your calculator. 

>>>NOTE: because sine is 1, cosine is 0, and tangent is undefined, what does that tell you about this angle???

         ***It is a quadrant angle!!! The reference angle of 450* is 90* :)

Hope this helped! If you need any clarification or for me to further describe verbally leave a comment down below. If you would prefer a video instead, also leave a comment. :)

Pythagorean Identities {Part 1 of 3}

Pythagorean Identity #1

The example below depicts how this identity came about. Using an angle, in reference to the Unit Circle, you see how I prove that sin^2(x) + cos^2(x) is ALWAYS going to equal to one.

Pythagorean Identity #2

This is derived in reference to sin^2(

𝞡

) + cos^2(

𝞡

) = 1 and dividing it by cos^2𝞡

Pythagorean Identity #3

This pythagorean identity is also derived in  reference to sin^2(

𝞡

) + cos^2(

𝞡

) = 1 and, but instead you are dividing it by sin^2𝞡

Unit Q. Concept 2 Problem Solving Methods {Part 2 of 3}

Solving Method #1 (Unit Q- Using Identities)

Solving Method #2 (Using the Unit Circle)

Solving Method #3 (Unit N- Using Right Triangles)

Unit Q. Concept 4 (Levels I,II,III, & IV) {Part 3 of 3}

Math Analysis Reflective Blog Post

1. How have you performed on the Unit O and P tests?  What evidence do you have from your work in the unit that supports your test grade (good or bad)?  Be specific and include a minimum of three pieces of evidence.

RESPOND HERE: For the duration of the past month, I have scored fairly well on my tests. Unit O I got an 82%. Despite the fact that our WSQ's were not signed by Mrs. Kirch, I continued to do all of the work. The test didn't consist of many problems and so each problem was worth a lot of points. The problems that I did incorrectly were some of the few that were high in value. Unit P I received 100%. This just goes to show that hard work pays off! At first I had trouble with concepts 6 and 7, but with practice, those types of word problems became easy. My WPP #13 and #14 demonstrate [through my detailed explanations] that I understood the concepts. I also practiced additional practice problems from the text book. Furthermore, I helped a few of my peers for this unit and while teaching them, I was also reviewing the material.

2. You are able to learn material in a variety of ways in Math Analysis.  It generally follows this pattern:

→ Your initial source of information is generally the video lessons and SSS packets followed by a processing and reflection activity via the WSQ
→ individual supplemental research online or in the textbook before class
→ reviewing and accessing supplementary resources provided by Mrs. Kirch on the blog
→  discussion with classmates about key concepts
→ practice of math concepts through PQs
→ formatively assessing your progress through concept quizzes
→ cumulatively reviewing material through PTs
→ Final Assessment via Unit Test.

Talk through each of the steps given in the following terms:
a. How seriously do you take this step for your learning?  What evidence do you have to support your claim?  Make sure to make reference to all 8 steps.
b. How could you improve your focus and attention on this step to improve your mastery of the material?  What specific next steps would this entail?  Make sure to make reference to all 8 steps.

RESPOND HERE: 

• STEP 1: I hold this part of my learning the concept with great graveness. I always put a lot of effort on my WSQ's because I want to comprehend how well I know the concept. If I have do a poor job on this step, not only to I not have clarity about what I am dealing with (as far as the math concept), but I am also left with a guilty conscience. I always highlight important information and write down the key pop-ups that you include during the video. I tend to finish the additional practice problems [depending on how much homework I have that night]. I could improve on this by watching your entire video lesson. The times when we have to complete extra practice problems, I don't watch the entire video. I skip to the answer to check my accuracy, but that is as far as I go. I should take the time [even if I have a load of other homework] to watch them. It is helpful to verbally hear your thought process in solving the problems. Looking at the big picture, I will develop the skills to explain the steps to solving for the problem [for which ever concept it should be] and move me further to mastering the concept.
• STEP 2: A lot of the time, your SSS packets along with the videos for each concept provide us with more detailed information than the textbook ever could. It is not right to say that I don't take this step seriously, because I do understand the value of having tools besides being dependent on the tools you already hand to us. If I need further practice, I do use the textbook for the countless problems they have. I could improve my attention to this step by completing more practice from the textbook on weekends. It wouldn't kill me to open the textbook, turn on the TV, and for an hour watch a re-run of glee and solve problems during the commercial. This would add to my mastery of the material by doing additional work to the extra practice I normally do. In the end, with more practice comes better results [mentally, on tests, and with the creation of WPP's].
• STEP 3: I don't always look at the supplementary resources. It is as simple as this, I am swamped with other homework and struggling to finish so that I can at least have a solid five hours of sleep. Thus far, I haven't needed to use them because I understand the material and I can do well without the extra. Despite this, I do appreciate the fact that you take the time to make these supplementary playlists. Everyone has their own way of recieving further mastery of the material and while I may not use it, this playlist may be of great use to someone else. I do look at these when we have to (ex. Fibonacci's sequence), but other than that I don't. I will try to pay more attention to these, but I have already seen great improvements on my comprehension of the concepts by helping my peers and making WPP's.
• STEP 4: I have found this to be VERY important to my understanding of concepts. There are always days when at least one of use from my group of three are confused about something and we talk it out. When we discuss, we find that the person who is verbally explaining how the concept works puts things into persepective for the other. Moreover, that person who is explaining further masters the concept because she knows what she is talking about and how to explain it so that it makes sense. We do put a lot of our attention on this and I am not sure how to improve it (even though there is always room for improvement). If all three of us are clueless, or if only one of the 3 grasp what is going on, we ask Mrs. Kirch for further explanation and then we try to explain what she said to ourselves in terms of our own words and this always seems to help us in gaining knowledge about the concept.
• STEP 5: I love, love, LOVE the PQ's! They are so useful and helpful and they are the practice that we all need! Frankly, I don't understand why some students do a crappy job and just scribble an answer or copy the work because when testing comes, they will not have adequently prepared and not do so hot. When they get a C or below, they will have to re-take that test and then comes their complaining about having to get a required hours of tutoring. [If the people really tried and still had trouble with the unit and asked you for help and still didn't do well, then that is a whole other circumstance.] PQ's are what I need to master the concept. It is as simple as this old saying: "Practice makes better." Yes, this is a major cliché, but it is so true! How can you expect to amster something if you don't put in the effort? You can't pray at night and sit there, God is only going to meet you halfway so you have to take a step by completing the necessary work. Besides this, there isn't really much of an excuse because we do have time in class to finish our PQ's. I like to think that I focus a lot of my attention on this. If you have any suggestions on how I can improve. please let me know. Most of the time, the quality of my work speaks for itself and I don't have an issue with integrity. In fact, the unit circle is not my BFF, but integrity and I go way back! ^.^
• STEP 6: Another thing that I enjoyed (yes, I did enjoy this!) were the quizzes. Now that they are gone, I do miss them because they were an indicator of how well I understood the concept. They differed greatly from the PQ's and the PT's because the quizzes put me in a testing atmosphere and gave me an idea of what it would be like when testing time would come: how long did I spend on solving a problem? should I speed up or can I take a little breather? what materials would I need (highlighters, ruler, etc.)? how well did I understand the concept? All of these factors would always come across in my head and they prepared me for the unit test. I could improve my attention on this step by re-taking the some of the few quizzes that I didn't get an 8 on. Normally I have been pretty good about this, but there were weeks when school and other tests to study for, other homework to study for, community service events to go to, family consumed my mind and pushed this priorty aside. If I have put in more of an effort on this, I know that I would have a higher A in this class. Sadly, we will not have these types of assessments anymore.
• STEP 7: The Pt's were another thing that I adored. Yes, it's more work, but its also more practice. It always gave me a chance to go back and review the concepts that I hadn't touched on in a few days because simultaneously we were prgressing forth with the unit. I was able to understand where I lacked comprehension and where I had mastery. I was able to target these components and practice even more So that when the test came, I would be confident in that I was fine with the unit. Improvement wise, I actually like doing my PT's all at once so that I can have a fresh look at all of the concepts and make sure that I can do them rather than doing the PT's along with the PQ's. On the other hand, I have come to realize that doing the PQ's and PT's simultaneously along with each other is a major time saving technique so I tend to do this. [Now, our Pt's and PQ's are incorporated into one]. I now look at the textbook for additional problems to do for practice and this helps me grasp the concepts well.
• STEP 8: The final assessment is what put all of my hard work into play. It validates to me that yes, my WPP's helped me really comprehend what I was doing, or yes, all of the time I spent on math homework was worth it, or yes, I am glad I helped him/her on the concept because I am repeating that thought process in my head. Of course, there are weeks where life or other subjects get in the way, but I tend to be pretty well on my tests. [This is my comparison to myself, not the people in my group or to anyone else in the classroom.] It was easy for me to accept the 82% on the Unit O test because I knew that it was challenging and I had other things occupying my mind. It was also easy for me to accept the 100% on the Unit P test because as challenging as the last couple of concepts were, I worked me patutty off to understand them [As I stated before, this was shown on the quality of me WPP's, PQ's, PT's, and quiz scores.] Yes, there are times when you work super hard and then don't get the score you want because you overlooked a few things or for some reason, you and the concept weren't clicking, but at least you know that you did all you could do. You can be proud of yourself and not have regrets. All of the factors of the above seven steps I do well on following with integrity and a positive attitude that I will understand it soon. It is because of that, generally speaking, I walk in on testing day confident that I did all that I could and come what may, I will be proud of myself.

3. Reflect on your learning this year thus far by considering the following questions:
a.  How confident do you generally feel on the day of a Unit Test?  Give evidence and specifics to back up your answer.
b.  How well do you feel you have learned the math material this year as compared to your previous years in math? Give evidence to support your claim.
c.  How DEEPLY do you feel you have learned the math material this year as compared to your previous years in math?  Give evidence to support your claim.
d. Do you normally feel like you understand the WHY behind the math and not just the WHAT/HOW?  Meaning, do you understand why things work, how they are connected to each other, etc, and not just the procedures?  Explain your answer in detail and cite specific evidence from this year.
e. How does your work ethic relate to your performance and success?  What is the value of work ethic in real life?

RESPOND HERE:
a. On the day of the Unit test I feel confident in what I know and that I have done all that I could have possibly done in order to well on the test. I have finished all of my PQ's and PT's as well as other practice problems from the textbook. I have asked questions to either my peers or Mrs. Kirch when I was confused on something. I have gone for help. I have helped my peers. I have done a good job on my WPP's I have gotten all 8's on the quizzes. Nonetheless, I am still bit nervous because my fifth period class is far away so I have to run to your class in order to try to get seated early and start the test ASAP so that I give myself enough time to finish a bit early and go back to check my work.
b. Elementary school all the way until the 8th grade I was exceptional at math. When ninth grade came, I did well on Geometry which was shocking because I was so used to being great at math. Sophomore year I understood what I was doing when I did it. I never felt that I got enough practice. When we did our homework, that was the practice in itself. I didn't go past that because I figured that once I knew the lesson, that it was drilled in my head and that it wasn't possible for me to forget it. Little by little, I would go into the test day not feeling prepared because I didn't go past my homework. I wasn't used to going past what was given. This year, that has all changed. I have never walked into a math class feeling this prepared and its really nice. I don't have to worry about anything because we are constantly practicing the concepts within the unit while simultaneously reviewing past concepts from past units.
c. I feel that my work ethic and mastery of math have greatly improved from the past. I can remember how to do problems from the very first units without looking at our SSS packets for help because that is how well I have comprehended it and practiced it. For this, I greatly thank Mrs. Kirch for all of the hard work she puts into giving us all of these resources and being consistent with the flipped classroom. In past years, as I stated before, I knew the material well enough for the test day and the day after the test, the week after the test, the month after the test, it left my brain (not everything, but some of the solving processes became very, EXTREMELY vague to me). This doesn't happen to me anymore. I feel very proud of this! :D
d. I feel that the reason I DEEPLY understand the material is because I understand the HOW. Before it was just memorize, memorize, memorize. Now that I can derive the meanings of some of the formulas, its not memorizing the formula, its more like knowing the background of it and being able to go based off of what I know to get to the formula. The songs that you have provided for us are fun and catchy and I don't think that I can ever forget them and also help me remember the formulas.
e. If I were to go solely based off of my work ethic, it is quite evident that I spend a lot of time on the work you give us. I am a neat freak and that adds to the quality and correctness of my work. I like things to be as accurate as possible and so I always check my answers. I think that my work ethic demonstrtaes how serious I take this class. It also shows why I tend to perform well on the quizzes and unit tests we take. Moreover, it is a good indicator as wo my understanding of the what, the why, and the how of each concept. I feel that it is because of my work ethic that I deserve an A in this class. After all, the two go hand in hand.