Student Problem 3 Unit R Concept 3

What is this problem about?
This problem goes over an example from Unit R Concept 3 which deals with trigonometric function of an inverse trig. function. When solving this we also have to use the sum and difference formulas to help prove that these values for the functions are true. This problem also makes use of previous units such as the unit circle to figure out the actual ordered pair for the trig functions used. Also in this concept our knowledge of the trig functions is able to be put to use such as knowing that sin is equal to the y value of the point and cosine is the x-value of the ordered pair used. 

What does the viewer need to pay close attention to in order to do the problem correctly?
In order to solve this problem correctly the viewer must pay close attention to how we are able to derive u and v from the original equation give; canceling out the inverse to just get the value on one side. This is the most important part because we then need to be careful when identifying our ordered pair that is used. Another spot where kids make mistakes is making sure that they use the correct equation to plug the values into. It is important that the viewer is familiar with the sum and difference formulas to pick the correct one when substituting and simplifying. 

Student Problem #2 UNIT R CONCEPT 2

What is this problem about? 
This problem goes over one from Unit R Concept 2 that is dealing with using sum and difference formulas when given values of right triangles. We have basically put together all of our knowledge from the unit circle, right triangles, SOHCAHTOA, and sum/difference formulas in order to solve a problem for this concept. This problem is mainly dealing with recognizing/identifying the trig functions with the limited information given of the triangle so that later we are able to use this information to solve for the exact values using the new sum and difference formulas we have. We then use substitution  to plug in the values into the equations, which leads us to finally simplify the fraction and get our final answer. 

What does the viewer need to pay special attention to in order to do the problem correctly?
When solving a problem like this from unit R the viewer needs to pay special attention to how we are able to identify the values of each trig. function from the two separate triangles. It is important that we do so becuase if you identify one trig function incorrectly that can result in your whole answer being wrong. The viewer also needs to pay special attention to the equations used to solve for the exact values of the trig. functions. Also, it is imperative that one is very careful when solving and simplifying, making it even more necessary to check your answer afterwards. It is also importnat that the viewer realize the difference in the quadrants in which these trig. functions fall in due to their signs (+/-).

Student Problem Unit R Concept 1

Using the difference Formula

using the Sum Formula

• What is this problem about?

This picture goes over a problem from Unit R concept 1, dealing with finding the exact values of sums or differences. This is also an example of finding exact values for angles that aren't on the unit circle, but rather now we are able to use our knowledge from the angles given on the unit circle to solve for other angles that are not put on the circle. By using the sum and difference identities we are able to solve for these non apparent unit circle angles and use the other angles given on the unit circle to help figure out the exact values. We basically are trying to figure out which angles on the unit circle can add or subtract to the angle in which we are solving for, and in this concept we can see that there are various ways in which we can go about solving these problems. 

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

In order to solve the problem correctly the viewer needs to make sure that he/she is very familiar with the unit circle. Due to the knowledge we take away from the unit circle it is imperative that we know the ordered pairs for the magic five and are familiar with identifying sin, cos, and tangent using the ordered pair that corresponds with the angles used. It is also important that we are all familiar with SOHCAHTOA and what that stands for in order to identify the trig functions for each of the angles used. Also, very important is that we need to focus on memorizing the sum and difference formulas so that we don't have to keep referring back to the notes. The last crucial thing one must do to solve correctly is checking your answers and also making sure that it is all plugged in correctly. 

Unit Q Concept 2 using identities and right triangle

Unit Q Concept 2 Example Explained using identities




Same Example worked out in the video above but now using right triangles:

Concept 4 Problems Each Level Explained

Derivation Of Pythagorean Identities

Derivation of Pythagorean Identities:

1. The illustration shown above helps support the derivation of the two Pythagorean identities in which we come up with. To help explain this process of deriving I am going to begin by explaining how the pythagorean theorem is derived from the unit circle as shown above. So as we know from our previous knowledge of the unit circle we are allowed to draw a right triangle. This allows for us to then label the sides of the triangle with different variables, which leads us to write the pythagorean theorem a^2+b^2=c^2. Being that we know that this triangle is within the Unit Circle we can use previous knowledge of how the hypotenuse will equal 1 which in this case is represented by the letter c. In this example, the side of a will be equal to cosine theta and side b will be equal to sine of theta. After figuring out which variable stands for what trig function we can see how the pythagorean identity is derived, because the values are substituted into the original equation (a^2+b^2=c^2) to produce the identity which is cosine theta^2+sin of theta^2=1. It is easy to follow how this equation was derived from the original pythagorean theorem in the illustration above. Then, to derive the other identities from the Pythagorean identity all it requires you to do is divide. For the next pythagorean identity what you have to do is divide the original equation by cosine^2 theta. As shown in the illustration above, when doing this process in number 2 the cos^2theta from the original equation will cancel out to just leaving a 1 and the sin^2 theta divided by a cos ^2 theta could be substituted in as tan^2 theta. We can also see how the 1 divided by cos^2 theta turns into sec^2 theta, which is because it is a reciprocal identity. When gathering up this information and putting the equation back together it is easy to understand how we come up with the identity to being 1+tan^2 theta=sec^2 theta. The last pythagorean identity follows the same process except now we are dividing by sin^2 theta from the original Pythagorean identity. When dividing the entire equation by sin^2 theta we are able to recognize other identities, which allow us to substitute such as cos^2 theta/sin^2 theta that equals cot^2 theta. After getting this we can easily see in example 3 how the sin^2 theta cancels out to leave a +1, and how csc^2 theta gets substituted in for 1/sin^2 theta which is a reciprocal identity. Once we have simplified this equation, we then are left with our third pythagorean identity which is cot^2 theta+1=csc^2 theta.

Math Analysis Reflective Blog Post

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.
On the past Unit O and P tests I have performed very well. Even though I could have done better in reaching higher quiz scores for the concepts I was very confident going into these tests due to the extra work I did, such as the PQ's and PT's. To also grasp the understanding of some of the more tougher concepts I also attempted in completing extra practice problems that were available. I feel like without the WSQ checks I have been doing the same and still feel on task because I complete my work as scheduled on the WSQ chart.

RESPOND HERE:

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.

• I take this step of watching the video lessons, filling out the SSS packets, and WSQ, very seriously because in order to basically learn the concepts these three steps are what it consists of. If I don't complete these steps I know that I will be so confused in knowing what to do in these concepts which is why it is important that I make sure to complete each of these steps knowing that it will benefit me in the end.
• Honestly, this supplemental research online and in the textbook are something in which I don't take advantage of. I feel like just by completing what is required I have enough knowledge to go on and figure out the concept for myself. I could make more use of the internet when stuck on a concept but I choose to just discuss with my group members and if not ask Mrs. Kirch for help.
• Reviewing and accessing the resources provided by Mrs. Kirch on the blog is very helpful and necessary to my learning in this class because it helps further my understanding about each concept. For instance, if I am stuck on a concept I am able to refer back to the blog for extra practice videos or problems and learn how to do the problem. This is a critical part to my success in this class because this is always where I refer to, to become more informed of what I am learning. It is a source where I go to for extra help and extra practice.
• Discussion with classmates is also a very imperative part to my learning in this class because this is where we all are able to share shortcuts and knowledge about what we have learned. This is a very important step in all of our learning processes because we can share out loud with each other and learn each others way of doing things.
• The PQ's are also a very imperative part of my learning in this class because when PQ's are done in class it is required that we seek help from our group members and even Mrs. Kirch. This is very helpful when learning new concepts because when we learn a new concept we start our practice problems and are able to ask for help when I get stuck. This is important to my success in this class because I am able to ask for help when I need further explanations of concepts.
• Quizzes are also a part of my learning process that helps me succeed in this classroom because it is important that I assess my self so that I know where I am struggling at. These quizzes are very significant in my learning also because I can ask questions and feel confident about what I am doing, so that when the chapter test comes I am able to score high and know what I am doing.
• Reviewing things in my PT are also very important in my learning because when doing my PT I am able to review and get extra practice in. This part is very important in my learning process because it gives me an overview of the whole unit, with problems of every concept, so that by the time the test comes I am able to complete every concept on my own.
• The final unit test is another assessment of my knowledge learned throughout which tests my skills. This is an important part of me learning in this class because I am able to put everything I have learned to work and finally put everything together. Obviously, this is a test of all my knowledge which helps me once again to review and see how well I am doing in each unit.

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.

• I think that in order to improve my focus and attention on this step of completing the sss packets and wsq is that I should manage my time more wisely. Managing my time is very important in this case so that I can spend more time on this step and think things through thoroughly. The specific steps that this would entail is making sure that I am not distracted so that everything gets done with maximum effort. 
• I could improve my focus and attention on this step of researching online and what not is that I should actually make use of the internet. I feel like I don't make use of it as well as I should, which can really help me if I do need further help. 
• I could improve my focus and attention on this step of accessing the blog to improve my mastery of the material by actually making time to search for more sources. I feel like I use the blog well enough already to help my mastery of the material which does help me a lot. The specific steps that this entails is that I spend time actually looking up resources to help me master this material. 
• The discussion part with my classmates can be improved by engaging more in these discussions myself. I need to make more of an effort to explain myself and share my knowledge with the people around me. I just need to focus more on making an effort and improving myself to actually discuss things with my peers. 
• The PQs we have to complete can also be improved by making sure I make use of my time in class. I need to make sure that each day in class I don't waste my time, rather I need to make use of my time so that if I need help with a concept I am able to ask questions. The steps that this would entail is that I need to be able to make sure that everything is completed on time and I don't fall behind. 
• My quizzes can also be improved by again managing my time wisely. I need to make sure that my quizzes should be taken a day after we learn each concept so that I have time to retake if there is something in which I see I am struggling with. This is important in my mastery of the material because I need to know and make sure that I understand everything before taking the unit test. 
• To improve my focus and attention on doing my PT's I feel like I need to make sure that this actually gets completed as I continue on through the process of learning the entire unit. Instead of leaving the practice test to the last day before the test, I feel like completing it as the time goes on it is important to my mastery of the material. This allows me to ask questions as well and seek help where ever I am stuck on. 
• I think my focus and attention for this step, the unit test, is pretty good the way it is because I do prepare myself and take time to actually study. I make sure that I am confident in every concept that is needed to complete the test before entering class on test day. This is very important to me so that I can succeed in this class and learn more. 

RESPOND HERE:


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.
On the day of a Unit Test I feel very confident because I make it a point to make sure that I am confident in solving problems from each concept. I make sure that I am prepared and ask questions before the test so that by the test day I am very confident. I also feel like with all the extra practice I complete it is hard not to feel so prepared and this has really helped me feel confident in what is on the unit tests.
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.
I feel like this year this new version of teaching math has been a great experience. As compared to my previous years in math I feel like now math is more of a class where we discuss and explain things with each other. I love that we now learn things on our own and this year it has taught me how to interact more and become involved in learning math. I have learned a lot in this class, beyond just simple mathematics. I have learned how to communicate and explain verbally how to explain concepts that we learn. Also, I feel like I have learned so much by how much time this flipped classroom gives us to ask questions and get help.
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.
I feel like this year I have been able to understand math on a deeper level. For instance, I have learned how to create my own math problems and tie them into what we are learning. I have also been able to explain and learn how to better my communication skills about math. Now, math is much more than just solving problems and learning step by step processes like previous years. It is now something in which we throw in all the things we have learned to create and communicate our understanding of the things we learn.
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.
Now that I have been apart of this flipped class room for this school year I feel like I do understand the why behind math rather than just the simple steps taken to solve the problems. It is easy for me now to pay attention to this important question because in math analysis it is not about the what and how. This class enables us to think more and think of the why by allowing us to communicate and share our knowledge with each other about each of the concepts. For example, in each unit we are required to learn we complete discussions within our groups which I feel help us discover the why in the concepts. I also feel like the blog posts that we do for some of the units/concepts allow us to also discover the why and further our knowledge about whatever we are learning. 
e. How does your work ethic relate to your performance and success?  What is the value of work ethic in real life?
My hard work ethic inside and out of class is very reflective to my performance and success because I make it a point to have each concept mastered before entering the test day. Meaning, throughout the entire unit I complete as much practice as I need to understand the concepts in each Unit. If  I have trouble with a concept, I then refer to my group and ask for help until I understand and am able to feel confident. Making sure I master the concepts and everything explain my work ethic well because until I feel confident and strong in doing a problem on my own, I take advantage of the practice provided so that I expand my knowledge and understand it well before taking the test. 

WPP #14

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WPP #13: UNIT P CONCEPT 6

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