letter to a Math Analysis Honors student in my class next year

Dear next years students,

I wants to inform you that if you are in Algebra 2 Honors, there will be two types of teachers that you are susceptible of getting for Math Analysis; Kirch or Tamaoki, that is a 50% chance of getting either of them. If you get Tamaoki, I will assure you that your math career for next year will be extremely easy yet monotonous, but if you have Kirch, you will have a rather difficult beginning (everyone does) and in the worst case scenario, the whole semester but you will tend to enjoy her class socially. I am trying to scare you because it is true, there are people there that could not adapt quick enough, with their grades in jeopardy, but that can be prevented. If you watch the videos and stay consistent you will do well. If you don't do that, well conflicts happen. If you do however get Tamaoki, I'd advice you to watch Kirch's videos as reference because her videos do help as Tamaoki's ways of teaching can be quite confusing otherwise that class is an easy A. Of all this boring dialog that you have comprehended, what I'm trying to say is consistency-not the bad one the good one- is key to doing well in Kirch's class and well... all your classes for junior year.

It's easy to adjust to the flip classroom honestly most of you guys talk in class and don't pay attention to the teachers in traditional classes anyhow and refer to the internet at home to understand how to do that "one assignment". It's the same with Kirch's way of teaching; go home and learn through videos, go to class and talk about it-with a couple of side conversations here and there- and if you're confident take the quiz in class. All you need is the computer. Now lets get to the facts, we live in Santa Ana. It's all middle class and well to do here in South Santa Ana, but most people in Segerstrom live up North and not everyone is rich and owns internet, much less a computer, but that's solved, Mrs. Kirch has computers available everyday and if you don't have internet she can provide you with actual tangible DVDs that she created just for you guys, just another heads up for those who aren't technologically updated and wealthy.

Aside from that, like I said from the second paragraph, the Flip Classroom is much different from traditional classes as you do the learning at home and you do the homework which is classwork in class, and you take the quiz at your own time. Mrs. Kirch may say that you can go on your own pace in the flip classroom, but that only works for those who are impatient with slow learners and wants to move ahead. For example, you can't take a Unit A test in December just because that is your pace, you have to go by schedule, if you slack off you will get a support (homework) card. In other words you can go ahead but you can't fall behind. What you can expect in your daily life in her class are walking to the door, saying hello to Mrs. Kirch, sitting at your table, talking with your table, taking a quiz, going on the computer, and talking some more, that's about it, or course it's a different atmosphere during test days. 

Your year with Kirch and Tamaoki may be somewhat different as there is that common core that Obama recently established for all public grade school classes in the US, but the changes shouldn't be too drastic. 

 You can learn some KIRCH terminology and some tips in this video

Unit V Big Questions

Explain in detail where the formula for the difference quotient comes from now that you know!  Include all appropriate terminology (secant line, tangent line, h/delta x, etc).  Your post must include text and some form of media (picture/video) to support.

The difference quotient has something to do with the slope formula.

The difference quotient formula comes from a point in x and then the height in the point that shows the functions. We can find the secant line with the 2 points with point x then the second with x+ a number in this case h because variables are easier when numbers are ambiguous and because we can note that the distance from x to x+# is h. 

To find the tangent line we need to use the difference quotient formula and get h to the limit 0 because tangent lines are derivatives.

Unit U Big Questions

What is continuity? What is discontinuity?
A continuity is something that goes on and on without getting interrupted by other things. Discontinuity means  that it goes on but it gets interrupted by jumps, holes, breaks.

Example of Discontinuity:                                                                    Example of continuity:



What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?
A limit is the intended height of a function. It exists when both the right and the left hand limit must be the same. A limit does not exist if the graph has a break, including jump, oscallating, and infinite discontinuities.  The difference between a limit and a value are that a value is the number that always belong and appear in the graph. A limit can sometimes exist and sometimes cannot exist. A limit can still exist if there is a hole in the graph.

How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numerically by creating a table that has x-value and f(x) value on the left side. We use the number that the limit as x approaches, which will be the value that we need to look for. We need to write the limit statement mathematically (lim x-># f(x) = ) and the limit statement that contains the limit as x approaches _______ of _______ is _______ and a short sentence explain whether or not the limit can be reached.
We evaluate limits graphically by looking at the given graphs and answer the questions.  If the sides do not meet, the limit does not exist. If the graph has a point discontinuity, the limit still exist. If it has jump, oscillating, and infinite discontinuities, the limit does not exist at all.
We evaluate limits algebraically by using direct substitution, dividing out/factoring method, or rationalizing/conjugate methods. For direct substitution, use the number, that the limit as x approaches to, and plug it into equation. There are 4 results from that method: 1. a numerical answer, 2. 0/# -this is 0, 3. #/0- undefined which means limit does not exist, and 4. 0/0 -indeterminate form. If one of the first 3 answer appears, it means that we are done. If the answer is indeterminate form,  we have to use dividing out/ factoring method. By factoring out both the numerator and denominator and cancel common terms to remove the zero in the denominator, we can use the direct substitution after to find the limit. The conjugate is where we change the sign in the middle of 2 terms. Using conjugate of the denominator or numerator depends on where the radical is. 1. Multiply the conjugate on top and bottom, 2. simplify by foiling 3. eliminate from the top and bottom by canceling out common factors, 4. base on the simplest form of the equation, use direct substitution to find the limits.

Unit T Big Question Blog Post #4

4) Why do sine and cosine not have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.

Sin and cos don’t have asymptotes but other trig functions do because cos is x/r which is x/1 on the unit circle. If r was 0 then it would be undefined but in this case r can't be 0, it is always 1. This is the same thing for sin which is y/r or y/1 and why they don’t have asymptotes. Sec and csc in the other hand do have asymptotes because sec is 1/cos and whenever cosine is 0 then it is undefined which makes it an asymptote. Csc is 1/sin and whenever sin is 0 the answer would also have an asymptote. For tan it equals sin/cos and whenever cos is 0 then the answer would be undefined which will have an asymptote. Cot it is cos/sin and whenever sin is 0 it will also be an asymptote. This is why sin and cos aren’t asymptotes but the other trig functions are.

https://wiki.eee.uci.edu/images/6/6e/Triangle_on_unit_circle.jpg

Unit T Big Question Blog Post #3

3) Why is a normal tangent graph uphill, but a normal cotangent graph downhill? Use the unit circle ratios to explain.


Normal tangent graph is uphill because tan is sin/cos and when cosine equals 0 it is undefined which is in 90 degrees (pi/2) and 270 degrees (3pi/2). Those are asymptotes and when we graph and label asymptotes and follow the Unit circle, tan is shown positive because the arrow goes up to the asymptote. Cot is cos/sin, when sine is 0 it is 0 degrees, pi, and 2pi which are asymptotes and when labeled, the positive is at the asymptote towards zero and once drawn we see that it is downhill.

cot

tan

http://www.cartage.org.lb/en/themes/Sciences/Mathematics/Trigonometry/differentfund/different%20Functions%20.html

http://jwilson.coe.uga.edu/emat6680fa06/crumley/unit/6_files/image010.jpg

Unit T Big Question #2

How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response. 

     a.Tangent? When cosine is zero, it makes tangent become undefined. It will have an asymptote on the points where cos = 0: pi/2 or 3pi/2.  

     b. Cotangent? 

    

Cot is the inverse of tan because tan is sin/cos. Cot is cos/sin. When sin is 0, there is an asymptote because the whole thing is made undefined at that particular part. Tan asymptotes are at pi/2 and 3pi/2 so, to contrast, this means that cot asymptotes are at 0 and pi. Using the graphs, the asymptotes of the graph y = tanx are the x intercepts of the graph of y = cotx. There are vertical asymptotes at each end of the cycle. The asymptote that occurs at pi repeats every pi units.

      

c. secant?

sec is the inverse of cos which means that it is 1/cos and when cosine is 0 in this case, sec becomes undefined as a whole since it is the reciprocal of cosine. When we have undefined as our answer there is going to be an asymptote. These are vertical asymptotes.  The asymptote that occurs at pi/2 repeats every  pi units. The brown lines are the sec graph. The pink line is part of the parent graph of cos.

d. cosecant?

csc is the reciprocal of sin making csc 1/sin. So, when sin is zero in this case, cos becomes undefined anytime sin is 0, thus resulting in an asymptote. The x intercepts of y = sin x are the asymptotes for y = csc x. There are vertical asymptotes. The asymptote that occurs at pi repeats every pi units  

the red is the csc graph and the blue is the parent graph of sin. 

  

    http://www.regentsprep.org/Regents/math/algtrig/ATT7/othergraphs.html

   

Unit T Big Question #1

How do the trig graphs relate to the unit Circle?

    a. Period?- Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

b. Amplitude?- How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?


a. sine and cosine has a period of 2pi because it takes 360 degrees (2pi) for the period to repeat again. Whereas it only takes tangent and cotangent 180 degrees (pi) for the period to repeat.

b. cosine and sine only have amplitudes of 1 because it has no asymptotes and in the unit circle they both can equal 1 depending which radians. Tangent can only equal 0 or be undefined.  For csc and sec they are the reciprocals of sin and cosine and when it divided by 0 then it is undefined.

Student Video 7: Unit S Concept 7

This video is about how to solve a problem from Unit S concept 7 using a variety of skills learned from the past and to this unit. This problem will solve number 5 in concept 7.

The reader should pay attention to the steps and listen to how certain identities seem to appear out of know where. The reader should have some Unit circle jnowlege and skills from the last two units.

Unit S. Concept 4 Student problem

Solving Sine, Cosine, and Tangent of 100 with Half Angle Formula

This picture show how to solve a unit S concept 4 level problem using half angle formulas.

Students must remember that determining if sin or cos is + or - depends strictly on u/2 not u, they should know that these problems need basic unit circle skills to solve these problems.

Solving sin cos tan with Sum Formulas

This picture shows that you can solve for the same answers using the sum formula, you can also use the difference formula and get the same answer.

The student should know how to use the sum and difference formula. They should also pay attention to the work done in this student problem.

In conclusion we know that the answers are the same because when plugged into the calculator they get the same answers.

Unit S. Concept 3 PQ #5

This video is about how to solve a specific problem using power reducing formulas.

The viewer needs to understand how I used the power reducing formula to solve the specific problem. Note that even though the variables may be similar does not mean they can simplify. The power should be 1 that is the point of power reducing.

Student Problem Unit R Concept 3

This is a type 2 problem from Unit R concept 3. There is ambiguity in the problem but using the first quadrant is always the most ideal choice. This type of problem requires knowledge on the unit circle as there are angles and radicals that are to be dealt with. This problem will show the reader how to solve a type of type 2 problem.

The reader must pay attention to the steps and understand how the formula got on there. The reader must understand the use of quadrant 1 as first priority and then they must learn to plug them together to get the final product.

Student Problem: Unit R Concept 2

This is about how to use the sum and difference formulas when given values for right triangles. First, you have to draw the triangles in the correct quadrants. Next you have to find the missing piece of the triangle. Now, the numbers need to plug into the sum and difference formulas. In this case, the formulas sin(u+v) and tan(u-v).

The readers need to pay attention to the quadrants of the triangles. They also need to pay attention to what is positive and negative. The hypotheses will be positive. There are two triangles that lie in two different quadrants, II and I.Quadrant II. To solve for tangent, the readers can multiply by common denominator and solve for it. 

Student Problem 1 Unit R

Sum Formulas 

Difference Formula

1. These pictures show two ways of finding the exact values of the sine, cosine, and tangent of 165 degrees. This is Unit R Concept 1 which is about finding the exact values of angles that aren't on the Unit Circle. We use the angles on the unit circle that can add or subtract to that angle. The answers of the trigonometry functions of that angle must be the same. Using the Sum and Difference Formulas, we can plug in the numbers to find the answers. 
2. What does the reader need to pay attention to?
The readers need to be careful when they are trying to solve when there is a negative. The solution is knowing which part is negative, and which part is not. When solving for tangent the reader needs to pay a very close step in solving for tangent because we don't solve it all the way through, we have to multiply by its opposite, then foil and solve.

Unit Q 3 Concept 4 problems

Level 1                  cosx=1

Level 2                                   2sinx-1=0

Level 3                     2sin^2x-3sinx+1=0

Level 4                              csc4x=2

Unit Q 2 Solve with identities and right triangle

How to solve using identities:

How to solve using identities:

How to solve using SOH CAH TOA

How to solve using the unit circle

Unit Q 1 Pythagorean Identities Derivation

The formula above comes from the Pythagorean Theorem (y^2+x^2=r^2).  The triangle would look like the picture on the right, which is labeled y, x, and r.  So, if we look at the triangle, we can see that sin^2 in the Pythagorean Identity is y because sin is opp/hyp.  Cos^2 is x on the right triangle because cos is adj/hyp, and 1 is r because r will always equal 1 in this case of the unit circle.

To get the second main (tan^2x+1=sec^2x) identity we do the following below:



















Now to get the 1+cot^2x=csc^2x identity we do the following below:

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.

I have preformed poorly on both tests, I have shown low quiz scores on certain concepts that I found were difficult. I've shown not to practice well in the quizzes or I just finish the quizzes for the sake of finishing.

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.

• I don't always put a lot of effort on my WSQ's because I want usually want to finish them as soon as possible. I always highlight important information and write down the key pop-ups that you include during the video. I tend to finish what I need to do in order to complete the WSQ. The times when we have to complete extra practice problems, I don't watch the entire video. I skip to the answer to check if I have the answers right, but that is as far as I go. It is helpful to see how the problem is done and hear why certain steps must be done in that way. Looking at the big picture, I will develop the skills to explain the steps to solving for the problem and move me further to understanding the concept.
• Your SSS packets along with the videos for each concept provide us with more detailed information than the textbook. If I need further practice, I use the extra practice problems from the blog for the countless problems they have. I could improve my attention to this step by completing more practice problems from the blog if I want to practice for the test. This would add to my mastery of the material by doing additional work to the extra practice I normally do.
• I don't always look at the supplementary resources. I am consumed with other homework and struggling to finish so that I can sleep. I do however look at them when I do have the time or when I need to study for the test. I will try to pay more attention to these if I can.
• I have found this to be important to my understanding of concepts. We help each other out when we don't understand how to do the concept. When we discuss, we find that the person who is verbally explaining how the concept works applies it in a way that makes sense. If there was a way to improve the group I'd say we should improve our communication skills when it comes to math material. If all 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.
• PQ's are so useful and helpful and they are the practice that I need. If I practice the PQ's I can get a good grip on the concept. PQ's are what I need to master the concept. I don't focus much on this even though I need to. I really need to do all my PQs to practice and improve myself.
• The quizzes showed if I understood how to do the concept. 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.
• I don't think I took the PT's seriously, Most;y I just tried finishing PT's for the sak of finishing, for almost every WSQ I would finish everything but the last part of the PT's and it would get me stuck. I was able to understand where I lacked and where I good at with PT's though. Improvement wise, I think by taking the PT's more seriously, I can get a feel of the test and gt a higher score on the test.
• The final assessment is what sometimes shows how well I put effort into trying to learn the Unit. When I get my quiz scores or when I do my PT's and PQ's reflects on the tests. There are times when I work hard and then don't get the score I want because I did dumb mistakes or something. When I overlook at the tests I begin to show mastery when I start studying to retake the test, seeing my mistakes and trying to fix them helps a lot. What I could improve though is that I need to answer the questions carefully and show every small part that will get my answer correct.

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?

• I normally feel scared on the day of the Unit Test. I'll get scared if there were pieces of things that I was suppose to turn in but I didn't and then trying to quickly finish the test. I normally study for the test, too by reviewing everything in the Unit.
• This is one of the first few times where I actually learned something in math and remembered it well. Compared with my other math teachers, there are times where I just learn something and I forget it the other day, but with this class I was able to understand it better.
• I believe that, out of all my years in Segerstrom, this has been the year that I have understood math the most. In all my other years of math, I understood the concepts, but it was only to an extent. I mean I understood it, but I wouldn't be able to explain it that well. This year, I can confidently explain the concepts to anyone who's having trouble.
• I do understand the "why" behind the math concepts, not just the "what/how." I understand the "why" because during the school year, many of my group mates would ask "why" in regards to the concepts and I would be able to give them the answer if I knew it. I was able to learn this through the videos that you made for us which explains why this is so.
• My work ethic is terrible. Normally I would do the homework and practice but then for some parts it was for the sake of finishing. Sometimes my work ethic does show my performance on a test, but other times it doesn't. I remember getting almost all 8's on my quizzes but then getting a C on Unit M, I even knew how to explain how to do it and why it is that, I helped people who needed the help and ironically they get higher scores than I did. Then there's Unit N where I did crappy on the practice and the quizzes, but great on the test. But I'd say my effort on the practice affects me a lot on the test. The value of work ethic is important in life because a lousy work ethic can affect everything I do in life.

WPP 13-14

wpp 11-12

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Unit N Concept 7: Derive the Unit Circle

These pictures show how the unit circle is derived from these special right triangles. As long as r=1, we can see that the points in the graph is the same as the unit circle showing that these special triangles are connected with the unit circle, which is why they are connected. As I derived the circle I can see that the angles match with that of right triangles, also as I derive the special triangles they are the same as from the unit circle.

So... about those conic sections

1. For the parabola the mathematical definition is any point on a parabola that is at equal distance from the focus, and the directrix. The definition of a parabola plays a role in the properties of the parabola because as the definition stated that the focus would be equal distance from the parabola and the directrix, gives the parabola the property that it has, such as if anything were to hit the parabola then it would go back towards the focus point. The reason the parabola has that property is because of that universal definition of parabolas. The definition of the parabola also affects on the shape because if the definition was invalid, then parabolas would either not exist and so we would not have satellites therefore more likely would not achieve the space era. 
2. The focus of the parabola affects the shape of the parabola depending on the number of the focus point such as if the focus point is farther away from the vertex then the parabola gets wider of fatter, but when the focus is closer to the vertex or the number gets smaller then the parabola is skinnier. By the law of Physics, if something were to hit a paraboliod, such as a transmitter, then the signals must always hit at the focus point. P is the distance between the focus and the directrix, so it makes sense that the farther the focus is the fatter and the closer it is, it's skinnier. Sometimes though P is a negative, so in that case the parabola will open down or left, while for p positive the parabola will go up or right. 

3. The properties of the parabola apply to real life because in a paraboloid shaped object, with the definition of a parabola being the same length, from focus to a point on a parabola and the directrix, we know that any part an object hits at a perfect paraboloid, it will accurately hit into the focus point, which is why we have satellites. With the paraboloid everyone can watch TV.
www.antenna-theory.com/antennas/reflectors/dish.php