"End of the Year Final Exam" Blog Post

Dear Student,

If you're reading this letter, I bet you've heard all about the flipped classroom. But whether or not these stories or rumors bring about positivity or negativity all depends on you. The flipped classroom is very different from a traditional classroom setting because it involves more independence and drive (as in get ready to manage your own work schedule). What you can expect to see is a lot more computer use in your life--not that you don't already do enough of that *cough* facebook/tumblr/twitter/instagram/flickr/youtube *cough*--through various websites that will be used throughout the school year. In the beginning you might feel very stressed especially once Mrs. Kirch asks for you to create your own blog or mentormob, but don't fret, in time you will get...it.
The first few weeks of the school year will be all about your rate of adjustment. Because the flipped classroom requires you to do nearly all of the learning at home, "homework" (which for your benefit to know is ACTUALLY, PQs, PTs, Quizzes, WPPs, Student Videos, and Student Problems) must be done in class. Go ahead, open your eyes a little wider, I know your freakin' out a bit because ALL of this is different from your previous math class. However, I assure you it is manageable as long as you stay motivated to do work day-in and day-out.

The best way to adjust to the flipped classroom is to master the inevitable struggle that all of us juniors/seniors face, time management. But of course! how could I forget, you're growing older, more independent, your parents finally let you have a boyfriend (girlfriend?)! and on top of that you've got all of your extracurricular activities and AP classes, well get ready to pile on the Math Analysis flipped classroom because this class is just as important--if not more useful--than going out to the movies on Friday. I remember at the beginning of my junior year when I struggled with this exact situation. If the task of creating a math blog and WPPs/Videos was not difficult enough, I had to stack that frustrating job on top of my AP classes, Red Cross Club, and Band activities. And well, knowing me, I am definitely technologically challenged. The first WPP I ever made took me all day (like hours on end) because I just couldn't figure out how to use the technology on my computer and phone!

But don't let my little mishap startle you, everything is rough at the start. With a little enthusiasm, a dash of dedication, and a load of hard work, you'll make it through this year before you know it! Just make sure to keep up with all of the videos and WSQs every night and ask plenty of questions! The work load that Mrs. Kirch gives may seem tedious and overwhelming but you certainly will learn A LOT :)


Sincerely,

               Tiffany Tran

Unit V Big Question

Explain in detail where the formula for the difference quotient comes from.

The picture above is just to refresh your memory on how to evaluate the difference quotient. (and for me to reference to the formula to instead of typing it out)

What is the difference quotient? The difference quotient is just a fancy way of writing out the slope of an equation. And in calculus it is used when introducing a concept called the derivative.

Because the difference quotient is just another way of saying slope, we know that this formula must relate to the slope formula:

Moreover, we know that the slope formula needs two coordinate points on the graph, this is much like a secant line. A secant line looks like this: (in red)

Using this information, lets figure out how the difference quotient formula came to be!
Watch this video up to minute 6:15 to see how the slope formula turns into the difference quotient formula.

As you can see, the difference quotient comes from using two points to find slope, and these two points on the graph form to make a secant line. But what about tangent lines? Well, we know that tangent lines only touch the graph once, so how to we find the equation of the tangent line of a graph?
First off, the slopes of tangent lines are referred to as derivatives.
To find the derivative, do the difference quotient formula normally, but take one more step, find the limit as "h" approaches 0
Here's an example: (watch starting at 2:30 to the end of the video)

Sources: First picture-http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/sld002.htm
Second Picture-http://math.about.com/od/allaboutslope/ss/Find-Slope-With-Formula-JW.htm
Third Picture-http://clas.sa.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm
Difference Quotient Video- http://youtu.be/1mlkc3Pfxu4
Derivative video- http://youtu.be/vzDYOHETFlo

Unit U Big Questions

What is continuity?

In mathematical terms, continuity occurs when a graph has no breaks, holes, or jumps. Basically, the graph must be continuous. We can identify a continuous graph, or continuous function, when we see that is makes a good bridge. A function is considered continuous if the limit as "x" approaches "c" of f(x) equals f(c). [f(c) is the height the function reaches at that point of x=c]
The picture below shows a continuous function as well as a bridge, to show and complete the analogy: a continuous graph makes a good bridge. *Notice there are no breaks, holes, or jumps.

What is discontinuity?

A graph is considered discontinuous when there are breaks, holes, vertical asymptotes, "wiggly" lines, and/or jumps.
When a graph has a hole, this is called point discontinuity. Point discontinuity is under the category of REMOVABLE discontinuities because a hole can easily be "fixed" by filling it in to make a real point.

When a graph has a jump, this is called jump discontinuity. A jump is when the graph breaks at a certain x value with two different y-value (different left/right) see below:

When a graph has a vertical asymptote, this is called infinite discontinuity.

When a graph is "wiggly", this is called oscillating behavior.

Important Note: NONREMOVABLE discontinuites are: jump discontinuity, oscillating behavior, and infinite discontinuity.

For further clarification, watch this video to see the difference between continuity and discontinuity :)

What is a limit? When does it exist? When does it not exist? What is the difference between a limit and a value?

A limit is the intended height of a function. A limit exists when the limit on the right and on the left are the same. Note: A limit can still exist if your final destination is a hole in the graph. A limit does not exist when the graph has a break, such as when it exhibits oscillating behavior, unbounded behavior, or a jump discontinuity. A value is a number that can always be defined on the graph. For example, with point discontinuity, there could be a hole and a point on the same x-value, although the intended height is where the hole is, the real value is where the defined point is. Lets look at a picture:

Looking at the point discontinuity at x=-2, we see that the LIMIT as x approaches -2 is 1, however the VALUE as x approaches -2 is actually 3.

How do we evaluate limits numerically, graphically, and algebraically?

To evaluate limits numerically, we must create a table that has rows for the x-value and the f(x) value (on the left side). The number we are trying to reach (which is the limit) will be in the middle. On the left side of the approximated limit, we will have numbers that are slightly less than the x-value and on the right side we will have numbers slightly larger than the x-value. Trace these x-values in your graphing calculator and record the f(x) values. Once you have found the limit, write out the limit statement based on what you've found. lim x-># f(x) = # and verbally this would read as "the limit as x approaches # of f(x) is equal to #" Be sure to include a short explanation as to why the limit can or cannot be reached.
 To evaluate limits graphically look at the picture of the graph. Now that you see the graph, place two fingers--one on the left side of the limit and one on the right side of the limit--then trace your fingers (follow the line) to have them meet where the x-value is. If your fingers meet, then that is where the limit is! If the fingers do not meet, the limit does not exist. And this can be due to unbounded behavior, jump discontinuity, or oscillating behavior.
To evaluate limits algebraically use direct substitution, dividing out/factoring method, or the rationalizing/conjugate method. For direct substitution, all you do is plug in the number that "x" approaches into the equation. NOTE: When using direct substitution, there are 4 types of answers we can get:
1. a numerical answer
2. 0/# -this is just zero
3. #/0- this is undefined which means the limit does not exist
4. 0/0 -indeterminate form (this one is special...meaning we must use another method!)
For options 1-3, this means we are DONE with the problem, but if we get indeterminate form (option 4), then we must use the dividing out/factoring method or the rationalizing/conjugate method.
Dividing out/Factoring-This process is done by factoring the top and bottom and canceling out any items that we can. By doing this, we cancel out the zero in the denominator. Only then, can we use direct substitution to find the limit.
Rationalizing/Conjugate- This process is where we change the sign--positive or negative--in the middle of the 2 terms. We use the conjugate wherever there is a radical in the expression, so that could be either the denominator OR numerator.
1. Multiply the conjugate to both the top and bottom of fraction
2. Foil
3. Simplify by canceling out common factors (be careful!)
4. Use direct substitution to find the limit

Works Cited: Bridge-Wikimedia Commons
Continuous function-http://www.conservapedia.com/Continuous_function
Discontinuity pictures-http://faculty.wlc.edu/buelow/calc/nt2-4.html
Oscillating graph picture- http://webpages.charter.net/mwhitneyshhs/calculus/limits/limits.html
Youtube Video- http://youtu.be/392XQbIl8TQ
Point discontinuity-http://www.zweigmedia.com/RealWorld/calctopic1/contanddiff.html

Unit T Big Question #4

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do?

Watch this video to find out why! :]

Unit T Big Question #3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

Watch this video to find out why! :)

Unit T Big Question #2

How do the graphs of sine and cosine relate to each of the others? (tangent, cotangent, secant, cosecant)

This video will go over how sine and cosine graphs relate to tangent and cotangent graphs.

Concept 1: Sine graphs relate to cosecant graphs because cosecant is the RECIPROCAL of sine. (sin=1/csc vice versa csc=1/sin) So, if sine equals 0, then cosecant is undefined. (1/0) This means that there will be a vertical asymptote where sine is 0.
Here is a picture that will help in further explaining this first concept:

Concept 2: Now that we can see what the asymptotes look like, the next part is that wherever sine reaches a maximum value of 1, cosecant will reach its minimum value of 1. And wherever sine reaches a minimum value of -1, cosecant will reach its maximum value of -1. (Purple Math)
Now lets look at this same photo but with the points marking 1 and -1:

Following the ideas learned in the second concept, lets apply what we know and fill in the graph:
(Cosecant kind of looks like a mirror image of sine)

Concept 3: Cosine graphs relate to secant graphs because secant is the RECIPROCAL of cosine. (cos=1/sec vice versa sec=1/cos) So, if cosine equals 0, then secant is undefined. (1/0) This means that there will be a vertical asymptote where cosine is 0.
Using these same ideas mentioned in concept 2, we can draw a cosine graph and use it to guide in making a secant graph. Look below:

This website will help understand why and how sine and cosine relate to secant and cosecant graphs.

http://www.purplemath.com/modules/triggrph3.htm

Citations:
http://www.purplemath.com/modules/triggrph3.htm
Mrs. Kirch's lecture on friday about all of the graphs

Unit T Big Question #1

How do the trig graphs relate to the Unit Circle?

a. Why is the period for sine and cosine 2π, where as the period for tangent and cotangent is π?

b. How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?

Studen Video #5: Unit S Concept 7: Assessment #4: Problem #5

What is this video about?
This video is about solving equations with half-angle formulas. We are given a problem that incorporates the use of half-angle formulas along with previous concepts learned in order to solve the entire problem. This video will go over step-by-step how to solve these kinds of problems.

What does the viewer need to pay special attention to in order to understand the concept?
The viewer needs to pay special attention to plugging in the correct formula. Also pay attention to the end when we determine the three answers. You must know your Unit Circle!

Unit S Concept 4: Student Problem: Assessment #2 (Parts 1-5)

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Student Video #4: Unit S: Concept 3: Assessment 3: #5

What is this video about?
This video covers #5 in the SSS packet from Unit S Concept 3: Using power-reducing formulas
It will go over, step-by-step, the process of solving #5 with the formulas given. We will also be using previous skills learned in algebra and previous units (half, double angles).

What does the viewer need to pay special to in order to understand the concept?
The viewer must pay attention to carefully plug in the correct power-reducing formula with the right positive/negative signs. Also pay attention to the distribution of the negative sign in one of the steps. The last thing to pay attention to is the denominator and how to get rid of it using reciprocals.
 

Unit R Concept 3: Student Problem 3

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Unit R Student Problem 2

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Unit R Concept 1: Student Problem 1

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Unit Q Blog Post: Concept 4 Problems

Solving Trig Equations that have more than one function in it.

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Unit Q Blog Post: Concept 2

Finding All Trig Functions Using Unit Q:

Finding All Trig Functions Using Unit N:

Derivation of Pythagorean Identities

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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.

RESPONSE: On the Unit O test I performed terribly. I got a C, granted a high C, but I got it for all the wrong reasons. It wasn't because I slacked off on any work, I was a matter of not reading directions completely. On the bright side, I did retake the test and got an A :) I did well on the retake because I went in for mandatory tutoring and talked through the problems with Mrs. Kirch. On the Unit P test I performed well with a 94%. I think I performed better with Unit P because I reviewed the day before with my PT's and looked over my SSS Packet.

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.

RESPONSE:
Step 1: a. I watch all video's excluding the extra video portions. I only watch those to check answers or for additional help. While watching the video's I copy down all of the notes Mrs. Kirch makes and then some because sometimes she says important things and I jot them down. During video's, if Mrs. Kirch advises us to try a certain set of problems on our own, I do follow these directions because this tactic does help us learn and it shows us just how much we understand the concept. I answer all the WSQ questions thoughtfully every time because this helps me remember the concept.
b.To improve my mastery and focus upon the material I could rewatch videos and ask more questions the next day in class. I think my HOT questions could be a little more insightful instead of being so easy.
Step 2: a. During video's if Mrs. Kirch tells us to look up a fact about a certain mathematician or fact about ellipses etc. I take it seriously because I do find it interesting on some levels. I think its great that she incorporates little facts for us to remember, because this makes us more well-rounded as individuals.
b. To improve on the online research portion, I could research a little deeper and share it on my blog. I have yet to look at my textbook this year.
Step 3: a. If we are given an additional playlist or link, I typically click on them to see what it is. However sometimes, I cannot see the image and I don't know why. I take this portion not as seriously as the others because I am more focused on the information given in videos.
b. To improve on this portion I could take steps to download the software that the website tells me ad proceed to look at and understand the material more thoroughly
Step 4: a. I take class discussion's very seriously because it is a chance to interact with your peers and share what you know and don't know about the material. Whenever we go over key concepts, this helps me remember what I learned the night before.
b. I don't think there is any room for improvement besides asking more questions, however, I do feel like I always ask Tien, Arlene, and Mrs. Kirch questions every time I am confused or unsure about a problem.
Step 5: a. I take my PQ practice very seriously because it is a chance for me to try the problems on my own and think with my own thought process without listening to Mrs. Kirch give me the answer. I always do all of my PQ's regardless of Mrs. Kirch signing off our WSQ charts. When I am done, I always check my answers in the back of the SS packet.
b. To improve I could go online and do extra problems to master the material more accurately and quickly.
Step 6: a. During the in-class quizzes, I always do those the day after watching the concept after doing PQ practice first. The quizzes truly test my understanding of the material come test day. I like the in-class quizzes because it's like a preview of the test.
b. To improve on quizzes, I could study the SSS packet right beforehand and if I dont' get an 8, I should always retake it until I get a perfect score.
Step 7: a. I always do all of my PT problems however I do not do them throughout the course of the unit--concept after concept--I prefer to leave it until the end close to the test date because that way I am given a chance to refresh my memory on all of the concepts and test what I know. And if I get problems wrong I watch the PT Answer Key videos or the work shown online.
b. To improve I could work on PT's throughout the Unit that way I get extra practice and don't have to have a lot of homework the night before a test.
Step 8: a. I think I take my final assessments seriously because they show us what we know. If I do poorly on a test, I always look at my classmates tests' and figure out where I went wrong.
b. To improve I think I could ask questions on why my work is wrong compared to the way other students approached the problem.

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?

RESPONSE:
3a. On the day of Unit tests, I usually feel very confident because the night before I usually do my PT's or review a few problems form each concept. Also throughout the day, I study my SSS packet and remind myself about tricky problems.
3b. I feel like I have been more involved with this years math material than any other year. This is because the activities (blog posts, WSQ's, and WPPs) have forced us to learn the material even more by creatin out own problems and solving them or even explaining a concept through and in depth blog post.
3c. I feel like I have learned my math material MUCH more in depth than any other year. This is because the flipped classroom is set up in a very interactive environment. it a place where we are very independent and are expected to know the material through lots of practice and determination.
3d. I feel like first semester I only understood the WHY sometimes, but this semester, since the material is getting harder, Mrs. Kirch emphasizes the derivation of certain formulas and encourages us to be able to derive it ourselves to better understand the material and relate it to other formulas we know. For example, this semester, we learned the derivation of the Law of Sines and Cosines, this was a challenging concept but we related the properties of trig functions to it and I thought that was interesting. I also find the "real life" examples to help us understand WHERE we can find these problems in real life.
3e. My work ethic is very strong. I do all of my work without being told because it has been a goal I ahve set for myself to do well. I always try my best no matter how challenging. I think the value of work ethic in real life is very valuable because It shows the kind of character a person posses. If you have a good, constant work ethic, It shows that you fight for your achievements and deserve it well enough.

Tiffany T.'s WPP #14: Unit P Concept 7

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Tiffany T.'s WPP #13: Unit P Concept 6

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Tiffany T.'s WPP #12: Unit O Concept 10

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Tiffany T.'s WPP #11: Unit O Concept 9

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Unit N: Special Right Triangles Blogpost

This video will go over example #1 from the "Derive the Unit Circle Activity"
worksheet :)
http://www.educreations.com/lesson/view/unit-n-triangles/4989843/?s=MscJyh&ref=appemail

Unit M: Conic Sections: Ellipses

ELLIPSES!

1. What is the mathematical definition of this conic section and how does that definition play a role in the properties of the conic section and how it is shaped or formed?
   The mathematical definition of an ellipse can be determined by two points, each called a focus (plural: foci). An ellipse is considered an ellipse by taking any point on the ellipse, and seeing that the sum of the distances (from the point) to the focus points is CONSTANT. The sum of these distances is equal to the length of the major axis, which is 2a. The position of the foci determine how "skinny" or "fat" the ellipse will be.
   
2. How does the focus (or foci) affect the shape of the conic section?The position of the foci determine how "skinny" or "fat" the ellipse will be. The position of the foci also attributes to the eccentricity of an ellipse. The eccentricity of an ellipse is between 0 and 1. So 0<e<1
   http://www.mathopenref.com/ellipse.html
   
   Click on the link above and move the focus around, notice what happens to the graph!
   The trend here is that the closer the foci are to the center of the ellipse, the more round the ellipse will become (much like a circle) This means that the ECCENTRICITY of the ellipse is getting closer to zero. As the foci move further away from the center, the ellipse becomes more squished (like an oval). This means the ECCENTRICITY of the ellipse is getting closer to 1.
   
3. How do the properties of this conic section apply in real life?
   An example of an ellipse in real life is "The Whisper Chamber" at the United States capital. If you stand at one focus you can perfectly hear someone else whispering at the other focus. This is due to the reflective properties of ellipses regarding their focus points.
   
   
   Another example is a lithotripter. It is a machine used in medicine to destroy kidney stones and gallstones. Shock waves are sent into a person's body to crush the stones into smaller pieces making them easier to pass.
   
   The last example is orbital paths of planets. They orbit in an elliptical shape! Not circular.
   
   This video will explain how Pluto's elliptical orbit around the sun works and is proven through an equation:
   http://www.youtube.com/watch?v=YBbsb4N-KWM
   
   CITATIONS:
   First ellipse image: http://img.weburbanist.com/wp-content/uploads/2009/08/felice-varini-ellipse-rouge.jpg
   Mathematical definition: http://www.mathopenref.com/ellipse.html
   Website explaining position of foci: http://www.mathopenref.com/ellipse.html
   Real life ellipse images: https://sites.google.com/site/arinsellipse/real-life-examples
   http://illuminations.nctm.org/Lessons/MarsOrbit/2MarsOrbit-MarsEarth.jpg
   Real life ellipses video: http://www.youtube.com/watch?v=YBbsb4N-KWM