Letter To Future Math Analysis Student

Dear Future Math Analysis Student, 

 To have the most successful year possible in your future math analysis class be ready to adapt to change. You have to go into the classroom expecting things to be different and keep a positive attitude. You should no give up simply because you do not like the flipped classroom method. Start the year off right by doing your work on time and take advantage of your class time to complete assignments like student problems or WPP's. With such a filled agenda of work for the class, you should plan accordingly to complete all your assignments on time. It can be tough to manage if you have sports and have late practice, so you should complete some assignments early when you have free time. Also, if you work very efficiently in class you should never have any extra work besides watching the next concept. Make sure to come to class prepared every day by watching the videos and completing the wsqs. I think that if you do all your work, and plan out your busy weeks, you should survive math analysis. 

Adjusting to the flipped classroom is essential. You will need to take on responsibility for your own learning. So you must watch every video lesson everyday before class and pause and rewind during a lesson when you do not understand. It is not like a traditional classroom and you will not be able to get help right away, you should ask questions in class the next day. Also, ask classmates around you, everyone is there to help you. For the flipped classroom, you are required to film yourself solving a problem for student videos or WPP's. You should get familiar with using a basic camera or an ipod that will be available to you in class.  Also, if you do not have a filming device at home, make sure to plan to use one during class. Do not be shy when making your videos and make sure to speak clearly. The first video will most likely be the hardest to do because you may find it a bit weird, but soon you will have a blog filled with almost 20 self created videos of you solving problems. Make sure that you have a reliable internet connection or plan to use the schools internet when you need to submit something online. 

The flipped classroom will be very different from your previous math classes because you are responsible for your own learning. You will not sit in class and take notes on a concept, you will have guided SSS packets that you will complete every night while watching the video. Also, you will most likely never have homework for this class besides watching the lesson video. Opposed to traditional classrooms, you will do most of your work during class. Also, this class has quite a bit more work than previous math classes so be ready to work hard. You will definitely go deeper into math concepts and understand where things come from and why we do certain things in math. Also, you will create your own blog that is a online portfolio of blog posts and self created videos that explain math concepts.

Unit V Big Question

Where does the formula from the difference quotient come form? 
1. A derivative is the slope of a tangent line to a graph. A tangent line is a line that touches the graph at one point. A function can have different derivatives at different values of x. To try and find the slope of the tangent line we first have to find the slope of a secant line similar to the tangent line. A secant line is a line that touches the graph at 2 points. We find the slope of the secant line using the slope formula using the 2 points on the secant lines. This is shown in the picture below. When we plug in our points we get m= f(x+h)-f(x)/ x+h-x. So when we simplify the denominator we are left with 
m=f(x+h)-f(x)/ h. So this is what we know as the difference quotient and is also the slope of the secant line. But remember we want to find the slope of the tangent line. So what we will notice is that as you make h(which is the change in x or the difference between the two x points) smaller, the secant line begins to resemble the tangent line better. So we want h(or delta x) to be as small as possible. So we use a limit as h approaches 0. We cannot just say that h=0 because then we will have a 0 in the denominator, so this is why we use a limit. By using a limit of 0, we are now finding the slope of the tangent line. 

Image: http://en.wikipedia.org/wiki/Numerical_differentiation

Unit U Big Questions

1. What is continuity? What is discontinuity?
Continuity demonstrates that a function is predictable and has no breaks, holes, or jumps. A continuous function can be drawn with a single, unbroken stroke of a pencil. You can tell if a function is continuous by imagining that you are driving a car on the line. If the imaginary car meets anything that makes the car fall off the line such as a hole or jump, the graph is discontinuous. Discontinuity or a discontinuous function has either a jump discontinuity, oscillating behavior, or a vertical asymptote. In reference to the imaginary car, a discontinuity would make the car fall off the line. 

2. 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.  A limit exists when the graph reaches the same height from the left and right side. This can be tested by placing two fingers on the graph and tracing the line towards a certain point to see if your fingers meet. If they meet, the limit exists at that point. A limit doesn't exist when your two fingers do not meet on the line or in other words when the left and right behavior of a graph do not match and when their is unbounded or oscillating behavior. The value is the actual height of a graph, while a limit is the intended height. 

3. How do we evaluate limits graphically, algebraically, and numerically? 
We evaluate a limit graphically by determining left and right behavior. We do this by using two fingers to trace to a certain point and of our fingers meet at the same point, the limit exists. If they do not, then that is most likely due to a hole or jump and the limit doesn't exist. We evaluate limits algebraically by using either direct substitution, dividing out/factoring method, or rationalizing/conjugate method. These methods can all result in indeterminate form which is 0/0 and so we have to figure out which  method works best. Direct substitution is substituting the limit into the equation.  Dividing/factoring out method is factoring the numerator and denominator and canceling terms then substituting in the limit. Rationalizing/conjugate method is multiplying by the conjugate of the term with the radical in the function. We evaluate limits numerically by creating a table with values that get closer and closer to a certain value on the x axis and notice that the graph of f(x) get closer to a value on the y axis. Our table consists of x values that approach the limit from the right and left and after plugging the function into our graphing calculator we can trace to the x values and record the y values. With our table we can determine if the limit can be reached or not. If the f(x )value doesn't match the limit on the table, then the limit cannot be reached. 

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

 This has to do with the shift in the asymptotes between tangent and cotangent and the pattern for cotangent and tangent. Cotangent has asymptotes where sine is equal to zero. So its asymptotes are located at 180 degrees(pi) and 360 degrees(2pi). Tangent has asymptotes where cosine is zero. So its asymptotes are located at 90 degrees(pi/2) and 270 degrees(3pi/2). Now we pay close attention to tangent and cotangent's pattern + - + - dealing with ASTC. Once we draw our asymptotes our pattern guides us to whether our graph should be above or below the x axis. If it is positive, it is above the x axis and if it is negative then it is below the x axis. When we draw our graphs, we will see how the placement of the asymptotes affects the shape of the graph, even though they follow the same pattern of + - + -

Why do sine and cosine not have asympotes, but the other four trig graphs do?

Sine and cosine will never have asymptotes because of their trig ratios. The ratio for sin= y/r
and the ratio for cos= x/r. According to the unit circle, r will always equal 1. Asymptotes exist when it is undefined, meaning it has a 0 in the denominator. So this means that sine and cosine will always be over 1, making it impossible for them to have an asymptotes. Tangent,cotangent,cosecant, and secant can have asymptotes due to their trig ratios. Their ratios are as follows:
 tan= y/x
cot= x/y
csc= r/y
sec= r/x
Because they have an x or a y value in their denominator this opens up the possibility to be undefined and therefore have an asymptote.  For example, tangent and secant will have an asymptote wherever the x value equals zero, which is at 90 degrees and 270 degrees. Cotangent and cosecant will have asymptotes where the y value is zero so this is at 0 and 180 degrees.The image below displays the trig functions according to the unit circle.

picture:  http://htmartin.myweb.uga.edu/6190/resources/unitcircletrig.gif

How do the graphs of sine and cosine relate to each of the others?

Tangent and cotangent have asymptotes where their respective ratios are equal to zero. We know that Tangent=sine/cosine. So when cosine is 0, tangent will have asymptotes. So this means that tangents asymptotes will be at pi/2 (90 degrees) and 3pi/2 (270 degrees). These are the points where cosine, or the x value, is equal to zero. This can also be thought of as cosecant is the reciprocal of sine.  Cosecant is 1/sinx and we know based on the unit circle that sine is 0 at 0 and 180 degrees, so its asymptotes lie there. Whenever we have a zero in the denominator, it is undefined, and undefined means asymptote!  Secant is also the reciprocal of cosine  and its ratio is 1/cosx. Cosine equals 0 at 90 degrees and 270 degrees, so its asymptotes are at pi/2 and and 3pi/2. We know that cotangent=cosine/sine so when sine is 0, cotangent will have asymptotes. So cotangents asymptotes are at 0 and pi(180 degrees), the points where sine is equal to zero. The images below show that cotangents asymptote is in the same place at pi for cosecant.
photos:  http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg94.gif
http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg2.gif

How do trig graphs relate to the unit circle?

Why is the period for sine and cosine 2pi,whereas the period for tangent and cotangent is pi?
 Trigonometric graphs are cynical which means they repeat themselves over and over again. One time through their cycle is called a period. The period for sine and cosine is 2pi while the period for tangent and cotangent is pi.  The pattern for sine is ++-- (meaning it is positive in the first quadrant and second quadrant while being negative in the third and fourth quadrant) This cycle is one period. It takes one full revolution around the unit circle for this cycle to repeat, so one revolution is 360 degrees or 2pi. This is the same for cosine because its pattern is +--+(meaning it is positive in the first quadrant, negative in the second and third quadrant and positive in the fourth quadrant) and it would take one revolution or 2pi for the cycle to repeat. The period for tangent and cotangent is pi. Tangent's cycle is +-+- . This means it is positive in the first quadrant , negative in the second quadrant, positive in the third quadrant, and negative in the fourth quadrant.  Notice that it only takes half the unit circle for its cycle to repeat from positive to negative. Therefore it only takes 180 degrees or pi units to repeat its cycle, making its period pi. The image below gives you a visual of what quadrants the trig functions are positive or negative.

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? 

 The  amplitude is half the distance between the highest and lowest points on the graph.  Sine and cosine have amplitudes of one because based on our knowledge of the unit circle it only goes 1 unit in every direction. So the furthest points it can go to is 1 or -1. So the distance from 1 or -1 to the x axis is 1 unit, making our amplitude 1. Cotangent, tangent, cosecant, and secant do not have amplitudes, however they do have asymptotes. They have asymptotes where sine and cosine equal zero. 

Photo:  http://www.oojih.com/show/trigonometry/generalangle/

Student Video Concept 7 #3

This video explains step by step how to solve an equation with half angle formulas. The problem that we will solve is sin(x/2)+cos(x)-1=0. When solving this problem you need to make sure to check for extraneous solutions. Because we squared both sides, we may have created answers that are not really solutions to the problem. To check for extraneous solutions, plug your answer values back into the original formula, and check if the answer is true.

Student Video Concept 3 #5

This video explains how to use power reducing formulas to reduce our problem so that the problem has the highest power of 1. The problem that we will be reducing is sin^2(x)cos^2(x). Pay close attention when foiling that (2x) is a variable and does not get foiled. Also, pay attention to when we substitute (m) for 2x and know that you will have to plug 2x back in for the value of m. Lastly, you may have to use the power reducing formulas more than once to reduce everything down to the highest power of 1.

Sum and Difference formula in relation to the half angle formula

 The first picture depicts using the half angle formula to find the sine,cosine,and tangent for the angle 105 degrees. The second picture shows using the sum formula to find sine,cosine, and tangent for the same angle. When using the half angle formula, you need to pay attention to what is positive and negative based on the quadrant that your half angle is in. When we solve with the half angle formula and the sum formula we find that our answers are the same for sine, cosine, and tangent. Our answers are written differently when we solve using both formulas, however they are equivalent. We can check by plugging in our answers for sine,cosine,and tangent for both formulas in our calculator , and we find that our answers are the same. The two pictures above depict how to plug our answers into our calculator and shows that they are the same.

Student Problem #3

Student Problem #2

Student problem Unit R concept 1

Difference Formula

Sum Formula

Blog Post #3

Blog Post #2

Blog Post#1: Deriving Pythagorean Identities

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 performed well on the Unit O and P tests. I have done well on my unit o and p quizzes. I received all 8's on Unit P and almost all 8's on Unit O. Also, i received an 86% percent on Unit O and a 81% on Unit P. 

 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.

1) I believe watching the videos and doing the SSS packet is very beneficial and so i take it seriously. I watch every video and submit every wsq before class. I refer back to my SSS packet when i need extra examples or more help. I could improve my focus on this step and improve my mastery of the by giving more detailed answers on the wsq and trying to understand why we do things in math. This would require me to seek other resources like the textbook or be active in my classmates group discussion. 

2) I do not usually look in the book or research on the concept online before class. I could improve this by accessing the additional resources Mrs. Kirch provides on her blog. This would help me understand why we do things and not only that we do them. 

3) I look at the additional resources provided by Mrs. Kirch sometimes. These additional resources help explain the concepts in relation to real life, and i tend to slack off on that part. To improve this, I should start accessing Mrs. Kirch's resources to further my understanding of the math concept. 

4) I think the discussion in class is semi-beneficial. Most of the questions are the ones on the wsq, but in some cases they are not. I think the questions that differ from the wsq are most beneficial to help us further our understanding. I could improve my focus on this step by being more active in class discussions and providing interesting hot questions. 

5) I take the pq's seriously because they test you understanding of the concept and give you examples to try on your own. In some cases, the pq's can be a bit excessive when you understand the concept thoroughly. To improve my mastery of the material i can do the pq's completely, so i have more practice on the concept, thereby improving my mastery of the concept. 

6)I take quizzes seriously because not only do they affect your grade significantly, but they also test t=your knowledge on the concept. They are usually just a couple questions and if you have trouble on the questions then that shows you need more practice on that concept. To improve my mastery of the concepts i can make sure to get 8's on all my quizzes and retake quizzes when necessary. 

7)I think the PT is excessive. It provides additional problems and a cumulative review of all the concepts but i think we should only have to do the concepts we need more practice on, and the review portion from other units. Completing all problems even the concepts i know well will help me master the concepts. 

8) I take the Unit Tests seriously because it is a test of your knowledge. Also, tests are a great reflection of your grade. To further my mastery of the concepts, i can seek additional help from students and find other resources on Mrs. Kirchs blog. 

 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 usually feel for the most part confident. However, sometimes go into the test knowing that i still struggle in some areas. My grades reflect that i know 80% or more of the material.
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.
Compared to past years, i would this that this has been the year i have struggled most in math. I received my first C in all my years of learning in math this year. I think this was just a matter of adapting to the flip classroom. However, i have learned math in a deeper sense this year compared to previous years.
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 i have learned math deeper than any of my other previous years of learning. I learn more of why things happen and real life applications this year, than other years. Some examples include deriving laws like the law of sines and cosines and the applications of shapes like parabolas in real life.
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.
I feel i understand why, after Mrs. Kirch gives a further explanation or asking a classmate. I understand why we do things, and how they are applied in real life. I have certainly have gone deeper into math this year, than previous years.
e. How does your work ethic relate to your performance and success?  What is the value of work ethic in real life? 

I believe you get out, what you put in. Meaning your work ethic reflects your success or failure. My work ethic reflects my grade because i give an average effort, but in some areas i should be putting a little more effort. Work ethic in real life determines your position in life. You can work really hard and become successful or you can slack off and fall behind others. You can either be in charge, or have someone in charge of you.

WPP#14

WPP#13

Angie, Charlie, and Billy are playing tag in the wilderness. Angie and Billy are separated on opposite sides of a river. Angie measures the distance from where she is standing to where Charlie is standing to be 250 feet, and they both happen to be on the same side of the river. Angie then measures the angle from herself to Billy to be 60 degrees and measures the angle from Charlie to Billy as 53 degrees. Find the distance from Angie to Billy to the nearest tenth of a foot.

WPP#12

Create your own Playlist on MentorMob!

WPP#11

Peter's Planned Prank

Peter plans to make his way on top of the field goal post during his high schools football game dressed in a chicken mascot suit. He plans to dance and get the crowds attention as a prank. He is contemplating the best way to boost his way up to the field goal post and decides using a ladder will be the most  efficient way. If the post is 12 feet high and Peter plans to place the ladder 9 feet away from the base of the post, how long will the ladder have to be in order to get up onto the field goal post?

Unit Circle Right Triangle

The unit circle consists of four right triangles. since we know our radius is equal to one, this is like our hypotnuse of a right triangle. Since our hypotnuse is equal to 1, and our hypotnuse is also equal to 2n, when we set 2n eqaul to one we find out that n is equal to 1/2. When we set it equal to n radical 3 on the x axis, and multiply it by 1/2 we find out our x is equal to radical 3 over 2. This is like a point on a unit circle being (radical 3 over 2, 1/2 ).

Understanding Conic Sections

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?
Answer: The mathematical definition of a parabola is that the distance from the focus to any point on the parabola will be the same as it goes straight down perpendicular to the directrix. Because the distance from the focus to any point on the parabola is the same, it creates sets of points that creates a "U" shape which is symmetrically cut in half by the axis of symmetry. 

2.How does the focus (or foci) affect the shape of the conic section?  (If you choose ellipses, you should include information about eccentricity in your response; if you choose parabolas, "p" should be a big focus... haha, get it? "p" is the distance from vertex to focus and it should be a big focus. Ok, moving on...) 
Answer: In a parabola "P" is the distance from the focus to vertex and the vertex to the directrix. The greater the value of "P", the wider the parabola will be. The smaller the value of "P", the skinnier a parabola will be. This is because the greater the value of "P", means that their is a greater distance from the focus to vertex and vertex to directrix. Because each point on a parabola is the same distance from the focus, as p increases, the farther out the points are and so the the parabola is wider. The smaller the value of "P", the smaller the distance from the focus to vertex and vertex to directrix. So the smaller the distance from the focus to points on a parabola , the skinnier a parabola is. 

3.How do the properties of this conic section apply in real life?  (While using the exact examples I gave you may be acceptable, I will be looking for some research, creativity, and thought.  There is a lot out there! 
Answer: The properties of Parabolas apply to real life in a automobile headlight. The light from a bulb at the focus point of  the metal back of the headlight reflects and is sent outward to light our way in the dark. Also, cable TV dishes are shaped like a parabola. Their shape allows them to absorb parallel waves and reflect them to a specific point (focus). A TV dish is like a parabola because it absorbs the satellite waves and focusses them on a certain point to allow someone to receive a strong enough signal to watch television. 











Citations:
Parabola Image pasted from: http://www.mathsisfun.com/geometry/parabola.html
Focus image paste from http://en.wikipedia.org/wiki/File:Parabolic_reflection_1.svg
Direct tv image from http://www.post-gazette.com/stories/business/news/shopsmart-sorting-out-cable-satellite-phone-web-bundles-377744/
Information from http://www.intmath.com/plane-analytic-geometry/4-parabola.php