Letter to Future Math Analysis Students

1. What do you want to say to them to help them have the most successful year possible?  

To have the most succesful year you must watch the videos when you get home right away, and do all your math homework FIRST. MATH IS A PRIORITY! Also, do all the quizzes and tests that Mrs. Kirch gives you, even though they are extremely tedious. When you do all your work, you will have the best test results. If you do not, your test scores will be below average.

1. How can they best adjust to the flipped classroom and learn to work with all the technology I require of them?  (do you have any specific tips for them or experiences you could share?)

Just follow instructions, and don't question why you are doing it. For example, when you make a blog, make sure to follow the directions as stated. I used to question why we did all these pointless exercises during class and pointless blog posts, but I soon learned that it helped me with not only math but also developed other skills that I could possibly use in my future life. For example, when you make a video it is required that we thoroughly analyze and explain a problem to a fellow classmate. I learned how to talk without being nervous which helped me with my public speaking. Math Analysis does not only teach math, but also teaches good ethics and morals that we need to begin creating in our lives. It is not simply remembering a bunch of methods for a test but to learn how to apply it when we encounter a problem.

1. What can they expect to be different from their previous math classes?

The workload is especially different, and the way it is done is also completely altered. Students will no longer be doing all their work in class, but also work at home. Blog posts and videos will be extremely different from what they did at home last year. In previous math classes, you sit in school and learn about boring lectures, but in Mrs. Kirch's class room, you learn the material at home and go to class and ask all the questions you want about math. Asking a lot of questions will get you to ace the test so ASK QUESTIONS.

Unit V: Big Question

Q: 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).

First of all we have our graph. In our graph, we can find the distance between two points by finding the slope of the secant line. To find the slope of the secant line, all we must do is plug in y2-y1/x2-x1 which is f(x+h)-f(x)/x+h-x  as you can see we can cancel out stuff in the denominator. If we do so, we would be left with our difference quotient

Unit U: Big Questions

Q1: WHAT IS A CONTINUITY? WHAT IS A DISCONTINUITY?

A continuity is a function that can be drawn as a graph CONTINUOUSLY without anything from stopping it. In simpler terms a continuous graph or function can be drawn without lifting our pencil from the paper because it doesn't have any breaks or jumps in it.

A discontinuity is a break, hole, jump, or whatever in the graph that causes it to be discontinuous. It's easy.

Q2: WHAT IS A LIMIT? 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 graph can approach two different directions, left and right. If the height of a function does not reach in the same place from left or right, that is when the limit does not exist. A limit, is the height that a graph needs to reach, and a value is an actual number that a graph reaches. If a graph reaches a hole, that is a limit but it is not a value.

Q3: HOW DO WE EVALUATE LIMITS NUMERICALLY, GRAPHICALLY, AND ALGEBRAICALLY? 

Numerically, we can do it on a table with the x value on top, and the limit of the function on the bottom. Graphically, we are just looking at a graph as shown above, and from there we can determine the jumps and discontinuities.Algebraically we use direct substitution and plug in whatever x is approaching.

Unit T Big Question 4

Unit T Big Question 3

Unit T Big Question 2

Unit T Big Question 1

BIG QUESTION 1a




 BIG QUESTION 1b

Student Video #2 Unit S Concept 7 - Solving Equations w/ Half-Angle Formulas

      This video is about solving equations by replacing them with Half-Angle formulas and such. The whole goal of this concept is to find out how to solve for x. We mainly use the zpp to find our answers. In this concept we have to remember our identities from Unit Q to replace, and figure out our answers.
      What the viewer must pay attention to is factoring and combining terms. When combining terms we cannot combine something with a power to one without a power. For ex: cos^2 cannot combine with cos.

Student Problem (Assessment #2)

I know that these answers are the same and equivalent to each other because first of all, they look identical, and secondly, if they were not to be identical, we can plug them into a calculator and compare the exact values.

Student Video #1 Unit S Concept 3 - Power Reducing Formulas

           This problem is about using power reducing formulas. What we do in this concept is substitute a given trig function with a power reducing formula to create a whole new equation. In the new equation, we will create a reduced formula. In the reduced formula, the highest power should be to the first power. If it is not, we keep reducing until we do so.

           When watching this video, make sure that we reduce the equation fully. Also make sure to foil correctly. When foiling, multiplying the same trig function with itself, ADDS the exponents. Some make the mistake of actually multiplying them which is not correct so be careful. Also make sure to reduce it fully before giving the final answer.

Unit R: Student Problem 3 (Concept 3)

Unit R: Student Problem 2 (Concept 2)

Unit R: Student Problem 1 (Concept 1)

This problem is about using sum and difference formulas in concept 1 of unit R. In this concept all we must do is plug in our exact values of an angle, and we will get our exact answer.

We can always check our answers by just plugging them into our calculator.

Unit Q Blog Post 3: Concept 4

Unit Q Blog Post 2: Concept 2

Triangles

Ratio Identities

Unit Q Blog Post 1: Pythagorean Identities

Deriving Pythagorean Identities

In this chapter we have 3 Pythagorean Identities:

1. sin²x + cos²x=1

2. 1 + tan²x=sec²x

3. 1 + cot²x=csc²x

Pythagorean Identity #1

For our first Pythagorean Identity, we can derive it from a unit circle.  As you can see on the right, we have a right triangle labeled x, y, and 1 (hypotenuse is always 1).

From here, we can substitute or x and y value. 'x' will be substituted to  become cosx, and 'y' will be substituted to become sinx.

We label 'x' value as cosx and 'y' as sinx because of the ratio identity, tanx. As a ratio identity, tanx=sinx/cosx. As a trigonometric ratio, tanx=y/x therefore, sinx  will be in place of 'y' and cosx  will be in place of 'x'.

Now that we have labeled our triangle, we can integrate the Pythagorean theorem, x²+y²=1.

Since we have replaced 'x' and 'y', we now have cos²x+sin²x=1.

Pythagorean Identity #2

For our second identity, we start with our first identity cos²x+sin²x=1. From here we can derive it and retrieve our second one.

First things first, we have cos²x+sin²x=1. Now what we must do is divide the WHOLE equation by cos²x.

Once we do so, we can cancel out the two cos²x and it will become 1, sin²x/cos²x will become tan²x as a ratio identity, and 1/cos²x is equivalent to sec²x. So now we have our equation 1 + tan²x=sec²x.

Pythagorean Identity #3


For our second identity, we start with our first identity cos²x+sin²x=1. From here we can derive it and retrieve our second one.

First things first, we have cos²x+sin²x=1. Now what we must do is divide the WHOLE equation by sin²x.

Once we do so, we can cancel out the two sin²x and it will become 1, cos²x/sin²x will become cot²x as a ratio identity, and 1/sin²x is equivalent to csc²x. So now we have our equation 1 + cot²x=csc²x.


All Pictures taken from here. I do not take credit for making any of these pictures.

Math Analysis Reflection

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 Unit O Test, I feel like I did extremely well until I got the test results back. I made extremely minor mistakes. As I prepared for the retake, I didn't feel like any of the work I did was necessary. I retook the test and got a way higher score from learning from my mistakes. Overall, I felt like my test grade was very good for the Unit. For Unit P, my test grade was actually better than I expected. For this test I really gambled my luck. I got an 85 expecting to get something way lower. My quiz scores demonstrated that I made minor mistakes as well and fixed them on the assessment. 

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.


The first step I take very seriously. I try to focus as much as I can to learn the material that I need to. I don't know how to improve my focus but I already try my best. For the second step, I do not take seriously at all. I feel like that is not necessary, I only use the videos and the packet. Same with step three and four. I try to do my PQs as much as I can until I feel very confident. The quizzes are pretty much a simulation of the test so try to take them seriously as well, If I feel like I am not confident, I would know what to study before the test day. For the test, I take my time and go over each problem slowly and this sometimes leads to conflict because the bell rings and I run out of time.

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?

On the day of the test, I feel somewhat confident. Realistically, the test is always intimidating, however I have learned to conquer my enemies, and step on the field as I if I am always going to conquer. Starting off with the first test in semester 2, I always circle the A+ on the corner of my test, believing that I would get that. Realistically I never earn the perfect score but I always earn a score that is fairly close to that. One time I did not circle the letter grade I predicted to get at all and I got a C, which was very coincidental, but now on every test it is like a ritual for me to circle A+. I feel like this year of math I have learned WAY more than last year. I feel like this class is more organized than any of my other classes from previous years. In my opinion, the practice quizzes help a lot. If I get something wrong, I can learn from my mistakes and never make that same mistake again on the test. I feel like the PQ, and PT Fall under the same category, and are just there for practice. Math this year has help me to understand why something happens in math. It is not just the procedure anymore, but the reasoning behind it. By deriving formulas from triangle, it has helped me understand that all these formulas do not simply pop up from the shadows. My work ethic is very inconsistent. There are weeks were I try very hard, and then there are weeks where I am so tired from how hard I've been trying. However, after the many months of junior year, I've started to prioritize math over my other classes. Work ethic is very important in real life. One slip off the track and it could be possible to never get back on. It is important to stay consistent through the whole year no matter how hard it is, it will eventually pay off. As of right now, I am currently trying my best to practice what I preach. I am still young and learning and math analysis this year has taught me a lot.

WPP #14 Unit P Concept 7

• This concept incorporates law of cosines in word problems
• Listen carefully as I demonstrate how to find sides and angles using the law of cosines

WPP #13 Unit P Concept 6

• This is about incorporating law of sines in word problems
• Watch carefully as I find the missing angles and sides of the triangle. 

WPP #12 Unit O Concept 10

Create your own Playlist on MentorMob!

This video is about translating word problems in math.
Watch closely as a explain how to use trig functions to find the missing sides of a triangle.

WPP #11 Unit O Concept 9

Create your own Playlist on MentorMob!

This video explains to you how to use the Pythagorean theorem.
Pay attention as I translate the word problem in to a 2d visualization.

Unit N Concept 7: Derive the Unit Circle

Focusing on the top right part of a circle we have two special right triangles inscribed: 30-60-90 and 45-45-90. Within the first quadrant of a circle, we have are three main points. They can be found and explained by these two special triangles.

Triangle #1

As we can see this triangle is formed by creating a 30 degree angle. To find our point on the circle, we must know the rules of these special right triangles. Each side of a triangle is labeled with a missing variable. For example, for this special right triangle, the hypotenuse is equivalent to 2n, the shortest side of the triangle is equivalent to n, and the longest side before the hypotenuse equals n radical 3. All we need to do is find the lengths of x and y of the triangle. All we have to do is set the hypotenuse (2n) equal to 1 and we can solve from there on. work is shown below.

Triangle #2

 The next special right triangle is created by a 45 degree angle. What we must know is that for this triangle the hypotenuse is equivalent to n radical 2, and the the two other sides are equivalent to n. Again all we must do is set the hypotenuse equal to 1 and solve. From there we can get the lengths of x and y, and eventually our point on the circle.

Triangle #3

For this special right triangle, we can see that it is pretty much the same as the first triangle. Once we do this last one we can see that we have found three coordinates, and these coordinates correspond on a unit circle. This occurs throughout the unit circle except in the other quadrants we must face the signs of the coordinates. We can see that these three points are the same throughout the unit circle and how it relates to it.

Conic Section (Ellipse)

Conic Section

(Parabola)

Q #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?

A #1: A parabola consists of a focus point, and a directrix. The focus is just a point inside the parabola, and the directrix is a fixed line beneath the parabola to help draw the curve. We can see that the length between the focus and directrix are equidistant when connected to touch the parabola. The eccentricity of a graph is usually 1. This is because if it were less, the curve would close in, and if it were greater, the parabola would eventually open wider and create a hyperbola. Click here to get a visual understanding of eccentricity. 


Q #2: How does the focus (or foci) affect the shape of the conic section?
A #2: The distance of of the vertex to the focus point named as the term 'p'. The 'p' is equidistant to the length of the directrix connected to the graph. The distance of the focus and directrix to the vertex can fluctuate the size of the parabola. If the focus and directrix were farther away from the vertex, the graph would open much wider, otherwise it would be very thin. Click here to experiment with a live parabola. 

Q #3: How do the properties of this conic section apply in real life? 
A #3: Here we can see that the cables of the S.F.G.G. Bridge is very close to a parabola. It is shaped this way because the two towers holding the cables are very close to each other. Therefore, the cable must equally distribute its weight from the distance of one tower to the other. The closer it is to the tower, the higher the curve. This creates the shape of a parabola. 

Citation

http://www.mathsisfun.com/geometry/eccentricity.html

https://www.google.com/images

http://www.intmath.com/plane-analytic-geometry/parabola-interactive.php