To the incoming Math Analysis students, to make sure that you have a fun
and successful year you must be responsible. This is your
responsibility because it was either your own choice or your parents'
that you are taking this class the following year. This class is a rigorous and difficult course, however it is easy if you follow instructions and complete your required assignments on time and legitimately. Do not procrastinate otherwise you will be left behind and clueless when you enter the classroom the next day. Be sure to ask plenty of questions if you are confused about something! Remember it is not bothersome to ask the teacher, they are there to help you.
Adjusting to the flipped classroom can be confusing at first. You will make several accounts, most likely Edmodo, Mentormob, and a Blogger Be sure to keep track of Edmodo as you will receive important updates for class assignments. There will be instructions on how to use and upload the assignments.
The Flipped Classroom is an ongoing experiment. Instead of learning in the classroom you will have time at home to watch tutorials on the concept you are learning about and upon the next day you will be expected to have watched the video(s), be exposed to the material, and have questions on what you have trouble on. Good Luck and Have Fun~
1. Unit V “Big Questions” Blog Post - 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 your writing.
The Difference Quotient can be explained as we are attempting to find the tangent line in a function on a graph. First we will have a curved line that has a secant line that will touch the graph twice and a tangent line that will only touch it once. At the farthest point of intersection, we will use dx to solve for the secant line and labeling it as h. As we see that the secant line is no where near the tangent line, the equation is f(x+dx)-f(x)/dx. We need to find the value for the secant line that is closer to the tangent line. We can make dx smaller so that the secant line is barely touching the tangent line. The slope of the secant line can be described as the difference quotient, we can find the slope by using the slope formula which is m=y2-y1/x2-x1. Then we can plug in the x-value (x+h)and then the y-value (f(x+h)) the new equation would be (f(x+h)-f(x))/(x+h)-x)). As we simplify this function we see that f(x+h)-f(x) does not simplify. Then we use lim-->0 to show that the secant line and the tangent line cannot touch.
Here is an example of how the difference quotient turns out to be, A being the tangent and B the secant:
Unit U Big Questions
1) What is a Continuity? What is a Discontinuity?
A Continuity is a graph that can be drawn without having to lift your pencil off the paper, meaning that the graph will have no holes, jumps, or breaks within the line. While a discontinuity is a graph that will have breaks, holes, and/or jumps within the graph.
Continuity Graph:
Discontinuity Graph:
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, it exists when the line on the graph from the left or right meets up in the same place.and the same value of x and has no discontinuity whatsoever. Limits exist only when there are discontinuities, the difference between a limit and a value is that the value is an exact answer while the limit is just a notation explaining the discontinuity on the graph.
Example of Values:
Example of a Limit:
3) How do we evaluate limits numerically, graphically, and algebraically?
We can evaluate a limit numerically by making a chart which represents our x-values F(x). We then write our limit statement: lim-->x f(x)=x while subbing the x when necessary. We can evaluate the limit graphically by looking at the graph to check and see if there are any points of discontinuity, if there are no points of discontinuity then the graph will have no limits. We can solve a limit algebraically by substituting x from our limit statement to our functions. There are a total of 4 different answers that we can receive from doing this: A number, 0, undefined (DNE), and 0/0 which means that it cannot be found there. This means that you will have to multiply the conjugate to get rid of the 0 in the denominator and simplify, then start from there again.
1. Show and explain how to derive the two remaining Pythagorean identities from sin^2(x)+cos^2(x)=1. Make sure to include in the beginning where sin^2(x)+cos^2(x)=1 comes from to begin with (think Unit Circle!).
The reason is that sin^2 + cos^2 = 1 is written differently from cotangent and tangent. So dividing the first equation by sine^2 or cos^2, they will then cancel out. This will then leave you with an answer related to your ratio identities and your reciprocal ratios which as you know will be: 1/cos^2(x) is the same as sec^2(x), 1/sin^2(x) is the same as csc^2(x).
2. Chose a Concept 2 problem to solve in two ways: with identities (Unit Q) and with right triangles (Unit N). If you find another way to solve it, include that as well.
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. The Unit O and P tests were not that difficult to me. However I did struggle with Concepts 6 and 7 on the Unit P test. The Unit O test was fairly easy for me. The matrix however was still tricky for me on the concept review. I studied for the concept O test and that helped me a lot. The worksheets and pq's based upon the concept helped me and the SSS and videos I had watched also helped me a great deal while doing the test. 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. In the WSQ I will do most of the problems that I deem necessary. I only do supplementary research if required for blog posts. I have touched on some supplementary info that is on the blog. In class, in our group we do discuss concepts that we thought to be difficult. PQ's are really helpful, but kinda tedious as they are lengthy. I do well on quizzes, I make sure that I can get an 8 on the first try. The PT's are a good assessment that I should take seriously, the only reason is because of all the work that is done in the PQ's that I deem them unnecessary. I need to do my WSQs more earlier than waiting. Supplementary info is always good, I can touch up on a little more info in the future. I can well discuss concepts with my group in class. I need to review what I get wrong on quizzes so that I can apply that problem to future problems. PT's are a good extra help for me. The assessments after the tests when I get it back help me. I can see what I did wrong and compare it to other people's. 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 feel generally confident that I can at least pass the tests after rigorous studying with the SSS and the PQ's. I feel that it is a tougher course than my other math classes, the material that I learn and review based on my previous years show that it is well established. Rather than learning from the textbook and in class we can go at our own pace.
I feel that the content within this math class is very interesting. Compared to other math classes there is more interaction with other students rather than learning by yourself listening to lectures.
Yes, I've seen connections to previous years and the higher level of thinking with other assessments is also a great and new learning experience.
My ethnicity does not really affect my performance other than comparing to other and how they do I feel that I must not let my peers down and work so that I will not be the lowest level in the class.
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 a parabola is shown by referring to the focus, drawing a line from the focus to any point on the parabola, should be equal to a line drawn straight down in a perpendicular matter starting from that same point to the directrix. "P" is the key component that describes how the parabola will look like in shape, form, and direction it opens up to.
This will determine in which direction the graph will open up:
+p x^2 Graph opens up
+p y^2 Graph opens right
-p x^2 Graph opens down -p y^2 Graph opens left
2) How does the focus (foci) affect the shape of the conic section?
- "P" determines a major portion of the graph. It shows where the parabola opens up and shows how "fat" or "skinny" the graph really is. If "p" equals a large number then the graph will be fairly large and "fat", if it is a smaller number then it determines that the graph will be small or "skinny".
3) How do the properties of this conic section apply in real life?
-With its parabolic shape and a focus a parabaloid can be used in many different situations.One that I found interesting was the parabolic mirror, this mirror can generate almost 2,000 kilowatts of solar energy. Although it is impractical the mirror can heat objects quickly due to its concave shape and focus in the middle. Solar energy is concentrated in the center and is emitted forward and focused. This video explains its many uses:
