Dear future math analysis student,
Do not take advantage of the extra freedom that is privileged in this class. Hold this advice above all else. Do you want a successful year? If your answer is no, then you simply cannot have a successful year. Somewhere in the classroom there is a poster that says "If it is important to you, you will find a way." This is possibly the most important lesson in this classroom. Take a step back and forget math for right now. Think about the last time you did something, anything. Now realize that at one point in time, this "something" was exactly what you needed and wanted to do. Now that you understand this, apply it to not just this math class, but to your entire schooling. "What do you want to say to them to help them have the most successful year possible?" Simply, make yourself want the successful year.
Take it from me, if you won't listen to Mrs. Kirch, have an open mind. This is the most practical way to adjust to this classroom. Yes, you might miss the lectures from Mrs. Basu (or whoever was your Algebra 2 teacher), but now you can skip ahead if you understand and not have to sit through those boring minutes. Or probably even more common, you can ask questions without having to feel like you are slowing the rest down. Just always keep in mind that the technology is there to help you, not stop you.
Expect having your homework checked thoroughly. Thoroughly. Expect work that seems to be "extra" or however you want to call it, but above these things, know that they are for your own good. Actually, the the greatest tool will be your opportunity to collaborate on the material. That's really what got me through the year, and now on the last test of the year (calculus material) I received a 91%. DO NOT BE AFRAID TO collaborate.
-Ben Camacho
Any further questions? Consider this video!vvvvvvvv
Thursday, June 6, 2013
Tuesday, June 4, 2013
Unit V: Big Question
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.
A regular positive parabola, a slope is found using m=y(sub2)-y(sub1)/x(sub2)-x(sub1). This is basically what we are doing again, only for a tangent line for a single point. This is important to remember! Two points on the parabola are plotted, the second point must be a tad bit to the right and up of the other. Make sure to label the points. In the original formula, "x sub 1" becomes "x" and "y sub 1" because "f(x). Next we draw a line that connects the points, and call that distance "h". So then it becomes "x+h" and "f(x+h)". Next we plug in the new points. Next we just cancel out the +/- x values and we get the difference quotient, or "derivative".
source: sciencehq.com
A regular positive parabola, a slope is found using m=y(sub2)-y(sub1)/x(sub2)-x(sub1). This is basically what we are doing again, only for a tangent line for a single point. This is important to remember! Two points on the parabola are plotted, the second point must be a tad bit to the right and up of the other. Make sure to label the points. In the original formula, "x sub 1" becomes "x" and "y sub 1" because "f(x). Next we draw a line that connects the points, and call that distance "h". So then it becomes "x+h" and "f(x+h)". Next we plug in the new points. Next we just cancel out the +/- x values and we get the difference quotient, or "derivative".
source: sciencehq.com
Tuesday, May 28, 2013
Unit U: Big Questions
1.What is continuity? What is discontinuity?
Continuity is a concept in calculus that deals with graphs, it means that there is something that goes on (continues) without being interrupted. Discontinuity is along the same lines, but instead of being continuous, it is interrupted by jumps, holes and breaks in the graph.
<----CONTINUITY
<----DISCONTINUITY
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 graph/function. The limit exists only when the right and left limits are equal. On the other hand, the limit does not exist if the graph has a break (jump, oscillating, infinite discontinuities). The difference between a limit and a value is that the value is the number that belongs and appears in the graph.
3.How do we evaluate limits numerically, graphically, and algebraically?
Numerically: create a table that has x-values and function of x values, use the number that the limit as x approaches, which is what we will find. Write it mathematically (limx-x>#f(x)=).
Graphically: analyze given graphs. Notice if sides do not meet, a limit does not exist. If the graph has discontinuity, the limit is there. Jumps, oscillating and infinite discontinuities = DNE.
Algebraically: we use direct substitution, dividing out or factoring method, rationalizing/conjugate methods.
Continuity is a concept in calculus that deals with graphs, it means that there is something that goes on (continues) without being interrupted. Discontinuity is along the same lines, but instead of being continuous, it is interrupted by jumps, holes and breaks in the graph.
<----CONTINUITY<----DISCONTINUITY2.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 graph/function. The limit exists only when the right and left limits are equal. On the other hand, the limit does not exist if the graph has a break (jump, oscillating, infinite discontinuities). The difference between a limit and a value is that the value is the number that belongs and appears in the graph.
3.How do we evaluate limits numerically, graphically, and algebraically?
Numerically: create a table that has x-values and function of x values, use the number that the limit as x approaches, which is what we will find. Write it mathematically (limx-x>#f(x)=).
Graphically: analyze given graphs. Notice if sides do not meet, a limit does not exist. If the graph has discontinuity, the limit is there. Jumps, oscillating and infinite discontinuities = DNE.
Algebraically: we use direct substitution, dividing out or factoring method, rationalizing/conjugate methods.
Wednesday, May 1, 2013
Unit R: Student Problem 1
Tuesday, April 30, 2013
Wednesday, April 24, 2013
Unit T Big Question #4
The Fourth Big Question:
Why do sine and cosine don't have asymptotes, but the other four graphs do?
Sine and cosine do not have asymptotes because of their ratios. Cosine is x over r or x over 1. So when a number is input as X, then the value of the ratio will be that number (because the denominator is 1) even if zero was inputted. The same rule applies for sine, which is y/1. So when a value is inputted for Y, the value of the whole thing will be that number.
So what does all this mean?
Sine and Cosine cannot be undefined in these situations.
Notice all the other trig functions involve cosine or sine being the denominator, and if they end up equaling to zero, then it makes that certain function undefined and creates an asymptote.
Unit T Big Question #3
THE THIRD BIG QUESTION:
Why is a normal tangent graph uphill, but a normal cotangent graph downhill?
The normal tangent graph is uphill for several reasons. Let's start with identities. Tangent equals sine over cosine, and when cosine equals zero, the value becomes undefined. This occurs when the degree value is 90 (pi/2) and 270 (3pi/2). So this means that at these values, there will be asymptotes. Now let's recall ASTC. In the first quadrant, tangent is positive, so the line will begin from the y-axis going upwards (because it's positive) towards the asymptote located at pi/2. The opposite happens in the next quadrant, then the pattern repeats (remember this from previous blog posts). Cotangent is equal to cosine over sine, so let's go with the same logic we used with tangent. If sine were to be 0, then it would become undefined (aka, an asymptote would be present). This occurs at zero degrees, pi, and 2pi.
Why is a normal tangent graph uphill, but a normal cotangent graph downhill?
The normal tangent graph is uphill for several reasons. Let's start with identities. Tangent equals sine over cosine, and when cosine equals zero, the value becomes undefined. This occurs when the degree value is 90 (pi/2) and 270 (3pi/2). So this means that at these values, there will be asymptotes. Now let's recall ASTC. In the first quadrant, tangent is positive, so the line will begin from the y-axis going upwards (because it's positive) towards the asymptote located at pi/2. The opposite happens in the next quadrant, then the pattern repeats (remember this from previous blog posts). Cotangent is equal to cosine over sine, so let's go with the same logic we used with tangent. If sine were to be 0, then it would become undefined (aka, an asymptote would be present). This occurs at zero degrees, pi, and 2pi.
Unit T Big Question #2
How do the graphs of sine and cosine relate to the others?
Tangent: We graph tangent using cosine to show us where the asymptotes will be located. This is because a tangent line can only be placed when cosine equals 0 (aka, asymptote, because tan=sin/cos)
Cotangent: Graphing cotangent is easy when we have sine because it will show us our asymptotes. For cotangent, sine is the denominator in the ratio so if sine is zero, then there will be an asymptote at that spot for the cotangent graph.
Secant: The secant graph comes from the parent graph of cosine. The cosine graph will show us where the asymptotes of the actual graph will be. Another obvious connection is that secant is the reciprocal of cosine.
Cosecant: Drawing a cosecant graph requires the use of the sine parent graph (the reciprocal). The sine graph will easily show us where the asymptotes of the cosecant graph will be placed, thus showing us how the cosecant graph will look like.
Big Questions Blog Post #1
The 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?
Periods are cyclical, they are referring to revolutions of the unit circle. A period is only one time around the trig graph (one revolution around the unit circle).
This cycle repeats until the end of time, that's why it is called a cyclical graph. The difference between the periods of sine/cosine and tangent/cotangent is accredited to ASTC. Sine's order is positive positive negative negative. The signs repeat at 2pi. Now if we look at tangent, the ASTC is positive minus positive minus. For this, we only have to go to pi to allow the pattern to repeat itself.
B) Amplitude: 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?
(This is our friend)
Amplitudes are found when sine is pi/2 and 3pi/2, which are the quadrant angles of 90 degrees and 270degrees. Cosine is similar, however this occurs at zero degrees, pi, and 2pi.
The rest of the trigonometric functions (tangent, secant, cosecant, cotangent) are similar. They have asymptotes when the denominator of the ratio is zero (this makes it undefined and it cannot occur).
google images used.
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?
Periods are cyclical, they are referring to revolutions of the unit circle. A period is only one time around the trig graph (one revolution around the unit circle).
This cycle repeats until the end of time, that's why it is called a cyclical graph. The difference between the periods of sine/cosine and tangent/cotangent is accredited to ASTC. Sine's order is positive positive negative negative. The signs repeat at 2pi. Now if we look at tangent, the ASTC is positive minus positive minus. For this, we only have to go to pi to allow the pattern to repeat itself.
B) Amplitude: 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?
(This is our friend) Amplitudes are found when sine is pi/2 and 3pi/2, which are the quadrant angles of 90 degrees and 270degrees. Cosine is similar, however this occurs at zero degrees, pi, and 2pi.
The rest of the trigonometric functions (tangent, secant, cosecant, cotangent) are similar. They have asymptotes when the denominator of the ratio is zero (this makes it undefined and it cannot occur).
google images used.
Tuesday, April 16, 2013
AS #4 Unit S Concept 7, #5
AS #3 Unit S Concept 3 Number 5
AS # 2 Unit S Concept 4
In these pictures, I solved the same problem twice. However, each time I used different methods: the sum and difference formulas and half-angle formulas. Even though the angles used are not included in the unit circle, this process shows how we can still solve the problem. We know that the answers I ended up with are the same in both instances because when we plug them in, they give the same decimal number.
Wednesday, March 20, 2013
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Unit Q: Blogpost for concepts 2-3
tanx = rad3 / 3; cosx = -rad3 / 2Special right triangles apply here.
Understand that:
tan = y/x
cos = x/r
when tan = rad3 / 3, the angle is 30 degrees.
when cos = - rad3 / 2, the angle is 30 degrees.
Now that we understand this, we have to find the value of the ratio when the angle is 30 degrees, so we must see that the point on the unit circle will be (rad3/2,1/2). We use this for all of the ratios.
Understand this next:
sin=y/r=1/2
csc=r/y=2
sec=r/x=2rad3/3
cot=x/y=rad3
Now that we are this far, we must remember that the angle could be in multiple quadrants. Tangent is positive so we automatically know it could be in the first or 3rd quadrant. Cosine is negative, it could be in the 2nd or 3rd quadrant. They both have the 3rd quadrant in common, so it has to be there.
sin = -1/2
csc = -2
sec = - 2rad3 / 3
cot = rad3
Using identities to solve the same problem.
We must understand all this:
sin^2 = 1 - cos^2
sin^2= 1 - (-rad3/2)^2
sin^2 = 1 - 3/4
sin^2= 1/4
rad sin^2 = rad1/4
sin = 1/2 or -1/2
csc = 1/sin
csc = 1/ (1/2)
csc = 2
sec = 1/cos
sec = 1/(-rad3/2)
sec = -2 / rad3
AT THIS STEP WE RATIONALIZE
sec = -2rad3 / 3
cot = 1/tan
cot = 1/√3/3
cot = 3/√3
RATIONALIZE THE DENOMINATOR
cot = 3rad3/3
cot = rad3
Deriving Pythagorean Identities
X is the adjacent side of the triangle, while Y is usually the opposite side, and the R equals 1.
With this information, we can substitute in values, where sin = y/1 = y and cos = x/1 = x
Now thinking back to the Pythagorean Theorem, we know that it says if you square the sides of the triangle, add them, we end up with the hypotenuse. In other words (or variables, I should say), x^2+y^2=r^2
NOW REPLACE THE VARIABLES!
cos^2+sin^2 = 1
It has now become one of the three Pythagorean identities, the other two can be easily derived from this one. Let's have a look now.
The next identity is 1+tan^2theta=sec^2theta
So how do we get this? Well, take the first formula and divide it by cosine. The first part should look something like sin^2/cos^2 (THIS IS Y/X, AKA, TANGENT). The second part is cos over cos, which is equal to 1. The last part is 1/cos (THIS IS SECANT!)
So now we simplify it and get:
1+tan^2theta=sec^2theta
Pretty simple right?
Now I will show you how to derive the third identity!
Goal to make
1+tan^2theta=sec^2theta
look like
1+cot^2theta=csc^2theta
This is accomplished by going back to the first identity and instead of dividing by cosine, we will divide by sine.
Following the same steps as before, we should end up with sine over sine (1) and cosine over sine (RECOGNIZE THIS AS COTANGENT) and 1 over sine (RECOGNIZE THIS AS COSECANT)
Now we simplify, and end up with:
1+tan^2theta=sec^2theta
Monday, March 18, 2013
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 received a C+ on the unit O test, and retook it. I had done all my work correctly the first time, but the reason I received a C is because of small mistakes. Unit P I did bad on because I did not get enough help with law of cosine.
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.
step 1: I use the SSS for everything, I always keep it with me and make sure I have highlighted what is important.
step 2: I don't use the text book at all, and only videos if I need help on something.
step3: I don't use resources on the blog at all.
step4: I depend on the PQ's to give me practice and allow me to understand the material, this is something that is important to me.
step5: Discussion is very important to me too because I learn and ask questions what I am confused on.
step6: Quizzes let me see what else I need to study, they are important to me.
Step7:The PT is important to me too becuase it's basically one of the last study tools that I use right before the test.
Step 8: The test itself is important because it's worth a lot and can make or break me.
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.
step1: I could avoid just doing what is needed and fill in all the boxes or problems with work. I usually just do what is needed.
step2: I think I am doing the most i can with this step, the videos i use and learn from very well now.
step3: I could use the extra resources now because sometimes the videos are not clear, I suppose this is why those resources are there, to clarify things.
step4: I believe I am doing everything correctly and efficiently with this step, the PQs are important to me.
step5: I believe that the discussion that happens with my group is also done very well. I learn things from others that I may have been confused on.
Step6:The quizzes could be retaken for full credit, sometimes I do not do this. I could retake those which do not get high scores on.
step7: I believe I do the PT to the best of my ability. I always spend a lot of time on it.
Step 8: I should review the material prior to the test from quizzes which I know i struggled on.
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.I usually feel nervous, but confident at the same time because I study the night before. Anything that I'm still confused about I basically ask other students or go at lunch to watch a video on it.
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.
I feel like I haven't learned as well as last year or the year before. I never struggled this much in the algebra 2 honors course. I think it's only specific areas where I have trouble in.
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 definitely have learned the material with a greater depth. Basically, I know how or why something is in mathematics now. It's not just a mindless following of steps. Just like we derive formulas, we never did that in other years.
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 normally do understand this, just sometimes it's hard to follow when someone explains it. I usually learn by learning the "what and how" part first, and then learning the why. Just like when we derived formulas (law of sine/cosine), I first learned how to solve a problem using the formula, and then i learned to derive it.
e. How does your work ethic relate to your performance and success? What is the value of work ethic in real life?
I have bad work ethic. I have a bad grade. Work ethic shows the ambition someone has for something.
I received a C+ on the unit O test, and retook it. I had done all my work correctly the first time, but the reason I received a C is because of small mistakes. Unit P I did bad on because I did not get enough help with law of cosine.
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.
step 1: I use the SSS for everything, I always keep it with me and make sure I have highlighted what is important.
step 2: I don't use the text book at all, and only videos if I need help on something.
step3: I don't use resources on the blog at all.
step4: I depend on the PQ's to give me practice and allow me to understand the material, this is something that is important to me.
step5: Discussion is very important to me too because I learn and ask questions what I am confused on.
step6: Quizzes let me see what else I need to study, they are important to me.
Step7:The PT is important to me too becuase it's basically one of the last study tools that I use right before the test.
Step 8: The test itself is important because it's worth a lot and can make or break me.
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.
step1: I could avoid just doing what is needed and fill in all the boxes or problems with work. I usually just do what is needed.
step2: I think I am doing the most i can with this step, the videos i use and learn from very well now.
step3: I could use the extra resources now because sometimes the videos are not clear, I suppose this is why those resources are there, to clarify things.
step4: I believe I am doing everything correctly and efficiently with this step, the PQs are important to me.
step5: I believe that the discussion that happens with my group is also done very well. I learn things from others that I may have been confused on.
Step6:The quizzes could be retaken for full credit, sometimes I do not do this. I could retake those which do not get high scores on.
step7: I believe I do the PT to the best of my ability. I always spend a lot of time on it.
Step 8: I should review the material prior to the test from quizzes which I know i struggled on.
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.I usually feel nervous, but confident at the same time because I study the night before. Anything that I'm still confused about I basically ask other students or go at lunch to watch a video on it.
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.
I feel like I haven't learned as well as last year or the year before. I never struggled this much in the algebra 2 honors course. I think it's only specific areas where I have trouble in.
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 definitely have learned the material with a greater depth. Basically, I know how or why something is in mathematics now. It's not just a mindless following of steps. Just like we derive formulas, we never did that in other years.
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 normally do understand this, just sometimes it's hard to follow when someone explains it. I usually learn by learning the "what and how" part first, and then learning the why. Just like when we derived formulas (law of sine/cosine), I first learned how to solve a problem using the formula, and then i learned to derive it.
e. How does your work ethic relate to your performance and success? What is the value of work ethic in real life?
I have bad work ethic. I have a bad grade. Work ethic shows the ambition someone has for something.
Thursday, March 7, 2013
Monday, February 25, 2013
Friday, February 1, 2013
Unit M: Conic Sections
Parabola
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?
This conic section is shaped like an arch, when a right circular cone is cut parallel to the edge of a cone. Any point on the parabola is an equal distance away from the focus or directrix, this allows for the "arch" shape of the parabola. 2. How does the focus (or foci) affect the shape of the conic section?
3. How do the properties of this conic section apply in real life?
A real-life application of this conic section is the satellite dish that television companies use to send and receive signals. A satellite is in orbit, sending out signals to Earth, our dishes are facing towards the signals, and the paraboloid shape of the dish creates a perfect angle for the signals to come in, at the focus is the "arm" of the dish, which captures the signal.
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