A little irrelevant

The United States is home to the greatest education institutions in the world. Nobody is going to argue that MIT, Caltech, Harvard, Princeton, Cornell, Yale,  and Harvard are the top dog institutions on our planet. And, given our increasingly technologically oriented society, the need for students who are technically adept is absolutely necessary. 

Yet, the trends in education is disturbing. Some highschools grad cannot even grasp basic algebra:

http://www.nydailynews.com/ny_local/education/2009/11/12/2009-11-12_cunys_got_math_problem_many_freshmen_from_city_hs_fail_at_basic_algebra.html

In my opinion, math is as important as English for university. If highschool grads are ill-equipped for the rigor of university, what then? Will the university be forced to water down their curriculum in response to the influx of bad students?

Not to imply that our curriculum is bad; it isn't.  But when the United States decides to walk down a new path, Canada isn't very far behind. This article addresses those issues:

http://www.osstf.on.ca/Default.aspx?DN=76e080a5-0ec6-4448-8bba-67494d2add93

More conics!

Assignments: exercises 37, 38, 39. This was assigned previously, but after today's lesson, we can do more.

Today, Mr Max expanded our knowledge on conics. The central idea of today's lesson is that we can change general form conics into standard form conics.


Conversion between general form conic and standard conics.

Ax^2 + Bxy + Cx^2 + Dx + Ey + F = 0


Remember how to complete the square?

Take: x^2 + 4x + 4

We can take a square and
(x+2)(x+2)
(x+2)^2

or:
How do we find x? Well, take b/2, and square the result. sqroot(3/2)







If we try to enter (X^2 + y^2 + 6x - 8y) = 11 into graphimatica directly, it fails to map the equation because it cannot parse the data. So, we use this form after we complete the square

(X^2 + y^2 + 6x - 8y) = 11
(x^2+6x+9)(y^2-8y+16) = 11 + 9 + 16
(x+3)^2 + (y - 4)^2 = 36

Using the general form:

(x-h)^2+(y-k)^2= r^2
(h,k) = center; r = radius
we enter (x+3)^2+(y-4)^2= 36
and get:








c)4x^2 - y^2 - 8x - 4y - 16 = 0
4x^2-8x -y-4y = 16 + 4 -4
4(x^2-2x+1) -1(y^2+4y+4) = 16 + 4 - 4
(4(x-1)^2 / 16) - ((y+2)^2 / 16) = (16 /16)
(x-1)^2/(2^2) - (y+2)^2/(4^2) = 1
(x-h)^2 / a^2 - (y+2)^2/b^2 = 1








The final is only ~27 days away. I'm going to start studying today! Yes, that sounds crazy, but the exam is difficult, and this class is pretty well perpetration for the exam.

Assesment.

Yeah, time for some introspection. Last Friday, we had a test -- I didn't do too well. The problem is that it was half Trigonometry and Logarithms; those are two areas I'm very weak with. I'll confess that I didn't know the trig circle until I noticed the numerous patterns in it. Now, I can derive all the values from it. Also, I'm going through the math 40s website sheets. I'm going to so some of the green book to complement that material with the main exercise sheets. It's obvious that I am not doing enough work. And, to remedy this, I''ll do more work.

The good news? I'm actually doing it. I got through many of the trig sheets and I'm onto the log sheets. I'm well off when it comes to Compressions with Expansions and Perms & Combs. I feel that I need to do more work in all those areas, but the areas that I am well off in do not require much focus, for the material is familiar to me.

Another thing . . . I was in this class as a means to an end -- to get a very important credit. I strongly disliked the material, but for some reason, I really like doing it now. (Perhaps that is the sole reason I started doing the work in this past week?) At any rate, I think that I only have good news to report. As for accelerated math, I'm full steam ahead on that project. It's something I'm behind in, but given this recent turn of events, I'm fully confident that I will catch up and finish all of the exercises.

My reason for this post it really not for anyone else except myself. It's more or less something that forced me to think about my situation.

Thoughts on homework article.

A response to:

http://grade12precalculus0910.blogspot.com/2009/11/to-homework-or-not-that-is-question.html

Would you like a no-homework agreement?

No. My reasons for this is explained below.

Would you benefit from a no-homework agreement?

No. Someone with a part time job will, but I think that it wouldn't really benefit me. I'd like to study certain things that I am interested in, but I have some catching up to do.

Would you learn enough with no homework?

That depends. For the most part, I don't get much class time to complete the assignments. Assignments reinforce the material, so they are necessary.

Do you have enough time in school to both learn and practice?

During the class, no. The instructor time / assignment time is too unbalanced.

Is the subject area a consideration when discussing such agreements?

Math and English comes to mind. Science also comes to mind as well. Those are subjects that generally require a great amount of practice in order to master. Social studies can take up a lot of time as well.

Here are my thoughts on the subject. If you do not want to do your work, too bad. Let's take a look at our situation: given the material that we have to go through, we need to spend some extra time on our work. As far as I am concerned, this guy just doesn't want to spend some extra time after school doing his work (assuming his schedule is similar to ours). As for the instruction quality, I'm not going to gripe about our school -- I think it's been great this semseter. It's either the classroom or correspondence.

Yes, there are cases where the school assigns and inordinate amount of work. However, in those cases, completely getting rid of homework is not the answer: one should reduce the workload to something more reasonable. Another thing: we can't take a black and white perspective on a situation like this (or any situation, for the world is not a black and white place; it's more like many subtle shades of grey with many nuances.)

Tthe workload should also depend on the given subject that you're trying to learn. Math is a subject that certainly merits more practice then many other school subjects, because, let's face it: math is cognitively intensive and requires the student to hone their problem solving skills. Some people are better at math than others, but you are never ever going to pick up on a subject by looking at one example. Even the best artists had to go through years of study and practice. You might have the occasional Murry Gell-mann or John Von Neumann, but it will be painfully obvious to the teacher if a student is truly that exceptional: in those cases, you give them special instruction. So bringing examples like that up is moot.

In summary, some subjects will require a fair amount of study, and given the breadth of subjects studied in school, one can expect to do homework. I believe that homework should be reduced to be as little as possible, and students who need extra work should be given the material to better understand the subject.

Those are my thoughts on the subject. Take it from someone who is insanely behind on his work: one can turn the leaf and finish the work, and I intend to get this credit. There are those cases where they will assign too much work, but for the most part, I don't think it's much of a problem for me at SVRSS.

Bonus material (though a lot of this stuff is common sense):

http://myfishbowl.ca/high-school/303-how-to-stay-awake-in-class
http://myfishbowl.ca/high-school/297-top-5-time-management-tips

Natural logarithms,

We did mental math. I didn't do too well; because of that, I'm going to study logs and do some questions in the green and white book. Mr Max decided to axe our third test and to defer the next onw. That comes as a great relief to me. He also gave us the answer key for the test, so we can see where we messed up.

Today's subject was natural logarithms. We approximate "e" and the natural logarithm(ln) A natural logarithm as a logarithm to the base e where e is ~ 2.718281828.

(1 + 1 / 100000) ^ 100000 = 2.71826237

We also messed around with other sequences like the Fibonacci sequence.

These strange mathematical constants are interesting.

Monday, November 9th: change of base and solving exponential equations.

Today we learned about the change of base formula; also, it was semester day 40, so we are halfway in. Mr. Max believes that we should work hard at our accelerated math and do the relevant sheets before the test. If we were to do that, we'd do so much better on the test. If you're a little behind on the sheets for our upcoming test, that's fine. You could always get caught up before the next test, and keeping on top of accelerated math would make things less stressful.

logbn = logan / logab

As long as we follow the pattern of the formula, you can make it workable on any base you choose.

given y = log2 3

can be rewritten as:

y = log10 3 / log10 2

How to we evaluate? We can either use our calculator and get y~ 1.585 . . .

Or, we can use logs. (No kidding, eh!)

y = log2 3

change to an exponential expression.

2^y = 3

then we apply the def of logs: log(2^y) = log3

y = log10(3) / log10(2)

ex. Solving exponential equation.

Solve:

2(3^x) = 5
CAVEAT! (A latin word that means "warning" or "beware"):
log (x + 3) != (not equal to) logx + log3
instead, it becomes:
log(2(3^x)) = log5
log102 + log103^x = log5
log2 + x * log3 = log5
x * log3 = log5 - log2
x = log5 - log2 / log3 (at this point, we cannot continue without a calculator)
x = .8340437671
to check out answer: 2 * 3 ^ .8340437671 = 5

We just solved our very first exponential equation!

Let's solve another one:

19^(x - 5) = 3^(x+2)
log 19 ^ (x - 5) = log 3^(x+2)
Our reason for doing the log is to bring the exponents down.
x * log19 - 5 * log19 = x * log3 + 2 * log
(x - 5) * log 19 = (x + 2) * log 3
x * log19 - 5 * log19 = x * log3 + 2 * log3
x * (log19 - log3) = 2 * log3 + 5 * log19
x = 2 * log3 + 5 * log19 / (log19 - log3) (now we use our calculator)
x ~ 9.166

log2 (x - 2) + log2(x) = log23
log2(x - 2)(x) = log23
x - 2(x) = 3 (we can remove the logs since they have the same base)
x ^ 2 - 2x - 3
(x + 1)(x - 3) = 0
x = -1, x = 3

STOP!

x = -1 doesn't solve the equation.

x = 3 works, though

(On a wholly unrelated note, they killed of cookie monster and turned him into . . . veggie monster. Since when did political correctness have to start and destroy our childhood memories? Is nothing sacred?!)

log5(3x + 1) + log(x - 3) = 3
log5[(3x+1)(x-3)] = 3
use definition of logs:
5 ^ 3 = (3x + 1)(x - 3)
logab = n
a^n = b

125 = 3x^2 - 8x -3
0 = 3x^2 - 8x - 128
(384: 16 and 24 as factors)
0 = 3x^2 - 24x + 16x - 128
0 = 3x(x-8) + 16(x-8)
0 = (3x + 16) (x - 8)
3x + 16 = 0, x - 8 = 0
x = -16/3 and x = 8

check for error:
- 16 /3 isn't a solution because after we plug it in, there is no value that 5 ^x that would be equal to a negative argument.


That's all for today

More logarithms

We were taught even more rules and identities. Though Mr. Max doesn't want to prove the identities to us, he wants us to use them.

Here they are:

log a (MN) = log a M + log a N

You factor the logarithm and it's equal when it's factored out.

log a (M/N) = log a M - log a N

It's anoglous to subtracting powers. With x^a / x^b, it becomes: x^ a-b. That is why it's the same here, I think.

log a M(exponent x) = (X)log a M

X is multiplied by log a M. This one I don't quite understand. . . .

That is as far as I remember.

Logarithms.

y= logb(x) is the inverse function of the exponential function y = b^x.

y = 4^x
turns into log4y = x

and

Also, here is logarithm log(x) mapped in graphimatica:














All logs are, are inverse exponential functions. That's all I know thus far.

Yikes!

It's been way too long since I lasted posted here. That's quite embarrassing. I'm going to stay on top of things from now on, ugh.

Anyway, I've been sick these last few days. We have a test next week coming up on Thursday next week, and I have to write a test tomorrow. I don't want to recap all the material that we learned in class between my last blog post and today. Rather, I'm going to try and make a good effort to improve my posts and post more frequently. That is a promise.

Trig Identity

I think that trig identities are equations that take the sin, cos, tan, cot, sec, and csc functions and arranges them into equations using variables, constants, and powers to demonstrate that some variations of these functions are equal to other variations.

And identity is an equation that always works no matter what your input is.

Identity - and equality, trigonometric in this case, that evaluates as TRUE for any value of input; that is both sides of the trig-equations are true for ALL possible variables.

Trig equations that are not identities are conditional equations.

You cannot prove "indentityness" with a graph, but graphs are adequate for disproving it. I suppose what you see is equal, but elsewhere on the graph, there will be places where the the equations aren't if they're not identities.

This was the most interesting class thus far.

Test

I was sick on Tuesday, and that deprived me of seeing the pretest answer key. That hurt me, but what the heck -- bite the bullet. I think I didn't do too good, but not as bad as the last test. Anyway, I don't feel too choked up over it.

I used to take graphs at face value. Then I discovered a bunch of cool things. Graphs help us visualize numbers. And, with graphs, you can do cool things.

f(z) = x^2 + y^2.

http://www.math.uri.edu/~bkaskosz/flashmo/graph3d/

What does this look like? Well, it's a 3D graph. I tried to visualize it, and it was hard making a three-dimensional leap from the two dimensions, but it sets in once you realize that there are two x^2 parabolas, so, if we twist one of parabolas 90 degrees from the vertex, we create a bowl like structure in 3D:




















What does this have to do with what we do in class? This made me realize that the reason they teach us graphs is that these graphs help us visualize the answers to these equations. In the end, these are number solutions that we're looking at. This really puts things in perspective for me.

Reciprocal functions & rational expressions

This post is a gamut of what we did during this week thus far.

On Monday, we did some mental math. Once again, I made some stupid arithmetic errors -- that's something I badly need to improve upon. Ugh.

As for the material we learned in class, it was about transformations using various stretches of functions:

y = a * f(x)
y = f(x * b)

We were also introduced (reminded?) to absolute value functions and their graphs. So, it was about manipulating functions and viewing graphs so we can visualize the solutions to the problem.

If y = abs(x) then we get something that looks like a big V on the Cartesian coordinate plane.

These transformations can be summarized as:

y = a * f(x) | vertical stretch a > 1 | y = 2f(x)
y = a * f(x) | vertical compression 0 < a < 1 | y = 1/2
y = f(b(x)) | horizontal compression b > 1 | y = f(2x)
y = f(b(x)) | horizontal stretch | y = f(1/2x)

On Tuesday, we learned about reflections. No, not the kind that we do on this blog but the kind that pertains to functions.

A function can flip across the x-axis when f(x) becomes -f(x) and f(x) flips across the y-axis when f(-x). We also messed around with inverse functions.


Reciprocals . . . stuff we learned way back in grade five, and stuff we messed around with in grade 9. For example, a/b --------> b/a

so f(x) --------> 1 / f(x)

For inverse functions, the denominator can never be zero (since division by zero is undefined). This is something useful to keep in mind, if we are to try and visualize the function in our head.

Well, that's about it.

Mr. Max talked about the test and success. He believes that a re-test would be great, since he wants to cultivate the idea that we can be successful at this course by tackling the material. I certainly found it to be quite motivational, and I do confess that I haven’t been taking a proactive approach to learning the material in this course. Well . . . I know I did bad on the test – I didn’t learn the material.

Anyway, today’s math topic is transformational geometry. All we are doing is moving and changing graphs around. Altering values in equations can cause translations like this: X^2 + 2

Well, it shifted downwards twice. We did something like this back in grade 10, I believe. One could do the same to trigonometric equations.

More reflections.

Spur of the moment. I'm trying to memorize the charts, but I think it's easier to use these two triangles to derive the values:

Some people take this for granted, but this sure is going to help me along in this course.

Motivation.

We did mental math today, and we got an xtra salty motivational speech from Mr. Max. ;)

We should not be afriad to fail; we learn from our mistakes, after all. Also, my mark was lacklusture on the mental math sheet, so it tells me that I need to practice more. Good thing it didn't count!

Relfections for monday and tuesday.

I was absent on Monday, so I came back and copied the lecture notes to my USB Storage Device.



Tuesday was a work period. Of course, I gave thought a lot about why I'm in this course. For one thing, I need it for university; also, I want to understand mathematics, and this is the most hardcore course that the school offers in that subject. ;)

It's a real hurdle for me to learn a topic like this: you are given a method for carrying out and solving a problem. But, you don't understand the "why" part. I have to accept the fact that I need to take a "leap of faith" and bite the bullet in order to learn this, but in the case of mathematics, it isn't too hard to explain the why part.

Why radians instead of degrees? Well, degrees is a system based on an arbitrary constant, so we need something a little more precise. Why the number base 10? Well, we evolved to have 10 fingers, and we very well could use a system based in two (binary), 8 (octal) or 16 (hexadecimal), but that doesn't matter too much.

Eh, I don't know why I'm getting so philosophical about the subject; I should be focusing on the subject matter, but math can be a really dry subject or it can be really damn interesting if we are given the "meat" along with the basic material.

Yes, that's my biggest problem: I'm thinking too much about the wrong things. I don't know. I'm kinda like a fish whose at the bottom of a boat, flopping, slapping, struggling to get back into the water. How am I going to get back into the lake? I guess I'll have to struggle through the material.

And another thing: I have to teach this to myself. Mr. Max is providing the material for me (pay attention during the lectures!); I have to take the material and teach it to myself. That's why there's so much homework I guess. . . .

Hey, hey!

I've assessed myself mathematically, and I'm right about here: the only thing I really know is that a radian is 180/PI.

3 * 180/PI = PI.

Or, three radians is equal to 180 degrees.

Also:

Sin = y/1 Secant = 1/x

Cos = x/1 Cosecant = 1/y

Tan = y/x Cotangent = x/y

Well, the change to radians just teaches me to view the measurement of a unit circle (or polygon) in a different way. Speaking of unit circles, I found this lovely diagram on wikipedia:

and :

As an aside, I found a really awesome website:

http://mathworld.wolfram.com/Trigonometry.html

This isn't as important, but partaking in this would really put you ahead since the stuff here is harder than the highschool ciriculum:

http://cemc.uwaterloo.ca/contests/contests.html

I might enter, but only if I show Mr. Max that I really care about the competitions and practice for it.

Alright, my last posts were junk. Hopefully, this will be the exception. ;)

Back to the math!

Work.

Today was a work day, since our teacher was absent for the day. I did learn one thing: I have to work harder. I'm in no danger right now, but I could fall behind.

I really need to get my act together. Heck, this post wasn't even done yesterday, like it should have been.

But, I have my goal in mind. I'm going to get that 90 percent.

Radians

-3 and some radians (3.14) is 180 degrees.
-One radian is ~57.3 degrees and 180/π

Reflections: 11/9/09

Today, we learned about radians. I knew about radians beforehand, so that made things easier today. One thing that was discussed in class was a simple algebraic trick where:

(a/b)/(c/d) = (a * d) / (b * c)

A radian (180/π) is about 57.2968 degrees. I'm going to memorize that, and I will do some math problems over the weekend in order to "get into the curve" of doing things.

Goals

My goal for this semester is to get at least 90 percent in pre-calculus math. I want to get a high grade so I get scholarships when I go to the U of Manitoba. I'm contemplating the idea of getting a geology, geophysics, or mathematics degree, and science requires good math skills.

Worthy of note is that I own a book on computer graphics, and I need to learn a fair bit of math in order to go through it:

















I'm also taking game design, so the skills I aquire in pre-calculus math will help me write the code for the game. (If all goes well, I think I'll show the results of my programming off to the class.)

Lastly, it is my goal to create a 3D program of some sort in C. It will most likely end up being a game engine, but I won't know for sure. One of my bigger inspirations for this is Ken Silverman, the programmer who built the game engine for Duke Nukem 3D.

As he said on his build page:

How do I get started in 3D programming?

Know your math! For 3D engine programming, you will need to master: algebra, trigonometry, and geometry (feel free to sleep through calculus). You also should learn how to manipulate vectors, such as dot and cross products.

http://advsys.net/ken

Hi.

Why hello there.