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.