NUMBER 11

Editor's Note: This post is reprinted from an email that has been circulating around the internet. Some of the things stated as fact are false. It is a hoax. Please read the comments.

Mr. Kuropatwa

THE SECRET BEHIND NUMBER 11

I figured this has something to do with Math. HAHA! I think this is interesting as well. I don't know if all of these are true though. Maybe some of you have already read this...

TODAY'S DATE: JUNE 5, 2007

06/05/07

6+5 = 11

Pretty Chilling - read to the bottom. Try it out.
If you are a sceptical person - still read on as it's actually very interesting!!
This is actually really freaky!! (Mainly the end part, but read it all first)
1) New York City has 11 letters2) Afghanistan has 11 letters.3) Ramsin Yuseb has 11 letters. (The terrorist who threatened to destroy the Twin Towers in 1993) 4) George W Bush has 11 letters.

This could be a mere coincidence, but this gets interesting: 1) New York is the 11th state.2) The first plane crashing against the Twin Towers was flight number 11. 3) Flight 11 was carrying 92 passengers. 9 + 2 = 114) Flight 77 which also hit Twin Towers, was carrying 65 passengers. 6 + 5 = 115) The tragedy was on September 11, or 9/11 as it is now known. 9 + 1+ 1 =116) The date is equal to the US emergency services telephone number 911.
Sheer coincidence..?
Read on and make up your own mind:
1) The total number of victims inside all the hi-jacked planes was 254.
2 + 5 + 4 = 11
2) September 11 is day number 254 of the calendar year.
3) The Madrid bombing took place on 3/11/2004. 3 + 1 + 1 + 2 + 4 = 11. 4) The tragedy of Madrid happened 911 days after the Twin Towers incident.
Now this is where things get totally eerie:

The most recognized symbol for the US, after the Stars & Stripes, is the Eagle. The following verse is taken from the Koran, the Islamic holy book:
"For it is written that a son of Arabia would awaken a fearsome Eagle. The wrath of the Eagle would be felt throughout the lands of Allah while some of the people trembled in despair still more rejoiced: for the wrath of the Eagle cleansed the lands of Allah and there was peace."
That verse is number 9.11 of the Koran.
Unconvinced about all of this still ..?

Try this and see how you feel afterwards, it made my hair stand on end:
Open Microsoft Word and do the following:
1. Type in upper case Q33 NY. This is the flight number of the first plane to hit one of the Twin Towers.2. Highlight the Q33 NY.3. Change the font size to 48.4. Change the actual font to the WINGDINGS……………………
What do you think now????

BOB-ing 8

WOW, I can't believe that this is the last time I'll BOB. Such a tremendous experience and a really interesting journey I had in this course. I've been struggling, really... This unit is easy though. It's pretty straightforward. There's nothing much to say about this short unit. With its low level of difficulty, we are able to catch up on our schedule and still retain those 3 days of review. I hope everyone does well on the test. I know Mr K. will find a way to complicate this simple unit but I think we'll be prepared by then. Just remember the formulas.

Also, remember:

n - the rank

d - common difference in an arithmetic sequence

r - common ratio in a geometric sequence

t₁ - first term in a sequence

a - also indicates the first term in a sequence

t_n - n^th term

S_n - the sum of the terms until n^th term

GOOD LUCK ON THE UPCOMING TEST and THE MOST AWAITED PROVINCIAL EXAM!

BOB-ing 7

I'll go straight to the point. The Probability Unit evokes chaos in my mind. In short, probability is one of those units in Math that makes me go, "WHAAAT?!" It may not seem like it because I don't do it in class. But seriously, in my head, I do it a lot. I grasped the concept easily. However, the application of that concept is the real puzzle to deal with. Maybe, the struggle originates from lack of practice and absence of confidence with myself. Like in the last quiz, I doubted myself so much that I erased the solutions for two questions that transpired to be correct. That was really... KJ (kill joy). Another frustration dawned upon me as the week concluded and the weekend commenced. I promised that I would be practicing during the weekend, but I failed to such a thing. Some problems throw me off because of too many disturbances. I really hope that I do well in this test, but not without a lot of practices. Maybe I should start after this BOB! Anyway, this year is about to end and I couldn't ask for more with a fun class already at hand. But, it's so fun that sometimes my attention linger over other topics. My BOB ends here and I hope you all had a great weekend and good luck on the upcoming pretest and test! See you guys tomorrow!

I hope that all my practices will pay off. Good luck guys! I know I need that too! HAHA!

Congratulations!!

Well.. since our homeroom does contain 3 out of 4 candidates for the school elections this year... I think this blog deserves this post.

Congratulations to Richard who is next years wonderful President!
and... Congratulations to Vincent who is next years wonderful Vice-President!

You guys worked hard and you both deserve your positions on the executive council. I'm definitely looking forward to next year with you two. (:

So again, congratulations to the both of you. =)

BOB-ing 6

I really like this unit... That's all I have to say. I grasped the concept pretty much faster than I did with the other units. I'm more focused than ever on Math mainly because I know that I have to strive harder to accomplish my goal in this course. I like this unit too because it's a breath of fresh air. I mean that we've never encountered ellipses and hyperbolas. Now, I've learned something different and interesting. They're easy to manipulate too! The equations are pretty much the same and that's what's going to confuse the minority. Also, those applications seem difficult but not really... They're okay... I think I'm going to do well on the test. I hope! So, good luck to me and to everyone else!

One HYPER-bola morning...

One HYPER-bola morning, still worn out from the exhausting activity of origami, we immediately began identifying the similarities and differences of the equations of horizontally-oriented hyperbola and the vertically-oriented hyperbola. We enumerated the similarities first then we moved on to the differences.

After each discussion about the similarities and differences of each equation. We talked about what if the equation is equal to - 1. Well, it's simple... YOU JSUT CAN'T HAVE NEGATIVE 1 ON THE RIGHT SIDE. Therefore...

THEN, we tried a couple of problems... PIECE OF CAKE!!!

The graph looks like this... and the asymptote is determined using:

SECOND PROBLEM...

This is the graph of the hyperbola described above.

This concludes our lesson for today. Gladly, I didn't have to make a scribe post for two classes, considering I have been busy lately... That was easy! Anyway.. the next scribe is JOHN!

BOB-ing 5

WOW! Another BOB! This only means one thing. TEST!!!! Gee.. This unit has been different. I've had so many confusions with this one. There's a pick formula, choose formula. Those words are synonymous! LOL! Maybe... So yeah. The binomial Theorem startled me in the beginning but I know that with a few more practice, I will be comfortable with this! But, my quizzes have been lower than what I used to get. It's all of these DEV stuff that stress me out. I couldn't do my homework because of that! Anyway, Permutations and combinatorics are easy. It's just the matter of remembering which is which, which formula to use, in which circumstances! WOW! That's hard! Nevermind. So is anyone ready for the test? I'm not. To tell you the truth, I don't feel good about it. Pre-emptively, I will feel something about an upcoming test and for this one, I know it wouldn't be a good one. Sad... So, instead of rambling all over the place here, I should continue my DEV project! It's due on Monday! See you guys! I'm out!

BOB-ing 4

WOOWW! I haven't noticed we're already halfway through the semester because of the pile of assignments and projects that teachers give us these days. i suppose this unit is easier than the first ones. However, I'm not used to the questions Mr K. includes on the tests and quizzes. They are harder. Well of course, they would be. But I just hope we practice with hard questions so that when we encounter them, we know what to do! LOGARITHM IS AN EXPONENT is the idea that we all SHOULD remember. I just hope I do because it will propel me to achieving success in this unit! HAHA! Well, I'm going to review now. Good luck everyone!

SsSsSs-Spring-cribe

MORNING CLASS

--- Hey guys, I’m Thursday’s scribe and I am obliged to reiterate and discuss what happened in today’s classes. Before we dove into Logarithms, Mr. K. had some announcements. One of the announcements pertain to the FLICKR assignment. All pictures’ link must be posted up on the blog and the evaluation sheet regarding the assignment should also be submitted. He pointed out the random pictures that some people posted, which were all related to the particular theme --- 300. After the FLICKR assignment, Mr. K. discussed the Developing Expert Voices Project. He showed us several examples of different presentations of a project. One was when the student incorporated SIMS into his project. One sang the project which I thought was cool. I was going to do something like that but not for the entire project. MAYBE! HAHA! They certainly inspired the class and further increased the level of the bar that we have to overcome. Definitely, we are pressured, but I am excited about DEV. We also learned today that our blog is viewed by different people all over the world.
Now, with the real stuff --- LOGARITHMS! Let’s start by remembering what a logarithm is.

LOGARITHM IS AN EXPONENT!!

--- Then, say that five more times! Drill it in your head so that you will do well in this unit. Mr. K. displayed a few questions as warm-up questions that concern the Laws of Logarithms we learned before the Spring Break:

log_basePOWER = Exponent

log_b MN = log_b M + log_b N

log_b (M/N) = log_b M - log_b N

We were asked to simplify the following logarithms:



* It can also be answered it like this :
which is also correct. Although, it would have looked better with BRACKETS.

* Do not neglect the BRACKETS, as indicated in this question partly because it does not confuse the person and also because it looks better =).

* The negative part of the expression can be anywhere. It doesn't have to be at the end of the logarithm. Mr. K. pointed out that it is clever to solve and simplify in only a few steps but also it will help a novice to understand what's happening through the display of the intermediate step.

TAKE NOTE: Proportinal reasoning does not work with EXPONENTS because exponential functions increase exponentially as oppose to straight lines.

Next, we were given values of some logarithms with specific values for the following logarithms.
GIVEN : log_a 2 = 0.3562; log_a 3 = 0.5646; log_a 5 = 0.8271

• Expand logarithms using the Law of Logarithm.
• Re-write the expanded logarithms using the Law of Logarithm.
• We now see either log_a2, log_a3, or log_a5. We already know those values so we just substitute.
• Then perform the required operation.

• First line, let unknown value be x.
• Definition of a logarithm - log_basePOWER = Exponent.
• Indication of log₁₀ if there is no base. It doesn't show because "10 is conveniently programmed in our calculators" - Mr. K.
• Isolate x on one side (either left or right; there is no left-hand prejudice rule).
• You end up with an equation in the CHANGE OF BASE FORMULA.
• x is approiximately 1.5850.
  

Change of Base Formula : ( c is the new base of 10 )
log_ba = log_ca / log_cb

--- So then Mr K. began, "WHEN WE SOLVE LOGARITHMS, WE USE 4 DECIMAL APPROXIMATION." Then he reiterated that concept so many times in different ways because he knew a few of us will always forget it. He reminded us that if did not round off up to 4 decimals, it would be a full mark deduction because it is a concept error. Then, we tackled questions involving approximate values of logarithms. Although, it was unfortunate that the class had to end.

AFTERNOON CLASS

--- Fairly frustrated with how the experiment went in Chemistry, we went to Pre-Cal. From there, we carried on with the problems given to us in period 1.

• Rewrite the power using the LAW OF LOGARITHM with a base of 10
• Bring down the exponent using
• Isolate x and using the CHANGE OF BASE FORMULA
• Make sure subtracting 1 is done after the division.

--- Up to this point, everyone was dedicated in solving the problems on the SMARTBOARD. It was that time when Mr. K. showed us another unbelievable project. The project contained a lot of animations and was incredibly done. Again, we were amazed and at the same time, pressured due to the fact that our projects must meet that level. Well, not really. But still, who didn't feel pressured at that time? Yeah Yeah, maybe a few people didn't but I was! Then, we solved this problem.

--- Mr. K. also quoted, "There's more than one way to skin a cat, but don't literally skin a cat because that's mean." He meant that there's another way that we could have solved logarithmic problems. In the previous question, we could have multiplied both sides by 1/2, resulting in the isolation of 3^x on the left hand side and 5 / 2 on the right hand side. From there, we can easily use the CHANGE OF BASE FORMULA, and skipped two intermediate steps. Also don't forget to apply brackets around the numerator.

• It is an intermediate step.
• Bring down the exponents using log_bM^k = klog_bM.
• Subtract two from both sides.
• Use the distribution process to expand (x+1) log6.
• Isolate all the terms with x.
• Factor x out.
• Multiply both side by 1 / (log3 - log6) then evaluate.
• Shortcuts or other ways to do the problem.

• Bring down the exponents using logbMk = klogbM.
• Isolate like terms.
• Factor x out.
• Evaluate.

• Use log_b MN = log_b M + log_b N to combine all xs.
• Use FOIL, then rewrite the equation in log_basePOWER = Exponent form.
• Evaluate 5³ then subtract 125 from both sides.
• Since the quadratic equation does not factor nicely, we will use the quadratic formula.
• Then, x = 6.8226 and x = -6.1560 .
• However we cannot use the negative value because it is an extraneous solution. Although, we have to include both solutions and we have to STATE THAT THE NEGATIVE VALUE IS REJECTED.

The reason for the rejection of the negative solution is that:

--- Log_baseARGUMENT = Exponent or log_basePOWER = Exponent, if and only if BASE^exponent = POWER. The base has to be positive in that law of logarithm. If the base is negative, a funky dotted graph is produced. If we accept the negative number, then log_base negative number = Exponent. Therefore, this statement must also be true: BASE^exponent = negative number. However, it's not correct because if we substitute -6.1560, we will have:

log₅[3(-6.1560+1)(-6.1560-1)] = 3

log₅(125) = 3

5³ = 125 (but, 125 isn't negative at all like what we had earlier : If log_base negative number = Exponent, then BASE^exponent = negative number. Obviously, we are not given a negative number as a power at all.

--- On the other hand, if we substitute a positive number, it will all work out!

--- This concludes the day's lessons. It was so intense having to cover two-days worth of material. Now, We have to do Exercises 22 - 24 by Monday. Just a reminder to all of you that he will not be in class on Monday and Tuesday. So then, after the class, a few people styeyd behind to find out what our marks so are. And continuing on with the second cycle of the Scribe Routine, the next scribe will be ROBERT!

SCRIBE?

Hey guys! I just want to say that I think I should not scribe for today primarily because there is nothing to scribe about. The purpose of the scribe is to review something and to help someone catch up if he/she was absent that day. I want to ask you guys if it's okay for me to be the scribe for thursday instead when we'll have a real class. That's all.

FLICKR ASSIGNMENT:

TRIGONOMETRY

Dentist'sOffice<

BOB-ing 3

Honestly, I find this unit very difficult. I don't know how I will pass the next test considering, I might have failed the last two! Seriously, I get the concept and the overall jist of it but when it comes to like problems and proving, I can't seem to put everything together. It is so east as we discuss it in class but my mind seems to let go a lot of things after. I just hope I'll be able to learn from my mistakes. I know there's another test on tuesday and I can't seem to relax for it nor at least, be prepared for it. Questions are unexpected. So I'm going to have to expect that! I'm not doing good in this course, at least not as good as I want to. This unit has been easy in the beginning and the part where we simplify a complicated question is where i seem to struggle. I don't know why but I guess I panick too much. This is my early BOB because I want to get this out of the way with all the projects, homework, and work that I will have during the weekend. Good luck on the test guys!

BOB-ing 2

____ Well, I'm BOB-ing again! This time, it's for the second unit, TRANSFORMATIONS. In this unit, we meticulously discussed the "translations" and "transformations" of different functions. In the first transformation, we learned that 'a' in the standard form f(x) = af(bx) controls the vertical stretch or compression of the function. It alters the y-values of the coordinates of a point. If a is greater than 1, then the y-values are multiplied by that value and stretches vertically. If a is greater than 0 and less than 1 (or a fraction), the function compresses vertically. 'b' in the standard form f(x) = af(bx) controls the horizontal stretch of the function and alters the x-values of the function. If b is greater than 1, the function compresses horizontally and "speeds up". Mathematically, the x-values are multiplied by the reciprocal of the b values greater than 1. On the other hand, if b is a fraction, the function stretches horizontally, and the function's x-values are multiplied by the reciprocal of those fractions. Therefore, the graph "slows down". The second transformation we learned in this unit is the shifts. 'a' in teh equation y = f(x-a) + b alters the x-coordinates of the function and determines the new coordinates of the function by shifting it a units horizontally. 'b' in the equation y = f(x-a) + b alters the y-coordinates. It determines new coordinates for a point by shifting the fucntion vertically. If it's negative, fucntion shifts b units down. If it's positive, fucntion shifts b units up. The same thing goes for a. If a is positive, function shifts a units to the right, and negative a units to the left (shown as (x+a)). Overall, I believe that this unit is easier than the first unit. Although, there has been a few times where I keep forgetting the phrase, "STRETCHES BEFORE TRANSLATIONS". But now, I know what to do. Also, we learned a few ways on how to transform a graph. Either you start with the x-coordinates first then the y-coordinates OR you can simply do the stretches of x and y coordinates, then their shifts. In short, do whatever is comfortable! So, days pass by our knowledge about transformations increases. We learned how to graph reflections over y and x axes, and inverse functions where the graph is reflected over the y=x line, by switching the x and y values. We covered EVEN FUNCTIONS where the graphs are symmetrical over the y-axis, and ODD FUNCTIONS where the graphs are symmetrical about the origin. Odd Functions will be the same if the graphs are rotated 180 degrees. They are determined algebraically by replacing -x where x exists. Graphing RECIPROCAL FUNCTIONS was a tricky one. This time there are terms that we had to know such as INVARIANT POINTS (points that never move such as 1 bcz the reciprocal of 1 is itself). and ASYMPTOTES (where the function crosses the x-axis; the reciprocal function cannot cross the asymptotes). But, as we did more practice with them, I grasped the concept for sure! Absolute Values Functions was easier! All you have to do is re-draw the negative part of the graph above the x-axis because of the mathematical reason that y = x. Therefore, all y-values or OUTPUT must be positive. Those that are positive already remain where they are, but those that are negative change their signs to positive, thus, reflects themselves over the x-axis. Well, this was fun! HAHA! In general, I think this unit is easy and fun bcause of the moments where we could all draw on the PRO SMARTBOARD! See how technology evolves and aids us in our daily lives! Gotta love it! Pre-test will be on Tuesday? I think and there will be another group work in there too, definitely. This will help us a lot too! All I have to do is do the exercises which I seemed to have neglected! Now, I'm done! GOOOOOOOD LUCK!

BLOG ON BLOGGING

____This Pre-calculus40S class is awesome. I haven't worked with an intelligent group of people in my life that day by day challenge each other and ME. As a reflex, I have to keep up with the pace of learning that the course requires. The first unit is CIRCULAR FUNCTIONS. In my opinion, this unit can be so easy if you kow your 'thing'. Otherwise, it will be confusing and difficult to grasp. Indeed, converting radians to degrees and vice versa, finding the sector, arc length, etc. are easy. However, when it comes to graphing equations and deriving a SINE or COSINE equations from graphs, I can be perplexed. Currently, I am worrying with the fact that I can mess something up with silly and simple errors prior to solving equations or graphing functions. Today's class and the pre-test last week sort of set the blueprint of my own building of learning. Now, all I have to do is work my way to the top and to nail the answers for every single problem thrown at me. I will never forget the minor errors that our group made today when it appears on the test. Everyone learns from their mistakes. The dictionary helps me a lot too and so are the scribes that are posted every night. They help me refresh my memory about the day's lessons. Groups definitely help mind enrichment due to the fact that integration of minds exists and therefore enhances every member's knowledge about a particular topic. like today, my group members and I shared answers and opinons about our answers. It felt good! Though, I don't want to get to dependant on groups because most of the course will comprise of individual work like tests and quizzes. Our first test is on wednesday and no group member will save anyone's life there, will there? So, that's my perspective. Also, good luck on the test!

February 8, 2007 - secant, cosecant, cotangent, arc functions...

Hey guys, sorry for the late post. I suppose. Well, our class today was not much of a ROCKET-SCIENCE (well rocket-math). Well, you know what I mean. It wasn't difficult, YET! First, we started of with the 'BLOG issue'. Have you read the guidelines? Do you agree with them? Are you sure? If not, go ahead and read them and make sure you follow them.

Let's get down to business. What have we learned in this class so far anyway? Let's refresh our memories...

We know that degrees can be converted to radians and Mr. K. has shown us the method of "Proportionality" to perform the mentioned operation. Furthermore, we can convert back and forth from radians to degrees and degrees to radians. The circle above depicts all the radians that will be in most use during the course. They are in multiples of pi or (3.14 ..... ). 'Four corner points' sounds 'weird', because a circle doesn't have corners. They are the points on the unit circle (a circle that has a radius of 1) that intersect with the x-axis and y-axis.

The chart above shows the angles in degrees and in radians, the quadrants they are in, and most importantly the coordinates (x, y) on which they are exactly located on the unit circle. We know that 0 and 360 degrees are on the x- axis on the positive side and 180 degrees is on its opposite side. 90 degrees is on the corner point where it meets the y-axis and 270 degrees on the negative y-axis.

Mr. K. also has taught us a way to remember the exact values and coordinates of each angle (30, 45, 60, and their related angles, 90, 180, and 360). Tan is calculated by the equation:

TAN θ = SIN θ divided by COS θ


... and also has a pattern of:

• all the radians with denominators of 4, will have 1 as their tangent value, depending on where they are on the cartigean plane.


• the radians closest to the x-axis will have 1 over root of 3 as their tangent values, and of course, the signs ( + or - ) depends on their location


• the radians closest to the y-axis will have ROOT of 3 as their tangent values, and of course, the signs ( + or - ) depends on their location

Okay, that is the review. Terms of the day are as follows:

RECIPROCAL - the "flipped" values of a fraction ( ex. the reciprocal of 1 over 2 is 2 over 1 or simply 2 )

SECANT - the reciprocal of cos θ

COSECANT - the reciprocal of sin θ

COTANGENT - the reciprocal of tan θ

INVERSE FUNCTION - it is not the reciprocal but the inverse, it is then calculated using the calculator

So pay attention to this diagram.

The first problem asks: If sin θ is - 2 over 5, and tan is positive, find: cos, csc, sec, tan, and cot θ... AND they are simple to calculate. Just use the length of the sides, set up an equation like the basic ones we used way back in Jr. High for trigonometry ( Soh Cah Toa ). Although this time, once you obtain the values for sine, cosine, and tangent, you have to find their reciprocal values in order to get the values for secant, cosecant, and cotangent.

NOTE: It is okay not to rationalize the denominator!

The second problem asks to solve for x. I KNOW! It is weird because that is quite simple. However, we were just used to punching in values on our calculator and never knew how inaccurate it could be. I wonder why. So, Mr. K. explained how the calculator became unintelligent to the point that it only provides us with one value for sine (for example) instead of 2. We know that sin of 1 over 2 is 30 degrees but wait... as I can recall, our unit circle tells us that there are 2 possible and TRUE sin 1 over 2. They are 30 and 150 degrees, because 150 has a related angle of 30 degrees, which is the only value that the calculator provides us with.

NOTE: arc functions - input of length and output of angles

A couple more problems show the fact that the calculator provides us with insufficient information about the inverse values of sine, cosine, etc. because it only gives us the related angles. Refer to the problems that we took down in class to see more of these arc functions. Note that the quadrants help us determine the exact value for which they indicate whether it is positive or negative. I did not have time to put them all up since this post is already late. Maybe, we all can elaborate from one example and extend that knowledge further to solve similar problems about arc functions and reciprocals. YES, solving for a variable in a normal algebraic equation requires less work and less skills. However, the fact that the calculator is somehow misleading, we have to use our own package of brain cells to determine the exact values.

I hope this post has slightly increased your understanding about what we have learned so far in this course. I apologize for the lateness of my post and my reasons are valid. If I made any mistakes, please feel free to notify me as it can contribute not only to your benefit of learning but also to mine. Thank you. Anyway, to conclude this post, I have to pick the next scribe and she is AICHELLE!

DO NOT FORGET EXERCISE 4 AS YOUR HOMEWORK!