Prologue to the Secret of the Universe

Unfortunately today, Mr. K had to leave for a meeting about curriculum changes so we were left to fend for ourselves. Luckily, we packed some survival tools: pencil, eraser and a calculator! We would need them to fight off the Fundamental Principle of Counting Sheet.

In the afternoon, with the return of our teacher, we wrote a quiz. Apparently, it consisted of the stuff we had been learning so far.Then, we went over the results of our efforts and assembled into 4 groups and continued to advance down the slides one by one solving each problem that came with each slide. Though as we may, we only managed to complete three but a surprise was in store for us next class. As Mr. K described the sneak preview of what would happen tomorrow. The next lesson will blow our minds out, teach us about the universe and poker...? Sadly that is all we did in two periods. Hopefully we can pick up the slack in the future.

NO HOMEWORK! well except the for the Fundamental Principle of Counting sheet

Tomorrow's scribe: Benofschool (upon request)
Next time in class: One of the ways god intended the universe and the game of poker??? Good luck Ben! you're gonna need it...

Set Theory and Binary Code

Imagine we have a set of numbers: {1, 2, 3}. How many subsets can we make out of that? We can take none of the numbers as a set, this is known as the null set and is represented by this symbol (∅). Then we can take each single number as set. Since there are three numbers, we will get three sets. Another three sets can be produced if we take two of the numbers like so: {1,2}{1,3}{2,3} and finally if we take all the numbers as a subset of the original set. This is known as the power set and has more things in it than the original set or we can say that it has a higher "cardinality". That brings our total to 8 subsets from the original set. What if we take a set of all the numbers from 1 to infinity. What we would get is an infinite set. The hebrew letter Aleph (א) with subscript zero is used to denote this. We can take the power set of that and have a set with a greater cardinality. This would be א with a subscript of 1. We can infinitely take the power set and get infinities larger than the previous infinity.

Why does 123 equal to 123? How can such a number exist when we only have 10 numbers. We use place value in base 10 to conjure up such numbers as 1596, 45 ,100, etc. Why base 10 though... according to Mr. K it's probably because we have 10 fingers but what if we use another base like 5. If we use this base 5, what would 123 be? Well the 3 would be the ones place, 2 would be the fives place and 1 would be the twenty-fives place. 123 in base 5 would actually be 33 back in base 10. Unlike us, computers calculate numbers in base 2 also known as the binary numeral system. There are only two numbers in binary, 0 and 1. Any number can be written in base 2 or any base for that matter. Even though we can do all these magnificent things with computers, at the heart of the matter, all the computer really does is just decide if a switch is on or off as in a computer chip there are little switches.. This is all the computer is actually able to do and understand.

After all that business, we took a crack at solving two logarithm problems. Eventually it all came down to just applying the logarithm laws. Following that, we looked the properties of a exponential function. The domain, range, x-intercepts, y-intercepts, concavity, if it was increasing or decreasing and the asymptote(s).

My BOB for Identities

Once again it is time to write a BOB on one of our units. It so happens this time around, the lucky winner is Identities. This unit allowed us to sharpen up our algebra skills(I liked that) as we were "algebraically massaging" expressions.
This was rather an intriguing unit because it made us think "outside the box" and try to be more "elegant" with our solutions. When I say elegant, I don't mean we draw a nice unicorn beside our problems. I mean that we should try to solve the identities with the least number of steps without skipping any.
I found it easier when writing everything in terms of sine and cosine. Although this might not be the best way to come at things because we want to be more elegant.
A big struggle in the unit for me was my own work ethic. Sometimes you think you are ready, but you're not... I feel as though I didn't get enough practice in proving identities. Everything else was a breeze. Well to solve my dilemna there is only one thing to do: practice makes perfect.

That is all.

Sum and Difference Identities

Hey Class, it's me again scribing for you. In this particular class, we took a kinesthetic approach in learning about identities. We were given 4 formulas, although Mr. K prefers not to use formulas and would rather have us understand where those formulas are derived from instead. The formulas seemed like quite a task to memorize so... we had a little help. From the greatest mathematical dance of all time:
THE SINE DANCE.

Sin(α + β) = SinαCosβ + CosαSinβ
Sin(α - β) = SinαCosβ - CosαSinβ
Cos(α + β) = CosαCosβ - SinαSinβ
Cos(α - β) = CosαCosβ + SinαSinβ

This dance contains three easy steps which are called: SINE, COSINE AND BUST A MOVE! The order of which the steps are performed are located in the formulas shown above. In the first step sine, which is used to represent all the sine functions, we stick our arms out with one behind our back and bend them so that it forms the letter "s". The next step cosine, is even more simple than the first and is used to represent all the cosine. All you have to do is make the letter "c" with your arms. The final step is used to represent the sign change that cosine makes. To perform this step, you rotate your body 180 degrees and then hold your arms in front of your chest and make the letter "x". Once you have those down, feel free to use it as often as you like to aid you prove the sum and difference identities because that's what we did next except without the dancing.


Well, my scribe post is over now so that means I get to choose the next scribe. The next scribe is Paul. This begins cycle three.
By the way, I found about that writing your name in red ink thing. It turns that that superstition is Japanese. Still not sure if bad luck comes to the writer or the person who's name is written.

Nothing Less Than The Best....Group

SLIDE 18

On a typical day at an ocean port, the water has a maximum depth of 20 m at 8:00 a.m. The minimum depth of 8 m occurs 6.2 hours later. Assume that the relation between the depth of the water and time is a sinusoidal function.

Let's draw a graph!


a) What is the period of the function?

From the information we have been given...
* We can set the 8am as t = 0 hours.
* A maximum value is when t = 0 hours and when d = 20.
* A minimum value is when t = 6.2 and d = 8.

We can see from the graph that the period is 12.4 hours.

b) Write an equation for the depth of the water at any time, t hours.

cosine equation's parameters...
A = 6
B = (2pi) / 12.4 = pi/6.2
C = 0
D = 14

To get A, amplitude, calculate the distance from the sinusoidal axis to a maximum value or minimum value.
sinusoidal axis = (20+8)/2 = 14
amplitude = 14-8 = 6

B = pi/period = pi/6.2

C, the phase shift, is 0.

D is the sinusoidal axis, 14.

D(t) = 6cos [(pi/6.2)t] + 14


c) Determine the depth of the water at 10:00 a.m.

10:00 am = 2 hrs from when t = 0 or 8:00 am. Plug in the 2 as t into the equation to get the answer.

D(2) = 6 cos [(pi/6.4)2] + 14 = 17.1738 metres

d) Determine one time when the water is 10 m deep.

The wave is 10 metres deep, so the qestion is asking for what the time is when D = 10. Plug in 10 as D, then solve for t.

10 = 6cos[(pi/6.2)t] +14
-4 = 6cos[(pi/6.2)t]
-4/6 = cos[(pi/6.2)t]
arc cos(-4/6) = (pi/6.2)t
2.3005 = (pi/6.2)t
2.3005/(pi/6.2) = t
2.3005 x 6.2/pi = t
14.2632/pi = t
t = 4.5401

Convert the 4.5401 into "actual time" because we use hours:minutes:seconds to show time, so...

4.5401 hrs + 8am = 12.5401 hrs

Obviously, its not efficient to say .5401 hrs so we convert that to minutes.

0.5401 x 60 = 32.406 min.

12:32:24pm

We can round that to 12:30pm.

My BOB for Transformations

Ahhh yes our second unit... TRANSFORMATIONS. I know what you are thinking: Transformers, sorry but nope....
This is so much better than those silly robots. Unlike the Transformers who are only limited to a few or several forms (Optimus Prime always has the most), you can take a function and transform it in an infinite amount of ways by changing the values of the parameters: A,B,C and D. Anyways... I do not really have any beef for transformations, in fact I sorta enjoyed it. Looking at graphs in four different ways: numerically, graphically, visually and symbolically, my favourite being numerically since that is way I usually think. Carrying on, as I said before, this unit was not really that complex. A few bumps down the road such as reciprocal functions. Biggering...? Smallering...? What happened to the good old terms like increasing and decreasing... That kinda threw me off. Guess I should of read more Dr. Seuss back in my childhood. At first, I did not really get it, but now I do. After those examples we did today, it all became clear. Find the roots, variant points and graph away! Piecewise functions wasn't really a snag but I still don't know what an open circle means... Well that brings my BOB to a close. Can't wait until pi day, the anticipation is boiling within me.
Until next time, LATER

Tuesday, February 26

My Turn to Scribe 2 ~ HOOOOOOOOOO!
It was my turn to scribe so...

At the start of class, Mr. K started by showing us a few websites with varying purposes. First he showed us skrbl.com which is basically an online whiteboard. Then he presented twitter.com which you can use to communicate with people but you have to sign up with an account.
After viewing the features of those sites, we finally got to buckle down and write our Pre-test. It was basically a scaled down version of a normal test. There were 2 multiple choice, 2 short answer, and 2 long answers. We were alloted 15 minutes. After our time limit was up, we assembled into groups of about 3 people. Each group would have to hand in the a paper with the best possible solutions. Following that we went over the answers to the test. Bell Rings*
ＴＨＥ END。

The next scribe is kristina.

Monday, February 25

MY TURN TO SCRIBE ~ HOOOOOOO!
It was my turn to scribe so...

First of all, we started out by going over the test about maximum area of a triangle. So we have point P on the unit circle. B on the x-axis and it has to be a right angle triangle. What's the maximum area? Well we know that the area of a triangle is base x height divided by 2 and the base and height is equal to the x and y components of the triangle. We also know that the x and y components are equal to the sine and cosine of the central angle. We tested out the values and found that as the angle goes up the area would also increase but after π/4 radians the area would go down again. That means the sine and cosine of π/4 radians will give up the largest area.

After that, we went over question N on the sheet. Turns it out it was not a typo and we actually had to factor out the 2 pi. This will give us a period of 1. After that we went over some more transformations and Ｉ a few important things we should know. For example, parameter B is not the period itself, but actually helps us calculate the period. If we substitute B in the formula: period = 2π/b. Another important tip we should know is if we are given parameter A = -1/2 and we are asked for the amplitude, then the answer would be 1/2 and not -1/2. This is because the amplitude is the distance from the sinusoidal axis to the maximum or minimum points. Therefore, distance cannot be negative because this would destroy all logic. In other words, read your questions carefully, if it asks for amplitude, then discard the negative sign but if it asks for the value of parameter A then you include the value of A.
A few minutes later, we were given a wave graphed on the Cartesian Plane and were asked to write the equation of the graph in sine and cosine. Turns out there are INFINITE ways of writing the equation so... there's nothing to complain about there and plus we will only be required to only write one or two (max) equations on tests and the provincial exam.
Somewhere in the class, we looked at the infamous tangent function. Finally after a few of us were dying to see it, it was revealed. The tangent function is kinda special. It appears to be like x³ but IS NOT. The curvature is slighty different. Another feature of the wonderful tangent functions is asymptotes. Asymptotes are areas where tangent cannot exist. Since tangent is a trigonometrical function it has to do with triangles. At certain angles, tangent does not form a triangle therefore tangent cannot exist. Tangent can also be infinitely large unlike sine and cosine which are confined to -1~1.
That pretty much wraps up the class. Until next time.

My BOB for Circular Functions

The test is coming up soon so I thought the time was ripe for a BOB. Our very first unit is circular functions. At first, it was relatively easy. Nothing too difficult, I would say it was exactly like Grade 11 but units in radians instead of degrees. Took me a while to adjust to radians since we have been using degrees for most of our lives but as Mr. K always proclaims "FRACTIONS ARE OUR FRIENDS". Casting away CAST took zero effort whatsoever. As so, the memorization of the unit circle. Since it always has to be a root of 1, 2 or 3 over 2. Converting radians to degrees and vice versa was cakewalk since setting up a proportion is easy for me. All thanks to my Grade 8 Math teacher: MR. TRAN!

As things progressed on, we were exposed to more things that I found familiar from Grade 11. For example, solving for trigonometry functions of X was one of them. The only difference is that the angles are in radians now. Another thing was sine and cosine waves and their transformations.
I liked a few things about this unit. The fact that radians have no units was a load off my mind (in case I forget to write the unit on a test and lose a 1/2 mark). Solving for exact values of stuff was also enjoyable.

That brings my BOB to an end. Good night and Good Luck!