PC40SW07.BLOGGINGONBLOGGING.FINAL

WoW! (Acronyms for World of Warcraft FTW) I can't believe this is my last bob of pc40s!

Sequences! What a unit that was; the unit really displayed math (Patterns was a large header: Math is the science of patterns!). At first, I found the unit quite easy. Then, it became a tiny bit more complicated (but not really) and was still without a doubt easy.

In my whole opinion, I think that this unit was just about remembering the equations (that are given to us anyway?). Arithmetic sequences, Geometric sequences, Series, Series, Series, Sequences and more Sequences! Also don't forgot about infinite series, the Sigma notation (I think that's what it's called) and Converging/Diverging.

Yes yes. This unit includes it's own puzzles but overall was less difficult than the other units.

Good luck to everyone on the Test Tomorrow morning! Sleep early and drink lots of water =)

PS: I almost forgot to blog 3 times today and 2 over the weekend (I would not remember at all, then I suddenly have it in my head. The next moment, the idea was gone again) Good thing I finally did it before I forgot AGAIN!

LOTD(Lesson of the Day): DO YOUR BLOGS WHEN YOU REMEMBER (**The first time)

Bye bye, no more blogs fr: ME <---

B O B

Hrm, yet again it's time to blog. This unit went by quite fast. During the beginning of this unit, the problems were without a doubt easy and included no difficulties for me. However, as we progressed, I found that there were questions (such as those on the quiz) that surprised me.

On the otherhand, I should've been more responsible for completing more of my exercises than I have. I found that since I did not practice as much, I struggled at times and found that I was stuck; not knowing what to do next on certain questions.

Truthwise, I will "try" to do more practice questions tonight and we'll see how I fair on the pre-test and test.

Good luck everyone on the pre-test and test this coming week!

SCRIBE! ( ^.^) Introduction to Probability

Hello fellow classmates and readers! I'm Tim-Math-Y your scribe for today! Today we looked at the beginning of our new unit: Probability. We talked about things such as favourable outcomes over total possible outcomes, certain events, impossible events, sample space and other interesting things.

We started off the class by looking at this interesting question:

• There was no right or wrong answer in this exercise.
• Mr. K was looking to see what we thought of probability and why we chose each numerator or denominator.

While going over each solution that the class came up with, we found that 1/(infinity) is pretty much equal to zero since it comes so close to zero.

• Here we find that 4/9 (0.444 repeating) + 5/9 (0.555 repeating) = 9/9 (1)
• It is quite interesting to see that these two repeating numbers added together gives us the value of 1.
• What we were trying to get to here was the fact that if a number comes so close to another number, we can assume that it takes that form (Ex. 1/infinity = 0)

Next we took a look at the meanings of probability and it's characteristics, including sample space.

• Notice that the "Compound Event" examples are involved in one of our last units, Counting. Specifically, the Fundamentals of Counting.

Next we observe carefully more characteristics of a probability.

• The slides today are quite self explanatory.

Next we looked at some examples of probability.


Next we looked at examples that involved finding the sample spaces.

• To find the sample space of the problem, we find the total possible outcomes either through a tree diagram or a chart/table.
• We are usually only asked this when there aren't a large number of possibilities (will be too much work)

Now we look at more examples:

• To find the probability, we divide the total number of favorable outcomes by the total number of possible outcomes.

Now we look at a really hard example:

• This is one of the attempts of solving question 1a, by one of our fellow students, John

Running out of class time, Mr. K steps in to give us the solution:

• Remember that making mistakes benefits the class in learning*
• Mr. K makes a chart with the times the bus will arrive at the top and the times of the train will arrive on the left.
• Doing this, we find the total number of ordered pairs: also known as the sample space :)

Well that was all for today guys. I know it isn't as flashy and isn't that long or awesomeness to the max but yeah, I was sidetracked a lot today haha.

Well, that wasn't too bad. Most of it was self explanatory.

Tomorrows scribe is... hmmmm...

Grey-M..

Sorry, didn't know who to pick, and it's almost SAMS BDAY HAHAH!

B.O.B.

It seems this unit went by quite quickly. There isn't too much to say other than it was simply straightfoward; parabola, circles, hyperbolas and ellipses.

Though this unit is quite simple and easy to understand, I am uneased by the way this unit seems "too" easy. I know there's going to be something that I'll miss but that's only because that's been happening in the past.

Well, the only way not to become a victim of worrying is to practice!

By the way, nice review posted Mr. Siwwy! hehe

Goodluck to everyone tomorrow! On the pretest and test! =)

SCRIBE: Properties/Equations of the Parabola+

Hello, I'm Tim and I'm your scribe for today.

Today, we went over the parabola and it's properties: these include the focus point (the point that the parabola "wraps" around), the directrix (line that is related to the position of the vertex), the vertex (coordinates h,k) and of course there is more, described throughout this post.

Slide 1:

• This is an equation of a parabola. Notice that in this equation, the exponent 2 is on the x-variable. This causes the parabola to open up or down (depending on the sign of the value p, described later)
  
  
• This is another equation of the a parabola. Notice that in this equation, the exponent 2 is on the y-variable. This causes the parabola to open left or right (depending on the sign of the value p)
  
  
• This graph represents the equation where the parabola opens up. This is due to the positive coefficient (4) of the variable "p."
  
  Note that the distance from a point on the parabola to the directrix is equal to the distance from the same point to the focus.
  
  Also note that there are 2 points on the same parabola that will have the same distance from the directrix and focus EXCEPT the vertex which is special.
  

• This graph represents the equation where the parabola opens right. This is due to the positive coefficient (4) of the variable "p."

In both of these graphs, notice:

• The vertex has the coordinates of (h, k)
• The focus has the coordinates of (h, k+p) for the parabola opening vertically, and (h+p, k) for the parabola opening horizontally.
• The directrix has the equation y=k+p opening vertically, and y=h-p opening horizontally.

Slide 2:

• This is the newly derived equation of the parabola, from yesterdays lesson, in standard form.
  
  
• This is the old equation of the parabola that we used in grade 10/11, in standard form.
  
  
• In this step, we simply rearrange the new equation.
  
  
• On the other side, we also rearrange the equation to try and find similarities between the two equations that both represent the parabola.
  
  Remember that when "a" or the amplitude is greater than 1, we have a stretch in the parabola horizontally. When "a" or the amplitude is less than 1 but greater than 0, we have a compression of the parabola horizontally.
  
  Now notice that since the equation is rearranged so that the coefficient is (1/a) we come to understand why this happens.
  
  Ex: if a=2, substituting it into (1/a) gives us 1/2 which signifies a compression.
  

Notice:

• 4p = (1/a)
• "p" is a factor in the compression or stretch horizontally of the parabola
• "p" is also the distance from the vertex to the focus and to the directrix
  

Slide 3:

• Sketch this equation.
  
  
• We know that the coefficient is equal to 4p so we solve for p, to find the amplitude of the parabola.
  
  
• We find that the coordinates of the vertex is (2, -1), the focus is (2, 0) and the directrix is
  y = -2. Also since p is equal to 1, the amplitude is equal to 1.

Slide 4:

• Sketch this equation.
  
  
• Find p by creating the equation -8 = 4p (since -8 is the coefficient).
  
  
• Since the exponent 2 is on the y variable, the parabola opens horizontally. Also, since the sign of p is negative, the parabola opens left instead of right.
  
  Because the absolute value of p is greater than 1, in this case 2, the graph is stretched by a factor of 2.
  
  We also find that the coordinates of the vertex is (3, -1), the focus is (1, -1) and the directrix has an equation of x=5. Remember to always draw the directrix with a DOTTED LINE only.

Slide 5:

• In this slide, we just sketched more examples of parabolas.
  
  
• Since John didn't have the chance to outline his sketch like Vince (Left), I shall:
  
  This parabola opens up/down because x is squared. It opens down because the coefficient of y is negative (in this case -4)
  

Slide 6:

• In this step, we generalize the equation by expanding.
  
  
• After simplifying and moving every term on one side, we have the end product of the general form equation.
  
  
• After a handful of moans from the class, Mr. K attempts to show us that it is easily converted back into the standard form, first by completing the square.
  
  
• Simplifying the equation occurs.
  
  
• 4 is factored out of 4y-8.

Slide 7 Last:

• A Circle is a LOCUS (A set of points that follow a certain rule) that are equidistant from a central point. In this case, the center point "O" (h, k) is "r" (radius) away from "P" (x, y) a point on the circle.
• The length from O to P is equal to "r"
  
  Noting this, Mr. K attempts to find this distance by using the distance formula and leads us to the end of the class with the standard form of a circle.

At last I finish my seemingly long post on my laptop. However I have good news! I just fixed my computer by replacing the network adapter (Now I can return to gaming :) ). I hope this scribe post helps whoever reads this and, I hope it is simple and easy to understand as that was and is my goal.

Hmm.. it seems like the scribe list hasn't been updated so I will choose...

...

..

.

Miles! (I don't think he's been scribe yet, this cycle)

Have a great night people!

BOB TIME

Seeing that the test is coming up, it's time to blog on the blog once more.

This unit included its share of interesting and difficult times. I must agree with Mr. K that homework was definitely essential for this unit, in my opinion. Most of the questions had their own elements that may have related to other questions but were not exactly the same.

I find that the most difficulties I have with this unit is the race against time. There are so many ways to think about each question that it seems as though I'm playing against time to find the path that will lead me to the solution.

Also, besides the more simple combinatorics word problems, I don't think that I'm extremely comfortable with the binomial theorem yet. Only time will tell because I have yet to finish my homework and we still have pretest day tomorrow.

Well that's all I have to reflect on, Good luck to everyone on the Pre-test and Test this week!

Bob..

This unit was filled with questions where I found that the answer was much more simple than I was thinking. I tend to make things complicated and the real problem is finding out if I'm on the right path or not. However, this unit wasn't too difficult. Ofcourse there may be unexpected questions on the test but all I can do is hope I'll know how to answer them. That, or practice more. Otherwise it's like all other math units, practice makes perfect!

Logarithms are exponents!

Goodluck!

Flickr.com

Click on the photo =)

BOB

This unit on Identities was quite interesting. I find that I favour these types of problems over any other problems, especially word problems. I understand that there are no exact guidelines as to do what next in any given problem, but that doesn't stop me from solving it.

Not knowing what to do next is quite the mystery at times and I find that amusing. The more you struggle to find the answer, the greater you feel when you solve it.

Overall, this unit was quick and simple. I'm not sure if I have any problems or had any problems over this unit. I'm guessing I might run into some eventually, so the best I can do is try to study. I will admit that the Sine Dance helped me remember the Sum and Differences but when I'm trying to recall it, I don't imagine the dance, I retrace its rhythm.

Good luck to you all on the test and pretest!

Simple Bob

Most units are the same to me; they usually include a hill that you must climb over to achieve your goal. An example of this is like the trig problem solving. It's hard to think about all at once, especially by yourself, but quite easy to understand when the class or Mr. K is going over the question or working as a group. All of this narrows down to the solution: using your new found knowledge to practice independently. I think that is one of my weak points.. not practicing enough.

Another thing that I think would help a lot are math dictionary notes. They are concentrated knowledge all in a notebook. It's easy to access and would aid any misunderstandings greatly. I'm personally an independent learner, with the occasional team efforts. My strong suit is usually by myself.

I am not confident enough to say that I'm ready for the test. However, since we still have time to practice and we have a pretest before the actual test, I'm sure it's enough to get me over that hill.

Good luck to you all on the test and pre-test! Stretches before Translations!

TIM BOB

Circular Functions. Quite the unit. This unit was like most units that you learn in math. It may take time to soak into your head and understand the concept. However, once you've seen enough examples, heard enough lectures, and had enough practice, it's .. less complicated (let's put it that way). It's kind of like learning how to ride a bike, if you know what I mean.

I believe that I have a good understanding of the unit. I will admit that at times I was puzzled until I seen how it was done. I'm trying to think of a specific topic that I struggled with, but I can't seem to pull one out. Overall, each lesson included struggles for me; finding the right path that will lead you to the final answer. I have pulled through and out of the tangle often enough though and so I think I'll be fine.

What I found to be interesting was mainly the radian angle measurements. It was something completely new to me and made me think. Now understanding radians, I know what that "radian mode" in my calculator means =). Also, I like the fact that converting to radians and backwards, finding arc length and all of that stuff was quite interesting. They all have to do with proportions like Mr. K stated earlier and that's something that definitely helps me understand things (quite logical if you think about it).

Blogging Prompt:
Now, thinking about how I could relate radians and degrees, I can generally say that.. (thinking of the right words) they both can identify specific angles and the relationships between them, such as coterminal angles, number of rotations, related angles, and all that fun stuff.

Thinking about the differences, the first obvious difference would probably be the fact that radians usually consists of fractions and often have something to do with the value of "pi", whereas degrees are whole numbers + decimals and include the degree unit. They have different values that determine the same point?

Well those were off the top of my head.. I think I'll end this here.

Scribe: (DABC) Sine/Cosine; Relations and Modes of Representation

As your scribe for today, I will put my efforts into defining today's lesson. I hope that standards will be met and that this may be used to a greater extent as a future reference.

Today, we learned more about trig functions. We learned how to sketch the graphs, interpret the information, and undo our steps to either find equations from analyzing the graph or vice versa. There were many things that we went over today, upon which I will try to re-explain.

Class 1: Morning

To start the class off, we put our thoughts into questions that were written on the board.

1.) On the same cartesian plane, using 2 different colours, sketch at least 2 cycles of:

f(x) = Cos(x) and g(x) = Sin(x)



As Mr. K stated during the class, graphing these functions are as easy as counting from 1-4. In this example, upon which the graph is in intervals of Π/2, you would label your scale as Π/2, Π, 3Π/2, and 2Π on the positive side of the x-axis. On the negative side of the x-axis, you would label the scale with the same values except that they are negative: -Π/2, -Π, -3Π/2, and -2Π.

There are things to remember when graphing in order to achieve full marks. Some of which are quite simple:

- label your axis
- add arrows to your axis and your curves
- **make certain that the curve arrows either point up or down, NOT STRAIGHT

Back to the question, there are things that you should notice about the function of Sin(x) and Cos(x).

- both "wrap" around a line, known as the "sinusoidal axis" or the "average value of the function"
- Cosine starts at its max value
- Sine starts at its sinusoidal axis

Exploring further into the concept, we find that we can rewrite the function of Sin(x) in terms of Cosine, and rewrite the function of Cos(x) in terms of Sine.

Sin in terms of Cos:
-> Cos(x- Π/2) = Sin(x)

Cos in terms of Sin:
-> Sin(x+ Π/2) = Cos(x)


NOTE: Π/2 in both of these equations are the phase shift a long the horizontal axis, either left or right, depending on its sign, which will be discussed further on in this post.

2.) Without using a calculator, sketch each of the following graphs:


a.) y = Sin(x) - 1


b.) y = -2Sin(x)

Notice that the amplitude, in this case (2) is not negative because amplitudes are described as distances and therefore should not be negative.


c.) y = 2Sin(x) + 1


In graphing these functions, there are certain steps that may be followed to help make graphing easier also explained later in this post.

- - - - - - - - - -

-In general or standard form, the equation of sine functions are:

• f(x) = ASinB(x-C) + D

-Cosine is affected by the same transformations:

• f(x) = ACosB(x-C) + D

A- The value of A relates both to the amplitude and whether the function will be inverted or not over the y-axis. The amplitude (which is the absolute value of A: |A|) of the graph is the distance from the sinusoidal axis. Its sign influences whether the curve will "flip" over the y-axis or if it retains its normal position. Further explained later on.

B- Parameter B is not the period of the graph but helps determine the period. This is also explained further into the class.

C- The phase shift/ horizontal shift of the graph.

D- Parameter D is the sinusoidal axis, average value of the function, or the vertical shift.

- - - - - - - - - -

Now, having discovered the properties of the transformations, we dive deeper into the concept and talk about how these variables effect the graph and how graphing can be put into an easier form of remembering.

Ex. Sin2x

• the value of "B" causes "everything to happen twice as fast"

• the angle is "doubled" before finding the value of sine

Ex. Sin-2x

• Inputs are made negative first

• Note: x-coordinates are changing sign

• Note: the negative signs do not flip the graph vertically over the x-axis but horizontally over the y-axis

**To help us improve our understanding of these examples, we compared them with Sin(x) on the same Cartesian plane.

Now diving even deeper with this new knowledge, we compare more examples to notice occurring patterns (mathematics is the study of patterns). We do this by observing sketches of functions.

Sinx = 1/2 <-- 1 wave between 0-2Π
Sin2x = 1/2 <-- 2 waves between 0-2Π
Sin3x = 1/2 <-- 3 waves between 0-2Π Sin4x = 1/2 <--4 waves between 0-2Π So, we find that the value of "B" is somehow related to the number of waves between a given interval (in this case 0-2Π). However, when the word "Period" comes to mind, people ask themselves, is the value of "B" in fact the period? The answer to that is no. "B" is in fact not the period of the graph, but indeed helps determine it.

• A period is the distance of a hill and a wave.

Using this mode of representation, we can find the relationship between the value of "B" and the period.

Now discovering all of this in the first class of the day may be quite a load. However, one thing that should definitely not be forgotten is the concept of the mnemonic: DABC. At first, the abbreviation may look unfamiliar. However when you peer closer, you notice that A, B, C, and D, are part of the transformational equation of Sine and Cosine! DABC is actually that, except in the form of order upon which can be helpful in remembering how to sketch trig functions.

D - Is the first step in sketching the graph of a trig function. D is the vertical shift or sinusoidal axis of the graph and should be found first, as you should know that the graph "wraps" around the sinusoidal axis.

A - Is the second step in sketching the graph of a trig function. A is the amplitude and determines the stretch of the graph. Also important about the amplitude is its sign. If it is negative, the graph appears to be inverted; it flips horizontally over the y-axis. If positive, Sine graphs will start at zero, and Cosine graphs will start at its max value, which is one.

B - Is the third step in sketching the graph of a trig function. B represents a factor that influences the period of the graph. This is used to determine the scale values of the x-axis.

C - Last but not least, the last step in sketching the graph of a trig function. C represents the horizontal shift of the graph.

When thinking of graphing one of these monsters, you may curse in the form of DABC (pronounced: Dah - Bick!). Then all of a sudden, a flash of insight washes over you and you suddenly remember how to sketch the graph! Isn't that amazing everyone?

Finally, we reached the last couple minutes of a long class (or short?). Mr. K put up an example, which he ended up sketching quickly on the board:

y= -3Cos2(x - Π/4) + 1



For indication purposes, during the class, Mr. K compared graphing trig functions to the "etchisketch" which is quite the analogy. He stated that one dial of the etchisketch could be compared to Sin and the other dial could be compared to Cos. Imagine both of the dials being turned simultaneously and the result is quite frankly circular functions displayed in a graphical manner or atleast visual.

Class 2: Afternoon

We started off the afternoon by taking a look at these two previous equations:

f(x) = AsinB(x-C)+D
f(x) = AcosB(x-C)+D

Discussing briefly about the two, we move quickly to an example.

Ex. y = -2sin3(x+Π/6)+1

• Note: The vertical shift is to the left in this example. The positive sign is due to a negative value input replacing the variable C. Since the formula has a negative sign, and the example has a positive sign, the horizontal shift is to the left because a (-)(-) = (+).
• Note: When solving this equation, for any reason, simply use the order of operations: BEDMAS

Now having graphed this function, Mr. K talked brought up a great topic. He said that, if you know how to do something one way in math, you should know how to undo it. So, that's what we ended up doing. Instead of converting the equation to a graphical mode of representation, we reversed the method and converted the graphs into symbolic modes of representation: in the form of an equation.

So you might ask, how do you do that? Well, we started off by find the values ABCD of the transformational equations of sine or cosine.

Note: For this one graph, there can be a large (when I say large, I mean LARGE) quantity of equations that when graphed, will all look similar or are exact replicates.

We decided to rewrite the equation in Cosine:

• A= -2

• B= 3 (B= 2Π/p = 2Π/2Π/3 = 3)

• C= 0 (normal cosine curves start at its max, whiel this graph starts at its min due to the negative amplitude sign)

• D= 1

By substituting the numbers back into the framework equation, we get:

y=-2cos3x+1

Next we decided to rewrite the equation 3 more times: 2 in terms of cosine, and 1 in terms of sine.

Sine

• A= 2
• B= 3
• C= Π/6
• D= 1

y=2sin3(x-Π/6) + 1

Cosine1

• A= 2
• B= 3
• C= -Π/3
• D= 1

y=2cos3(x+Π/3) + 1

Cosine2

• A= 2
• B= 3
• C= -Π/3
• D= 1

y=2cos3(x+Π/3) + 1

After all those nice examples, we look at where the quadrants are located in the graph. To put it short, depending on where your starting point is, the period of the curve is divided by 4 pieces, quadrant 1, 2, 3, and 4 before repeating itself again.

After this, we quickly went over number 15 of exercise #5, and number 12 of exercise #6.

We are now DONEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE the unit and our pretest is on its way (supposedly thursday). Tomorrow, we are taking notes into our super duper "Math Dictionary of Power" (lol) and are going over any issues, concerns, and questions about the unit! Make sure to ask some questions or we'll just have wasted a class that could've been used towards our exam studying time near the end.

WOW THAT WAS A DOOZY. Now we near the end.

Any comments, suggestions, error fixing, criticism, appreciation, is accepted :)

Homework for tonight: Curve sketching posted by Mr. K (so far has not appeared)

Scribe for tomorrow: Bertman! (sorry Sam, Bert asked me first :P)

Have a nice day. Night.