june/4/09

Arithmetic and Geometric series.

We started the class with a quick probability question, one that was brought up by a student.

When solving questions like these, always try to go back to the basics. Probability is the number of favorable outcomes over the possible outcomes. From that, we have something to work with.
Finding the possible outcomes, or Sample Space.

• The question states that "4 men and 4 women" would be chosen. This means that out of the 7 men, 4 will be chosen (7C4), and out of 10 women, 4 will be choose (10C4).

Finding the favorable outcomes.

• Since the question said that, "Allen and Bridget will be among these 8 chosen people", we already know that they are part of the possible outcomes. They are represented by the green and blue "1"'s.
• Out of the 7 men, only one has been chosen, (Allen). This leaves us with 6 more men, and 3 more spots for men, 6C3.
• Out of the 10 women, only Bridget has been chosen. This leaves us with 9 more women, and 3 more spots for women, 9C3.

As a result, you would get what is shown on the image above. (:

---------------------------------------------

Today's class was focused on:

1. Arithmetic and geometric sequences
2. Series
3. Arithmetic and geometric series.
4. Sigma Notation

Arithmetic sequences
An arithmetic sequence is a sequence (ordered list of numbers), where a fixed number(common difference) is found between two consecutive terms. (negative numbers are added too! )
This means each term is going up or down by the same number.
When wanting to find the nth term in an arithmetic sequence, refer to the equation below. (use Carl Friedrich Gauss's 7year old story to help you remember the equation.)

Geometric sequences
Geometric sequences are like arithmetic sequence, but instead of adding its multiplying. This means instead of a common difference, there's a common ratio.
When finding the nth term in a geometric sequence, refer to...
Series
Series is defined as, the sum of terms in a sequence. (Sn, where S reads as "sum of" and n would be the rank of the nth term. ex. S4 = sum of the first 4 terms.)

If we were to have a sequence of 1,2,3,4,5,6,7 etc, the series would consist of 1,3,6,10,15. Why? Well the first term is a given, 1. The second term would be 3, because that was the sum of the first and second term from the sequence, (1+2). The third term is the sum of the first, second, and third terms from the sequence. (1+2+3). The orange circled numbers in the above picture are the ranks.

Arithmetic series
Arithmetic series is the sum of numbers in an arithmetic sequence. This would be helpful if you are asked to find the "sum of integers from 1 to 5000" for example. Where n would be 5000, because there are 5000 terms, a would be 1, because that is the value of the first term, and d would be 1, because that's the common difference. (numbers are going up by one)

Or you might be asked "What is the sum of all multiples of 7 between 1 & 5000".
What you know:

• From the sequence of multiples of 7 between 1-5000, first term is 1.
• Last term is 4998
  
• common difference is 7.

What we need to know:

• number of terms within that sequence.
  

By using the arithmetic equation, tn = a + (n-1)d, plug in known values, do some grunt work, and you'll be able to find n. (or you can just take the last number, 4998, and divide by 7, to see how many times 7 can go into it.4998/7 = 714) Once you've found the number of terms, plug it into the arithmetic series equation. ta-da~
Now don't get too carried away with questions a and b. These kind of questions wont be asked on the exam, but you'll need to know the methods of a&b in order to solve c. (c = a question likely asked on the exam).

Logic:( The sum of all integers 1-5000) - (sum of all multiples of 7) = sum of all integers not multiples of 7. This "build up" to a question, is called scaffolding.


Geometric series.
Sum of numbers in a geometric sequence.

Sigma Notation
Is the shorthand way of writing a series, also known as the weird looking "E". Sigma is really confusing, if you don't know how to read it. The n=1 tells you the value of the first term, which is 1. The 4 on top of the sigma is nth number of term to stop at. The (2n-3) is the "rule" or equation you follow.

bye guys! good luck on the exams and your DEVs!
mary
The next scribe is jonno!

stress impedes learning?!

"What’s your sine? It must be pi/2 because you’re the 1"

HAHAH XD
just to lighten up the moods for all those who are studying hardcore. As pacifico said,"stress impedes learning", so cure it with laughter!

source

DEV sneak peak

Hello!!! Here is a sneak peak into our DEV!! Enjoy! CLICK

Conics BOB~

The conics section is one of my favorites. There weren't any big surprises and it seemed "short and sweet".

What we learned:

Parabolas

• vertex (h,k), a directrix (straight line, fixed distance from vertex), and focus(fixed distance from vertex)
  
• Standard form VERTICAL:

• Standard form HORIZONTAL:

• When 4p, also known as a, is greater than 1 the parabola is wide.
• When 4p is less than 1 the parabola is skinny.

Circles

• focus= center (h,k)
• Standard form:
• remember, the r is squared.

Ellipses

• center (h,k), Vertices's (endpoints on Major axis), major axis (length of 2a), semi-major axis (length of a, center to one of the vertices's), Minor axis (legnth of 2b), Semi-minor (length b),Foci (c units from center), Focal radi (distance b/w Foci and point on ellipses.
  
• The midpoint of two foci would center of ellipses.
• Standard form HORIZONTAL:
• Standard form VERTICAL:
• a^2 is the bigger number.
  
• c^2 = a^2 - b^2

Hyperbola

• center (h,k), Transverse axis (the distance b/w the vertices's, which are also the vertex. Is the difference of foci to pt, foci2 to pt. length of 2a), semi-transverse axis ( a),conjugate axis (length of 2b), Sem-iconjigate (b), Foci (c units from center), asymptotes.
  
• Standard form HORIZONTAL:
• Standard form VERTICAL:
• c^2 = a^2 + b^2
• When given slope of asymptotes, you can figure out the value of b or a. b/a = rise over run.

When changing equation from general to standard, you'd need to complete the square.

study hard guys!

Combinatorics BOB!

This unit was all on Combinatorics. We were exposed to:
Fundamental Principle of Counting, Factorial Notation, Permutations, Permutations of Non-distinguishable objects, Circular permutations, Combinations, and Binomial Theorem. (so much for a short unit, huh Mr.K? XD)

(8) Break it down!

Fundamental Principle of Counting: When you have Q ways to go a first thing, and P ways to do a second thing, than you'd have QP ways of doing both things. Genius!

Factorial Notation: When you multiply consecutive numbers. example: n! = (n)(n-1)(n-2)(n-3)..etc. To solve a Factorial Notation, 3! for example, you can multiply (3)(2)(1) or punch in 3, [MATH], left arrow key, [4].

Permutations: Ordered arrangement of objects, without repetition. Its when you "pick", meaning order matters. nPr = (n!)/(n-r)!. where n = # of objects to pick from, r = # of objects to be arranged in, and read as n PICK r. [MATH], left arrow key, [2].

• example: [LINK]

Permutations of Non-distinguishable objects: When you arrange objects n ways, where some of its objects are non-distinguishable. This means you can't tell the objects apart from each other, and it wouldn't really matter which one of those are which. In cases like this, you'd have repetition of the same result. To get rid of these repetition, you would divide by them. (n!)/(k1!k2!k3). The the numbers beside the K's are subscript.

• example: [LINK]
  

Circular permutations: When you have # of ordered arrangements that can be made of n objects in a circle. (n-1)!. Since its in a circle, there is no end or start. So she'd have to make one.

• SPECIAL CASE!: some circles can be flipped over, so the # of arrangements would be divided by two. (so no repetitions). [(n-1)!]/2
• examples: [LINK]
  

Combinations: When the order of chosen objects does not matter. The CHOOSE formula: (n/r) or nCr = (n!)/[(n-r)!(r!)], where n = # objects to choose from, r = # objects to be arranged.[MATH], left arrow key, [3].

• examples: [LINK]
  

Binomial Theorem: There's a super super long formula for this...and i refuse to type out. You can find it in May 11 slides, 15/17. Its what helps you figure out binomials, (a+b)^n, but remember the patterns and you're set.

• The coefficients in each term can be found by Pascal's Triangle, nC(i-1).
• b's exponent always starts out as b^0 in the first term, and b^n last term.
• n is the sum of the exponent's on a and b.
• The number of terms is n+1, because

• examples:[LINK]

k im taking a breather guys, this took longer than expected. XD
examples will be scanned and uploaded, brb!

so much yellow....

Overall, this unit was tough. I would always have trouble telling if the question was a permutation, or combination, etc. and then you have questions like [THESE], who fall into more than one category. D:
Hopefully we'll work on more questions tomorrow in class, and have things straighten out.

good luck!
mary~

exponents and logarithm BOB!

For this unit, we focused on exponents and logarithms. Its straight forward, as long as you remember that a LOGARITHM IS AN EXPONENT, the laws, and how to apply the formulas. So far, this would have to be my favorite unit. Probably because its the only unit i did good on the quiz. XD
The thing that could bug me on the test, would be those "model" questions....just because i haven't done many of them. But i'll be sure to concentrate on those when studying tonight.

Like always, I suggest you guys to do those questions left behind on today's slides for practice, and past provincial exams. links are found on the side of the blog. and dont forget to do your delicious tags!

okay, good luck guys!

D.E.V. Timeline for...well.........us!

Heeeey there all you matheees! Here's is our ever so short and ever so vague (well not really) TIMELINE for our D.E.V. (which stands for DEVELOPING EXPERT VOICES!). Haaah as you can see I am veeery excited to do this project and so are my two amigos, Dion and Mary! (Okay well one amigo and one amiga.) So anyways, I'll stop myself before I ramble on.

This timeline is brought to you by me (jayp) on behalf of my group members (Dion and Mary). It's also brought to you by the letters D, E and V and the number 6! Here it is!

Apr.24-30: Figure out/Finalize our EPIC storyline.

Apr.30-May 13: Figure out/Answer/Finalize all 6 questions and add them into the storyline.**

May 14-16: Make a video filming plan so we aren't filming around aimlessly.

May 17-June 2: Film and edit video so that it looks beee-yoou-tee-full!

June 3-4: Make sure everything is "movie" perfect and upload.

June 5: ABSOLUTE DAY OF DOOM!

**We basically gave ourselves 2 days/question (2 days*6 questions=12 days) plus one day just in case. We didn't give ourselves days for each question because we know that each question may take a little more time than others so we just thought we'd give ourselves around two days for each question. If we have extra time, then we'll use that for the tougher question(s).

So there it is! In black, Arial font, size "normal" open for all of you guys to see! Good luck with all of your project guys! And also good luck on our next test!

April.15/09

heeey guys!

For today's class, we all worked long and hard on a handout called "exponential equations exercises". Unfortunately, Mr.k wasn't able to enlighten us with his presence today, WHERE DID YOU GO MR.K?!, but luckily we had a substitute, Mr.N, for guidance!

From questions 1-10 on the handout, it was pretty straight forward, questions 11-24 were more interesting. As a class, we decided to leave question 21, 22 and i think 24 for Mr.K to explain. (Some of us got the correct answers, but not with the methods Mr.K would probably recommend.)

Before you get started...
Things to remember:

• Change the terms so they all have the same base.

• Multiplying powers with the SAME BASE = adding exponents.

• Dividing powers with the SAME BASE = subtracting exponents.

• Powers with negative exponents = reciprocal of the number. Example: 1/5 = 5^-1
  

• Square roots = are fractions.(such horrible wording xD) example: \sqrt[5]{a} is the same as a^1/5

Here are the questions, and answers from 1-10. [LINK]
Here are the solutions and answers from 11-24.[LINK1] [LINK2]
If there's any questions or problems, i'll try my best to resolve/fix them.
**the answer for 24 is not -1/2, but -1 and 2. i'll upload a writen soultion later. thanks sheldon!


ohh and before i forget! There wont be any slides from today's class. I guess we decided to pull old-school and dusted out the white bored BOARD. haha spelling errors are always on purpose. :D
Today was a pretty productive class, don't you think?

mary~

and because i had ginger ale, next scribe will be ale! :D

BOB = bob

This unit we focused on PROVING IDENTITIES. BOB = bob, get it? xD

Unlike what we did last unit, where we solved equations, here we proved each side to be equal. This meant that, the same expression looks different. Remember, we're not solving equations, we are proving them. (changing how they look). So never "cross the great wall of china"!

I was okay with this unit, nothing to complain about. But it would always take me awhile to see the "cleaver" CLEVER idea. This would be a problem when time plays a big role. (TEST) If you're going through the same situation as me, i suggest you do a lot of practice examples. There are links found on the blog to other references. AND Mr.K has left a bunch of practice on the previous slides. I'll be doing them tonight, so i'm down with comparing answers! :D

good luck on the test guys!
mary

bob bob bob (8)

hey guys!
Another unit has pasted...meaning bob makes a reappearance!

So through out this whole unit, there were plenty of things i found confusing.
~QUE MUSIC~ <-click
It was very frustrating for me, especially during the quizzes. (ambush) But I guess from the frustration, i learned a very important lesson....always do your precal homework D: And after a few youtube videos, and a long over due of Pjarry session, its s'all good. ;)
~STOP~


What really threw me off was graphing the inverse and reciprocal functions. ughh and how \sqrt{-a} would be completely different from -\sqrt{a}. The trick was noticing where the negative sign was, but it still gets to me. If the negative sign is inside the \sqrt{a} than its just a reflection across the yaxis. If its outside the \sqrt{a}, than its across the xaxis. (please correct me if I'm wrong! I've always seem to mixed them up. -___-)


I suggest you watch the youtube videos provided on the slides, those cleared some questions i had. ohh and when you go into a study group, see if you can teach what you learned to your members. only then have you truly understand the lesson. (: "does that give me cool status?" -niko

just like the first, I've enjoyed this unit as well. Maybe even more, since there's pi approx. day coming up.

Ever since I've found out about this built in spell check on Firefox, my spelling's at its best! no more yellow guys ;DLOL
good night~

PRETEST

I've had some people asking for the pretest questions...so for those who weren't here today for the pretest, this is what you missed out on...
prepare yourself for bad quality pictures!

http://img11.imageshack.us/img11/5710/pretestcircularm.gif
http://img3.imageshack.us/img3/7052/pretestcircular2.gif

the answers are on the slides! but if you're able to make out the chicken scratch that i call "writing", then whoaa.. XD

and am i allowed to randomly post a blog like this? Mr. K? LOL

oh bob~

hey guys! yeh its me again. don't worry, i'll keep this one blunt and short.

Recapping on your last unit, circular functions..
What i really liked about this unit, was how we were able to see the different methods of answering one question. Along with that, a little background info on how that method was discovered was interesting as well.
What i found hard about circular functions was and is the translations. The whole amplitude, period, and what-not got to me. maybe this is a side effect from physics? Ohh and the review questions from each exercise. i had allot of trouble recalling what i learned in grade 11... haha nonetheless, at least i know what i'll be studying tonight. What was easy for me, was most likely converting degrees to radians and vice versa. XD

Tips? i honestly don't know. i suggest you go over the slides, and redo the questions from the exercises that confused you the most.

okay im off to study, and i hope guyd are too!
byebye (:

UGHH DABC!

EDIT~

3.14159..

hey guys! It's mary here and as your scribe from yesterday, i'll be summarizing the wonderful moments of precal class...today. Sorry about leaving you with that cliff hanger last night. But as promised, i will finish the post.

For yesterday’s class, we learned..
~ What the sine and cos graphs looks like and why the way it does.

~ The difference between the sine and cos graph, along with the meaning of amplitude and period.
~ What a sinusoidal axis is.
~ The 4 ways you can alter a sine or cos graph. (Translations)




1a) Sine Graph:
wow, the image quality sucks. LOL bare with me!
Okay, so in order to explain the graph, you must understand the unit circle. As you go around the unit circle, starting from zero(or 2/pi), the sin value increases than decreases. The Sine value goes from 0, 1, 0, -1, and back to 0 again as it makes it full circle. It has a max value of 1 (when reaches /pi/2, and a min value of -1 (when reaches 3/pi/2).Regarding the unit circle, the sine graph displays the sine values as you travel around the unit circle, in a way where we can see that pattern. (of 0, 1, 0, -1, 0). This means the y-axis on the graph shows the sin value, while the x-axis shows the distance traveled on the unit circle. (2/pi, /pi/2, /pi, 3/pi/2 etc). Understanding this would come in handy when asked to draw a sine graph. 10 points for those who can understand the relationship! (brownie points) if not, ask yourself this: "where does sin equal zero on the unit circle?" look at the sine graph, "where does sin equal 1 and -1 on the unit circle?" then look at the graph. Your answers should be the same as the answers on the graph. "zero,2/pi, /pi, /pi/2 and 3/pi/2".


1b) Cosine Graph
Again, just like with the Sine Graph, the y-axis on the graph shows the cos value, while the x-axis shows the distance traveled on the unit circle.

2a) Sine Graph vs. Cosine Graph

The Sine Graphs and the Cosine Graphs are pretty much the same. The only difference is, cos is horizontally shifted, /pi/2 units to the left, on the x-axis, from the sine graph . This means, if you were to take a Sine Graph, and push it /pi/2 units to the left, you would have a Cos graph. And if you were to take a Cos Graph and pull it /pi/2 units to the right, you would have a Sine graph. "They are identical in amplitude, period, domain and range". I hope you guys can understand the picture below. I was trying to show you visually that the two graphs are /pi/2 units away from each other.

2b) UH OH, new terms?! don't worry, i got this! ;)

--------------------------------------
mary says:
whats Period and Amplitude again? -__-
pj says:
Period is how long it takes for 1 cycle to repeat

pj says:
Amplitude is the distance from the highest points to the average
mary says:
so 2pi?
pj says:
it's 2pi for the regular sine/cosine curve
-----------------------------------------

Sinusoidal Axis: "Sinusoidal axis: is the line which passes directly through the center of the sin curve horizontally. This line is always parallel to the X axis if it is not the X axis to begin with." The sinusoidal axis is used as a reference line when altering your sin or cos graph. Remember, its not actully part of the graph, so don't draw it on a test, unless told to.

ALMOST DONE, STAY WITH ME GUYS!

3a) Translation:
When graphing trig functions, there are four ways it can be altered. (from its regular form, sin(x) or cos(x).)
Shifting it up/down. (vertical shift)
Shifting it left/right. (horizontal shift)
Stretching it vertically.
Compressing it vertically.

~Vertical shift (Shifting it up/down) this would go the same for cos and sine

example: (because its more effective than me blabbing away) sin(x)-2
the negative 2 would move the max and min values, of the graph, two units down. Another way of looking at it, would be having the new sinusoidal axis two units down from the previous. In this case, the sinusoidal axis would go from 0 (x-axis) to -2. Then you would just draw the graph as you would from the new sinusoidal axis.
example: sin(x)+2
Just like before, but the positive 2 would move the max and min values, of the graph, two units up.

~Horizontal shift(Shifting it left/right) this would go the same for cos and sine
example: sin(x-2)
the negative 2 would move the of the graph, two units right, from the regular sin graph. Notice how i said right, not left.

example: sin(x+2)
the positive 2 would move the of the graph, two units left, from the regular sin graph.

~Stretching&Compressing it

example: sin(2x). sin(1/2x)
The 2 would compress the graph, twice as much. While the fraction, 1/2, would stretch it twice as much.

3b) Translation order; DABC!
Most of the time, you will be faced with a combination of these translations. (vertical/horizontal shift, compress/stretch). And there is a order you go by, when you have to draw them. D A B C

From AsinB(x+C)+D or AsinB(x-C)+D, you would use DABC. Meaning you would do the vertical shifts first, since D is first in order.

Okay, enough of the snore-fest!

IMPORTANT NOTICES!
1) Mr. K will not be in class for the rest of this week, and the following week after. Reason: because of a family emergency. But show no fear,(that one was for you pj!), Mr. k had said he’ll keep in touch with us through the blog, and I'm sure everything will work out with Mr. K's situation.
2) Also, there is still an unit test on Friday, FEBRUARY 20th. Study hard and good luck :D

Resources:
http://www.cbv.ns.ca/mathhelp/P66.htm
http://www.clarku.edu/~djoyce/trig/
http://www.fooplot.com/

Next scribe will be laika~