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Monday, October 31, 2005

Scribbinggg .

In the beginning of class Mr. K gave us three questions to solve on the board but only explained how to do the first two because class was almost finished.

Question 1:
Find the area of ∆ABC
with A(3,0) B(-1,2) and C(-4,-3)

First make a sketch to help you determine what you're trying to find.

Since you know you're trying to find the area and area of a triangle is BASE x HEIGHT. You have to find the height of the triangle using the slope formula

Then use the point-slope formula to find out what you can use to substitute into the Distance formula.

Now substitute

Now since you have the information you need to answer the question substitute the numbers in the formula.

Question 2:
Prove that quadrilateral ABCD with A(-2,-2) B(-1,2) C(8,6) and D(7,2) is a parallelogram. Also prove that its diagonals are not equal in length.


To find out if the quadrilateral is a parallelogram it must have two sides with the same slope.

Therefore it is a parallelogram.

Now to find out if the diagonals are equal find the distances between them.

The diagonals are not equal in length.

** For full marks you need to have statements or words to show what you did or you will lose marks.

Question 3: Draw each pair of lines on the same cartesian plane.
We did not go over.


Then we put some notes in our dictionaries and class was over.

"Never stop and give up. There are always pauses to stop and look at what you did, keep your eye on the destination and keep your mind on the goal."

Scribe for tomorrow is JonathanJ

SORRY FOR THE LONG WAIT .. it's halloween yeeeno !

Analytic Geometry Assignment

Here it is!

If you need a little help with this stuff here is another tutorial for you. (Important: Gradient is another word for slope. And the stuff about the area of a polygon is cool, but we don't study that in our course -- go ahead and learn it if you like!) If you think you understand this stuff take this quiz from the end of that tutorial.

Also, check out my previous post if you haven't already.

Mr. K.

Sunday, October 30, 2005

What Mathematicians Think...

Earlier this month Jan Nordgreen at Caymath posted about an interview of a couple of professional mathematicians talking about their work. Here's one quote.

Isadore Singer: ... when I try out my ideas, I’m wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I’ve done something different.

Another quote:
Michael Atiyah: My fundamental approach to doing research is always to ask questions. You ask “Why is this true?” when there is something mysterious or if a proof seems very complicated. I used to say — as a kind of joke — that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture — it may be a beautiful result but with terrible proofs — you spend your time trying to find better ones; you do not listen to the lecture. It is all about asking questions — you simply have to have an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material.

The full interview is right here.

These are two things I find myself constantly belabouring in class when teaching problem solving:
  • Take risks! Experiment, play, try something out and see where it takes you. Good math isn't knowing what to do with any problem -- good math is knowing what to do when you don't know what to do. ;-)

  • Ask questions! If you don't ask questions then I can't tell whether you understand or not. I'll either go on to something new, leaving you confused in the dust, or go over and over something you already understand trying to help you but really just wasting our time.

Food for thought ...

Hare and Hounds

In this game you have to "run for it!" You can be the Hare or the Hounds. The hare has to escape; the hounds are trying to corner him.

The real question is: Are you an expert Hare or expert Hounds? ;-)

Have Fun!

Friday, October 28, 2005

Gimme Your Coordinates!

Here is a review of what we learned in class today.

Practice your test taking skills here (every time you [Refresh] the page you'll get a new set of 5 questions).

Thursday, October 27, 2005


wow sorry im posting this up so late and no i didn't forget i was the scribe as you can see i'm on here right now haha but anyways we started class off today with Mr. K talking about pie day and the mystery coin hunt and explaining to us how fun and exciting it is haha, but yeah afterwards Mr. K gave us a problem to solve which i thought was kind of difficult...hope i'm not alone on this one but yeah we talked about how lines that are perpendicular have negative recipricals and that parallel lines have the same slope, rise over run.

Then we learned what a lattice point is, which is the point on a plain that has numbers which are integers. Don't forget that it doesn't matter what line you use for the lattice point because its all the same.

rise/run = ∆y/∆x = slope
y=mx+b slope intercept
y-y1=m(x-x1) point slope
****REMEMBER the better choice is the point slope formula****
∆y y2 - y1
── = ────
∆x x2 - x1
graph ----> rise/run
values ----> slope
equation ----> point
I'll try explaing or showing the problem that we did in class again hope it helps some of you guys out, but as i said earlier didnt' really do that great with these problems so just bare with me...
Before we started solving this equation and looking for the smallest point, it's a good idea to draw the graph first cause it does help you a lot to see how it looks like. Then what you do is you use the distance formula to see how far the point is...and notice that while your looking at the graph and at the point, you notice that 2x-3 and d both live on the same line. Also the lattice point is on the origin. Okay so md = 1/2 and so now you can solve the equation;
2x-3 = -1/2x
4x-6 = -1x
-6 = -5x
6/5 = x ----> x = -1/2(6/5) ----> x = -3/5
Now once you've done that you can now use the distance formula to solve it...
d = √ (0-6/5)² + (0+3/5)²
d = √ (36/25) + ( 9/25)
d = √ 45/25
d = √ 9.5 / √25

d = 3√5 / 5 <---- so that's the shortest point
After this question we did another one and that's how we ended the class and im so sorry if this doesn't help anybody but crossing my fingers and hope it does though.
tomorrow's scribe is ROBERT!

Wednesday, October 26, 2005

Blogging b4 Testing

Hey guys! Sorry, last night I completely forgot about posting because I was out. But, I hope this still counts as a post "before" the test.

First of all, I hope all of you studied for the test we wrote today beause if not, you might have got caught on a couple of tricky questions. I've always found algebra easy, but this unit has really got me thinking. I think it's that there are some major parts of the unit that we all focus on more like "Rational" or "Absolute Value" equations, but we forget about some of the small functions of the unit for example:

b^2 - 4ac = the disriminant

A couple nights ago, it took me a while to work out a couple of these questions so I went back and looked at it. In fact I looked at all of the small details of the unit that one might forget about. So I know it's probably too late, but ask yourself, "Diid i study all parts of the unit?" because if not, there might be a couple of questions that you lost marks on because of it.

Finally, I think this program is going well. I think Mr. K. is really great at explaining all topics. As well I think this blog is a great idea and thanks to all who participate. Well good luck on any up and coming tests and the rest of the semester.

Craig K.

P.S. once again very sorry about posting up late.

test blog =D

just got off work and remembered about my blog for the test lol =/. anyways algebra i don't know if its just me but it seems kind of straight forward to me. Like everything is like the old algebra we used to do when we had to find what x equals. I hope i'm not alone on this one, because honestly i hate it when i'm the only one who understands something because I don't like the feeling of knowing something more than another person because I believe that we're all equal in our own ways =/. ANYWAY the only hard things i'm worred bout going into this test is word problems. It's just how to interpret the question theat bother's me, like what i'm supposed to do and how i'm supposed to do it. hopefully i do good on this test so yeah good luck to all you guys =D

Tuesday, October 25, 2005

postblog before the test

Mmkay, so heres my post before the test. =P

Wow, this unit went by pretty fast. And we have covered alot. I think I'm okay with finding the sum & product of the roots, the quadratic formula (yes, i sing it alot too =P "pop goes the weasel :D"),how to tell if the equation has one, two or no roots (the discriminant), imaginary numbers? im not sure if that one is on the test. is it?? the one with the "i^2" and "i". well ya.. anyways.. im still having trouble doing some problem solving =P uh-oooh. yeah...i gotta go review that one again. okee im done here.


blog post before our test

well, the lessons we tackled for this unit seems to be easy because equations were quite simple than the first lessons we discussed. I only often got problem if the questions were different from the questions we usually solve. It confuses me. well, I hope tomorrow's test will be easy especially because we don't know yet the result of our lesson that gives us pressure cause we don't know if were still doing fine in this subject.
Good Luck for tomorrow. hope to do fine in the test.

Blogging For Test

So I thought this unit was okay. I agree with Jacky on this one, that the word problems confuse me. Like, I totally blanked on what to do when we had our pretest. The quadratic formula seemed pretty okay overall, when we first learned it, I was like, wow Mr. K, and you made us do all that other work to find the roots, when we could've just used THIS FORMULA? But hey? Good thing he made us sing that "x = -b + or _ the square root, of b squared minus 4 ac, all over 2a" like seven times in class. It's such a good song, and I practically sing it everytime I write down the formula. Haha, but yeah. For the rest of the things we learned, I was okay with them too. So, good luck guys on the test tomorrow!

blogging again before the test!

Good thing I remembered to do this, or else I would have lost a mark=s. Anyways this unit wasn't that bad. I think the things that bothered me the most is the word problems once again. I mean I know what formulas to use, because it's right there in the question, I just don't get how to put everything together to get the answer. Confused?!?..same here. For the other stuff that we learned in this unit, like imaginary numbers, and abosolute value is somewhat a review of what we learned last year and I was fine with that. It was fun having to memorize the quadratic formula to the tune of "pop goes the weasel." It actually does work!. Ok well good luck on the test!!=)

Blogging Mark For Equations

Hi guys! This unit went by fast. I can't believe we have another test. We didn't even get to see our mark on the trig test. We covered a lot in this unit. I remember the quadratic formula song. I'm probably going to sing that in my head tomorrow for the test. Haha. The easiest thing for me is finding the discriminant. The formula is short and simple. The thing I have difficulty with is PROBLEM SOLVING. I hope the questions on the test won't be too hard like the question on the back of the pretest. When he went over it the first time, I didn't understand but the second time, I got it. Isn't it weird when the teacher does it on the board or starts it off for you, you get the answer right away but when you do it by yourself, it's like you forgot everything. Well, that's how I feel. Well I think this is enough. I'm going to keep it short and good luck tomorrow!!!

Scribing For Circles

Hi guys! Today we started a new unit on equations of circles. The first thing we did was talk about the distance formula. You might remember this from last year. This equation is used to find the distance from one point to another.
The next thing we did was use this piece of information to find the radius of a circle. If the coordinates (h,k) is the center point of the circle and (x,y) is one of the endpoints then the distance from both points is the radius.

To get rid of the square root sign you have to square both sides. You'll end up with an equation like this. This is the equation of a circle.

Two things you need to know about a circle is its radius and its center point. From this equation, you can retrieve (h,k). Remember that the actual coordinates of (h,k) is the negative inverse or opposite from h and k in the equation. For the radius, remember to square root it for your answer because in the equation, it's not r.

ex.1: (x-3)²+(y+2)²=64

center=(3,-2) (negative inverses of h and k)

radius=8 (square root of 64)

ex.2: x²+(y+5)²=20

center=(0,-5) (opposite of h and k)

radius=2 root 5 (square root 20 then simplify)

The next thing we learned was how to write an equation if we were given the information.

ex. center=(-2,8)



To retrieve this answer the first thing you do is find the opposites of (-2,8) which is 2 and -8 then plug that in to the equation. To get the radius, change 3½ to an improper fraction, 7/2. The next step is to square it and you'll get an answer of 49/4. Leave it as a fraction. Remember, fractions are our friends.

When you have an equation like x²+4x+y²-2y=4, in order to find the center and radius, you must complete the square.

ex. x²+4x+y²-2y=4





After, we were given plenty of time to do exercise 21.

Well, that's all that happened today. Good luck on your test tomorrow. The next scribe is Pamela.


Well here's my post before the test. I think i have all the basic things downpat. When we were reviewing with Mr.K .. everything seemed to make sense and it seemed pretty easy. But when it comes to applying these things into a word problem, it seems so much harder? There's not much else I want to say in this blog because I need to get to studying and reviewing my notes for tomorrow. GOOD LUCK EVERYONE =)

bloggin post #3

So here my post before the algebra test tomorrow. I personally think some of the concepts we learned we're pretty easy like finding the sum and product, using the quadratic formula and finding the descriminant and all that. But there were some concepts that I had a lot of difficulty with such as the rational and radical equations. I did spend a lot of time trying to figure out and do the questions he assigned us. After doing that it seemed a bit more simpler but still confusing. Well that's my post, good luck on the test tomorrow guys, and keep on doing the work (Y).

Monday, October 24, 2005


I’m the scribe for today and I don’t know if I can explain everything well but I’ll do my best. Today we started off the class by Mr. K. telling us what we’ve learned this unit and what we learned was:

Quadratic Formula
Imaginary Numbers
Sum and Product of Quadratic Roots
Solving Equations-Radical / Absolute Value / Rational

He then put two questions of the board.
√-12 and (i-3) (i+2)
He gave us some time to solve them and once you solve them you get:

= √12(-1)
=√12 √-1
=2i√3 [We wrote it like that instead of 2√3 i so you can tell which numbers are under the square root.]

(i - 3) (i + 2)
= i² - i – 6

= -1 – i – 6
= - i – 7

Then Mr. K. asked a few people their favourite numbers from 1-10 and he came up with a sum and product of quadratic roots. From the sum and product we had to find an equation. We started by finding a common denominator which is 18. After that we wrote down what a, b and c were, then we plugged in the numbers for the equation.




a = 18 b = -4 c = 9

y = 18x² - 4x + 9

Mr. K. put up another question where the roots were x = 2 + √3, x = 2 - √3 and we had to find the equation. So to find the equation we found the root sum and the root product. From that we found what a, b and c were. Then we had our equation.

Root Sum= 2 + √3 + 2 - √3
-b/a = 4

Root Product= (2 + √3)( 2 - √3)

a = 1 b = -4 c = 1

y = x² - 4x + 1

Then Mr. K. asked if you can find a different equation and Graeme said you could put it in its factored form which is: y = (x - (2 + √3))( x – (2 - √3)). Then we solved it to see if it was the same. Then Graeme said you could multiply everything by 2. Mr. K. said that it wasn’t unique because you could have the same roots and have a different function with a new constant.

y = (x - (2 + √3))( x - (2 - √3))
y = x² -(2 - √3)x - (2 + √3)x + (2 + √3)(2√3)
y = x² - 2x +√3x - 2x - √3x + 4 – 3
y = x² - 4x + 1

f (x) = 2x² - 8x + 2

The last question from the pretest was the next question we went over and this is what we got in class:

I’m not quite sure what the answer is but to get the answer you are supposed to use the quadratic formula.Mr. K. put three more questions on the board and the first one was:

The profit, P, for publishing a book is given by the equation
P(x) = -5x² +400x - 3000
where x is the selling price per book.
a) Is it possible to set a selling price that will earn a total profit of $6000?
b) What range of selling prices allows the publisher to make a profit on this book?

The answer to a) is:
6000 = -5x² +400x - 3000
0 = -5x² +400x - 9000
0 = x² - 80x +1800 [We divided everything by 5.]
After that you use the quadratic formula. You end up getting a negative number as the discriminant so there is no sol’n. The answer is, no, you cannot get a profit of $6000.

For b) you find the roots of the original equation to find the range of the selling price. You end up getting [$8.38, $11.62] as the range.

2) Jason ran 4 kph faster than Gary walked. Jason ran 15 km in the time it took Gary to walk 9 km. What were their speeds?

When Mr. K. was doing the question he said he made a mistake somewhere so the answer was quite unclear.

3) The square root of three less than a number is 12. What is the number?
12 = √ x – 3

144 = x – 3

That was pretty much everything we covered in class. Mr. K. also said not to worry about the distance equations because there won't be a lot of those on the test. Remember we will have a substitute and the test is on Wednesday, so review your notes from your dictionary and go over the questions you had difficulties with. Tomrrow we will start on circles I think it was. Just tell me if I made mistake;).

***The scribe for tomorrow is Rosel S.


hi, its come to my attention I have forgotten to say who is scribe. Well it is Aichelle!

Sunday Funday (Oops, I missed a beat...)

Here are the rules.

Here is the game.

Have fun with it!

Sunday, October 23, 2005

Random Post. LOL

So. How's life. Fine. Thank you Very Much. How's math. Cool as always. How's everything else? Um... Great. I think. Hahaha. So the Scribe is Aichelle right? Yes sir. That's cool. How do you know. Secret. I see. Haha. Alright. So what you doing tomorrow? Um... I dunno. School? What about the day after tomorrow? Um. School. How bout. Um.. Nevermind. So when is this conversation gonna end? Now I guess. Hahaha.


Hahaha. That was totally random. Hahahahahaha.

Blogging Time

Okay this unit I think is probably the easiest we've done so far and I think I've understood everything we need to know. I know the discriminant, how to find the roots using the quadratic formula, and finding the sum and product of the equation. I get confused in the problem solving and it's something I really need to work on. Hopefully it won't be TOO hard, but we don't really learn anything if we don't challenge ourselves.
I really want to get this test over with so I won't have to worry about it anymore, which makes me remind myself about report cards coming out in a few weeks. I can't really find anything more to say, so I wish everyone luck on the test Tuesday!!!

Notes eh?

Alright, I tried to publish this once already and it didn't work (I'm really starting not to like blogger). So here are the notes, if you don't use 'em you won't gain anything from them. And one more thing, Mr. Kuropatwa do ya think you could skip a few steps when deriving formulas?

Notes Set 10 Deriving the Quadratic Formula

Notes Set 11 Imaginary Numbers and Working with Quadratic Roots

Bloggage before test

So...math eh? I get how to find sum of roots and product of roots.[-b/a, c/a] I know what a discriminant is and I can decipher if there is one, two, or no real roots. I understand absolute value and how you can get two different answers. Rational equations are okay. Something I'm having trouble with is solving radical equations, like when Mr. K. does it on the board I'm like o0o0o0o0oh but when I'm at home I'm like n0o0o0o0o0o because I don't get the right answer and I do the question again but I still don't understand. I got most of the questions on the pretest but the one question that threw me off and probably threw everyone else off was the last one with the triangle, I looked at it and was like je ne pas compris (I don't understand). I'm not really good at problem solving or those kinds of questions so hopefully tomorrow Mr. K. will go over more of those kinds of questions for us so we can understand how to do them. One last thing, good luck to everyone on the test and I hope you all do well.

Saturday, October 22, 2005

My Muddiest Concept..(blog)

Hey...its me agian. Well here goes my third bolg. As you can tell from my name this concept we are taking right now just so happens to be the muddiest we've taken so far for me. Its very confussing and so I'm rightly confuzzeled.
The pre-test was one of those things in life the tend to like to name "wake up calls". Our test is on Tuesday so I have some real work to do. The problem isn't the in class work. With Mr. K there explaining, it all just makes sense, it's when I get home and try to apply what I learnt in class to the quesitons that I get really confuzed..but only on this one concept so far. i don't kow if it was because of the many teachers that have been gone through in the past week or that I just have a harder time grasping the type of concept, but I hope I won't be stuck in this rut for very much longer. So for those that understand what I'm going through...good luck on Tuesday, and for those who have no idea what I'm talking about I'm glad there are some people who are understanding this,(You might be the ones I ask for help, lol) and good luck as well!

Scribes Scratches....

Hi fellow math people! Sorry this is late but the computer at my mothers house is down so I have had to wait until I got to my fathers. So ne ways here is what we accomplished in class on Thursday.
The first thing Mr. K went over was a review a quick review of everything Mr. Clark taught us. eg. sum =-b/c and the product equals b/a. Then he showed us case 1 and case 2.
He took an absolute value equation eg. ---pretend the brackets are straight lines please----(x+3)=7 Then he put the equation into the case 1 and case 2. x+3=7, x+3=-7 and solved coming up the answers x=4, x=-10.
Then came the pre-test which came as a major surprise to most of us I am sure. This pre-test included one multiple choice question on the sum and product of one equation. Where you use the sum and product equations themselves to figure it out. pretty straight forward. Then we had a second multiple choice question on wether the answers to an equation would have certain characteristics. The answer to this question was that the equation would have tow negative numbers in their solution sets. eg. they had four solutions for x: -4, 1, -3, 10. Within those four solutions there are two that are negative.
We also had a question were we had to find out if the equation had two answers in its solution set. To do this you must find use the discrimmatent. You plug in the numbers and if the answer was greater then 0 then x would have only two answers. If it equals zero then x would have only one answer. The last question was a problem solving question of which no one understood completely.

Sorry about how small the writing is. First time I have done this. Ne who. You want to find the distance of x. We know that the distance across the top is 10 miles. We also know that the person swam at a rate of three miles an hour and ran at a rate of five miles an hour. The last thing we are told whithin the question is that the entire trek from A to B to C took 3 hours and 20 minutes.

This was the question. The answer? Well that is a whole different story. I myself didn't quite understand it so I do not feel at all comfertable trying to explain it to my fellow classmates and everyone else on the world wide web. I am very sorry about this. If Mr. K after having read this would kindly go over this question one more time on Monday it would be a great big help, But for those out there reading this think of it as a challange to try and figure it out yourself.
After the pre-test we were split into groups and were to go over the questions amoungst ourselves and come up with what we thought were the correct answers and hand one of our sheets with all our names on it into the teacher. Mr. K then went over each question individually. Well that was our Thurdsday class and once agian I apologize for it being late and for not being able to properaly explain the problem solving question. I will finish off with saying that the test has been moved to Tuesday, Monday now being a question period, and to study and do your questions this weekend. (The reviews) Ciao guys.


Let us take advantage of this situation that we are in... So anyways... Blogging, Homework, Class, 3/6. We learned this material 3 times already. Now we need more. More help, more math, more math, more... oh I said that already. Hahaha. So anyways. You guys get it right. Same rules apply and this post will be the first post you'll see until Saturday night. If you guys remember how to do that. Haha so ya. To quote a famous math teacher.

"You can use your name or leave your comment anonymously, but, whatever you do, share your troubles here. Remember, not only can Mr. Kuropatawa help you but you can help each other too! Leave tips and advice in the comments for your classmates. And don't forget, you can form an online study group and "meet" in the chatbox of our blog! Unlike Sysiphus, you're being set up to succeed! Take advantage of every opportunity you've got! "

So ya. I'm out for the weekend. Later.

BTW... I hope this works and... Thanks Robert fot the idea in class. Hahahahaha

Thursday, October 20, 2005

Mr. K. This is for you.

Hey Mr. Kuropatwa,

Umm.. I know you're probably very very busy right now with this SAG thing going on but I was wondering if you've forgot about our Trigonometry tests. I'm sorry to rush you, but I get really nervous if I don't know what marks I have. Sorry, just a reminder, no need to rush.

Next Round of Bloggers...

Man... Man... Man... This is great... Test is now on Tuesday... Man... Ok it feels like I'm ranting... But I shouldn't be. Hahaha.

So anyways. This unit is really, really, really, umm.. How would I put this. Intermediate/HARD. Hahahahaha. That was strange... Haha.

Well anyways. I understand the basic concepts. Check.
I know how to apply them. Check?
I can do problem solving questions. D.N.E. Hahaha
Anything else... I am so screwed... Man...

Anyways. To my point of this whole, short & sweet blog post... I'm a bringing back "The MUDDIEST POINT" for this unit and calling it.

"THE SUPER MUDDIEST ALGEBRAIC POINT" Dum dum dum. And I'm going to set the date on the blog post so it's going to be the first blog post you see until Saturday night.

That should work. Hah. So if we have problems on this unit. Cause I know we do. And I know the class as well as I do. And If we do this Mr. K will give us help online. So it's a plus, plus, plus. HAHAHAHAHA

That was queer...

Well anyways. I'm a out. and I'm a back in. BTW I allways put this... If you need help blogging... Go to. BLOGGERS 'R 'US. Later///

Wednesday, October 19, 2005

AHHHHHHHHHH! I hate my computer!

Well guys I'm very sorry. When I published my blog i did not preview it first, big mistake!

Well as you might have read in my scribe post, I was having computer problems so I had to do all equations on the actual word box. it turns out all of my spacing got messed up.

I'm very sorry. I think you should just ignore the equations and just read the descriptions. :(

Once again very sorry I'll make up for it next time.

Uh-Oh This is going to be Difficult!!!!

Well folks once again I'm scribe and I think it'll be the hardest class to cover so far this year. These are the reasons:
-for those of you who were here today you know how complicated this class was
-for those of you who weren't here today I'm going to have to explain this in a very detailed manner
-and I do not have a Windows computer (I have an iMac) and my only drawing program isn't responding so I'm on my own with the examples.

Here we Go!

Today's class was with Mr. Clark and he taught us "Rational Equations". Right off the hop he went into just a little review of what we needed to remember from previous years of Rational Expressions.

The first thing that you must recall is that with Rational Equations there is an answer and a RESTRICTION. As long as there is a denominator with a variable there will be a value that "x" cannot equal. (Restriction or Non-Permissable Value).
To get this value you must make all the denominators with a variable in them equal to zero. Do this by solving the equation:

_7_ or ____3a____ = ____3a____
x-3 --> x-3 cannot equal 0 a^2-2a-15 (a+3)(a-5) --> a+3 c.e. 0 & a-5 c.e. 0
x cannot equal 3 a c.e. -3 a c.e. 5
a c.e. -3,5

Next thing you must recall about Rational Equations is that when adding and subtracting expressions with a denominator you must put all terms over a LCD (Lowest Common Denominator). Do this by finding the factors of the denominator. Once you have factored all of the denominators find the number(s) that are common and multiply them together and then find the numbers left over and multiply them into the product (if there is only one term with a denominator, use it for all expressions in the equation):

_x_ + _2x_ - _1_
10 12 50 --> {2,5,5} 2 is common to all three --------> 2 2,3,and 5 are left over:
----------> {2,3,2} 5 is common to two of the terms--> *_5_
------------------> {2,5} 10 2*3*5*10= 300
LCD = 300

5x^2 y 10x^3 y^3 15x^2 y^5 --> {3,5,x,x,y,y,y,y,y} 5,x,x,y are common to all three 5*x*x*y*y*y*1*2*3*x*y*y
------------------> {2,5,x,x,x,y,y,y} y,y are common to two =5*1*2*3*x*x*x*y*y*y*y*y
-----------------------------> {1,5,x,x,y} 1,2,3,x,y,y are left over =30 x^3 y^5

After this we were given four examples on the board, all of different difficulty. Yes, I will try to show how to solve them:

A) _x-2_ + _3x_ - _3x-8_ = 1 B) _2m-9_ + _m_ = _5_
6 8 24 m-7 2 m-7

C) __5__ - ____2y-4____ = _3_ D) ____4____ - ____5____ = ____3____
2y+6 y^2-y-12 y-4 x^2+2x-15 x^2-x-6 x^2+7x+10

First find the LCD of the equation which is. Now multiply each term in the equation by the LCD. Cancel out from the LCD with the denominator as needed then multiply the numerator by whatever is left. Then find and combine the like terms. Now solve for x:
*NOTE: factoring or quadratic formula may be needed to solve for x.

A) _x-2_ + _3x_ - _3x-8_ = 1 B) _2m-9_ + _m_ = _5_
6 8 24 m-7 2 m-7

24{_x-2_} +24{_3x_} -24{_3x-8_} = 24{1} 2(m-7) {_2m-9_} +2(m-7) {_m_} = 2(m-7) {_5_}
{ 6 } { 8 } { 24 } { m-7 } { 2 } {m-7}

4{x-2} + 3{3x} - 1{3x-8} = 24{1} 2 {2m-9} + (m-7) {m} = 2 {5}

4x-8 + 9x + -3x+8 = 24 4m-18 + m^2-7m = 10

10x = 24 m^2-3m-28 = 0

x = 12/5 (m-7) (m+4) = 0

m-7=0 m+4=0

m=7 m=-4

m=7,-4 ; m c.e. 7 .: m=4

C) __5__ - ____2y-4____ = _3_
2y+6 y^2-y-12 y-4

__5__ - ____2y-4____ = _3_
2(y+3) (y-4) (y+3) y-4

[2(y+3)(y-4)] {__5__} -[2(y+3)(y-4)] {____2y-4____} = [2(y+3)(y-4)] {_3_}
2(y+3) (y-4) (y+3) y-4

(y-4) {5} - 2 {2y-4} = 2(y+3) {3}

5y-20 + -4y+8 = 6y+18

-5y = 30

y = -6 ; y c.e. -3,4 .: y=-6

D) ____4____ - ____5____ = ____3____
x^2+2x-15 x^2-x-6 x^2+7x+10

_____4____ - ____5____ = ____3____
(x+5) (x-3) (x-3) (x+2) (x+5) (x+2)

[(x+5) (x-3) (x+2)] {____4____} -[(x+5) (x-3) (x+2)] {____5____} = [(x+5) (x-3) (x+2)] {____3____}
{(x+5) (x-3)} {(x-3) (x+2)} {(x+5) (x+2)}

(x+2) {4} - (x+5) {5} = (x-3) {3}

4x+8 + -5x-25 = 3x-9

-4x = 8

x = -2 ; x c.e. 3,-5,-2 .: x= NO SOL'N

Phew! Done.

Well that was today's class. BTW Mr. Clark assigned exercise # 19, a "Rational Equations" worksheet, and an "Algebra Review" sheet.

Once again sorry that my computer wasnt cooperating, I tried my best.

Oh Yeah! tomorrow' scribe is umm...

... Kasia (sorry I picked you again). Don't worry yours won't be as difficult as mine was.

Rational Equations

Review solving rational equations and look at these solved examples.

Try this quiz on solving rational equations. Here is another one.

Tuesday, October 18, 2005

Scribes Subscript

Alright well today in class we learned about absolute values with the substitute teacher. The lesson was really just a demonstration on how to solve absolute value equations after a to the point explanation on what the absolute value is. He explained that the absolute value is the positive of whatever real number you have within the two upright lines. Now when I first looked at what he had on the board I though, "Why does he have a one before and after that x?" I think that mathematicians could have chosen a better symbol out of the myriad of shapes that you can make with a pencil other than one that looks exactly like a one (or atleast the ones I write), except maybe slighter longer. Now that I'm done with that tangeant... Here are a few of the questions he put up on the board as examples.

This was the first question. Well it wasn't really a question, it was more of an example. It reads, ahem, the absolute value of x is equal to eight. So x can be equal to 8 or negative 8.

Alright well it seems blogger doesn't like my second image so ill try to do it correctly on here... (after I did this it seemed that blogger didn't like my absolute value symbols either, so in there place are braces { })

{3x - 5}= 7

3x - 5 = 7 or 3x - 5 = -7

3x = 12 or 3x = -2

x = 4 or x = -2/3


{3(4) - 5} = 7 or {3(-2/3) - 5} = 7

{12 - 5} = 7 or {- 2 -5} = 7

{7} =7 or {-7} = 7

Both answers work

Once again it doesn't like my images for any of the other questions and it doesn't like my symbols. So I'm just going to go to the last one, which was the only other one that was different.

{x - 2} = {3x + 1}

x - 2 = 3x - 1 or x - 2 = -( 3x - 1 )

2x = -1 or x -2 = -3x + 1

x = -1/2 or 4x = 3

x = 3/4


{-1/2 - 2 = 3(-1/2) -1} or {3/4 -2 = 3(3/4) + 1}

{-5/2} = {-5/2} or {-5/4} = {13/4}

Only -1/2 works as an answer. The other answer is extraneous.

Now with that question it seemed that there would have been four possible answers because of the ++, --, +- and -+. But..... The double negative and the double positive are the same equation and the positive negative and the negative positive are also the same equation. So you only need to write each of those once. If either of these examples is wrong please tell me because I'm still not 100% sure on these and there are about three different answers to the last one in particular on my paper. I chose the one that made sense to me but as I said I'm not sure.

I believe that I have covered everything and I am really starting to get angry with blogger, so I'm going to end now. The next scribe is................(drumroll please)...................... Craig K.

Radical and Absolute Value Equations

Review solving radical equations by looking at these solved examples. You can find more examples here.

Try this quiz on solving simple radical equations. Make sure you can do ones with two radicals in the equation as well -- like this one.

Here is where you can review how to solve absolute value equations. You can see some solved examples and you can take a quiz over there.

Monday, October 17, 2005

Solving with Radicals

At the beginning of the class Mr. K gave us three questions on the board, which were :

There are many ways to solving an equation with radicals in it. One way is to:
1. Balance the equation so that the radical is by itself
2. Square both sides
3. Radical then cancels

Example of that would be:

Anyways back to the questions, this is how you would solve question A.
**You can make sure it's right by substituting x for 3.

The second question gets a little bit more complicated:

**Normally you can square positive(+) and negative (-) numbers but in the given equation only use the principle root, so it can be a function.

The third question's getting even more harder because there are now two radicals:
First thing you would want to do for these types of questions is to make sure only one radical is on one side. Then solve the rest, like how it's been solved in the previous questions.

Then Mr. K gave us 3 more practice questions, I'm not gonna go show how its done (hehe to lazy), for the people who weren't in class today, you can also use them as practice. Here they are:

Then Mr. K told us he was going to be away 2 days this week, AND AND AND there will be an upcoming test, earliest is on Monday and the latest is on Tuesday. So expect a pre-test this up-coming thursday.

Anyways the next scribe is Graeme W

Sunday, October 16, 2005

Sunday Fun! (or Sunday Madness!)

I got this from a blog called Think Again! A great little math blog full of interesting puzzles. Lucky me, it took a while but I found my way out of the room ... can you?

You are trapped in a room. To get out requires some thinking. Good luck!

Over two million people have tried to leave the room already. No one knows how many are still stuck.

Friday, October 14, 2005

Quadratic Roots

So I’m the scribe today and hopefully you guys get something from my post. This isn’t going to be long but I will try my best to explain what we just have learned today :)

Today in class, Mr. K wrote these problems on the board for us to do:

Example #1:

For what values of k will the sum of the roots of the following equation be 8?

*Remember! To do this, first set up what’s a, b, and c.

*solving an equation always means finding the roots!

Well, I was having trouble on the first roots problem. But thanks to Mr. K for helping me out and refreshing my memory about the quadratic stuffs because to let you guys know, I really have a bad memory.

Example #2:

Find the one quadratic function whose roots are 3/2 and -1/4. Is the equation unique?

Example #3:

If 3x²-mx+2=0 can be factored, what values of m are possible?

Example #4:

For what values of k will the equation 2x²+4x+(2+k-k²)=0 have exactly one root?

And for the last thing we did in class was we took down some notes in our dictionary called WORKING WITH QUADRATIC ROOTS.

So..time to say bye now because I’ve done what I was supposed to get done :) and oh yah, the scribe on Monday is Jackie S. Have fun!

Thursday, October 13, 2005

Nature of Roots


Mr. Clark taught us how the sum and product of roots are related to the quadratic equation.

But before talking about that... how can we get the roots? Well, we can get the roots of a quadratic by:

1.) factoring

2.) using the quadratic formula

3.) or completing the square.

For most of us factoring is the easiest way to get the root.

(factoring review? *click here)

(the quadratic formula?*click here)

(forgot how to complete the square?*click here)

When a (leading coefficient) is 1:

the sum of the roots is the negative or the opposite of b (coefficient of x)

the product of the roots is the c (constant term)

Example #1)

If the roots are 4 and 8,then the quadratic is


The b (coefficient of x) is −(12), which is the negative of the sum of the roots. The c (constant term) is 32, which is their product.

Example #2)

a = 1; b = -2; c = -15

Sum of the roots is -b/a = -(-2)/1 = 2
Product of the roots is c/a = -15/1 = -15

That`s all for now. CIAO! :)

oh yah... ANNNNND tomorrow's scribe is Jamilyn G. ^_^

Wednesday, October 12, 2005

Scriber’s Revenge

Man...… You had to pick me...…
... ... ...…
Ok. Fine...…

Ok now that thats over with.…Today in class I get randomly chosen as scribe by Rannell. Thanks...… So anyways. First we had to do three questions like what we usually do.


If any of you forgot what the Quadratic Formula was...… Sing the Song. lol

a) f (x) = 2x2 - 3x - 4
b) f (x) = - 2x2 + 3x + 1
c) f (x) = x2- 6x + 10

Answer to A)

Answer to B)

Then Mr. K. Told us why it's negative over positve.

Answer to C)

Then we went back to factor trees. lol.

Fundamental Therom of Arithmetic. Only 12 can have the number 2,2,3.
We learned about a German Mathematician named GODEL.
He proved that certain things we know can be true but we dont actually know it's true.

For Example.
This Sentence is False.

He invented the GODEL number.
5 = 20*30*51

Fundamental Therom of Algebra

Any polynomial, whatever it is, it has to have the roots equal to the degree of the polynomial.

For example
f (x) = (x - 3)2
= (x - 3)(x - 3)

Then we learned that Some Roots can be imaginary. I'm like what...
Then we learned about a Swiss Mathematician called EULER and that he's responsible for PIE and E which is actually 2.17817... etc.
I wonder if Mr. K has a piece of E???
Hm...…Anyways then we get this whole explanation on how there are imaginary numbers outside the real numbers and stuff.

AKA It leads to this...

And This...

Some examples...

Then we finish question C. In the "Complex number system"

Then we learned about The Discriminant
b2 - 4ac <' 0

No Real Roots

(2 Complex or Imaginary Roots)

b2 - 4ac = 0

One Real Root
b2 - 4ac > 0

2 Real Roots

Then we get some notes in our Dictionary... I'll get Graeme to fill you in next time he has some notes. lol.

Mr. K. Then went over some question's on Exercise 14 & 15. Provided by Robert.
Homework if you didn't do it is Exercise 16 but leave out Questions 10, 11, 12, 13
Tomarrow we have a sub. Mr. Clark.


Yesterday's Scribe

First of all, I'd like to say sorry guys for not putting up a post yesterday. I know I was the scribe, but my internet wasn't working all that well. But, my post, for yesterday isn't going to be much, and I seriously apologize for it.

Okay so here we go:

Yesterday's class, Mr. K put up questions on the board to solve with our calculator to find the roots.

1. x² + 6x - 11 = 0

2. 7x² - 5x + 1 = 0

3. x² - √x-1 = 0

4. x³ + x = 30

5. Where does the line y = 2x + 3 cut the curve y = x² + 2x

(The thing is guys, I sort of got confused on which buttons to press on the calculator to find the roots. I'll try my best, but please, feel free to corect me).

So for the first question, x² + 6x - 11 = 0,
On your graphing calculators, you press the Y = button on the the above blue keys
And type in the the equation x² + 6x - 11, and hit the GRAPH button
Hit the TRACE button, and then 2nd CALC and choose 2: zero
It will say Left Bound? on the screen, move the cursor by hitting the left arrow key to bring you below the x-axis
And hit enter
It will then say Right Bound? on the screen, move the cursor by hitting the right arrow key to bring you above the x-axis.
Hit enter again.
It will then say Guess?, but Mr. K said don't guess, so at this point, hit enter.
And on the screen it will say
x = 1.472136

For the second question 7x² - 5x + 1 = 0, you practically do the same thing.
But the thing is, this equation, doesn't have roots.
You can type it in, and once you press GRAPH, you can hit the ZOOM key and choose 1: ZBox.
This will allow you to draw a box where the parabola curves, to see if it actually touches the x-axis.
And in this case, it doesn't. So we label this problem as no solution.

For the third question x² - √x-1 = 0, is like the second question. It has no roots, it's above the x-axis. So this problem is also marked as no solution.

Question four, x³ + x = 30, would have to be changed to: x³ + x - 30 = 0. And this is where Mr. K mentions that because it has a degree of 3, the most roots it would have is three.

And the last question, five. "Where does the line y = 2x + 3 cut the curve y = x² + 2x". Type both equations when you press the Y = button. And hit GRAPH, then you press the TRACE button. You then press 2nd CALC, and choose 5: intersect, and move the cursor the an intersection by moving the right arrow key. Press ENTER, it will then say First Curve? and press enter again, then it will say Second Curve? and press enter again, and it will ask you to Guess? but, you don't guess of course, and just hit enter. And it will give you the intersection.

Afterwards, Mr. K put us into groups of three, and we did a problem solver. Ones that we've done before. We struggled, and then he put the answers on the board. And that was about it for the class. Sorry if I missed anything, I really tried to re-cap from yesterday.

And Richard sorry I picked you, but hey, look at it this way, now Aichelle can win the race. And everyone else can comment on a wonderful post. ;)

Friday, October 07, 2005

my turn to be the scribe

haha.. i'm the last one to be pick out from the list to be the scribe. I hope you guys would not expect a lot from me because its hard to explain math terms but i'll try my best.

the class starts by solving some equation written by Mr. K on the board. It tells us to complete the square to find the roots.

It seems to be our topic before the trigonometry lesson but what Mr. K wants to show us is how to get the Quadratic Formula. The equations Mr. K gave us to solve were written in the general form. From the general form of ax²+bx+c=0 , we can get the quadratic formula.

Mr. K tells us some stories about where the numbers came from and developed to make man happy. I was suppose to create a link but I failed to figure out how to do it and that attempt erases some part of the words i had already typed so just go and visit http://mathworld.wolfram.com/QuadraticEquation.html

you can see from that link where some history about numbers.

After were done copying in our math dictionary, the fire alarm rang so we need to evacuate the building for the fire drill. most of us were unprepared so without wearing any jackets (cause most left it in their locker), we need to leave the building. What a bad time to conduct a fire drill. hehe..

when we cameback to our class we wait for the time while Mr. K taught us to sing a song so that we will remember the Quadratic formula. (maybe you are singing that right now and following the tune Mr. K. posted below this blog post).

Hope you guys have a nice long weekend.

the next scribe would be... Rannell d.

Pop Goes The Weasel!

Today we derived (watch this!) the Quadratic Formula in all its glory!

Here it is explained.

You can practice using it over there!

Now sing it with me folks!

x equals negative b
plus or minus the square root
of b squared minus 4ac
all over 2a

Thursday, October 06, 2005

Need Some More Notes?

Hello everyone! I have the notes posted in the links below. They might open a little funny and ask for a user ID, but if you just "x" that out it will still open just fine.

Notes Set 8 Trigonometry

Notes Set 9 Sinusoidal Graphs

Postage from a Scribe

Hello, everyone! I’m not very good at explaining things but I’ll try to explain things and show you what we learned in class. Today we started algebra, exciting isn’t it? Mr. Kuropatwa explained to the class that it’s an in depth analysis of solving equations and it’s about finding roots. We started class with questions from the board, where we had to find the roots, by factoring the equations:

x² + 7x + 10 = 0

2x² - 5x + 3 = 0
2x² + 3x = 2
3x² - 2x = 1

We then factored the equations and these were our answers:

1) x² + 7x + 10 = 0

(x+5) (x+2) = 0
x = -5 x = -2

2) 2x² -5x + 3 = 0

(2x-3) (x-1) = 0
x = 3/2 x = 1

3) 2x² + 3x = 2
2x² + 3x - 2 = 0 For number three and four we balanced the
(2x-1) (x+2) = 0 equation for the second step.
x = 1/2

x = -2

4) 3x² - 2x = 1
3x² - 2x - 1 = 0
(3x+1) (x-1) = 0
x = -1/3

x = 1

After that we had to solve these equations:

tan² + 7tanx + 10 = 0

2sin² - 5sinx + 3 = 0
2cos²x + 3cosx = 2
3sin²x - 2sinx = 1

Mr. Kuropatwa said that these equations with tan, sin, and cos (trig) are like the equations above. He said that it was in a pattern of a quadratic but isn’t a quadratic. Then he told us math is all about the science of patterns and how we have to remember patterns in order to solve problems. He also said if it fits the quadratic pattern, then you can solve complicated equations. We solved the trig equations like this:

tan² + 7tanx + 10 = 0
Let tanx = a
a² + 7a + 10 = 0
(a+5) (a+2) = 0a=-5 a=-2
tanx = -5 tan x = -2
X=281.3° x=296.6 °
X=101.3° x=116.6°

You are probably wondering how we got that…well I’ll explain it to you, well I’ll try. So first we let tanx=a and we rewrote the equation as a² + 7a + 10 = 0 and then we factored it and got (a+5)(a+2)=0. The roots we found were a = -5, a = -2 but since we let tanx=a we had to change it back to tanx. The roots were, tanx = -5, tanx = -2. Then we had to find the angles by putting arc tan into our calculators. Since it’s negative, we know that the angles are in quadrants IV and II. You then find out the angles are:

X=281.3° x=296.6 °
X=101.3° x=116.6°

After we solved that question as a class, he told us to do the second question by ourselves. After he gave us some time he went over the question and got:

2sin² - 5sinx + 3 = 0
Let sinx = a(2a-3) (a-1) = 0
a = 3/2 a = z
sinx = 3/2 sinx = 1
D.N.E. x=90°

Since sinx = 3/2 and you know it’s impossible you write D.N.E. (does not exist), Reject, undefined, or no sol’n (no solution).

***Make sure to write, D.N.E. (does not exist), Reject, undefined, or no sol’n (no solution) SO YOU DON’T LOSE A MARK otherwise you will lose a mark if you don’t write it. ***

That may be blurry but it says Do not let TANX=X! Make sure you switch the variable when you are looking for the angle. The bottom says Let TANX equal any other variable other than X.

After that Mr. Kuropatwa went over the other two questions real quickly.

2cos²x + 3cosx = 2
cosx = 1/2 cosx = -2

x= 60°, 300° D.N.E.

3sin²x - 2sinx = 1
sinx = -1/3 sinx = 1
x=340.5° x=90°

Once we were done all of that, Jacky reminded Mr. Kuropatwa that we had not finished taking down notes in our math dictionaries. So Mr. Kuropatwa finished the notes off by starting it off with the role of parameter B.

Here you go, knock yourself out:

The Role of Parameter B
In this course B=1, always. For more about B take Pre-Cal 40s.

The Role of Parameter C
C is called the Phase Shift or the Horizontal Shift.
C>0 The graph shifts right C units.
C<0 The graph shifts left C units.

The Role of Parameter D
D determines the Sinusoidal axis or Vertical Shift.
D>0 The graph shifts up D units.
D<0 The graph shifts down D units.

The Relationship Between
y=sinx and y=cosx

sinx = cos (x - 90) cosx = sin (x + 90)

After writing our notes Mr. Kuropatwa told us our homework was Exercises twelve and thirteen. Thanks for taking the time to read this and if I made a mistake just tell me=).

So tomorrow's scribe is Aldridge.

Factoring Review

Today I emphasized how to solve equations and trig equations by "recognizing the quadratic pattern". You can find a review of factoring here and practice your skills there and try harder questions here, here, here and there.

That review also explains why we call them quadratics -- "Doesn't quad mean four?" ;-)

Wednesday, October 05, 2005

Blogging MARK?!?!

If you guys were wondering about your one point blogging mark. Don't worry about it. It's covered. If your wondering why? Check the chat box. or just read this.

Originally Posted by Mr. Kuropatwa October 4th 9:42 pm On the Chat Box

Whoever reads this please tell the rest of the class tomorrow. I forgot to put "Did you blog?" as the 1st question on the test but your blog post STILL counts for 1 mark on the test -- I'll adjust it accordingly when I'm back at school. BTW, I posted a few practice quizzes online to help you study over here --> [link]

So ya. Blogging is Covered.

- Malcolm X
Hahaha... New Nickname... Hahaha

blogging before the test!

This is my third blog and got pretty scared cause my computer wouldn't let me access this page for some reason, but lucky me it finally worked today pheww... but yeah for me personally this unit went by so quickly it feels as if we just started working on it yesterday and now we have a test tomorrow, so far though i believe i've done pretty good in this unit, i still do get confuse here and there sometimes but who doesn't right...also i have a little problem with word problems never seemed to like them, but i know that if i started reading the question more carefully, maybe i can actually grasp what's being told and asked for, but i guess i just sometimes intrepret the question wrong either with what to draw or what's being asked for. But yeah today had a pre-test didnt' do bad but thought i could've done better well i hope everything goes great tomorrow for everyone and myself good luck to you all =)

Tuesday, October 04, 2005

another blog before the test...

we had a pre-test today and it was okay =) i actually get it now... yay!
I think i like this unit better than the last. haha i think i like trigonometry now.. ahaha the only thing that im having trouble with is uhmm...

the word problem and just drawing the diagram.

but everything else is okay ^_^ anyways... the test is tomorrow!

oh yah... before i end this post..when writing the test DO NOT FORGET:

-to put the degree sign when righting an angle.
-when graphing something, label your graph. (x&y axis, the points...etc.)

thats all for now... GOODLUCK GUYS!

My Blogging Mark Please....

Alright, well I found the beginning of this unit terribly hard to grasp at first. With the sign of the sine, and the sign of tangeant relating to the sign of sine and the sign of cosine, and so on and so forth. I got kinda lost in the jumble of sines/signs/cosines. But I found that once you do get what the man is talkin' about you don't forget it and the work becomes simple. Then he introduced the word problems which are extremely hard for a person to comprehend and draw (atleast for me). I still don't get how to do all of them, but I think I know enough to get myself through the test. If your wondering about the notes, I'll post them most likely tomorrow. Well good luck and godspeed on the test everybody!

bloggin' before the tesssssssst

hey everybody,

well today we had the pre-test and it went pretty good. well the the whole there was two cases to answer the last question was just weeirrdd haha... i never even saw that we can do that. well i hope the test is clear if there's a question like that hmmm.. what else to say. well if you think the pre-test was pretty good then you're probably ready but you might as wekk study riiighhht!?! you'll be twice as ready and there's no harm at that. well i hope you guys did the muddiest point comment i think it would help mr. k know where we are all at =)...

i don't know about you guys but....

well you guys i dont have anything else to day but....

STUDY ( awww.. look at him)