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Thursday, December 22, 2005

Morning before the test

Well isn't this strange that I nearly forgot to blog. Hahaha. Why does this keep on happening to me. Well I like to say that this unit was fun in the way that it was. Hahaha. Craig. Hahaha. What a guy. Finishes the logic puzzle and then the second logic puzzle next peroid. Hahaha. Wow. You have to do this to us you know. Hahahaha. Well anyways.... Um.. Back to studying I guess. Hahaha... and Good Luck you guys.

Wednesday, December 21, 2005

blogging before the test

wow this unit went by fast! it was pretty much all common sense after everything else fall into place. So basically this unit was pretty easy except for the Venn Diagram because I always get confuse on where to shade the parts that your suppose to do. The logic unit was pretty fun. It made us all think until our brains hurted lots! Unlike Craig who can figure it out in less then twenty minutes!! so yeah if i don't see you guys i just want to say:

merry christmas and a happy new year=)


and good luck

last blog before the break =)

WOW almost forgot to post a blog up...but good thing Craig reminded me haha well he didn't really but you know what i mean haha....all i have to say is that this unit went by so quick, and to me i found this unit to be really easy i understood practically everything that was goin on from the venn diagrams to those logic problems, and i really did enjoy this unit it was nice and simple haha...but yeah just want to wish everyone a good luck and to craig i will TRY to succeed in beating you tomorrow haha even if you are fast and real crazy at doing those logic problems haha

blog before the test

I think this unit is the shortest unit and the one that i enjoyed the most. I think its fun solving the Logic Problems. But unlike Crazy Craig,who can finish it in less than 25 mins., it takes me awhile to figure them out. The one that i liked the most is the one with the forensic experts. The uh, Kansas one. ^_^ pretty cool. I like this Unit because its the easiest unit so far. haha. At first, i didnt really understand the converse, inverse and contrapositive because i was away when they talked about it in class but Mr. K's post helped me alot. ahah. Doing the Venn Diagrams are fun too. The thing that im having a little difficulty with is the indirect reasoning but i still think that this unit is fun.


GOODLUCK ON THE TEST YOU GUYS. ^_^

blog post

so tomorrow will be our last test for this year in pre-cal. we had tackle up a lot of unit before this test. doing logic problems needs a lot of patience specially if only few clues are given. the part that i'm still having a problem with is the NOTATION part. i got screwed up when i see "union", "intersection", "compliment" or "exclusion" being used together in one problem. i still need to study up that part. i hope i'm doing good in consumer math part cause i did it last year.

Advance Merry Christmas to all and Happy New Year too...

blogging

wow this unit just flew by. wasn't as packed with so much info as i thought it would be when Mr. K talked about what we were going to do this week. From starting to relearn about venndiagrams!? Wow take me back to grade 6 and 7. From comparing characters from books. haha. I had the most trouble with the venn diagrams though because having to shade in which particular region got me very confused but after doing numerous question it seemed a bit easier. Also the logic stories and questions weren't as hard but your mind just has to be in that habit of getting the hang of thinking that way. Then the matrix questions didn't seem that hard to do because it's just the matter of reading clues and sticking the information you find from those clues to figure out what you're looking for. Well that's about it, good luck to everyone on the test tomorrow.

Blogging at its Best!

Hey, well its that time again. Man has this past month been psycho or what! Crazy. Anywho, the concept of logic is actually in my opinion a pretty simple concept. (That's going to come back to haunt me). But the thing is it is simple in a complicated way. At first glance my mind was careening in all directions! "What! No formula's! No rules! AARRGG!" but once you understand what your looking for and how to find it, things just fall into place. Not only that but this concept has sort of opened my eyes to the "fun" part of math. Or at least ways of making it fun and challanging. Well that is all for me tonight so good lucj everyone on the test tomorrow and Merry Christmas if I don't get the chance to mention it. :)

My Logic Acrostic

L ogic is a matter of controversy amongst philosophers.
O ther definition of logic is the study of the origin and natural sequence of fundamental ideas.
G uides reasoning within a given situation.
I s most often said to be the study of arguments.
C omplete, sound, and consistent, but not decidable.

Last Scribe Post before the Holidays !

WHY THANKS ALOT LIZ FOR MAKING ME SCRIBE .


Anyways today in class We first went over the "A Usual Day at Unusual School". Mr.Kuropatwa went over each person in the class and we figured out who was which.

Here are the answers :
Mrs. Smart - bright
Mr. Bumbleton - brave
Ms. Boss - brave
Jan - bright
Mike - bright
Pat - brave
Denise - bright
John - bright
Emily - brave
Billy - bright
Susan - brave
Sandra - bright
Pete - brave
Dan - bright
Alice bright
Tom - brave
Connie - bright
Mark, Pam , Tony and Becky are - unknown

To figure out this problem [or any other problems like this] you should assume one is either a bright or brave then if there is a contradiction for what you had assumed then the opposite must be true.


After that we went over Number 3 on the Matrix Logic Problems Sheet.
If you had trouble making the matrix remember the *staircase and the number of different categories that go across the top and vertically is one less than the number of categories. If you're stuck while trying to fill in the matrix look at the boxes that are completely filled for other clues to fill in the other boxes. AND once you've decided you can't find out any more boxes from the clues try and switch pen colors to help you determine who did what.

Next Mr. K explained some of the things we need to know for the test:
BE SURE KNOW YOUR DEFINITIONS [IN YOUR DICTIONARY]
KNOW HOW TO WORK WITH SETS [WE LEARNED FOUR OPERATIONS]
HOW TO SOLVE PROBLEMS WITH VENN DIAGRAMS [GOOD QUESTIONS IN "GETTING READY FOR THE TEST POST"]
BE ABLE TO KNOW HOW TO FIND COUNTEREXAMPLES
KNOW HOW TO SOLVE LOGIC TYPE QUESTIONS!

*** PRETTY MUCH JUST STUDY EVERYTHING WE'VE LEARNED, DO THE REVIEW =) and You'll DO GRRRREAT ! ***


OH, AND CONGRATULATIONS TO CRAIG FOR SOLVING MULLINER'S MIRACLE CURES LOGIC CHALLENGE IN LESS THAN 25 MINUTES!!! YOU ARE TRULY AMAZING hahah .

and the first scribe for 2006 is .. JACKY S.

HAPPY HOLIDAYS
MERRY CHRISTMAS TO ALL & A HAPPY NEW YEAR.

blogg'n for the test

Doing this early, before I once again lose an easy mark for the test. Well I thought this unit was easier than the last unit. It was also alot shorter. But I guess the thing I had trouble on in this unit was the matrix problems, so confusing! and the fact that everything in this unit doesnt have to deal with just one thing but a whole other stuff at one time. I think the most fun for me in Logics were the venn diagrams, only because their the ones I actually still remember doing from last year. Well I guess this is it for now, Good luck on the test everybody!!

blogging before the logic test

Well, this unit went by really fast. To me, this unit was a bit confusing. I thought the Venn Diagram problems were pretty easy. The thing I had trouble with was indirect and direct reasoning. It was pretty confusing at first but I think I got the hang of it now. I thought the matrix problems were fun. Even though some of them were difficult, it was fun to do. Well that wraps it up for today. Remember, the assignments for booklet 2 are due tomorrow. Study for the test and good luck!

Blogging Before The Test

Wow this unit went by QUICK . I don't think I can really sum this unit up in this post because it's logical, it's basically using some common sense and thinking out of thee box from time to time . Well just a few reminders .. remember to always put a Universe for your Venn Diagrams . Also an argument is ONLY valid if thee result is a direct consequence of it's premise. And a sound argument is when it's both valid and true . Also the difference between inductive and deductive . Well there's are just a few definitions from YOUR dictionary .. make sure you read over them . Good luck to everyone and I hope you all have a happy holiday =)

reflection

Logic...eh? haha..When we first started with the Venn Diagrams it reminded me of elementary...oh what memories...just kidding. I thought it was pretty straight forward and I understood it. The operations sort of get confusing when they are operations within operations but it's easy once you work it out. Conditional statements...those are funny haha "It will rain today if and only if I wear a hat" I still have to remember the notation for each of the types of statements because I don't know some of them, I'll just reread them until I'm comfortable---actually everyone should do that. Indirect reasoning...that can get quite confusing, it's also sort of funny. Matrix logic problems...those are quite fun because it's like a mystery that you need to solve..fun, fun, fun...except the super hard ones...or more challenging ones are...harder to solve..unless you are hardcore like Craig...he's a smart cookie..

I hope you all do well on the test tomorrow and don't forget to do a reflection..it's an easy mark, as well as your acrostic if you want to do one
study your defintions
do the review
and do the online quizzes that were provided for us..Good Luck!

Tuesday, December 20, 2005

A-Log-tic

•Venn stole Eüler's Circles and named them after himself.
•Each Venn Diagram has a UNIVERSE which includes all the numbers inside and outside of the circles.
•Null sets are a subset of every set.
•Not all objects in a universe are inside the circles


•Definition of a SET is a collection of objects
•In English, COMPLIMENT ( ' ) means "NOT"
•A complimented set excluding a regular set is equal to the compliment of the two regular sets unioned together. e.g. [ A'\B = (AUB)' ]
•Give all of your Venn Diagrams a universe around it and labels for each set when drawing them
•Reading "AUB" in English would be said "Set A UNION(or) Set B"
•A set must have at least two subsets: itself, and a null set
•Meaning of INTERSECTION (Ω) in English is "AND"

A-Log-tic



Scribe or what ?


Well we started off today with some questions on the board :
1) One of the Isle of view (say it aloud =P) knights always tell the truth and knaves always lie. You just arrived and met two people, Mr. A and Mr. B. Mr. A says, "At least one of us is a knave" . What are A and B ?

First make the assumption that the speaker (Mr. A) is one or the other. We'll pretend that Mr. A is a knave, meaning he only lies. If he was a knave and he said that he'd be speaking the truth and he can't do that. They both can't be knaves because then the statement would be a true statement. If we had one knave and one knight, the statement would be true, therefore Mr. A can't be a knave and he has to be a knight.

Mr. A = knight
Mr. B = knave

* Proof by contradiction
- assume the opposite
- if there is a contradiction, then the opposite must be true

2) A little later you meet Ms. C and Ms. D. Ms. C says "I'm a knave but D isn't." What are C and D?

C can't be a knight because she can't say she's a knave because that would be a lie. Therefore she has to be a knave. (indirect)
Assume : Ms. C is a knave (liar)
In logical a conjuction both ideas have to be true for the statement to be true. However, if one idea or part of the conjuction is false then the whole thing is false.

Ms C is knave meaning that she can't say that she's a knave .. BUT when she used a conjuction and put together a true AND a false statement .. meaning that she lied .

Ms. C is knave meaning that Ms. D must also be a knave because the second part of her conjuction has to be false "but D isn't" meaning the opposite.


Next we got a hand out called "
Matrix Logic Problems". We did that first one in class.

1) Three of our Olympic swimmers took 1st, 2nd, and 3rd in their race at the Olympic games. When they went to the victory stand each wore a different colored swimsuit. From the clues below tell the name, place, and swimsuit color of each swimmer.

a) The first place swimmer wore a
red swimsuit.
b) Tract took 3rd place.
c) Nancy wore
blue.
d) Mary and the girl in the white swimsuit were roommates.

Using the clues above and using a matrix, we get this :
Now using some thinking ..We get that Mary must be red because she can't be white or blue, and she has to be 1st because 1st place wears red. Nancy wears blue (given), which means that the only color that Tracy can wear is white. Also Tracy is 3rd (given) meaning that 3rd place winners wear white. Then the only place left is 2nd, and Nancy is the only one left and she wears blue.

Meaning:
Mary - 1st - Red
Nancy - 2nd - Blue
Tracy - 3rd - White


Also we got a story called "A usual day at Unusual School" BUT .. I won't give you my answers because it's due tomorrow and it's more fun figuring things out on your own =)


Monday, December 19, 2005

Getting Ready For The Test

A few links you may find useful in preparing for your test on Thursday:



And here are a few quizzes (refresh the page for more) ...



And finally, here are some logic puzzles to practice with ...



Study hard and do your best work; work you're proud of!

The Massive Notes Post

Allright folks, here are the up-to-date set of notes with the other notes in it just for the sake of easy access, I still need to put the Venn Diagram Notes and the Chart but I'm lazy. I'll do them during the holidays. Use them well.

Notes Set 1: Quadratic Functions


Notes Set 2:How to Write Interval Notation and Set Builder Notation


Notes Set 3:The Standard Form for the Equation of a Parabola


Notes Set 4: The Factored Form of a Parabola


Notes Set 5: The General Form for the Equation of a Parabola


Notes Set 6: Completing the Square


Notes Set 7: Completing the Square On the General Form


Notes Set 8: Trigonometry


Notes Set 9: Sinusoidal Graphs


Notes Set 10: Deriving the Quadratic Formula


Notes Set 11: Imaginary Numbers and Working with Quadratic Roots


Notes Set 12: Working with Quadratic Roots


Notes Set 13: Analytic Geometry



Notes Set 14: My Analytic Geometry Toolkit


Notes Set 15: Systems of Linear Equations



Notes Set 16: Circle Geometry


Notes Set 17: The Anatomy of a Circle


Notes Set 18: Circle Theorems


Notes Set 19: Logic

Venn Diagrams

Venn Diagrams consist of a Universe, which is the set of all objects being considered
Empty sets have nothing in it
Null set is another way of saying empty set
Numbers, people and things are examples of sets in a Venn Diagram

Definition for Union is when you gather all the objects in two or more given sets
Intersections are objects that are common to two or more given sets
A smaller collection of objects taken from a given set is called a subset
Gathering every object outside a given set is a compliment of that set
Remember "\" means exclusion, so that means you gather everything in a given set EXCLUDING other objects within another given set
All sets within a Venn Diagram are also their own subsets
Make sure you draw a Universe for all of your Venn Diagrams, it's very important!

Logical Acrostics

Here's the new set of acrostics for you.

Blogging Prompt
Your task is to create an acrostic "poem" that demonstrates an understanding of logic related to any one of these concepts:

REASONING
CONDITIONAL
COUNTEREXAMPLE
VENN DIAGRAM
DEDUCTION
INDUCTION

As an extra challenge (worth an additional bonus mark) try to make a Double Acrostic, that is, each line should begin and end with a letter of the word you are working with.

Remember, this is a bit of a race. Your answers have to be posted to the blog in the comments to this post. If someone has already used a word or phrase in their acrostic you cannot use the same word or phrase. i.e. It gets harder to do the longer you wait. ;-)

Here is an example of an acrostic that Mrs. Armstrong wrote:

Always in 2 dimensions
Region between the boundaries
Entire surface is calculated
Answer is in units2

Be creative and have fun with this!!

SCRIBE

I guess we did not do much in class today. Mr. K. Started up by giving us Consumer Mathematics Student Reference Book 2 and 3 other sheets which are Unit Price And Exchange Rate Assignment, Budgeting Assignment, and Budgeting Form. We are suppose to be working on this and hand-in the Assignment on Thursday. Then he asked us if we had question on Friday's assignment but then we were good. So Friday's assignment was suppose to be hand-in today but then he gave us an extra day, so that assignment is due tomorrow which is Tuesday. Yah and then he asked us something about grade 8, if we still remembered anything from grade 8. Mrs. K. Was there and they argued about this and I think Mr. K. Happen to remember most of things he did in grade 8 and I thought that was weird. And he explained the New Assignment which is to be hand-in on Thursday. And he told us a story about the bank which he said that banks round too much and we should REMEMBER THAT FRACTION ARE OUR FRIENDS. He told us that we should look at the numbers first which I did not get it. That was when he was explaining the second assignment. The last thing is that we wrote notes on "Conditional Statements" but the problem is that I don't how post it because of the table, so if you know how holla at me and I will try fixing it.

AND THE NEXT SCRIBE IS elizabethh

Sunday, December 18, 2005

Sunday 3x the Funday

A triple header this weekend.

Mr. Zhong Kui will make you laugh. I think his "problems" are the easiest ones to solve.

Rat is another "escape" puzzle. Every time you do something wrong he squeaks.

No. 5 is a set of three puzzle/adventures to get a little boy out of trouble.

Have Fun!

Friday, December 16, 2005

scribe post

today in class, we have a substitute teacher cause mr. k have a meeting with other math teachers. we started a new unit, Consumer Math. this unit will end as soon as we finish reading the "consumer mathematics book 1 & 2" and do the assignments about it. there is nothing much to talk about it because the hand-out have everything on it to describe how things work. about the 1st question in income and wages assignment, just remember that it works almost as same as the graduated commission. the only difference is that instead of percentage, rate per unit is given. income wage assignment and property taxes assignment are to be handed in on monday.

the next scribe is... Abdi.

Thursday, December 15, 2005

Scribe for Today

So the scribe today is me! Well today in class, we had a surprise quiz...as usual :)

The first question was:

1) Which of the following examples of reasoning are INDUCTIVE and which are DEDUCTIVE?

Every time we have a club meeting, I have a test in school the next day.
- INDUCTIVE
REVIEW * Inductive is when we observe several particular examples of something and then conjecture that it must always be the case.
Susan’s father brought her early to school each day. She noticed that Ms. Taylor, her math teacher, arrived at 7:30 each day for several weeks. Susan said, “Ms. Taylor always arrives at 7:30.”
- INDUCTIVE
All students in senior high must enroll in physical education. John is a student, so he concludes that he will take physical education.
- DEDUCTIVE
REVIEW * Deductive is when we argue from basic (inarguable) truths (like definitions) to a valid conclusion.
The sun has risen each morning from time immemorial. We can be certain it will rise tomorrow.
- INDUCTIVE
e. Anyone who likes to play football likes to play basketball. Sheeva likes to play football. We conclude that she likes to play basketball.
- DEDUCTIVE
f. Triangle ABC is an equilateral triangle. We can conclude that AB is congruent to AC.
- DEDUCTIVE
g. Joe counted the number of cars of different colors that passed his house in 15 minutes. More than half the cars were white. He decided that white is the most popular colors for cars.
- INDUCTIVE

The second question was:

2) Which of the above conclusions are VALID and which are INVALID?

A. INVALID
REVIEW * Invalid is when its conclusion is NOT a direct consequence of its premises---even if its premises (basic truths) and conclusion (the result of an argument) are true.
B. INVALID
C. VALID
REVIEW * Valid is when its result is a direct consequence of its premises.
D. INVALID
E. VALID
F. VALID
G. INVALID


After the quiz on logic, Mr. K gave us a Venn Diagram Problems sheet to work on the first problem which was called Music Survey and the problem was:

Recently several students at DMCI were questioned about their favorite music groups, with the following results:

- 22 like Hole
- 25 like U2
- 39 like Third Eye Blind
- 9 like U2 and Hole
- 17 like Hole and Third Eye Blind
- 20 like U2 and Third Eye Blind
- 6 like all three
- 4 like none of these performers

(a) How many students were questioned?


Also in class, Mr. K told us to write down the meaning of Counter Example on our dictionary. For those of you who missed the class today, here it is.

COUNTER EXAMPLES: given any logical argument of mathematical theorem, if you can find only one case where it is NOT true, then the entire argument of theorem is false and must be discarded---this is called a counter example.


Mr. K told us in class today about the “Fermat Theorem” which was really interesting and I even looked up on it on the internet to get more information about the Fermat Theorem. Here it is:

Pierre de Fermat died in 1665. Fermat is the most famous number theorist who ever lived. He was in fact a lawyer and an amateur mathematician. The fact that he published only one mathematical paper in his life and that was an anonymous article written as an appendix to a colleague's book. Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus' s Arithmetica.
Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.
Format wrote: “I have discovered a truly remarkable proof which this margin is too small to contain.” Fermat realized that his remarkable proof was wrong when he first studied Diophantus' s Arithmetica around 1630, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.
In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z withx2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat's theorem.
It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only. For if there were integers x, y, z with xn + yn = zn then if n = pq, (xq)p + (yq)p = (zq)p.
Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. Euler's mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form p2 + 3q2 and Euler shows that, for any a, b if we put p = a3 - 9ab2, q = 3(a2b - b3) then p2 + 3q2 = (a2 + 3b2)3. The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xn + yn = zn implies that one of x, y, z is divisible by n. Hence Fermat's Last Theorem splits into two cases.
Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible by n.
Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.
Case 2(i) was proved by Dirichlet and presented to the Paris Académie des Science in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 were published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).
In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lamé in 1839.He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers. Lamé acknowledged that the idea was suggested to him by Liouville. However Liouville addressed the meeting after Lamé and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true.
Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument He is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).]
Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.
Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA.
This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states
The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.
In March 1994 Faltings, writing in Scientific American, said
If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.
Weil, also in Scientific American, wrote
I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.
In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work.
Wiles worked on it for about two weeks, and then suddenly inspiration struck.
In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.
On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof. No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.


Click for the full information on Fermat Theorem

NEXT SCRIBE IS ALDRIDGE :)

SCRIBE.

late post? yes, i know... IM REALLY SORRY =(

hmm what happened in class yesterday?
okay,when we got to class Mr. K asked us to draw a BIG circle on a piece of paper. Before we started our activity he mentioned that "Math is science of patterns".

Then Mr. K started off by asking us... "If we connect two points (along the circumference of the circle) how many regions will the circle be divided into ?"

We all know that if we connect two points then it will be divided into 2 regions.

"how about 3 points connected all together?"
he drew on the board three points along the circumference of a circle then counted the number of regions. 4 regions.

and then he asked us to choose 4 points along the circumference of the circle, connect it and count how many regions the circle was divided into. 8 regions.

then we made a chart that looked like this:

looking at the table/chart, is there a pattern? yes there is..

what is the pattern? the number of regions goes up by multiplying 2 to it.

then Mr. K asked us to guess what the next number would be.

"16" the students answered...

is that correct?

well, we have to check if it is...so we drew 5 points along the circumference of the circle and counted the number of regions.

16! we were right !

Then Mr. K asked if the number of points was "n", can we come up with a formula for it?

one of the students said " if n represents the number of points then the number of regions will be 2n-1.

why "n-1"?

n-1 because the exponent of the number of regions is always one less than the number of points.

Then Mr. K asked what if we have 6 points connected all together? how many regions will the circle be divided into?

using the formula : 26-1 or 25 we got 32.

and just like before we have to check... so we drew a circle with 6 points connected to each other then counted the regions.


31 regions ?! only 31. That means that the formula we came up with is not true and must be discarded.(some may have gotten only 30 regions. why? maybe you just missed counting a little region or you drew your circle too many) notice that there`s a little triangle region in the middle of the above diagram? if you drew your circle big enough then it should show.

Therefore 6 is the counter example for the formula 2n-1.



coun·ter·ex·am·ple;

-an example that disproves a hypothesis, proposition and/or theorem.

"PRIME PRODUCING" POLYNOMIAL

F(x)=x2+x+41

is this really a prime producing formula? let`s check it out.

F(1) = 43 prime

F(2) = 47 prime

F(3) = 53 prime

F(4) = 61 prime

F(5) = 71 prime

okay, so far all the result are prime numbers but is it right to conclude that f(x) will be prime for ALL intergers?

answer: NO

why?: because if you try to solve it with x = 40

F(4o)=4o2+4o+41

F(40)=40(40+1)+41

F(40)=1681 which is also 412

MARIN MERSENNE.


(picture taken from:http://shl.stanford.edu/Eyes/kircher/mersenne.html)

Marin Mersenne is a french theologian, natural philosopher, and mathematician who tried to find a formula that would represent all primes but didn't succeed. Although he failed, his work on numbers of the form
2p - 1

p prime


is still of interest in the investigation of large primes.

He's name is best remembered today for Mersenne Prime.

Mersanne Prime: are primes of the form 2p-1.

For more about Mersanne Primes: *click_here* or here

(it includes a table of known Mersenne Primes, who discovered it and when)

So far they have only found 42? mersenne primes.

Wanna become world famous? or even win some cash? You can participate in what they call GIMPS. Great Internet Mersenne Prime Search. They have already found seven of them on GIMPS.

If you're interested you can CLICK_HERE. It tells you how it works, how long it will take, how much you can win, what you need to have, EVERYTHING you need to know. :)

for more about Marin Mersanne clickhere

VENN DIAGRAMS

Mr. K also posted up some questions on the board.

1.) In a group of students 12 are taking chemistry, 10 are taking physics, 3 are taking both and 5 are taking neither. How many students are in the group?

ANSWER: 24 students are in the group

2.)In a third-rate rock band, 3 people play guitar, 4 sing, 2 do both and 6 have no talent for singing or guitar so they do something else. How many are in the band?

ANSWER:There are 11 people in the band.

3.)There are 64 kids in the "Tiny Little Cherubs"(TLC) Daycare. At lunch 59 ate green beans, 56 cauliflower, 60 ate brocolli, 55 ate green beans and cauliflower, 54 cauliflower and brocolli, 56 green beans and brocolli and 53 ate all. How many ate none?

ANSWER: One kid ate none.

OKAY... maybe i exaggerated when i said it wasnt half way yet -_-". lol. hhm, thats all i could remember...so i guess this is it.

THE EEND :)

sorry again for being late!
*if i missed anything please let me know*

oh and i found a cool site about prime puzzles and problem connected. I don't have a delicious account so i guess ill just link it in here.

primepuzzles

There's a new puzzle every saturday and the solutions will be up one week later.



oh... the scribe today was jamilynG. check out her scribe post for tomorrow's scribe.

Tuesday, December 13, 2005

Venn diagrams

Today we discuss about Venn Diagrams and how they were originally called Eulers circles. Then we started talking about how 6 is the first perfect number. Then we talked about how Euler had 11 children. But Sadly only 5 survive because of all the deceases back then. So anyways we started to take notes about sets.

set: a collection of objects may be things, people, numbers, etc....

subset: a "smaller" collection of objects taken from a given set.
universe: the set of all objects being considered.
null set: (A.K.A. the empty set) the empty set (it's got nothing in it) is a subset of every set. It is written as {O}.

operation on sets:
union (U) aUb means gather all the objects in set A with all the objects in set B.
Union(U) means "or".





Compliment: (A') means everything that is outside set A
compliment (A') means "not".


exclusion(/) a/b means everything that is in set A excluding those things that are also in set B
exclusion(/) means excluding.


Intersection A (upside down U) B means gather all the objects that are in both set A and B
intersection A(upside down U) B


After we wrote in our dictionaries we had a worksheet that we worked on. Mr. K. gave us the answers for #1. By the way the answer to the second worksheet for #1 e) is wrong it should be (e,u) not (a).

The next scribe is............................


LATE POST SORRY!

Alright well i know its like 2 in the morning sorry you guys i had work today till 12 and got home around 12:30 i just want to apologize again but dont worry i'll let you guys know who the scribe is for tomorrow...well technically for today haha!

Well today Mr. K started class off with asking us what we thought average meant and what an average person would be like, and which one of us believed that we were average...and of course nobody raised their arm except for JENNIE haha its okay though because it was funny =) Oh yea another thing mr. k asked us during class was can you picture a triangle with angles but no lines? what a good question dont you think haha, and after that we discussed about inductive and deductive reasoning, and we also talked a bit and wrote in our dictionary, where we wrote and learned some new words...here's what we wrote;

logic: the branch of mathematics (and philosophy) that deals with sond reasoning.

agreement: a logical arguement consists of two (2) parts:
  1. a set of premisis
  2. a conclusion

premise: a basic truth

conclusion: the result of an argument

valid argument: an argument is "valid" when it's result is a direct consequence of it's premisis

example:

all men are mortal (premisis)
mr. k is a man (premesis)
therefore mr. k is mortal (conclusion)

An argument is not valid when it's conclusion is not a diret consequence of it's premisis even if its premisis and conclusion are true.

example:

all men are mortal (premisis)
Mr. k is a man (premisis)
therefore mr. k wears glasses (conclusion)

sound argument: an argument is "sound" if it is both valid and true.

induction: when we observe several particular examples of same things, and then conjecture that the "discovered" pattern must always be true.

example: all our "investigaion" activities in the circle geometry unit

deduction: when we are form a certain inarguble basic truths (like definitions) to a valid conclusion.

example: thales proof that every inscribed angle subtended by a diameter is 90 degrees.


Alright after we finished writing and discussing the different words and reasoning, Mr. k gave us some class time to do exercise 45, and that's how we ended our class i know this post isn't as long and im guessing it's cause we didnt' do as much today in class, but sorry again that this post isnt as great and detailed as my other ones, and sorry im trying my hardest right now im just really tired from work but dont worry i'll fix this up tomorrow night alright.

Till then the scribe for tomorrow is...i'll let you guys know in class haha

Sunday, December 11, 2005

blogging before the test

wow, almost forgot about this. well this unit wasn't as hard as i thought it was and i was pretty shocked on how much better i did on the pop quizzes. Somethings do still confuse me like the statement and reason part. Also knowing which theorem to use and how you have to find one part of an answer to figure out another part. There's so many steps to do and theres a big possibility to make one little mistake. I'm lucky enough to do my test on monday since i was excused on friday. I need to get studying, implant those theorems into my head =)

Sunday Jumping Funday!



'The goal of the puzzle is to switch the the pegs on the left with the pegs on the right by moving one peg at a time.

Move pegs by clicking and dragging them to open slots. A peg may only be moved to an open slot directly in front of it or by jumping over a peg to an open slot on the other side of it. You may not move backwards. The game ends when you win or get stuck.'

Play the game here. Can you win the 8 peg game? ;-)

(Thanks again to Think Again!)

Friday, December 09, 2005

blogging before the test

this unit, circle geometry, is interesting to do but i often got a hard time doing the statement and reason part. doing a question is quite complicated because there's a lot of possible different ways to solve a question but there are also lots of theorems to be consider in answering a question. it seems that i'm just the fourth one to post a blog and everyone may be busy doing the assignment.
just don't forget to blog, and goodluck on the test.

Thursday, December 08, 2005

blogging before test once again

well good thing i remembered to blog almost forgot, been too busy lately and yeah sorry i didnt go to class today haha and yea i know im the scribe for monday, but yeah this unit we just finished which was circle geometry i found was kind of confusing. i feel pretty good with a lot of the things in this unit but sometimes i just dont know how to explain myself when it comes to doing the statement and reason, im guessing that i should practice some more and hoping i do good on this test, since i've been doing pretty bad but yeah im trying to study haha anyways good luck to you all tomorrow...

Blog. Test tomarrow... Blog

Well I guess it's me who nearly forgot to blog. Hahaha. So lazy these days... So anyways. Well Let's see. This unit is... Confusing. Well sorta. I get it and all that but you know it's all good. Yeah. Well I know most of the concepts so I'll be fine. Becides this week has been so crazy. With the math Assigment and all that. Wooo Nearly forgot. Eh Robert. Hahaha. Well I hope that I do good on the test. Good luck to all of you too. Hahahaha. Well I got nothing else to do. Maybe some studying would do me good... Anybody up for an all nighter on the chat box. lol.

Blogging Before The Test

Hello once again! Wow this unit is a tough one! At first it seemed like Circle Geometry was going to be easy. Now that it’s coming to an end, I have realized that there is so much to it. There are many theorems that you have to remember and also how to use them. It suddenly becomes more difficult with the different problems that are challenging. Good luck to all and don’t forget to do your best.

Congruent

C hords that are congruent are equidistant from the center of the circle.
O pposite angles are congruent.
N o two tangents drawn from a common point exterior to a circle is different. They are always congruent.
G eometric term for equal.
R egarding the parallel chord theorems, the arcs between parallel chords are congruent.
U nanimous that congruent chords subtend congruent arcs.
E qual in shape or size.
N o two or more inscribed angles subtended by congruent arcs are different. They are always congruent.
T wo or more inscribe angles subtended by the same arc are congruent.

Chapter: Pre-Test, Scribe: Mr. K Late =/

Class started pretty strange, with no questions on the board. Except today we got to wait about 20 minutes for Mr. K to get into class. Side note: Mr. K, if you expect us to follow your guidelines to pass this class I think we ALL should follow them especially the BEING IN CLASS ON TIME thing ? Lol i'm not ranting on you I just thought it would be fun =D.

Today's class despite the wait, happened with a pre-test, OMG test tomorrow. Yeah that kind of thing lol, and here's the questions from the pre
test.


Question 1: The segment AB is tangent to the circle at point T. Which angle is qual to r ?

Choices: a) a b) b c) c d) d e) none of the these
Answer: a) a
Why you ask ? It's because of the tangent chord therom.





Question 2: Determine the value of x.

Choices: a) 50° b) 80° c) 90° d) 100° 3) none of these
Answer: b) 80°
Why you ask ? It's because of the central inscribed therom.






Question 3: Given the circle with centre O, which of the following best describes the relationship between x and y ?

Choices: a) x = y b) x + y = 180° c) x + y = 360° d) x + 2y = 360° e) none of these
Answer: d) x + 3y = 360°
Why you ask ? It's because of the central inscribed therom.






Question 4: In the diagram below, O is the centre of the circle. Determine the measure of angle x, in degrees.

Choices: a) 30° b) 40° c) 50° d) 60° e) none of these
Answer: c) 50°
Why you ask ? It's because of the central inscribed therom, but not really the therom it's better explained with the SPECIAL CASE OF THALES.






Question 5: In the diagram to the left, R, S, T and U are points on the circle with RS = RU = UT and SI = ST. Find the measure of LUST.

Choices: a) 30° b) 35° c) 36° d) 40° 3) none of these
Answer: c) 36°
Why you ask ? It's because triangle UST and RSU are both isoceles trianges LSUT and LSTU are both congruent and same goes with LRUS and LRSU. Which gives us something like this:






180 - 2b + a = 180
a = 2b

and

a + a + b = 180
2a+b = 180

With these two equations we can use substitution and like magic you get a number from a question that had no number. Ok Ok i'll do it for you guys don't have to get mad.

2 (2b) + b = 180
4b + b = 180
5b = 180
b = 36

And that's how question 5 was accomplishes have fun with it boys and girls NAR HAR HAR!



Question 6: Determine the measure of LECB, LBDC, LBAD and LDBE, where E is the centre of the circle. Justify your answers.

Answer:
Statment - Reason
EB = EC - Radius
LECB = 40° - Triangle ECB is isoceles
LBAD = 110° - Supplementary to LEBC
LBDC = 50° - Central inscribed therom
LDBE = 20° - Sum of triangle BCD




That was like whoa haha. Anyway that's what we did today in class hard questions ? Somewhat depending on if you know your theroms. Test tomorrow ? Shouldn't too much trouble why ? because EVERYONE should KNOW their theroms lol.

And guess what monday's scribe is PAMELA, haha see what happens when you miss class =P

Wednesday, December 07, 2005

My Acrostic (tangent)



Tangent is a straight line that touches a circle but does not cut it.
Any tangent line to a circle is perpendicular to the radius drawn to the point of tangency.
Not only a line but it is also a trig function.
GA (line) is a tangent. Therefore, the angle between tangent GA and chord BC (angle1) is congruent to the inscribed angle subtended by the opposite side of the chord(angle 2).
EG (line) and GB(line) are congruent because they are drawn from the same point.
Note that in the case of a circle, the tangent line will touch it ONLY at a single point.
Tangent and radius create a 90 degree angle when they intersect.


.

Tuesday, December 06, 2005

All Together Now!


OK folks, just like we did in class today. Everybody help each other and and solve this one in the comments to this post -- it's good practice for your test. ;-)

Scribe Scribles

Today in class, Mr. K. put us into groups to work on some problem solving sheets for circles. Sorry if some of this is wrong, feel free to correct me in the comments. (=

Problem Solving Circle 1:













Given:
- DC and DE are tangents to the circle at points C and E respectively
- F is the centre
- AE is a diameter


Prove: L1 = L2












This picture shows:
- Tangent-radius theorem
- ΔCFE is isosceles
- ΔCDE is isosceles

a + b = 90°
L1 = 2b
2a + L2 = 180°
a = 90° - b

2(90° - b) + L2 = 180°
180° -2b + L2 = 180°
-2b + L2 = 0
L2 = 2b


Problem Solving Circle 2:











Given:
- AE is tangent to the circle at A
- CE is perpendicular to AE
- CD is perpendicular to AB
- AC = BC

Prove:
- CE = CD












This picture shows:
- L2 = L6 tangent-chord theorem
- ΔABC is isosceles, L3 = L6
- ΔADC = ΔAEC by AAS
- DC is congruent to EC
Q.E.D. (added that in just to make Mr. K smile)

Problem Solving Circle 3:












Given:
- AB is a tangent
- BD = CD
- AB = BC

Prove:
- L3 = L6












This picture shows:

- ΔBCD is isosceles
- L4 = L5 + L7
- L2 = L5 + L7, by tangent-chord theorem
- L4 = L2
- ΔABC is isosceles
- L1 + L5
- L3 + L6

*Bell rings, and we're out of time folks!

The next scribe is... abriel_S. (=

Monday, December 05, 2005

scribin' at the last minute...

* sorry guys i did this at the last possible time i was suppose to work until 10:00pm ended up working until 11:00pm.

- Well, today we did a quiz on our circle geometry. Here are the questions on the quiz so you can try it again or for the first time if you were not there. In the end I'll post up the answers.

.............................................................

Circle Geometry Quiz 2

(1) angle BOC = 115'. Find the measures of arcs AB, BC, CD, and AD. Justify your solution (4 marks)

(2) If the measure of chord AB is 14cm and the measure and the measure of chord DC is 7 cm, how far from the centre of the circle is chord DC? Justify your solution. (4 marks)


(3) How far is a chord of length 8 cm from the centre of a circle with a diameter of 10cm? (2 marks)

(4) What is the diameter of a circle in which a chord 16 cm long is 15 cm from the centre? (3 marks)

..........................................................................................

ANSWERS:* i wish i could've made it into columns

(2) STATEMENT / REASON

AB = 14 cm / GIVEN
DC = 7 cm / GIVEN
OB = 7 cm & AO = 7 cm / RADII, AND AB DIAMETER
DE = 3.5 cm & EC = 3.5 cm / PERPENDICULAR BISECTOR THEOREM

OC = 7 cm / Radius
OE = 6.06 cm / PYTHAGOREAN THEOREM

a^2 + b^2 = c^2

3.5^2 + b^2 = 7^2
12.25 + b^2 = 49
49 - 12.25 = b^2
b^2 = 36.75
b = 6.06


(3)
PYTHAGOREAN THEOREM
a^2 + b^2 = c^2
4^2 + b^2 = 5^2
16 + b^2 = 25
25 - 16 = b^2
b^2 = 9
b = 3


The chord is 3 cm away from the centre of the circle.

(4) PYTHAGOREAN THEOREM
a^2 + b^2 = c^2
15^2 + 8^2 = c^2
c^2 = 289
c = 17
diameter = 17 x 2 = 34 cm

The diameter of the circle is 34 cm.

............................................................

* PHEW.... sorry again guys well next scribe will be....

rannell d.

hah =) have fun rannell.

Sunday, December 04, 2005

YOUR AGE BY CHOCOLATE MATH

Hey guys, I thought this was kinda cool so u should try it. all u have to do is follow this steps and u will see what happens (u r going to your age), and u might need a calculator.

1. First of all, pick the number of times a week that you would like to have chocolate (more than once but less than 10)
2. Multiply this number by 2 (just to be bold)
3. Add 5
4. Multiply it by 50
5. If you have already had your birthday this year add 1755, If you haven't, add 1754.
6. Now subtract the 4 digit year that you were born.
­_ You should have a three digit number.
_ The first digit of this was your original number (i.e., how many times you want to have chocolate each week).
_ The next two numbers are YOUR AGE! (Oh YES, it is!!!!!)
THIS IS THE ONLY YEAR (2005) IT WILL EVER WORK.

Saturday, December 03, 2005

Cubeoban Sunday



The objective of Cubeoban is to push/pull all the blocks to their corresponding lights. Do this by clicking on the blocks and drag them in the direction you want to push them. Play it here.

Level 1 was so easy that even I could do it. Level 2 (the image), started my thinking.

(Thanks again to Think Again!)

Friday, December 02, 2005

Straight From The Pen of a Scribe....

Well it's my turn for scribe again. Well I guess I should start with the beginning of class, which was a surprise quiz!!! How exciting! So here is the questions on the quiz and how they are to be figured out correctly, with a few helpful hints.
The first question:

In this question you obviously had to solve for x and y and use statement and reason to explain what you did.

Well first you should have been able to figure out that angle DEF= 70 degrees(red) because it is the exterior angle to triangle ADE (blue).





The two black dots indicate that those two angles are congruent. Creating a total angle that equals 140 degrees (yellow). Using that information you then can find the value of X. X=100 degrees because angle CEF is exterior to triangle ACE.





From this information we can derive that Y= 60 degrees because angle ACE (red) is exterior to triangle ABC (blue).

And that is pretty much question one for you. Question two was were you had to find two congruent triangles and then prove it using statement and reason. For this question, instead of demonstrating the diagram I will show you how to do the proper Statement and Reason set up. For this specific question there were two different ways of accomplishing the correct answer.The first is in blue and the second is in red.


The first way is: Statement Reason
angle AFD is congruent to Opposite angles
angle EFB.
angle FAD si congruent to Given
angle FEB
Line AD is congruent to Given
line EB
Triangle FAD is congruent to
triangleFEB AAS


The second way is this: Any of this look familiar. *Looks at previous post*
Line AC is congruent to
line EC Given
Angle ACB is congruent to
angle ECB Reflexive property
Line CB is congruent to line CD Given
Triangle ABC is congruent to
triangle EDC SAS

The third and last question goes like so:


For this question we were given that the diameter is 20 cm and that the chord AB is 12 cm. We were then asked to figure out the distant between the chord and the centre of the circle. Knowing that the diameter bisects the chord we know that each half of the chord is 6 cm. We also know that the radius is 10 cm. You then construct a radi to one of the chords points creating a right triangle. You are now given two of the sides and use the pythagorean to figure the third side out which comes to 8 cm.
*Don't forget that the hypotenuse is always C. Many people forget that small point and just add the two sides given (don't worry I did it myself today). Also don't forget to include the units.*

Last but not least we put more work into our dictionaries! YAY!!!! *Always read before bed*
Congruent Chord Theorem
Congruent Chords subtend congruent arcs. Congruent chords are equidistant from the centre of a circle.
The Tangent Theorem
Two tangents drawn from a common point, exterior to a circle, are congruent.
Tangent-Radius Theorem
A tangent and radius intersect at a right angle.
Tangent Chord Theorem
The angle between a tangent and a chord is congruent to the inscribed angle subtended by the opposite side of the chord.
Cyclic Quadrilaterals
A cyclic quadrilateral, all of whose verticies lie on the circumference of a circle. Opposite angles in a cyclic quadrilateral are supplementary. a+c=180 b+d=180
Interior Angle Sum of in Any Polygon
The sum of the interior angles in any polygon given by S=180(n-2)
S is the sum of all the interior angles.
n is the number of sides in the polygon.
Example: What is the sum of the interior angles in a 12 sided polygon?
S=180(n-2)
=180(12-2)
=180(10)
=1800
Alright that is really it guys. And the next scribe is......







My Acrostic

Circumscribed surface or space of plane curve everywhere equidistantfrom a given fixed point is a circle.
If only (part of) a circle is known, then the circle's center can be constructed as follows: take two non-parallel chords, construct perpendicular lines on their midpoints, and find the intersection point of those lines.
Ratio between the length of an arc and the radius defines the angle between the two radii in radians.
Circumference means the length of the circle, and the interior of the circle is called a disk or disc.
Line cutting a circle in two places is called a secant.
Every triangle gives rise to several circles: its circumcircle containing all three vertices, its incircle lying inside the triangle and touching all three sides, the three excircles lying outside the triangle and touching one side and the extensions of the other two, and its nine point circle which contains various important points of the triangle.