Wednesday, February 28, 2007

Probability: using the "choose" formula

Today in class we learned how solve probability questions using Combinations or the "choose" formula. Combinations (the "choose" formula): A combination of an arrangement of objects, where there is no particular order.

The combinations formula is:


n is the number of objects available to be arranged, r is the number of objects that are being arranged.


During our morning class, we started off with this question just as a warm up to what we learned during yesterday's class.

If 8 books are arranged on a shelf, what is the probability that 3 particular books are together.

What we had to find the probability that 3 particular books would be together when we arranged them in different ways.
P(E) = # of favourable outcomes
________Total # of outcomes


The total number of outcomes would be 8! because there are 8 books that are being arranged, that becomes your denominator. Then the number of favourable outcomes would be 6! and 3!. Mr. K started telling us that if we put the 3 particular books that we want to gether in a bag, it would count as 1 book. When we take 3 away from 8 you get 5, but since those 3 books count as one book you add the the 5 and 1, to get the 6! and then 3! would represent the 3 books you want together.

= 6*5*4*3*2*1*3*2*1
__8*7*6*5*4*3*2*1

The purple numbers: The 8 reduces to 4 because of the 2 and the 2 becomes a 1.
The grey numbers: The 6! cancels eachother out.

= 3/28 = 0.1071 or 10.71%

After that, Mr. K explained what the "choose" formula was and how we could do it on the calculator. Then we started to do afew more questions (which did not relate much to combinations), such as these:

a) How many numbers of 5 different digits each can be formed from the digits 0, 1, 2, 3, 4, 5, 6?

6*6*5*4*3 = 2160

The first 6 because, there are 6 numbers that could possibly become the first number minus 0 because then it would be a 4-digit number instead of a 5-digit number. The second 6 because you cannot repeat numbers, so you lose a number because you've already used one which puts you at 5 but you also gain a number because of the 0 that you haven't used yet. The 5, 4 and 3 are consecutive numbers that follow after losing one more number because of the no repeating number restriction.

b) If one of these numbers is randomly selected, what is the probability it is even?

_ _ _ _ 4
______________^ 0, 2, 4, 6

In the first slot, there can't be a 0 so there would be 6 other possible numbers that could fill in the slot. But then if one of the even numbers were used, then you would only have 5 numbers to choose from but if you used 0 as your last digit then you'd have 6 to choose from again because you gain a digit because you used the 0, which cannot be the first digit.

So, what we did was we split the equation and made it two equations.

First Equation: 6*5*4*3*1 = 360
Second Equation: 5*5*4*3*3 = 900
P(EVEN): 360 + 900 = 1260

Then you take the 1260 and divide it by how many different 5-digit numbers that could be formed. (As done in the part a).

1260/2160 = 63/108 = 7/12 = .5833 or 58.3%

c) What is the probability it is divisible by 5?

6*5*4*3*1 = 360
5*5*4*3*1 = 300
P(DIV. By 5): 360 + 300 = 660

Then like the part b, you put 660 over 2160 to get the probability of the numbers that are divisible by 5.

660/2160 = 11/36 = .31 or 31%

After doing these questions, we finally started to do questions that had much relation to what we started learning about the "choose formula. We started off with this question:

There are 10 football teams in a certain conference. How many games must be played if each team is to play every other team just once?

Most of us struggled with this question, but after a long agonizing session of figuring out the correct answer or just figuring out an answer Mr. K showed us two ways of doing this question. He showed us how to do it the "long" way by making a chart, of course we decided that it was just too much work and a big waste of time to show a 10x10 chart so we cut it down to a 4x4 chart. We chose to make the teams A, B, C & D.


The first diagram shows how we set up the chart. The second diagram shows us that Team A cannot play their own team, which is shown by the "X" but can play Team B, C and D, which is shown by the check mark. So, so far there has been 3 games played, Team A vs. Teams B, C and D.


The diagram on the far left shows that in the Team A row, there are X's because in column A, it already shows that A has already played Teams B, C and D. The diagram on the right shows that Team B cannot play itself, but can play against Teams C and D and has already played Team A, which is marked with an "X."


The diagram on the far left shows that Team C cannot play itself, but can play Team D and already has played Team A and B. The last diagram shows that Team D cannot play itself and already has played Teams A, B and C. Which completes the table, now we can determine the number of games that were played.

Looking at the last diagram you can determine the number of games played, in column 1 there are 3 check marks, in column 2 there are 2 check marks, in column 3 there is 1 check mark and in column 4 there are no check marks.

3 + 2 + 1 = 6 games were played.

We applied what we did in the above example to the question at hand and came up with a pattern and solved the question without drawing a chart.

9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45

That was the "long" way of solving the question. Using a chart is all fun and games, but is there a shorter way of finding the solution to a problem like this? Apparently so. So, here is how we solved this problem in the shortest way possible. This is the equation that we came up with in order to solve the problem.


The sub 10 represents the number of teams and the sub 2 represents the number of teams you need to play a game.




= 10*9*8*7*6*5*4*3*2*1
__8*7*6*5*4*3*2*1*2*1

The green numbers: The 10 reduces to 5 because of 2 and 2 becomes a 1.
The grey numbers: The 8! cancels eachother out.

And the answer is 45.

Then we got our final question of the day, which was:

Seven people reach a fork in a road. In how many ways can they continue their walk so that 4 go one way and 3 the other?

This is the equation that we came up with in order to solve the problem.


The sub 7 represents the number of people and the sub 4 represents the number of people that go one way.



= 7*6*5*4*3*2*1
__3*2*1*4*3*2*1

The orange numbers: The 6 reduces to 1 because of 3*2 and 3*2 becomes a 1.
The grey numbers: The 4! cancels eachother out.

And the answer is 35.

And so this concludes my scribe. Sorry guys, it's super long, i didn't really know what to scribe about so I just put up questions that we did in class. I hope it's not too confusing, there's a lot of stuff going on in my scribe. I put a bunch of "_'s" so that certain things align, but you can't really see them unless you highlight. But anyway, tomorrow's scribe is... *drum roll... Brittany =)

Today's Slides and Homework: February 28

Here they are ...





To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

And here is tonight's homework ...



... and the answers ...

Tuesday, February 27, 2007

Using Factorial Notations and Permutations










Today we went over on how to use factorial notations and permutations. So if anyone else missed the class here it is..



Factorial Notations
We learned that the symbol "!" can be used to multiply a number down to 1. So if we want to multiply 6 * 5 * 4 * 3 * 2 * 1 = 720, we can simply use "!" to make it easier to multiply. So it would look like this: 6!.



Examples for using factorial notations
So this is what we do:





So in the calculator it should look like this:



10!/(7!3!)*(1/2)^7*(1/2)^3



(and make sure that you put brackets in or it will make no sense at all)



When we equal it, it becomes 0.117 which is 11.7%.

Using Permutations
We use permutations to see how many ways can object be arranged and to do that we use the formula:



nPr = n!/(n-r)!



n is the number of objects available to be arranged.
r is the number of objects that are being arranged.


Examples using permutations


If we had 10 students in our class and we wanted to find how many ways can we change seating plans, there would be 3628800 ways to do it. This is how i know:






But what if there were 5 extra seats? We would do that same thing we did but we'd switch the "n's" to 15 so theres 15 vacant seats. So it would look like this:



Today's Slides: February 27

Here they are ...





To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

Monday, February 26, 2007

Today's Slides: February 26

Here they are ...





To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

Scribe Proability

Today in class we talked about different problems that can be solved,and different rules that apply.

If the question doesn't talk about telephones, then the First digit can't be a 0.

Example:
How many four-digit even number are there if the same digit cannot be used twice?



If the question talks about telephones then the first digit can be a 0.

Example:
2) The last part of your telephone number contain four digit how many such four-digit number there


10 * 10 * 10 * 10
__ __ __ __ = 10000

When the questions deal with Letters,In the rules , it can't be repeated nomatter what.

Example:

How many ways can the letters of the word FERMAT be arranged?

6 *5 *4*3*2*1

_ _ _ _ _ _ = 720

it can be display as 6!

Another faster way would be:

This way is called "Pick" Formula

nPr = n!/(n-r)!

n is the number of objects available to be arranged

r is the number of objects that are being arranged
6P6= 6!/(6-6)!= 6!/0!=720

When the question has the same letters, you would have to put them to together , because they can't be switch around, its impossible.

Example:

BOOK

4!/2! = 12

4!: is the numbers of letters.
2!: is the numbers of the same letters.

Remember when enter this question on the calculator you would have to be careful.

Example:

Putting this on calculator is correct:

11!/(2!4!4!)

Putting this on caluclator is incorrect:

11!/2!4!4!


Thursday, February 22, 2007

Today's Slides: February 22

Here they are ...





To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

Slide #3 is blank, it was the animated dice ... oops. ;-)

PROBABILITY

On Thursday, February 22, 2007, we started the class with learning the history of Probabilities.(refer to slides)

Things you MUST write in your MATH DICTIONARY. (refer to slides)
Terms you should know...
Theoretical Probability
Binomial Experiment & Probability

As the slides are a great source to refer to, here is one of the example that we have gone through...


In a family of 3, what is the probability that 2 of the children will be girls. Using a tree diagram we can determine the outcome by...

Also in class we covered, what sources we can go or use to generate random numbers to help figure out the probability. One of the sources we used was...

A. which represents the TOTAL number of people, items, objects, ect.
B. which represents one type of ________ you are trying to find out
C. which represents another type of _______ you are trying to find out
D. which represents the amount of ______ you are studying

in the case of the example above, we entered this...

120 which represents the TOTAL number of Kids
1 which represents: Boy
2 which represents: Girl
40 which represents the # of families

Another way of figuring the same information above, is by entering the information in your TI-83 calculator. http://www.stat.uconn.edu/~pepe/ti83binomial.html visit this page, it may help you understand computing binomial probability.

Mr. K always say that "MATHAMATICS IS SOMETHING TO SOMETHING PATTERN," and in saying that we also branched off on to Pathways & Pascal's Triangle and finding patterns. Paterns that was made were.. (can't really explain, need to get a clean blank triangle thing, ill get that at class.)

Congrats to...
TENNYSON
who will be the scribe for Friday. February 23, 2007

Wednesday, February 21, 2007

PROBABILITY

In todays class, we started our new unit Probability, but Mr.K was not there, so we had Mr. Harrison as our substitute teacher. We were told to read pages 4-9 of our book, and assigned to do page 10, #s 1-9.

Probability
> If an experiment has x equally likely outcomes of which y outcomes are favourable to event A, then the probability of event A is: P(A)= x/y.

Here is one example of questions we did on class (#8).
>On the game board shown below, you can move the game piece diagonally forward
.


















a) Determine the number of different pathways from the game piece to each of the squares on the other side of the board.


>There are 28 pathways to A.


>There are 34 pathways to B.


>There are 21 pathways to C.


>There are 6 pathways to D.





b) Assume that a player is equally likely to go left or right on each move.
What is the probability he will land on:
i) square A? ii) square C?

ANSWER:
Total number of pathways to exits=
28 + 34 + 21 + 6 = 89
i) 28/89 = .315 ii) 21/89 = .236


> The next scribe is just_in.... ~(_8(])

Monday, February 19, 2007

BOB

Our Matrix unit is almost over, and I feel confident going to our test tommorow. We started this unit by just adding and subtracting matrices, then we went on multiplying them. When we got to Transition Matrix, I didn't know what was going on because I was gone for three days and the topic was really confusing. But by doing some class works and group works I was able to catch up on our lesson.

Tommorow afternoon will be our test because we do not have class with mr.K in the morning.
Good luck on the test!

BOB

Oops i almost forgot to write my BOB.

I think the Matrix unit is not really a hard one, but it is tricky. The part where i had problems was the transition matrix. It's kinda confusing at the beginning, but the grouping thing really helped even though I like working alone, much. That's all, good luck on the test !

Sunday, February 18, 2007

BOB TENNY

Wow almost forgot about BOB ,would of lost a mark. Well i feel strong about this subject but their are tricky parts to it that you have to practice. But i like this Matrix subject, pretty cool, well im going back to studying later =P

BOB

Well we're at the end of our matrices unit. I was confused and lost at the beginning of the unit, but now it all makes sense. When we first started, I felt that adding and subtracting matrices was quite easy, then when we started to multiply matrices, and when we started to work with transitions matrices I was STUCK. But as our class worked together and asked questions, I started to get the hang of those transition matrices. And I LOVED how honest my classmates were when we didn't understand something.

Mr. K : Am I making sense to you guys? Or should I run that by you again?
Students: Nope!! We don't get it. Run that by us again.


Well now that I understand the things that we're doing I hope I get a good mark on this test. I really want to do well in this course. Okay. BYE BYE!!!



BOB

Well now that we have come to an end to this unit Matrices. When we first started this unit I was confused but I quickly caught on to adding, subtracting etc. But than we started "multiplying matrices" I was completely stuck, and the "fingering method", didn't help me out to much either. But then I started to ask some questions and bug my classmates , so I finally figured things out. With "transition Matrices" it confused me at the beginning, but then I finally got the nerve to ask Mr K for help and we found out where I was screwing up with those questions. I really want to do well in this course, and i really hope I get a good mark on the test... -Britt

Saturday, February 17, 2007

BOB

When we first started learning about matricies i was so confused. I didnt get how you had to multiply them with the rows and columns. But later i finally got it! What sparked that "moment of clarity" was Emannuel!! He helped me understand by writing what you had to multiply step by step. and thats how i finally understood matricies. I think i progressed alot since we started this unit. From not getting it at all, to only getting one question wrong from our last quiz is a big difference. I feel more confident to write the math test now than i was a few weeks ago. Hopefully i didnt jynx myself, and hopefully ill pass and get a good mark on the test!!

February 16, 2007

During our Friday class Mr. K inroduced to us the smartboard, then we had a pre-test in the morning & we were group into 4 & compared each others answer in the group. Then in the afternoon class Mr. K gave us some problem to solve. Obviously Mr.K didn't gave us any new lesson so im just going to solve one problem from the h.w. review from the book pg.89 #5.


- A young entrepreneur decides he will sell tank tops, short-sleeved T-shirts, long-sleeved T-shirts, and sweatshirts. The table below shows the cost to produce each item.





a.


b.
c. Determine the total projected costs.
-> [T] [S.] =

-> then add those two #'s to get the total cost.
-> 10606.25+1850 = 12456.25
-> so the total cost is $12456.25


THE NEXT SCRIBE IS AL

gOoD lUcK gUys On tHe tEsT!.... :D

Friday, February 16, 2007

How do YOU want to learn?

This is another remix of the Did You Know? video we saw at the beginning of the semester. Its interesting to see how different people remix similar (or the same) information ...



I could give you lots of reasons why I think collaborating with another class online is a good idea. I've shared some of my thinking with you in class. All I'm going to say here is: keep this video in mind ... and the title of this post.

Share your thoughts about how you feel and what you think about working with the folks in Mr. Maks class here in the comments to this post ... and if any students from Mr. Maks class drop in on this post feel free to add your comments here as well. ;-)

SmartBoard Notes

Here are the slides from this afternoon's class.

February 15, 2007

In today's class we just worked on another question about transition matrices. Mr. K did not teach us any new stuff today, so I am just going to answer one of our assignment from the other day.



page 83, #4
4.If Regan passes his math test, the probability that he will pass the next one is 97%. If he fails his math test, the probability that he will pass the next on is only 85%. Regan has worked hard on this unit so the probability that he will pass the first test is 95%.





a) Write the initial probability matrix and the transition matrix.



b) What is the probability that Regan will pass the fifth test.
[S] x [T]^5 = [.97 .03] (therefore Regan has a 97% on passing his fifth test and 3% on failing it)
I did not show my work because I know that you guys already know how to answer this kind of question.
We also talked about the up coming project on matrix. Mr. K just show us the different kind of presentation we can choose from. If you missed the class that day or you just want to know more about it, just visit our blog and read the post "PRESENTATIONS ......".
This is the end of my scribe.
The next scribe is Jowell

Wednesday, February 14, 2007

Presentations ...

Here are five examples of different web based presentations that make use of free web tools. None of them have anything to do with math. What I'd like you to pay attention to is the way the presentation is given; the style. You can copy a style that you like or mix and match styles to create your own style.

Example #1
In our first class you saw the Did You Know? presentation. It was made entirely in PowerPoint. The audio file (music) is embeded in the PowerPoint file. There is a command in PowerPoint, under the [File] menu called "Make Movie." Once you've done that you can get it online by uploading it to YouTube or Google Video.

Here's an example of another presentation made the same way, by the same guy (Karl Fisch). It's called What If?.

If you do go with a PowerPoint presentation you really have to see this. One of the two best PowerPoint presentations I've ever seen. (Here's the other one. You need to have Flash installed on your computer to watch it. It's a free download.)

And, as a last word about PowerPoint, this is a PowerPoint presentation on how to give a good PowerPoint presentation. It was created by a friend of mine from Moose Jaw, Saskatchewan. It's called: PowerPoint: Extreme Makeover Edition ...




Example #2
This next example is a presentation that I have given a couple of times. It's called Whiplash! It's on a wiki. I gave the presentation live and the wiki just held links to the things I wanted to talk about.

At the bottom of each page of the wiki is a link to an audio file. I recorded the audio of each presentation I gave so people could come back afterwards and "relive" it. It has grown into something that many people, who I have never met, use to learn different web based tools. Click on the link to the audio file (open it in a separate window) and then follow the presentation by clicking on the links to follow along as you listen at your own pace.


Example #3
coming soon ...


Example #4
The Flat Classroom Project is the wiki I was telling you about in class. Two computer science classes (Grade 10 in Camila, Georgia and grade 11 in Dhaka, Bangladesh) got together to produce this wiki. They had all read the book The World is Flat by Thomas Freedman and each group of 2-3 students had to put together an informative wiki page that explained a different part of the book. They had to discuss how the ideas in the book were making real changes in the business world and in education.

Example #5
This is a presentation from an online conference that happened in October/November 2006. It's all about how to use different (free) tools on the web. What's particularly interesting is that each of the 5 sections of the presentation highlights a different tool ... and uses that tool to give the presentation. This is one of the coolest online presentations I've ever seen.

“I Did Not Know You Could do THAT with Free Web Tools”


Resources
Here is the wiki I told you about that lists links to a whole bunch of free web tools you can use to create your presentations.

February 14, 2007


February 14, 2007


Hi everyone. Happy Valentines day. Today in the third period class we tried to do a question mr. k put on the board. my computer isn't working well so i can't put any pictures on or it will freeze.


In class we learned how to make equations out of the information given because we were supposed to make an equation out of a matrix.


ex. you have a matrix that is 3 by 3 and the info from left to right starting from the top, and it has these values,


f to f = .85, f to c = .13, f to e = .02 then the middle row has, c to f = .05, c to c = .91, c to e = .04, the the bottom row has, e to f = .03, e to c = .21, e to e = .76.


Then you have the initial state which was, f = .30, c = .20, e = .50.


So we had to make three equations out of this information.


ex. Make an equation out of the f column by mulitplying the f column by the initial state.


and it should look like this:


F= 0.85f + .05c + .03e.


and that would give you the new information for the f column.


Then to get the new info in column c, you have to do the samething as before but this time using the c column.


Then over again to do the e column.


In last period, we looked at a review for the matrices unit.


We were asked to do question 6 in the booklet.


Question from booklet:


Three children, Ann (A), Bill (B), and Carl (C) are throwing balls at each other. Ann always throws to Bill, Bill always throws to Carl, but Carl is equally likely to throw the ball to bill as he is to Ann. Write a probability matrix to represent this situation?


So we made a matrix with percents.


ex. matrix 3 by 3 had FROM A B C, TO A B C. In the A row we put 0, 1, 0. In the B row we put 0, 0, 1. In the C row we put 0.50, 0.50, 0.


The initial state had 1 0 0 starting with Ann.


And that was it but mr. k asked us to figure out the percentages for Ann, Bill, and Carl after the fourth throw.


So we used the initial state and multiplied it by the transitional matix to the exponent of 4 to get the percentages.


And we got: 0, 0.50, 0.50.


Meaning, if Ann started with the ball then she threw it to Bill, Bill then threw the ball to Carl, and then the ball ended up with Bill. The ball then had a 50-50 percent chance going to Carl and Bill again. So if you want to figure out the longterm percentages, you have to take the initial state and multiply it by the transitional matrix to the exponent of 50 and then you get Ann with a 20 percent chance to get the bal and Bill and Carl a 40 percent chance to get the ball.


So do you really think it's a fair game?


Also for people who were not in the afternoon class we had to tak these notes down in our dictionaries:


MATRIX:


A rectangular arrangement of numbers (or letters) in rows and columns contained in square brackets. The plural of MATRIX is MATRICES. They are usually named using capital letters.


ex. a 2 by 3 matrix: square bracket row (1 2 3) bottom row (4 5 6).


DIMENSIONS OF A MATRIX:


Determined by the number of rows and columns. Written as: r x c READ as "r by c"


ex. Matrix above is a 2 x 3 matrix.


ELEMENT OF A MATRIX A.K.A. ENTRY


One of the values in a matrix. The address of each value is given by its row and column position.


ex. A (sub) 23 = 6


The element in matrix , row 2, column 3 is 6.


We also had to finish the homework on p.83 numbers 6 to 9.


OH YEAH........mr. k kept om spelling matrices like this: matricies so whoever wrote it like mr. k you might want to change it.





AND SO THE NEXT SCRIBE IS CRIS


February 14, 2007


February 14, 2007


 


 


 


Hi everyone.  Happy Valentines day.  Today in the third period class we tried to do a question mr. k put on the board.  my computer isn't working well so i can't put any pictures on or it will freeze.


 


In class we learned how to make equations out of the information given because we were supposed to make an equation out of a matrix.


 


ex. you have a matrix that is 3 by 3 and the info from left to right starting from the top, and it has these values,


 


f to f = .85, f to c = .13, f to e = .02 then the middle row has, c to f = .05, c to c = .91, c to e = .04, the the bottom row has, e to f = .03, e to c = .21, e to e = .76.


 


Then you have the initial state which was, f = .30, c = .20, e = .50.


 


So we had to make three equations out of this information.


 


ex. Make an equation out of the f column by mulitplying the f column by the initial state.


 and it should look like this:


 F= 0.85f + .05c + .03e.


 and that would give you the new information for the f column.


Then to get the new info in column c, you have to do the samething as before but this time using the c column.


Then over again to do the e column.


 


In last period, we looked at a review for the matrices unit.


We were asked to do question 6 in the booklet.


 


Question from booklet:


 Three children, Ann (A), Bill (B), and Carl (C) are throwing balls at each other.  Ann always throws to Bill, Bill always throws to Carl, but Carl is equally likely to throw the ball to bill as he is to Ann.  Write a probability matrix to represent this situation?


 


So we made a matrix with percents.


 


ex.  matrix 3 by 3 had FROM A B C, TO A B C.  In the A row we put 0, 1, 0. In the B row we put 0, 0, 1.  In the C row we put 0.50, 0.50, 0.


 The initial state had 1 0 0 starting with Ann. 


And that was it but mr. k asked us to figure out the percentages for Ann, Bill, and Carl after the fourth throw.


 


So we used the initial state and multiplied it by the transitional matix to the exponent of 4 to get the percentages.


 


And we got:  0, 0.50, 0.50.


 


Meaning, if Ann started with the ball then she threw it to Bill, Bill then threw the ball to Carl, and then the ball ended up with Bill.  The ball then had a 50-50 percent chance going to Carl and Bill again.  So if you want to figure out the longterm percentages, you have to take the initial state and multiply it by the transitional matrix to the exponent of 50 and then you get Ann with a 20 percent chance to get the bal and Bill and Carl a 40 percent chance to get the ball.


 


So do you really think it's a fair game?


 


 


Also for people who were not in the afternoon class we had to tak these notes down in our dictionaries:


 


MATRIX:


    A rectangular arrangement of numbers (or letters) in rows and columns contained in square brackets.  The plural of MATRIX is MATRICES.  They are usually named using capital letters.


ex.  a 2 by 3 matrix:  square bracket row (1 2 3) bottom row (4 5 6).


 


DIMENSIONS OF A MATRIX:


     Determined by the number of rows and columns.  Written as: r x c READ as "r by c"


ex. Matrix above is a 2 x 3 matrix.


 


ELEMENT OF A MATRIX A.K.A. ENTRY


    One of the values in a matrix.  The address of each value is given by its row and column position.


ex. A (sub) 23 = 6 


    The element in matrix , row 2, column 3 is 6.


 


    We also had to finish the homework on p.83 numbers 6 to 9.


OH YEAH........mr. k kept om spelling matrices like this: matriciesso whoever wrote it like mr. k you might want to change it.





 


AND SO THE NEXT SCRIBE IS CRIS



BOB

Well when I was first introduced to matrices, it looked a little difficult until Mr.K showed us that we can use our hands to do this, it sparked some interest in me. I'm a kinesthetic learner, which basically means I'm a physical learning or I learn with my hands. So grasping how to add and subtract and multiply wasn't difficult at all because I could use my hand and memorizing the rules wasn't to bad either. That was UNTIL the latest homework assignment on the hand out "Transition Matrices." For some reason i can't get past creating the matrices, because of a one little word. PROBABILITY. It is almost like I have a phobia of probability, because any time i see it my mind goes blank. Which is understandable because that isn't one of my strengths in math. But that is OK because that tells me what I need to work on. Anyways, I may not be able to get them on my own, But it helps a lot to go over what i don't get in class. It wasn't until being in class and talking to people did i understand what I did wrong. Mr.K was so right, Learning is a conversation!

Tuesday, February 13, 2007

Feed Window


There's something new in the side bar ... way down towards the bottom ... it's a Feed Window.

A Feed Window is a window into another class blog that's learning the same (or similar) material that we are. Every time they publish a new post on their blog, our feed window to them will update automatically and create a link to the new post(s).

I also noticed some comments in the chat box from someone who felt we really need to catch up on some notes ... you're absolutely right! We'll do that tomorrow. In the meantime check out this post on Matrix Operations from Marsha, a student from the class on the other side of the feed window. It's got exactly what you're looking for ... it's also a great example of what an excellent scribe post looks like. ;-)

Friday Febuary 9th


Some things that were talked about before we got assigned work were; Blog before tests on what you are having troubles with and Mr. Kuropatwa will assist what ever way he can to help you understand better and get a good grasp of the topic. He wants you to feel free to e-mail him about anything we go over in class that your stuck with as well.



PoP QuIz On MaTrIcIeS! We did various questions on matrices. Some examples questions were as follows: Are the following Matricies possible or not possible?

[A] [B] [C]

311 21 32

021 01 01

101 32 21

3x3 2x2 2x2

A+C_______. Answer Not possible. You can't multiply The matrix unless the the rows match up with the column of what you are multiplying.

[B][C]_____. Possible. the 2x2 matrix matches up with the 2x2 matrix so the two are compatible.

C+B____. Possible. again because the 2x2 matrix matches the 2x2 matrix.


Shorty after we marked the quiz we were assigned numbers 1-5 to gather into groups to work on a complex matrix assignment to crack coding and decoding matrices. It was asked that we enter these into our handy dandy calculators as it makes it way easier. These codes gave a set of values for the letter and codes we needed to crack. Next class we are asked to have the cracked code to read out in class, and also do the homework assigned online.



Tags:

Matrices

Today's class was just basically a recap of last nights homework. The homework was confusing to a lot of us so here is one of the questions. I will try to see if i can give you a better understanding of this question based on what i learned.

1. If a train is late on one day, there is a 90% probability that the same train will be on time the next day, while if the train is on time, there is a 20% chance it will be late the next day. If in a given week it arrives on time on Monday, compute the probabilities that it will be on time or late for each of the subsequent days of the week. What would the corresponding probabilities have
been if the train had been late on Monday.



In this matricx (T) represents on time and (L) represents late. So as you can see the numbers stated in the problem have been inserted into matricx. Now because there is a .20 there has to be a .80 at the (T) (T) location. This is because each row must equal 100 or 1 depending if u are using whole numbers or percents. So following this rule there has to be a .10 at the (L) (L) location.



Now that we got the transition matricx we need what is called a state matricx. A state matricx is what happens first, because the train was on time on Monday the state matricx has to be;The (1) represents the on time colum and the (0) reprsents the late colum.

So to find the percentages of the train coming late or on time you would take the transition matricx and multiply it by the state martricx like this;


If u muiltpy these two matrices you will get the percents of the train being late or on time for the next day. If you multiply the result by the transition matricx you will get the percents for the third day and so on and so on ........

Now if the train happened to be late on that first Monday then the transition matricx would stay the same and the only difference with the state matricx is that the 1 and the 0 would be switched around.


Definitions:

Transition Matricx.....is the matricx showing how things change over time.

State Matricx.....is the matricx in which things begin

Another thing to remember is that if you are trying to find what the numbers stabilize at, rather then keep pressing enter on your calculator just rise the transition matricx to a high number and you will be able to see the numbers at which the amount stabilized. But remember this way only works if you are not interested in finding the exact time that the number stabilized.

I hope my scribe made it easier for you guys to understand things.

The next scribe is chris

Monday, February 12, 2007

BOB - Matricies

In the first week of class in Applied Math, learning about matrices is new to me. At first, it was a little bit confusing because it was my first time encountering this lesson.. but after some examples shown and discussed in class, I had that "moment of clarity" that Mr. K. was saying. I understood the lesson more when we were assigned to do problems involving matrices. I learned that matrices are used to solve real life problems, which I found interesting. We learned about multiplying matrices and transition matrices. About transition matrices, I did not get it at first because I missed one class, but i caught up and I am comfortable with it now. And now this unit is over, we are having a unit test on monday. Hope I get a good mark. =) >> Kristel

BOB

My first reaction to matrix was that it was going to be an easy subject since it looked and sounded simple, but I was wrong. To be able to figure out these matrices, must know what your doing and focus on it since you wont get lost without knowing it. The way these matrices are added, subtracted or times, i mean "multiplied", is not like normal addition or subtraction. You must know which ones go together and you must also satisfiy the rows and columns. In the beginning i had trouble with this part, but as the participation of the class increased, i started to understand bit by bit. So it really goes to show that "Learning is a conversation"

Transition Matrix

In the beginning of todays class, we had a review on last weeks lesson about multiplying matrices. Transition matrix was then introduced to us. Transition matrices involves probabilities and percentages. It describes change and can be symbolized with "[T]" To be able to work with these matrix, the columns and rows must have the same information.

This example I'm about to give is from the first transition matrices worksheet Mr K handed out. We had a hard time understanding example 2, which was the sports one, so ill re-cap on that one..

EXAMPLE 2, SPORTS
The annual Oxford-Cambridge boat race, has been rowed regularly since 1839. Using the data from 1839 up to1982, there were 57 Oxford wins and 67 Cambridge wins. If the relationship between the results of a given year and the results of the previous year are considered, the following table can be constructed.



**



**-This means that Oxford got 35 games won after another (two games won in a row) and 22 wins without a two game in a row streak ... This goes the same for Cambridge.

To be able to work with this matrix, we need to convert to a percentage then to a decimal. I'm sure we all know how to convert these into decimals. the result should be.










Question: If Oxford wins this year, what is the probability they will win next year? in two years? in three three years?

Answer: to solve this question we need to have a transition matrix and a state matrix.

The transition matrix is :










This matrix represents the change. for example, there is a 61% chance that Oxford will have a two game winning streak.


The state matrix is simply stated "how it begins". For example, in the question it stated that Oxford won this year

Therefore we put a 1 under the column of Oxford in a 1x1 matrix which the rows represent the win. There is a zero under Cambridge because this represents the first year and they lost the first year.


We then multiply the transition matrix and the state matrix to get the probability of each team.
It goes:

2









The Transition matrix has an exponent of two which will give us the result for a second year percentage. We all know by now how to multiply matrices. That is the next step. the result is:











This means that in the next year. Both teams will have the same chance of winning between themselves.


.




.i hope this is much better than my other post.. :)

BOB; Ivy; Matrices

Well, this unit was fun and I think this is only the unit in Applied Math that I can understand because the rest it's pretty confusing but anyways. I think I can do okay on our Matrices Test; Hopefully. Matrices was pretty fun doing the code, the multiplication, adding, subtracting, connectivity, but the transition matrix I kind of get it I'm 80 percent sure...I just need more practices and understand each different method how to do it, whether population, weather, sports or diseases. It's just freaking me out when Mr. K was telling us in our class that we actually use a little bit of probability, because seriously "I NEED HELP UNDERSTANDING PROBABILITY AND STATISTICS" I'm pretty low of understanding when it gets to that, but hopefully my second time in AM40s will be better for me. Okay that's it.

Matrices

When Mr. K first introduced the unit Matrices I felt like I knew what was going on, this unit seemed like it was going to be easy and we would just breeze right on through but I was wrong. The next day we were introduced to matrix multiplication and to be honest I was so confused but as the class went on I began to understand more and more. The finger method really helped me to be able to do the multiplying part. I also liked that assignment on decoding the messages, that was pretty fun. So I think if we all just buckle down and help each other out we’ll all be fine.

Sunday, February 11, 2007

BOB

When we started learning about Matrix Multiplication I think it was on Wednesday's third period class, I thought i was a little intimidating at frist. I was a little confused about how we were supposed to do it and what were the restrictions when doing it. But I think after doing some questions and doing the "fingering" method, i really got the hang of it. When we first learned about the addition and subraction in matrices, I thought it would be a breeze because there was less work and next to no restrictions, but I think that we've all got our work cut out for us in this particular area of the unit. One thing that I thought was cool so far in this unit, was the decoding and encoding. I think that it's the coolest things as of now, because it is a part of the unit and very educational and at the same time it's fun to decode and encode messages. I didn't know what exactly to blog on, so I sort of just jotted down a few thoughts. I hope someone else blogs as well, i'm really looking forward to hearing someone else's input on the unit so far. K, bye! - Grace

Friday, February 9, 2007

The Matrices Code

As promised, here is the assignment you have to complete for Monday's class ... and the secret message I have for you is encoded beneath it ... decode it and bring it in to class on Monday. ;-)



Your secret message ...

-6, -9, -8, 10, 23, -23, -8, -35, 75, 8, -8, 24, -23, 21, -51, 31, -26, 62, 4, -2, 10, -19, 8, -41, 31, -12, 75

Thursday, February 8, 2007

The Scribe List

This is The Scribe List. Every possible scribe in our class is listed here. This list will be updated every day. If you see someone's name crossed off on this list then you CANNOT choose them as the scribe for the next class.

This post can be quickly accesed from the [Links] list over there on the right hand sidebar. Check here before you choose a scribe for tomorrow's class when it is your turn to do so.

Cycle 4
Grace (2)
Donna
jhay-ar
Jennifer P
Ivy
jOweLL
Al Andrei
kyle
chris
CHRIS A.
Kristel***
nadiya -1
cRis_J
just_in -1
Brittany
tennyson
Network Matrices

Today in class we talked about NETWORK MATRICES. These matrices are used in the real world. Whether it is cables between computers, wries between telephones, railway networks, airplane routes between major centers, there is always a connection between places and things.

Connections may be achieved by "one-hop", "two-hop", "three-hop", "four-hop", etc.
I'll use airplanes as an example.

A "one-hop" is a direct connection between the two cities.

ex: one airplane trip (one hop) from Atlanta to Boston
A "two-hop" is the 2 different ways from one city to another with 2 different airplanes.
ex: one airplane trip (one hop) from Atlanta to Boston then another airplane trip (2 hop) either back to Atlanta, or Charlotte.
A "three-hop" is the 3 different ways from one city to another with 3 different airplanes.
ex: one airplane trip (one hope) from Atlanta to
Boston, then another airplane trip (2 hop) to Charlotte, then another airplane trip (3 hop) back to Atlanta.
And so on...



For example:




We can create a "connectivity matrix" showing the direct routes of the network on the left side of this page.

We can create a matrix that represents the connection between A, B, and C. In this matrix we will need 3 rows and 3 columns.






When there is one direct route, you place a "1" in any cell which indicates a direct connection between two places. For example there is one route from A to B, so a "1" could go into the cell that connects A to B. When there is an indirect route, you place a "0" in any cell which indicates no direct connection. In this case, there is no route from A to C, so therefore in the cell that connects A to C, you would place a "0"



The matrix for the image above should be:










We also learned how to indicate where there is a "2 hop" route between places. In our example, is there a 2 hop route between A and A? YES! Because you are able to take a plane to Boston, and take another plane back to Atlanta. So in a matrix that indicates "2 hops", you'd place a "1" showing that there is a "2 hop" route between A and A. In our case, there is no 2 hop route from A to B. Because there is a plane that takes you from A to B, but the next plane will take you somewhere else besides B. So you'd place a "0" in the cell connecting A and B for "2 hops"


The matrix indicating "2 hop" routes should look like this:












Whatever number a connectivity matrix is raised to, will tell you how many different routes there are.
For example, if the connectivity matrix is raised to the
power of 2, the result of that would tell us the number of ways from one place to another with 2 airplanes.
If the
connectivity matrix is raised to power of 3, the result of that would tell us the number of ways from one place to another with 3 airplanes. ETC.









And tomorrow's scribe is....... JHAY-AR!!

Wednesday, February 7, 2007

Feb. 6, 2007, Ivy's Scribe Post

Hello...Everybody... Ivy is here to be your class scribe for today (I just got off from work I'm sorry I will post my blog next time pretty on time).

Well, anyone who missed Applied Math 40S, Feb. 6, 2007. Here is your hope to know what happened.

Mr. K introduced us The Matrix unit, and read the following as I explain everything.

The definition of Matrix is an array of numbers or letters with dimensions, that can be organize to add, subtract and multiply. Also Matrix can't be add or subtract if they have different dimensions. We also name a Matrix by it's own rows and columns.

There are also two different type of Matrix; a Rectangular and a Square.

A Rectangular Matrix has a different rows and columns.
example. 3 x 4 or 3 x 2

A Square Matrix has the same dimensions.
example. 4 x 4 or 5 x 5

Mr. K also showed us an example of a Matrix. He also told us to look for patterns, Can you find one? Try to look for patterns before reading bellow.



The First Student saw the patterns of a Diagonal Zeros. It means in this type of Matrix it is counting by distance so Brandon cannot travel to Brandon, because it is in the same place no point of traveling.

The Second Student saw the mirror like of the matrix by triangular.

The Third Student saw in every city either row or columns it says the same thing.

Later on. Mr. K gave us another Matrix and compare the Matrix above and the Matrix below.




The difference of these two Matrices are:

- This is a Rectangular Matrix
- This Matrix has a 3 x 4 dimensions
- This Matrix has no Contex






- This is a Square Matrix
- This Matrix has a 5 x 5 dimensions
- This Matrix has a Contex





After comparing the two Matrices Mr. showed us this;

Later on...He challeged us to look for these elements in the following Matrices above. Try it!



Hint. To find an element. Look at the picture.

The answers are:

Plus he got another challenges: To add and Subtract it.



Yet he got another challege.

Hint. if you see a scalar always multiply that number to the matrix beside it.


Ok...That's it for Feb. 6, 2007 class. The scribe for tomorrow is Kristel.