What Mathematicians Think...

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

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

Another quote:

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

The full interview is right here.

These are two things I find myself constantly belabouring in class when teaching problem solving:

• Take risks! Experiment, play, try something out and see where it takes you. Good math isn't knowing what to do with any problem -- good math is knowing what to do when you don't know what to do. ;-)


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

Food for thought ...

Hare and Hounds

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

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

Have Fun!

Number Sets and Radicals

Over the last few days we have learned about the different sets of numbers and radicals.

You can review sets of numbers here and radicals here and here. This is a good review of rational exponents with lots of worked examples. Actually, ALL these tutorials have lots of worked examples to help you learn.

If you want something a little more visual choose from any of these videos to watch.

You can practice your test taking skills here and here (if you [refresh] the page you'll get a new set of 5 questions).

This is a little quiz on translating written expressions into mathematical expressions and equations.

Blogging on Blogging. Ndrew

The trigonometry curriculum was the most easiest unit of the 3. At fist it was easy until the problem solving questions then I sort of struggled on the last few. We learn sine law and cosine law it was sort of easy but did not like the homework that much when we did 50 questions for cosine law because it was very time consuming.

By Ndrew

Ndrew's internet connection is not working, he told me to post his blogging on blogging.

Non-right triangles...

This was the best unit for me, so far. It was easy to understand and it had lots to do with familiar stuff we did in trigonometry. I'm very happy that we spent a long time working on trigonometry in grade nine. To be honest, it seemed hard at first, but as we worked more and more into it, it was simple and fun!

What I didn't like from this unit was typing it all into my calculator. It was easy using the formulas and showing the steps into how I got my answer. But once I started typing the numbers into the calculator, it got tiring. I should do more finger exercises. And I do have fat or clumsy fingers, so sometimes I tend to type in the wrong numbers, and that really throws my answer off.

I did enjoy this unit because I actually understood it, and it was a short one. I really like math when I start to understand it. Makes me feel like I achieved something. Yes, well what I hope is that the test won't be that hard and I really really really hate word problems. They annoy me! That's about it, so I'll see you guys tomorrow in math class. And may the force be with you! STAR WARS!
-toodles

Blogging on Blogging. ALREADY!!

Today



Today’s class is cool in the beginning because we begin a new unit today and the until is about……… will are you sure your ready for this? Here I go - We are learning about NUMBERS Hahahah and I mean # like 1, 2, 3, 4. So ya, the first half an hour is all very easy and fun I guess but starting at about 2:50 Mr. Clark is making as work on squares and I don’t mean making squares out of papers but we are simplifying Radicals which is easy but then again there is some hard question that I don’t understand. Will tomorrow is the test on you know what so go to sleep at between 9-10 pm and never go later then 10pm or else you won’t get 100%, believe me.

My pre test blog!

Well today I am going to be talking about trigonometry. For me it is a pretty easy unit (most of the time), maybe just the word problem part and the angle of elevation and depression. I was a foggy with it last year and again this year. Let's hope by tomorrow the fog will clear up. Besides that there isn't that much you have to remember in trig. Mostly everything is pretty straight forward. I'm pretty sure that I like it last year too. It's kind of cool that you could take a triangle and have barely any information about it and then do some fancy stuff and end up with all the information. I think that is pretty DARN awesome.=) Even thought we got a lot of homework that one day it wasn't too bad because I understood it and it helped me practice when to use the right formulas. I think that I am decently prepared for this test and I wish GOODLUCK to all of you tomorrow!!!

my blog.

Well, this unit sure went by super fast. How long was this unit, three days or something? Next thing you know, the UNIT TEST IS TOMORROW! Anyways, though I failed my quiz, I'm sure that I'll do good on the test and so will everyone else because when you actually understand your mistakes, such as in the quiz you will see the light? bleh.
I have to admit that I did not fair well during the learning process on the first day but like always, you recover and start to understand what you do not (MR. Kuropatwa: Good math is when you don't know what to do but you find a way out). I probably got the wording messed up but you guys understand what I'm trying to say he said?
Well, I'll just wish you all goodluck on the test tomorrow :P.

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

Here are the rules.

Here is the game.

Have fun with it!

trig-er

This unit is like a breeze. It was easy working with triangles;finding the missing angles, the missing sides and solve for the unknown. The Sine Law is pretty simple, so as the Cosine law except for its longer to solve. Well in the surprise quiz was easy so I think I'm going to do well on this unit test. Overall it was easy, although the Cosine Law is pretty long to solve though.

Trig Word Problems and Related Angles Review

Practice trigonometry word problems here. Look for the button marked [Get a Different Set of Questions]. Every time you click it you'll get two new problems to practice with. Answers are provided with the click of another button.

You can also download this pdf file that explains how to calculate related (or reference) angles and has several practice exercises with answers.

The Cosine Law

Sometimes the Sine Law wont work, such as when the given information is one of these two cases:

1. side-side-side (SSS)


2. side-angle-side (SAS)

That's when we use the Cosine Law. You only need to use it once. After that you'll have enough inforamtion to use the Sine Law; which is much easier to use (less algebra). Watch this to see how the sides and angles in any non-right triangle are related.

Here you can review the sine law, watch a few animations of how it works and try a brief quiz (3 questions). You can try another quiz (5 questions) over there.

The Sine Law

We can use the pythagorean theorem or trigonometry (SohCahToa) to find the missing side or angle in any right triangle. But what about triangles that don't have a right angle? Well, that's when we use the sine law. Watch this to see how the sides and angles in any non-right triangle are related.

Here you can review the sine law, watch a few animations of how it works and try a brief quiz (3 questions). You can try another quiz (5 questions) over there.

Woohooo! The 100th Post!!

Why hello there. my name is wendy. i'm the scribe for today. so please sit and enjoy :) today's class started at 1:19 pm, just when lunch is over. we had a sub teacher. we started off talking about reviews on trig : SOH CAH TOA. we then wrote down notes about Sine Law. today's class we learned "Sine Law" ; Sine Law - is used for triangles that are NOT right angles. - the formula for Sine Law is a b c
---- = ---- = ----
sinA sinB sinC
( to solve a triangle means to find all the missing parts. )
Example : ∆WXY w = 18.3,
Solve for ∆WXY. first draw the diagram (which can't be drawn on this blog. :S)

w x y
----- = ----- = -----
sinW sinX sinY
---
--
-
18.3 x y
----- = ----- = -----
sin62° sin15° sin103°
---
--
-
18.3 x ----- = ----- sin62° sin15°
---
--
-
18.3sin15° = xsin62°
--------------- ------------
sin62° sin62°
---
--
-
5.4 = x


18.3 y
----- = -----
sin62° sin103°


18.3(sin103°) = y(sin62°)
----------- ----------
sin62° sin62°

20.2 = y

By the time it was 1:45 pm, the teacher handed out a worksheet (both side) on "The Laws Of Sines" **a quote from Sam. "best bath class she's had"**

Homework for tonight is Ex. 19 questions 1 - 10 & worksheet (both sides)

Muddiest point is... THE PERPENDICULAR LINES.... AND PARRALLEL LINES & ANY THING THAT HAD TO DO WITH IT :)

Tomorrow's scribe is going to be..... i .... chose... you .... PIKACHU! :) .. lol. no no . i uh ... im not really sure who i should chose? give me a second. MMMM let me think.... who should it be this time? someone that didn't get the chance to be the scribe? could it be T-sa? nope. she has been scribe already. could it be... steven F? nope. he was already scribe. darn it. i have no clue. i guess tomorrow's scribe will bee...... ** drumss....** ------ Natasha ------ ** -_- (t-sa's face) =p bleh!... GOOD LUCKK! :)

-->|SCRIBE|<--

Well I'm today's scribe. The class today we learned how to guess what the answer will be if the question is the sin, cos of a length. For example:

sin30 = 1/2
sin150 = 1/2
sin210 = -1/2
sin330 = -1/2

How are these angles similar? They are all in 30degrees from 180 or 360.

You can figure this out if you know what the sin30 equals to. If sin150 is (+), why is the sin210 and sin330(-)?? Its negative because its in the 3rd and 4th quadrant. In sin, you'll know if its negative because anything pass 180degrees is negative. In other words, the degrees that are below the line of origin or under zero(0) in the y-axis is negative.

Another thing that Mr. K talk about is about cos. Cos is similar to the sin, in cos if the degree is in the 2nd or 3rd quadrant of the graph, it is negative. To make it easier if the angle that you're looking at is in the left side of the graph, negative side in the x-axis, the number is negative.

EXAMPLE:
cos60 = 1/2
cos120 =-1/2
cos240 =-1/2
cos300 =1/2

In this example all of the angles are 60degrees for the line of origin. Cos60 and cos300 are positive because they are in the 1st and 4th quadrant.


Mr. K also talked about inverses.
The inverse of:
-addition is subraction
-subtraction is addition
-multiplication is division
-division is multiply

This connects with the lesson because to get the length of the side you have to use sin or cos depending on what side you are calculating.
The inverse of:
-sin is ARCsin or sin-1
-cos is ARCcos or cos-1

The sin function is turns angle in to length and the ARCsin turns the length in to angle.

ANNOUNCEMENTS:
+Homework-Exercise 18 #1-3
Exercise 19 #11-20
+Sub tommorow - Mr Clark
+Possiblity of having a TEST ON TUESDAY NEXT WEEK OCT.25

Thats it folks! oohh yea... the scribe tommorow is----------------------------------------------------------------------->WENDY =P

Sunday Fun! (or Sunday Madness!)

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

---

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

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

Scribe for Wednesday!

Well I am the scribe for Wednesday's class. I'm not to sure exactly what we went over so sorry if I don't explain everything correctly.

On Wednesday we started off the class with the usual questions on the board. But little did we know that the first question would last us almost the entire class.

The first question was something along the lines of: there are five lines that all pass through the point (2,1), find the five equations. Now looking back on it, it doesn't seem as hard as it was at the moment. We struggled struggled and Mr. K started to get worried because it was supposed to be a review. He tried to explain what formula would you use to get the product of 1. It was finally when Ian stated that you had to substitute in the point if you used the slope-intercept form. By then a few of us understood, but many of us were still struggling. We continued to talk about it and Mr. K continued to give examples and I can't speak for everyone but finally a light went of in my head. It finally made sense. I can speak for almost everyone (including Mr. K) that it was frustrating. Once we got it people started to give out answers. Some of the examples that i took down were : y=1, y=-x+3, y=x-1, x=2, y=2x-3, y=3x-5, y=7x-13. Those are to name a few.

The other two questions were pretty general so I won't bother get into detail. After going over the question we were separated into groups and had to answer a question in 15 minutes. Everyone tried to get their thinking caps on and some people understood what to do and were very close while others (like me) didn't have a clue what to do. I don't really remember what the question or answer was. It was a long answer, that's what I remember and that I was very confused.

Well I'm done here I guess since I can't really remember anything else, so the scribe for today's class is km! Goodluck!

Brittany's Pre-Test Blog

Brittany's computer isn't working so she emailed me her pre-test blog. Here it is...

---

Well so our test is tomorrow, it's weird but I actaully understood most things in this unit very well. But the whole "test" idea still makes me a wreck. I have a bad case of forgetting somthings when I see the test paper, but thankfully I'm getting over that habit slowely.

One of the only problems I had in this unit was a little one regarding finding point B when given the mdpt and point A, I just could never understand it. Somtimes I understood things really well and no offence mr.k but somtimes I tried not to remember what you were saying because the more explination you put on certain things made it more confusing. But it may have helped other so that was okay. My most memorable class probably had to be the one were we stood on the tables, I think i've only done that once in another class. It was interesting thats for sure.

I think I'll do a lot better on this test then the first one, well hopefully.

Goodluck

Blogging on blogging

Like my house at China?. Is not a apartment, is my HOUSE!
welcome to Mr. Lie’s after school Lessons on the DMCI. Today I will be talking about math, I am here interview a kid in room 66 AP math course “Jan Hong” and see how is A_Hong doing during mr. K’s class at Oct. 12. 2005. Today he have a really confusing class because he don’t seem to know the questions the teacher have on the broad and the question is “5 lines pass through (2,1) find the equation“. Wow what is such a easy question but it is good enough to stop A_hong, the teacher give the class a fell minutes to do the question and after that the teacher ask for the answer, after the class talk about it and give some answers on the broad, A_hong begin feel lot more better about the work and he started to know that really the question isn’t hard, is just that he don’t understand what the question is asking.


I think I have talk enough, lets see what A_hong say about what he think he will get on the test tomorrow. “tomorrow I think I will do “ok” on the test and which means that I will at less get over then 70% on the test but my goal is to get at less 90 but tomorrow I think I get over 70 because I can’t go over the reviews in the my computer because looking at the come computer make me go craze because my computer is so slow and it make my eyes hurt. But I will think of something after this test, I guess.

Blogging For The Test

One particular class I remember is the one when we stood on the tables. That was a bunch of laughs, huh? That class, we learned about the two right angle triangles.

My process in the class is processing. I am beggining to understand more. But I don't seem to remember how to use SOCSOATOA or how to do trigonomtry. That is what I am struck on, the trigonomtry problems.

Learning about parallel and perpendicular lines was "cool". They are so simple. All you have to know is parallel lines, the slope is always the same; perpendicular lines are the negative recipocal (the denominator and numerator are flipped).

I had problem use the different forms. I did not know when and how to use the different forms. I was very glad when Mr. K explain slowly that depending on what information is given how and what form to use. It was a reliefing feeling, I guess.

No offences, but my friend that goes to university says that calculus is not use much in life unless you are an arcitect or doing somethings with computers. And I agree, calculas is not used a lot in life, but it is used a lot the class. Like, in my geography, we had to find points and graph them and in my science class, we also had to find points and graph them.

Well signing off,
Thang

weblogging

Hmmph.. In the beginning of this unit, I must admit that I did not clearly understand 90% of anything that mr. K had taught or tried to incript into our minds.

After stressing the concept of focusing and thinking beyond the box, I've learned slowly but truly everything that I did not understand such as which formula to use, remembering the formulas, midpoint.. ALMOST EVERYTHING. I can surely say that I am ready for this test and that everything doesn't seem so complicated anymore.

I wish you all goodluck tomorrow as well and will remind you again that you should study :D because your mind needs recalling of memory :P

*+* =) Bloging the Blog =) *+*

I'm also nervous for the test too t-sa. On the last test we had, i barely past. i'm going to study very soon now. through this unit, i think it has been hard. learning about the slopes comfused me, and when we learned about the perpendicular lines. i knew i was totually lost. im still having a little bit of trouble on perpendicular lines but i think if i study and look back on the notes, and works we did, i think i'll understand it a little bit more. what i dislike most about this unit was, the perpendicular lines and anything that has to do with slopes. what i learned from this unit was learning the formulas. this is because i actually memorize the formulas. also the other thing i enjoy was learning how to find Distance. i remember the distance formulas, mid point, intercept, slope, and all the formulas's we have learned. GOOD-Luck To EVERYONE who needs it for the test tomorrow. [[ i'm going to bring m lucky charm. ]] [[ hope it actually works ]] :(

UNIT 2

Well this unit I learned about things that i havent learn before like the slopes, how to get the coordinates perpendicular line and other things like x and y intercepts, etc. When we started learning about slopes I had trouble with it. As the unit progesses i started to get it.

We also learned about formulas like point slope, slope intercept, general form and those other stuff. Making graphs, finding out what the coordinates are, perpendicular lines and parallel lines.

I think thats all I can say except the perpendicular still confuses me a little. Over all it was kinda easy. Hopefully I'll get an 80% or higher in this test.

Peace out.

Oscar Had A Heap Of Apples

Can you figure out that title?

If not, here's a hint and a review of some of what we talked about today. ;-)

So Many Lines!

When we first started doing this unit on graphing and lines. I wasn't all that excited. I don't think many of us were. Graphing is one of the units I don't enjoy doing because I really hate graphing lines, and I just don't like this unit. I really get confused and usually when I actually get what were doing in class, 20 minutes later I guarantee that I have already forgotten what we did and how we got the answer. That's what I predicted that would happen. But to my surprise I'm okay with this graphing unit.

The reason why I kind of like graphing is because of this blogger stuff. It's funny. When we did that whole division with polynomials, I didn't get any of it. And all my answers were thrown off because I didn't understand what we had to do. Two days before the test, I worked on the reviews that Mr. K posted up. After a couple and going step by step on how to get the product I understood what we had to do. That's the same thing with lines. I didn't understand how to get the standard form of things, but after the reviews I understand now. You guys don't know how happy I am. I'm practically in tears (there tears of joy)!

There's only 2 things that's bothering me. Perpendicular lines and remembering which formula to use and when. I don't like perpendicular lines because we deal with fractions. Fractions really bother me. I hope we review in class too. I'm still don't understand some stuff, especially perpendicular lines! Well, that's all I have to say. See you guys in class. And don't forget to blog! Remember that's worth a mark on the test! Haha.

-toodles!

scribe

I'am the oct.6 bogger scribe, iam sorry for being soo late.

During oct.6 we are talking about Formulas for lines:

here is the y=mx=b slope-int. formula which is use when we know the slope and the y-int.

note that m=slope b=y-int

we use y-y1=m(x-x1) is called the point-slope form, we use this when we know the slope and a point

Note that the one beside the y and x mean that the first y-int point and first x-int point

Here is the general form ax=by=c=0 "write this down in your dictionary"

The general (standderd) form

sometimes written as ax=by-c=0

useful for graphing lines when a and b are factors of c. we can easily find the int.

EX: 2x-3y=6 Ex: x-int., let y=0

find y-int. let x=0 2x-3(0)=6

2(0)-3y=6 x=3

y=-2

and anow we know the points for both int. so we point them and Oct.7th scribe is Parvin Sia. sorry for telling you soo late

My Mudiest Point

My mudiest point is just remembering all of the available formulas and when the appropriate time to use them. Like Mr. K said that you can't take a saw to put a nail in the wall. But, if I even manage to remember what formula I should be using I don't really remember all of the signs and exactly what goes where. For me that's when the math dictionary comes very in handy when I'm doing my homework and I can't remeber formulas. I just have to study really hard and maybe I could be a little more confident in my math skills. I think today's class helped because we were able to talk to others and see how they solve things, and maybe learn a thing or two that could potentially help us. We were able to put our minds together and try to answer a question. Good Luck to everyone on their tests and happy studying!!!

and don't forget to blog:)

Dr. Math

After our Muddiest Point exercise in class today I didn't get a chance to answer many of your questions about linear functions. I thought some of you might find answers to your questions in the Dr. Math archives so I've linked to some of them below. Take advantage of Dr. Math whenever you've got a problem or if you don't understand the answer I give you. You can always access it from the [Links] list over there --> in the right hand side bar. ;-)

• Finding the Equation of a Line


• Graphing Linear Equations


• Line Problems


• Find Slope, Equation, Midpoint

This is an interesting question and answer:

When two straight lines meet each other perpendicularly, the product of their slopes is -1. However, x and y axes meet at 90 degrees, but the product of their slopes is not -1. Why?

• Analytical Geometry

Don't forget to take advantages of ALL the reviews I've already posted to the blog. That is your only homework this weekend. ;-)

Getting Ready For The Test!

We've finished our unit on analytic geometry and the linear function. The test is next Thursday, October 13. Here is a list of online reviews and quizzes to help get you ready:

• Linear Equation Test (7 questions)


• Find Equation of Straight Line Given 2 Points (lots of practice)


• Describe a Linear Equation (lots of practice)


• Parallel and Perpendicular Lines (5 questions; refresh page for more)


• Graphing Lines (5 questions; refresh page for more)


• The Equation of a Straight Line (General Review)

You are also responsible for the distance formula, the midpoint formula and working with the slope of a line. Don't forget to take advantage of all the review material in my previous posts!

S-C-R-I-B-E

Hello there, as you can see today I am the scribe (lucky me).

Yes, well today's class wasn't long and we sure did get a big surprise. It's not one of those surprises where you find out what's happening and then your super happy. It's one of those surprises that make you go "ahwhwhwhw, WHY?!"

Well, surprise surprise, the class got a pop quiz today. It was only a page long but it was still difficult. The questions on the quiz was about our graphing unit. The first question dealt with finding the radius of a circle using either the distance formula or the midpoint formula.

The second question was about finding a coordinate because we were only given one coordinate and the midpoint of the line. Third question was about transforming information into a equation, but we call that the "slope-intercept form", I used that cause it makes Mr. K smiles =)
The fourth question required a ruler, cause we had to graph what was given and tell what the slope was, the x-intercept, and the y-intercept.

We then corrected the quizzes (it was for marks BTW). Mr. K told us how things will be marked on the exams and how we should answer the questions on it. Make sure that you use the right formula. ALWAYS put arrows, label your graph, and have the axis labeled properly. If you don't follow what was told, then you just lost 1.5 marks on simple little errors. Also, if your answering a question that requires only the stuff like the slope, x-intercept etc. Only put the number, not the coordinate. Apparently, there both right, but they will only except one way. That's bonkers! Mr. K got all worked up about that in class. Hah.

Yes, and that's all we did today in Mr. K's pre Cal class. Today's homework is exercise 13, do all the question. If you have any questions, just comment on this and I'll get back to you. Tomorrow's scribe is............ stevenF. Yeah, I think he knows, I told him today in class. Have fun being the scribe and good luck.

The Muddiest Point!

We've been studying linear functions for several days now. I'm away for the next two days. Use the comments of this post to share with Mr. Tram, my substitute, your personal "Muddiest Point". You can use your name or leave your comment anonymously, but, whatever you do, share your troubles here. Remember, not only can Mr. Tram help you but you can help each other too! Leave tips and advice in the comments for your classmates. And don't forget, you can form an online study group and "meet" in the chatbox of our blog! Unlike Sysiphus, you're being set up to succeed! Take advantage of every opportunity you've got!

Don't Read The Textbook .... Write It!

If you write it you probably understand it a whole lot better than if you just read it.

The internet has made all textbooks out of date. By the time a book gets published the world community has learned a whole lot more and shared it on the internet. Here's your chance to do the same. ;-)

First, the modern internet textbook is written in wiki format. A wiki is basically a website that can be created as easily as creating a blog post. Watch this to see what a wiki is and how it works.

After that check this out. It's a collection of text books that have been or are in the process of being written by an international community of ordinary people like you and I. As a matter of fact, YOU can add to any of them. Go ahead and do so if you wish. ;-)

What strikes me most powerfully about this latest development on the internet is the fact that anyone can write a textbook on any subject they wish! One of the textbooks being written is called How to pass a course. One of the things I really liked about this textbook was this:

Forming an understanding of the ideas behind each lecture requires active thinking. Try to think ahead of the professor: "What is he going to say next?". If the professor asks someone else a question, answer it in your head. If you answer wrong, try to think why it was wrong.

We will be using a wiki for our story project. Each participant will post their story problem and link to the solution which will be written on another page.

Do you think we should have our very own wiki in our very own webspace or should we add a textbook to the growing list of WikiBooks? Leave your thoughts in the comments to this post.

Cheers,
Mr. K.