| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on Oct 4. |
| Math Jam hosted by markan
(Sean Markan ). |
markan19:33:07
Hello, and welcome to the Introduction to Algebra math jam! Today we'll be discussing what this class covers and how it works, and answering any questions you may have.
markan19:33:14
My name is Sean Markan, and I'll be teaching the course this fall.
markan19:33:27
Introduction of the Virtual Classroom
Before we get started I would like to take a moment to explain our Virtual Classroom to those who have not previously participated in a Math Jam or one of our online classes.
markan19:33:35
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom. This helps keep the class organized and on track. This also means that only well-written comments will be dropped into the classroom, so please take time writing responses that are complete and easy to read. Also, only moderators can enter into private chats with other people in the classroom.
markan19:33:47
Note that it is not possible for the instructor to personally respond to every comment that you submit -- please do not take it personally if your comment is not posted or responded to! I will try to respond to all questions to the extent that I can.
markan19:34:06
In addition, the virtual classroom is LaTeX enabled. LaTeX allows users to make nice equations and other math expressions. If you would like to learn how to write in LaTeX, click on the tab on the left side panel of our site and there is a tutorial and reference guide there.
markan19:34:18
Using LaTeX in the virtual classroom is slightly different than using it on the message board or in a LaTeX editor. If anything you type up in a post uses LaTeX, then you must use a semicolon (;) to begin your post. For example, if you type
markan19:34:25
markan19:34:32
This message will look like this when posted in the classroom:
markan19:34:38
markan19:34:51
One last thing: we recommend not to use a wireless connection while in the classroom. These have a tendency to cause disconnections. Please use a wired connection if possible.
markan19:35:31
that's all we have to say about the virtual classroom, so now I'll start talking about the course
markan19:35:44
we're mainly discussing Introduction to Algebra (though feel free to ask general questions too).
markan19:35:58
As a reminder, our Introduction to Geometry and Introduction to Counting & Probability classes are starting soon. If you're interested in either of those, you can find more information here:
markan19:36:03
ttp://artofproblemsolving.com/Classes/AoPS_C_Enroll.php
markan19:36:12
There will be other Math Jams in the coming weeks to discuss other classes we offer this fall. You can see the full math jam schedule here:
markan19:36:22
All of our online classes take place in this Virtual Classroom.
markan19:36:33
Introduction to Algebra is intended to cover all the fundamental concepts of Algebra, including the following:
markan19:36:41
expressions
equations
quadratics and other polynomials
complex numbers
graphing
functions
sequences and series
exponents and logarithms
markan19:36:49
In terms of what schools usually call Algebra 1 and Algebra 2, our course covers
markan19:36:54
* essentially all the algebraic topics in Algebra 1
markan19:36:58
* most of the algebraic topics in Algebra 2
* certain advanced topics (like telescoping sums and piecewise functions)
markan19:37:08
The reason I referred to ?algebraic topics? is that schools often teach the basics of geometry and counting in their algebra classes. Instead of teaching those topics in our algebra classes, we teach them in our Geometry and Counting & Probability classes, as part of a deeper exploration of those subjects.
markan19:37:23
by the way, feel free to ask questions as we go along
markan19:37:33
Introduction to Algebra, like all our classes, emphasizes problem solving and conceptual understanding rather than rote memorization. So in addition to teaching students how to manipulate equations, we teach them why the techniques are logically sound, and we talk about general problem solving strategies. Our class meetings are largely interactive, meaning that most of the time is spent solving problems. As much as possible, the students do the solving; the teacher only guides them along and provides useful hints.
markan19:37:58
There are a couple major goals of Introduction to Algebra:
xjchen12319:38:06
I am sorry, I am new to math jam. Should there be voice besides the transcript? I couldn't hear anything.
markan19:38:16
nope, just text
markan19:38:17
there's no audio
virtuoso19:38:32
Will this algebra course by very helpful in MATHCOUNTS?
markan19:38:45
if you don't know algebra yet, it will be very helpful
avec_une_h19:38:57
What happens if you miss a class? Is it like at school where you have to ask someone what you have missed?
markan19:39:12
each week there is a transcript of everything that happened during class
markan19:39:15
we post it online
markan19:39:18
so you can just read it to find out what you missed
markan19:39:59
anyway, here's what we hope to teach in this class:
markan19:40:07
Students should develop the ability to translate a situation (which might be a real-life situation, a puzzle, etc.) into the abstract language of equations, and then manipulate the equations to develop insight into the original problem. This skill is the key that opens the door to numerous other fields of study, like higher math, chemistry, physics, and engineering.
markan19:40:21
The real world is the taking-off point which motivates many mathematical concepts. Once one begins to study those concepts, many new questions arise: Do all equations have solutions? Is every number a fraction? Can we invent a number whose square is negative? We delve into some of these questions so that students gain an appreciation for the structure of mathematics, the ability to think abstractly, and the confidence to tackle very difficult questions.
MoTheMan19:40:54
Will this class be helpful to people that take basic Algebra I
markan19:41:23
we cover much more than basic algebra 1, so probably
markan19:41:41
if you want to get a good sense of whether you should take the class, we have diagnostic tests on our website
markan19:42:41
(there are only two different tests there, I sent the same one twice by accident)
avec_une_h19:42:46
Are there any tests?
markan19:42:52
sort of
markan19:42:56
there is weekly homework, which is not collected or graded
markan19:43:12
and there are 4 "midterms"
markan19:43:22
which you print out, do at your convenience, and send back to us
markan19:43:31
there's no time limit on these
markan19:43:38
but you should send them in
markan19:43:48
we'll correct them and give you comments on them
toadoncart19:43:54
will there be MidTerm and Finals
toadoncart19:43:54
are there "finals?"
markan19:44:03
the last midterm is basically the same as a final
toadoncart19:44:07
so Algebra II AND I are covered?
markan19:44:32
yes, though there are a couple things from Algebra II that we don't cover (like ellipses, hyperbolas, and logarithm identities)
vzl1101419:44:37
Are you going to cover the whole book - intro to algebra?
markan19:44:40
yes
virtuoso19:44:51
What do I do if the class begins and I don't have the textbook yet?
toadoncart19:45:18
so I guess this class is the best for an 8th grader, right?
markan19:45:33
toadoncart: it really depends on what you know already
markan19:45:45
there's no fixed age/grade
yuria19:45:48
For the homework do the students go over the problems in class?
markan19:45:56
no, we have an online message board where we will post solutions
rrusczyk19:46:00
Don't worry about not having the text yet; the first class or two are largely intended to be review of basic pre-algebra concepts. You should be fine. (For those that already have the text, it's always a good idea to read ahead!)
markan19:46:03
students can ask questions and discuss the problems there
avec_une_h19:46:26
Do you have to have the book?
rrusczyk19:46:31
Students also can get together in the online classroom to work together on problem sets (this happens in most of our courses).
markan19:46:32
yes, the book is required
Nerd_of_the_Ages19:47:01
are there going to be example problems later in the math jam?
markan19:47:10
yeah, as soon as we finish this round of questions
yuria19:47:19
Will this class help me improve my score on the SSAT or SAT or ISEE?
markan19:47:55
it depends whether you know the material yet -- if you don't, it will probably help you
Rational19:48:01
How much should I study before the class
avec_une_h19:48:10
How will this class be different from my school Algebra I class?
rrusczyk19:48:11
This class will be a lot more challenging than your school course. Not only will it cover some topics that are "Algebra 2", but also, we will cover much harder problems within the topics of "Algebra 1". For example, the Intro Algebra will do much more to prepare you for, say, MATHCOUNTS or the AMC than a typical school Algebra 1 class.
markan19:48:48
Rational: you should be familiar with arithmetic with fractions and decimals, percents, and exponents
markan19:49:03
if you know those things, you probably don't need to study much before we start
markan19:49:06
but it can't hurt to read ahead
markan19:49:15
ok
markan19:49:23
now i'm going to do some example problems to give you a taste of the mechanics of the class
markan19:49:35
Here's a problem which involves some fairly sophisticated thinking.
markan19:49:40
What's the solution to the following equation?
markan19:49:47
markan19:49:56
How can we get started?
markan19:50:37
i see lots of ideas
markan19:50:43
Before doing anything else, it's usually good to simplify an equation as much as possible. How can we simplify this one?
Nerd_of_the_Ages19:50:46
silverspringmom19:50:47
the square root of x^2 is x
toadoncart19:50:53
the square of x and the square root cancel out
sy19:50:57
x-2x=1
Rational19:50:57
sqaure root of x square equal to x
markan19:51:07
Any time two expressions are equivalent, we can replace one with the other.
markan19:51:15
a lot of people are suggesting that we make the following replacement:
markan19:51:20
markan19:51:31
After all, the square root of the square of a number seems to be the original number. For example,
markan19:51:33
markan19:51:37
Let's try this and see where it goes. What's our new equation?
silverspringmom19:52:00
x-2x
MoTheMan19:52:00
x-2x=1
markan19:52:06
markan19:52:15
Now we can combine like terms (using the distributive law). What does that give us?
xpmath19:52:22
-x=1
MoTheMan19:52:25
-x=1
Rational19:52:26
-x=1, x= -1
markan19:52:43
markan19:52:47
So it looks like x = -1
virtuoso19:52:54
x=-1
yuria19:52:54
x= -1
markan19:53:00
Alright, now, it's always a good idea to check your work by plugging your solution into the original equation. Let's do that. If we plug x = -1 into
markan19:53:05
markan19:53:09
what do we get on the left hand side?
markan19:54:48
it's 3
markan19:54:29
Uhoh! Is x = -1 a solution?
markan19:55:15
so what have we done wrong?
markan19:56:18
remember: we tried to solve this equation:
markan19:56:30
markan19:56:37
and we took seemingly legitimate steps and got x = -1
markan19:56:42
but if we plug in x = -1, we get 3 = 1
markan19:56:51
so somehow we made a mistake
avec_une_h19:56:57
if x is negetive square root of x square isn't the same is it
virtuoso19:56:57
\sqrt{x}=\pm1 with a ;
krakola4519:56:57
square root of x^2 is absolute value of x
markan19:57:02
yeah, that's the problem
markan19:57:09
we made the assumption that x was positive
markan19:57:14
markan19:57:15
isn't always true
markan19:57:24
it's true if x is positive, but not if x is negative
markan19:57:36
if x is negative, what does
markan19:57:41
markan19:57:44
work out to?
MoTheMan19:58:12
|x|
samsun5188219:58:12
|x|
elan19:58:12
-x
silverspringmom19:58:12
|x|
markan19:58:18
right, -x, which is the same as |x|
markan19:58:40
when we solved our equation before, we assumed x was positive, but there's another possibility
markan19:58:41
x could be negative
markan19:58:53
Let's try solving the equation again, this time assuming x is negative.
What equation do we have now?
TY35819:59:08
-3x=1
markan19:59:11
markan19:59:16
This reduces to
markan19:59:19
markan19:59:22
markan19:59:35
Does this solution work?
samsun5188220:00:02
yep
MoTheMan20:00:02
yes
elan20:00:02
yes
virtuoso20:00:02
yes!
toadoncart20:00:07
hmmmmmmmmm............ -1/3+2/3=1!
TY35820:00:07
yes
samsun5188220:00:07
1/3 + 2/3 = 1
markan20:00:14
so now we've found the real answer
markan20:00:17
This problem illustrates the importance of not making any hidden assumptions when you attempt to solve an equation. You must keep *all* possible solutions under consideration rather than assuming the solution is positive.
markan20:00:40
I'll do one more problem.
markan20:00:46
Bottle A contains 10% alcohol. Bottle B contains 20% alcohol. In what ratio do I need to mix liquid from the two bottles if I want to create a liquid that is 16% alcohol?
markan20:00:54
How could we approach this?
virtuoso20:01:16
Set up an equation
Nerd_of_the_Ages20:01:16
we could write an equation
markan20:01:25
even before that?
virtuoso20:01:39
Set up our variables
markan20:01:44
yeah
markan20:01:51
Sometimes it's good to be a little more concrete than a problem asks you to be. In this case, the problem only wants a ratio, but let's imagine we're going to mix specific amounts of the two liquids to get a specific mixture. In other words, let's find *all* the ways we can get a 16% alcohol mixture, and then see what that tells us.
markan20:02:01
I'm going to let a be the number of liters we take from bottle A and b be the number from bottle B. We're going to make sure we get a 16% alcohol mixture at the end.
markan20:02:10
Ok, so how can we express our goal?
markan20:03:08
this is the "write an equation" part
markan20:03:09
what things can we set equal?
avec_une_h20:03:35
A liters from bottle A and B liters from B=16 percent
Rational20:03:35
(10%a+20%b)/a+b=16%
markan20:03:45
yeah, that's the idea
markan20:03:56
we want to make sure that16% of (a + b) = amount of alcohol in our mixture
markan20:04:05
But we can rewrite the "alcohol" part:
markan20:04:08
16% of (a + b) = 10% of a + 20% of b
markan20:04:23
can you rewrite this a little more in pure math?
virtuoso20:05:09
.16(a+b)=.1a+.2b
silverspringmom20:05:09
.1a+.2b=.16(a+b)
MoTheMan20:05:09
0.1a+0.2b=.16(a+b)
TY35820:05:09
0.16(a+b)=0.1a+0.2b
markan20:05:18
.16(a+b) = .10a + .20b
markan20:05:41
How do we solve this?
virtuoso20:05:58
FInd a relationship between a and b.
markan20:06:03
how?
Nerd_of_the_Ages20:06:14
move all of one variable to one side
markan20:06:17
ok, what does that give us?
yellowlime_220:06:38
.6a=.4b
TY35820:06:38
0.06a=0.04b
markan20:06:41
yeah
markan20:06:48
so what's the relationship between a and b?
TY35820:06:57
a/b=0.04/0.06
TY35820:07:03
2/3
markan20:07:05
a/b = 2/3
markan20:07:13
so what does this mean in terms of our original problem?
MoTheMan20:07:42
60% is bottle b and 40% is a
TY35820:07:46
thats the ratio. 2 litres from A and 3 litres from B.
markan20:07:49
We need 2 liters of bottle A for every 3 liters of bottle B, and we'll have a 16% alcohol mixture. The ratio is 2:3.
silverspringmom20:07:52
40% A and 60% B
markan20:08:01
Now it turns out you can sometimes just look at a problem like this and know the answer! I'm going to show you a trick for this one.
markan20:08:10
First let's ask ourselves: is the answer we got, 2:3, a reasonable answer?
Remember:
bottle A = 10% alcohol
bottle B = 20% alcohol
mixture = 16% alcohol
markan20:08:17
Just looking at these numbers, would you expect to use more of bottle A or bottle B?
markan20:08:23
(without even solving any equations)
MoTheMan20:08:29
B
yellowlime_220:08:29
b
toadoncart20:08:29
B
markan20:08:33
You want more of bottle B, because your "target" percentage, 16%, is closer to bottle B's percentage than bottle A's.
markan20:08:43
In fact, 16% is 4% away from bottle B, but 6% away from bottle A. 4% to 6% is a 2:3 ratio, and so the answer is 2:3!
markan20:08:53
Chances are you don't immediately see why that works; it's hard to see it with out actually going through the math (which I encourage you to do!). Unless you understand the trick, you're safer doing it the long way. But once you build up enough intuition about ratios and percents, you'll be able to make leaps like that.
markan20:09:11
In these two problems, we moved rapidly through many ideas: solving equations, substitution of equivalent expressions, hidden assumptions, mixtures, qualitative reasoning, etc. In class we will spend much more time developing these ideas, so if you felt lost just now that's ok.
markan20:09:20
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
markan20:09:42
i'm going to post some more logistical information
markan20:09:58
but feel free to ask any questions you have now, about algebra, our classes in general, or AoPS
markan20:10:06
DATES AND TIMES
The course will meet for 24 weeks on Wednesdays, starting October 17, at 7:30 PM Eastern / 4:30 PM Pacific. (There are no classes the week of Thanksgiving or December 17-January 4.) Each class is 90 minutes.
markan20:10:12
INSTRUCTOR
I'll be teaching this course; if you're so inclined, you can find a short biography for me and all the other AoPS teachers here:
http://artofproblemsolving.com/Classes/instructors.php
markan20:10:20
TEXTBOOK
This course will use a textbook in conjunction with the course: our own Introduction to Algebra book. The material covered in the textbook is roughly equivalent to the material covered in the course. You can see the table of contents and some excerpts from the book here:
markan20:10:31
The book is required for the course. Students will be able to read additional material that complements the lectures, and will have access to a large number of practice problems at varying levels of difficulty. We are recommending that students read the corresponding chapter(s) in the book before each lecture, and attempt some of that chapter's Review and Challenge Problems after each lecture. We also expect to spend some class time answering students' questions about problems from the textbook.
The homework for the class consists of weekly problems that will be posted to the class message board -- for these problems, you do not turn your solutions in, but you may post them to the message board if you like. The class also between 2 and 4 (exact number to be decided) longer problem sets (?midterms?) for which you should write up your full solutions and submit them. These solutions will be read, and you will receive detailed feedback.
Nerd_of_the_Ages20:10:33
is there any idea of when the next intermediate number theory is held?
markan20:11:21
starting in a couple weeks
markan20:11:25
and after that, probably summer 2008
virtuoso20:11:29
Is this the first time that there has ever been an Introduction to Algebra Class?
markan20:11:32
yes
toadoncart20:11:46
is this ur first time teaching
markan20:12:02
it's my first time teaching a full course for AoPS, but not my first time teaching
Rational20:12:25
what classes are ready to take after this class?
markan20:12:41
any of our other intro classes: geometry, number theory, and counting & probability
markan20:12:47
and also intermediate algebra
krakola4520:13:04
is there calculus
markan20:13:14
there's a good chance we'll eventually offer it
markan20:13:17
but not yet
toadoncart20:13:41
what classes have u first taught?
markan20:14:43
i've taught a class on linear algebra, TA'ed one on artificial intelligence, and done lectures on lots of math topics
elan20:14:47
will you have extra help?
rrusczyk20:14:48
There will be assistants in the class - the assistants will help take questions from students, and to work with them privately if they are falling behind or racing ahead.
avec_une_h20:14:57
Are all the classes at the rate of the example problems?
markan20:15:25
more or less
markan20:15:33
however, we spend much more time on each individual topic
markan20:15:49
so it won't be as difficult to keep up as it was just now
vzl1101420:15:53
is there audio and video for the online class?
rrusczyk20:15:56
No; we have found that the text-based environment is much more effective. Here is a description why: http://www.artofproblemsolving.com/Classes/classroom.php
toadoncart20:16:26
shouldn't calculus be covered in WOOT?
rrusczyk20:16:28
Calculus is not required for the USAMO, so it is not part of WOOT.
krakola4520:16:37
will we do any more problems today
markan20:16:41
no, that's all for today
MoTheMan20:17:01
how many people are you supposed to have in class?
markan20:17:08
it varies a lot
markan20:17:12
we expect 40-50
elan20:17:30
if somebody is having problems understanding, how will you adress them?
Rational20:17:32
Should we do all the exercizes from the book?
rrusczyk20:17:32
Our assistants will work with students on a 1-1 basis if they are having trouble during class. Students who have questions after class can ask questions on the class message board and instructors will answer them.
markan20:17:45
Rational: you should do all the ones we assign
markan20:17:57
and any others that you think will help you understand the concepts better
markan20:18:02
you don't have to do all of them
yuria20:18:20
can we do more?
markan20:18:26
of course!
MoTheMan20:19:03
are there any more books that you recomend?
markan20:19:13
any of the AoPS books
markan20:19:36
for algebra level, the Introduction series (Introduction to Geometry, Introduction to Counting & Probability, Introduction to Number Theory) and Art of Problem Solving Volume 1 are most appropriate
Rational20:19:40
what if you miss a class?
markan20:19:46
we have a transcript of each class posted online
markan20:19:51
so you can just read it to see what you missed it
markan20:19:58
*what you missed
pdgcss220:21:19
Are there any workbooks recommended/required/available for supplemental exercises?
rrusczyk20:21:21
Only the Introduction to Algebra text is required for the course. We don't believe you should expect to need further supplemental material in general, as there's a lot in the text. But if there are topics that you need a little extra help on, we can provide that if you ask on the course message board.
Rational20:21:48
How do you you write math symbols from the message board?
rrusczyk20:21:51
We use LaTeX. You can learn about LaTeX by clicking the LaTeX link on the sidebar of the website.
yellowlime_220:22:02
will everyone including people who are not in the class be able to see it?
markan20:22:15
transcripts are only available to people in the class
avec_une_h20:22:30
what's the difference between intro to algebra and intermediate algebra?
markan20:22:41
intermediate algebra is just more advanced algebra topics
markan20:23:03
you can see the syllabi by going to the classes sections of our website and visiting the pages for the two classes
markan20:23:49
there was a question about math symbols on the message board
markan20:23:59
you would just enclose them in dollar signs
markan20:24:00
like
markan20:24:04
$45^\circ$
markan20:24:07
for 45 degrees
elan20:24:12
what if you have already covered some material?
markan20:24:50
we tend to cover material in more depth, and with harder problems, than many schools
markan20:25:02
so you may learn something new about the topics you've already covered
markan20:25:22
if you think you may know too much to take Intro Algebra, you should try out our Post Test and see how you do
helagha20:27:26
Do you attach links if you decide to show some graphs?
markan20:27:29
yes
Rational20:27:33
what's AMC8 or AMC 10 basic requirements?
rrusczyk20:27:34
AMC 8: you should probably have a basic algebra background, and some exposure to basic geometry. For the AMC 10, you're expected to have finished algebra 1 and geometry, and probably have done much of Algebra 2.
rrusczyk20:27:59
There is probably a more thorough list on the AMC site.
rrusczyk20:28:26
Moreover, they expect you to have covered basic number theory and counting concepts that are not typically taught much in the standard curriculum (but are taught at AoPS!)
markan20:29:50
in a second i'll show you an example picture
Rational20:30:13
Do you have on line classes after this class finish? I mean
Rational20:30:13
in spring and summer?
rrusczyk20:30:15
Yes; we'll have our Spring schedule up soon, and our summer schedule will be up in a few months.
rrusczyk20:30:30
We'll offer Introduction to Number Theory in the spring, which would be a good follow-up to the Algebra course.
rrusczyk20:30:43
We'll offer Introduction to Counting & Probability in the summer.
markan20:30:52
markan20:31:02
this is an example from one of our other classes
Black Minx20:31:38
What is the diagram for?
markan20:31:53
just an example from our trigonometry class
markan20:32:00
it's showing how the sun lights up the earth
helagha20:32:07
do you provide links to short videos?
rrusczyk20:32:10
We don't have any videos in the course currently, but will be experimenting with them over the next year.
Black Minx20:32:41
So they will be coming soon?
markan20:32:42
alright, thank you all for coming
rrusczyk20:32:42
I hope so!
markan20:32:52
We hope to see you in class!
