| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on Oct 7. |
| Math Jam hosted by DPatrick
(Dave Patrick ). |
DPatrick19:28:29
Hello, and welcome to an Art of Problem Solving Math Jam.
DPatrick19:28:38
Today we'll be discussing the Introduction to Algebra course, both MATHCOUNTS/AMC 8 Problem Series classes, and the Intermediate Trigonometry/Complex Numbers course. We will go through a couple example problems from each class, and discuss both what these classes cover and how they work.
DPatrick19:28:51
My name is Dave Patrick. I wrote both of Art of Problem Solving's "Counting & Probability" books, and I teach many classes here at AoPS.
DPatrick19:29:03
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.
DPatrick19:29:16
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom. So, when you send a message, it will not appear immediately, and may not appear at all.
DPatrick19:29:34
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.
DPatrick19:29:52
Note that it is not possible for me to personally respond to every comment that you submit -- please do not take it personally if your comment is not posted or responded to!
DPatrick19:30:05
I will try to respond to all questions to the extent that I can. I will let you know when to start asking questions about the classes.
DPatrick19:30:22
Tonight, we'll be discussing the Intro Algebra class, then the MATHCOUNTS/AMC courses, and then the Trigonometry/Complex Numbers class.
DPatrick19:30:40
(If you're here just to learn about the trigonometry class, we should be getting to that in about 45 minutes...you can leave and come back then if you want.)
DPatrick19:31:02
The first class I'll discuss is our Introduction to Algebra class.
DPatrick19:31:15
Introduction to Algebra is intended to cover all the fundamental concepts of Algebra, including the following:
DPatrick19:31:25
expressions
equations
quadratics and other polynomials
complex numbers
graphing
functions
sequences and series
exponents and logarithms
DPatrick19:31:42
In terms of what schools usually call Algebra 1 and Algebra 2, our course covers
* essentially all the algebraic topics in Algebra 1
* most of the algebraic topics in Algebra 2
* certain advanced topics (like telescoping sums and piecewise functions)
DPatrick19:32:02
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.
DPatrick19:32:17
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.
DPatrick19:32:35
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.
DPatrick19:32:45
There are a couple major goals of Introduction to Algebra:
DPatrick19:32:52
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.
DPatrick19:33:04
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.
DPatrick19:33:17
I'm now going to run through a couple algebra problems to give you a taste of how the class with run. After that I'll spend some time taking questions.
DPatrick19:33:26
Here's a problem which involves some fairly sophisticated thinking.
DPatrick19:33:30
What's the solution to the following equation?
DPatrick19:33:33
DPatrick19:33:48
How can we get started?
ReseesPieces19:34:16
add 2x to both sides
davidd219:34:22
move 2x to right side of equation
AALAP19:34:29
add 2x to both sides and then square both sides
DPatrick19:34:49
We could do that: add 2x to both sides, with a eye towards squaring both sides, but instead...
asethi0119:35:00
basically simplify the equation
DPatrick19:35:05
...yeah, it's usually good to simplify an equation as much as possible. How can we simplify this one?
mathfreak7719:35:26
take x^2 out of the square
asethi0119:35:26
square root of x^2=x
emcat19:35:26
replace sqrtx^2 with x
ReseesPieces19:35:26
square root of x2 is -x or x
DPatrick19:35:33
Any time two expressions are equivalent, we can replace one with the other.
DPatrick19:35:49
davidd219:36:06
x may be negative
ero--senin19:36:22
nope sqrt of x^2=|x|
emcat19:36:22
so + - x
DPatrick19:36:30
Good! As you (and ReseesPieces a moment earlier) pointed out, we might have -x instead!
DPatrick19:36:39
Let's first see how we could have discovered our mistake.
DPatrick19:36:46
Suppose we replace sqrt(x^2) with x.
DPatrick19:36:51
What's our new equation?
emcat19:37:15
x-2x=1
isiahbunshart1019:37:15
x-2x=1
jennysoh19:37:21
-x=1
simo1419:37:22
-x=1
Math4619:37:27
-x = 1
DPatrick19:37:36
Right, we get x - 2x = 1.
DPatrick19:37:42
This simplifies to -x = 1, so x = -1.
DPatrick19:37:52
It's always a good idea to check your work by plugging your solution into the original equation.
DPatrick19:38:02
If we plug x = -1 into
DPatrick19:38:06
DPatrick19:38:11
what do we get?
isiahbunshart1019:38:40
3=1
simo1419:38:40
3=1
Godzilla!19:38:46
false
binmu19:38:46
3=1
DPatrick19:38:51
DPatrick19:39:09
So we get 3=1. Therefore x = -1 cannot be a solution.
DPatrick19:39:32
As some of you mentioned earlier, we made a faulty assumption:
DPatrick19:39:36
DPatrick19:39:48
We can only make this assumption if we know that x is positive.
DPatrick19:40:00
What should we do now?
ReseesPieces19:40:21
plug -x
Twin Prime Conjecture19:40:21
-x-2x=1
simo1419:40:23
make the square root of x^2 = -x
DPatrick19:40:31
DPatrick19:40:43
So we can try solving the equation again, this time assuming x is negative.
DPatrick19:40:58
LK119:41:13
-3x = 1
Twin Prime Conjecture19:41:21
x=-1/3
jennysoh19:41:21
x=-1/3
DPatrick19:41:33
This reduces to -3x = 1, so x = -1/3.
DPatrick19:41:49
This is indeed negative (as we assumed), so we've found the solution.
DPatrick19:41:57
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.
DPatrick19:42:27
We'll do one more problem.
DPatrick19:42:37
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?
DPatrick19:42:52
How could we approach this?
tenisu_no_oujisama19:43:39
set a variable first
DPatrick19:43:56
DPatrick19:44:21
How do we express our goal?
simo1419:44:37
(.1*A + .2*B)/(A+B) = 16
binmu19:44:37
(.1A+.2B)/A+B=.16
DPatrick19:44:52
Right.
ReseesPieces19:44:58
oh so .1a+.2b=.16(a+b)...i think
melvinmoose19:45:02
.1a + .2B = .16c
LK119:45:02
10%a + 20%b= 16% ... ?
DPatrick19:45:10
We could right it that way too.
DPatrick19:45:34
When we mix, we get .1a litres of alcohol from bottle A, and .2b litres of alcohol from bottle B.
DPatrick19:45:46
We need this to be .16 of the combined total, which is a+b.
DPatrick19:46:00
So our equation is .16(a+b) = .10a + .20b
DPatrick19:46:18
Now what?
asethi0119:46:30
expand the left side.
binmu19:46:30
so .16A+.16B=.10a+.20b
jennysoh19:46:33
multiply by 100
DPatrick19:46:49
Let's multiply through by 100 (to get rid of the decimals) and expand:
DPatrick19:46:54
16a + 16b = 10a + 20b
DPatrick19:47:03
Now what?
binmu19:47:12
Then you simplify
ReseesPieces19:47:21
6a=4b
LK119:47:21
6a=4b
Math200819:47:21
6a=4b
Twin Prime Conjecture19:47:21
6a=4b
emcat19:47:21
6a=4b
DPatrick19:47:30
Simplifying gives 6a = 4b.
DPatrick19:47:43
What does this mean? What's the answer?
simo1419:47:54
6a/4b=1
Math200819:47:54
a/b=4/6
simo1419:47:54
a/b=4/6=2/3
ReseesPieces19:47:54
2/3
Twin Prime Conjecture19:47:57
a/b=2/3
DPatrick19:48:06
We rearrange to get a/b = 2/3.
DPatrick19:48:11
So 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.
DPatrick19:48:30
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.
DPatrick19:48:38
First let's ask ourselves: is the answer we got, 2:3, a reasonable answer?
DPatrick19:48:42
Remember:
bottle A = 10% alcohol
bottle B = 20% alcohol
mixture = 16% alcohol
DPatrick19:48:47
Just looking at these numbers, would you expect to use more of bottle A or bottle B?
mathfreak7719:49:04
b
isiahbunshart1019:49:04
bottle b
alexperkowski19:49:04
B
asethi0119:49:04
B
LK119:49:04
b
Math200819:49:04
b
emcat19:49:06
yes, because it's closer to 20% than to 10%
DPatrick19:49:12
You want more of bottle B, because your "target" percentage, 16%, is closer to bottle B's percentage than bottle A's.
DPatrick19:49:24
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!
DPatrick19:49:39
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.
DPatrick19:49:53
In these two problems, we moved fairly rapidly through many ideas: solving equations, substitution of equivalent expressions, hidden assumptions, mixtures, qualitative reasoning, etc.
DPatrick19:50:07
In class we will spend much more time developing these ideas, so if you felt lost just now that's OK.
DPatrick19:50:24
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
http://www.artofproblemsolving.com/Classes/IntroAlgebra/PostTest.pdf
DPatrick19:50:47
(Or go to the "Online Classes" section of the website and follow the links to "Introduction to Algebra")
DPatrick19:50:59
Here's some practical information about the class:
DPatrick19:51:04
The course will meet for 27 weeks on Thursdays, starting October 16, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and each is 7:30 - 9 PM ET (4:30 - 6 PM PT). There is no class on the following holidays: Nov 27, Dec 25, Jan 1. The last day of the class is May 7.
DPatrick19:51:15
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:
http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=200
rikkilay19:51:29
for those who have popup blocker... hold control while you click the link!
DPatrick19:51:38
Right...thanks for mentioning that (I forgot to say that).
DPatrick19:51:52
The textbook 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 recommend 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.
DPatrick19:52:11
The course will be taught by Richard Rusczyk. Richard is the founder of Art of Problem Solving, one of the co-authors of the Art of Problem Solving textbooks, author of Art of Problem Solving's Introduction to Algebra and Introduction to Geometry textbooks, and co-author (with Mathew Crawford) of AoPS's Intermediate Algebra textbook. He was a participant in National MATHCOUNTS, a participant in the Math Olympiad Summer Program 1987, 1988 and 1989, the only perfect scorer on the 1989 AIME, and a USA Mathematical Olympiad winner (1989).
DPatrick19:52:36
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 has 4 Challenge Sets for which you should write up your full solutions and submit them. You will receive thorough feedback for your work on these Challenge Sets that will comment both on your mathematical accuracy and how well you write solutions.
DPatrick19:52:59
Are there any questions specifically about this class or the textbook? (You'll also be able to ask questions at the end.)
mitlh_219:53:09
is there a class transcript?
DPatrick19:53:20
Yes, all of our classes have transcripts that you can access at any time.
isiahbunshart1019:53:42
How do you submit solutions
DPatrick19:53:59
By mail, email, or fax. You'll be given detailed instructions when they're assigned.
kidsrok19:54:08
Is this class ok for a 7th grader?
DPatrick19:54:34
Check out our diagnostic pre-test (labeled "Are you ready?") in the Online Classes section of the website.
DPatrick19:55:01
I'm going to go on with the discussion of our next class...I'll take more questions at the end.
DPatrick19:55:19
This fall, we are offering two different MATHCOUNTS/AMC 8 classes: MATHCOUNTS/AMC 8 Basics and Mastering MATHCOUNTS/AMC 8.
DPatrick19:55:27
The MATHCOUNTS/AMC 8 Basics course will meet for 12 weeks on Wednesdays, starting October 15, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and each is 7:30 - 9 PM ET (4:30 - 6 PM PT). This class is for students just getting started with the type of problem solving required for success in MATHCOUNTS and the AMC 8.
DPatrick19:55:46
Our Mastering MATHCOUNTS/AMC 8 is for more experienced students, such as those who are training for State MATHCOUNTS, with hopes of qualifying for National MATHCOUNTS. The Mastering MATHCOUNTS/AMC 8 course will meet for 12 weeks on Tuesdays, starting November 11, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and each is 7:30 - 9 PM ET (4:30 - 6 PM PT).
DPatrick19:56:01
Both classes include some material from our previous MATHCOUNTS Problem Series class. The Mastering MATHCOUNTS/AMC 8 class will be a little more challenging than the old class, and the MATHCOUNTS/AMC 8 Basics course will be a little easier.
DPatrick19:56:13
Each class will be taught by Joshua Zucker. Joshua has been a Math Olympiad Summer Program invitee, a member of the first US Physics Olympiad team, and a top-10 scorer on the Putnam. He holds a BS in physics and an MS in mathematics from Stanford, as well as an MS in astrophysics from UC Berkeley. He has taught at levels ranging from summer camps for gifted elementary school students through remedial arithmetic at community college. He was a middle and high school teacher in Palo Alto, CA and was formerly a problem writer for MATHCOUNTS.
DPatrick19:56:35
While there is overlap in topics between the two classes, there will be almost no overlap in problems. Topics covered including methods of counting, probability, algebraic techniques, geometry, word problems, number theory, and more.
DPatrick19:56:54
This class is a Problem Series class, meaning that the major focus of the class will be working through various contest problems. Although there will be weekly problem sets for each class posted on the message board, students do not submit their homeworks to be graded, and there is no personalized instructor feedback on the solutions. (However, the instructors will be monitoring the message board to answer questions and comment occasionally on the students' discussions there.)
DPatrick19:57:17
Each course will also include a whole class that is a single giant Countdown Round contest!
DPatrick19:57:26
We'll now look at a few sample problems from the MATHCOUNTS/AMC 8 courses. Hopefully these problems will give you an idea of which course is right for you.
DPatrick19:57:40
We'll start with a couple problems that are examples of easier problems. These are problems that would be included in the Basics class, but would only be used as a warm-up (or not at all) for the Mastering MATHCOUNTS/AMC 8 course.
DPatrick19:57:47
simo1419:58:36
express as 2^40 * 5^36
DPatrick19:58:55
DPatrick19:58:58
How does that help?
isiahbunshart1019:59:19
2*5=10
Sychiri19:59:19
pairs of 10 can be made
Twin Prime Conjecture19:59:23
10^36*2^4
jennysoh19:59:25
2^36*5^36 = 10^36
DPatrick19:59:32
Aha. Number of digits means we care about powers of 10. Let's rewrite our product so we can see them:
DPatrick19:59:41
jennysoh19:59:47
2^4 = 16
DPatrick19:59:58
Right...so what can we conclude? What's the answer?
MATVP20:00:12
38
Twin Prime Conjecture20:00:12
so the answer is 38
Godzilla!20:00:12
16 36 zeroes
isiahbunshart1020:00:12
38
mitlh_220:00:12
38 is the answer
BOGTRO20:00:12
1600000000000000000000000000000000... so 38 digits
binmu20:00:12
2+36=38
LK120:00:12
38
DPatrick20:00:20
Right: our result is 16 followed by 36 zeroes, which clearly has 38 digits.
DPatrick20:00:32
Here's another basics-level problem:
DPatrick20:00:36
rikkilay20:01:03
what does ! mean?
DPatrick20:01:11
Good question: The exclamation point is a 'factorial'. 4! = 4 x 3 x 2 x 1, and factorials for all other positive integers are defined similarly - we multiply all the numbers from the given positive integer down to 1.
DPatrick20:01:38
How can we simplify this so we don't have to multiply it all out?
simo1420:01:44
express as 5!6!/(6*5! + 5! + 5!)
Twin Prime Conjecture20:01:47
denominator=5!*8
Godzilla!20:01:52
6!/(6+1+1)
MATVP20:01:55
6!/8
DPatrick20:02:06
MATVP20:02:20
90
BOGTRO20:02:20
and we are left with 720/90=8
Twin Prime Conjecture20:02:20
90
simo1420:02:20
which is 90
DPatrick20:02:30
So the answer is 720/8 = 90.
DPatrick20:02:37
Clever factoring is extremely useful in math problems, and often is helpful in MATHCOUNTS.
DPatrick20:03:08
The next two problems are examples of problems that would be on the harder end of the Basics class and the easier end of the Mastering MATHCOUNTS/AMC 8 class.
DPatrick20:03:12
BOGTRO20:03:40
5! and above have last digit 0
DPatrick20:03:48
Right. We don't get intimidated by the 99!. We only want the last digit. Most of these terms have last digit of 0.
jennysoh20:04:02
1
BOGTRO20:04:02
So, 1-2+6-24=-19. Therefore, 1.
DPatrick20:04:08
Only the first 4 terms have a nonzero units digit:
1 - 2 + 6 - 24 = -19.
DPatrick20:04:21
But the answer isn't 9 because our sum is clearly positive. All those terms with a zero units digit clearly have a positive sum, so our expression equals -19 + (something big that ends in zero). Thus we have a final digit of 1, not 9.
DPatrick20:04:44
Here's another "hard Basics" but "easy Masters" problem:
DPatrick20:04:48
DPatrick20:05:11
Where can we start with this problem?
lifeisacircle20:05:34
note that p=2
BOGTRO20:05:34
p and p+1 are prime, so p must be 2
jennysoh20:05:34
2 and 3 are prime numbers
Twin Prime Conjecture20:05:34
p=2
LK120:05:34
the only nums that match are 2 &3
Sychiri20:05:34
only value for p is 2
ReseesPieces20:05:34
i mean 2 and 3
binmu20:05:34
only 2 and 3
DPatrick20:05:41
Right: the only consecutive primes are 2 and 3.
DPatrick20:05:42
So p = 2.
DPatrick20:05:48
Now, how do we find the smallest composite number that is neither a multiple of 2 nor 3?
jennysoh20:06:13
We then get the next prime number, 5 and square it. 5^2 = 25
BOGTRO20:06:13
Therefore, there must be at least 2 factors, neither 2 or 3, so 5*5=25 is the smallest
Ma4Gramp20:06:13
5 x 5 then
Twin Prime Conjecture20:06:16
5*5=25
DPatrick20:06:21
DPatrick20:06:46
Again, those last two problems would be considered on the hard end of the Basics course but on the easy end of the Masters course.
DPatrick20:06:52
Finally, here are a couple problems that are on the harder end of the Mastering MATHCOUNTS/AMC 8 class.
DPatrick20:06:57
BOGTRO20:07:24
Prime factorize it
binmu20:07:24
factor 792
DPatrick20:07:36
DPatrick20:07:50
So how do we count the even divisors?
BOGTRO20:08:03
There are 24 total factors
DPatrick20:08:13
DPatrick20:08:24
How do we count only the even divisors?
jennysoh20:08:35
3*3*2
binmu20:08:35
3*3*2
isiahbunshart1020:08:35
we have to have 2 to a power greater than 0 included]
lifeisacircle20:08:35
any number with at least 1 power of 2 in it's prime factorization is even
DPatrick20:08:48
Right: the even divisors have a positive exponent for the prime 2.
DPatrick20:08:54
So for even divisors, a can be 1, 2, or 3.
DPatrick20:09:03
DPatrick20:09:25
(You could also count the odd divisors and subtract them; I'm going to omit that solution for time.)
DPatrick20:09:31
One more hard MATHCOUNTS problem:
DPatrick20:09:36
DPatrick20:09:55
How can we tackle this problem without going to base ten first (which would take a long time)?
LK120:10:11
how do u do bases?
tenisu_no_oujisama20:10:11
what do you mean by base number?
Math4620:10:11
what's a base?
mathfreak7720:10:11
whate does basea mean?
DPatrick20:10:30
Unfortunately I don't have time to explain it tonight; you might get some idea from the solution though.
jennysoh20:10:36
9 is 3^2
binmu20:10:36
9=3*3...
DPatrick20:10:47
jennysoh20:11:02
The base 3 number needs to get paired starting from 2.
DPatrick20:11:14
jennysoh20:11:30
2612
isiahbunshart1020:11:30
2612 base 9
DPatrick20:11:52
Indeed, if we pair up the numbers 2(20)(01)(02), we get 20_3 = 6, and thus 2200102_3 = 2612_9.
DPatrick20:12:02
Or, to write it out more thoroughly:
DPatrick20:12:09
DPatrick20:12:36
If the last 3-4 problems we did seem impossible to you, then the MATHCOUNTS/AMC 8 Basics class is probably the right class for you. If the first 3-4problems seem way, way easy to you, and you understood the solutions to the last couple of problems, then the Mastering MATHCOUNTS/AMC 8 class is the right class for you.
DPatrick20:12:49
Are there any questions about either of the two MATHCOUNTS courses?
ReseesPieces20:12:57
what if you are inbetween
DPatrick20:13:18
You'll have to decide on one or the other. It definitely won't make sense to take both.
DPatrick20:13:38
We'd probably recommend starting in the Basics course, and if you find it way too easy, you can switch to the Masters course.
Math4620:13:47
Is everything about mathcounts problems or is there any focus or the AMC 8?
DPatrick20:14:00
We generally focus on MATHCOUNTS, but much of the same material is also on the AMC 8.
LK120:14:30
so amc8 and mathcounts are similar?
DPatrick20:14:57
They cover similar areas of math. AMC 8 is in the fall and is multiple choice. MATHCOUNTS is in winter/spring, is short answer, and has a team component.
DPatrick20:15:22
I'll take more questions at the end, but let me now move on to the Intermediate Trig/Complex Numbers course.
DPatrick20:15:51
First, a disclaimer: if you're here primarily for Intro Algebra or MATHCOUNTS, then the next 10-15 minutes or so is probably going to make no sense.
DPatrick20:16:12
If you have more questions, you can stick around to the end; but please feel free to leave know if you have all the info you need.
DPatrick20:16:22
The Intermediate Trigonometry/Complex Numbers course is for students who have mastered basic algebra and geometry, including: the Pythagorean Theorem, solving basic equations and systems of equations, graphing simple functions, the geometry of triangles, circles, and arcs, and the basic arithmetic of complex numbers. We do not assume any past experience working with trigonometry.
DPatrick20:16:36
The course will meet for 12 weeks on Mondays, starting October 13, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and each is 7:30 - 9 PM ET (4:30 - 6 PM PT)
DPatrick20:16:51
The instructor for this course is Sean Markan. Sean participated in numerous math and science programs in high school, including the Math Olympiad Summer Program in 2001 and the US Physics Team in 2000 and 2002. He also won the Mandelbrot Competition in 2002. He graduated from MIT with a degree in Physics in 2006. Sean has previously taught Art of Problem Solving's Introduction to Algebra and MATHCOUNTS Problem Series courses.
DPatrick20:17:09
Each class consists of the instructor leading the students through a series of increasingly challenging problems. An additional assistant instructor is present to help students who need extra attention.
DPatrick20:17:26
After each class, extra problems are given to the students on a course message board for full-class discussion. The course also includes 2 Challenge Sets, one for the first half of the course and one for the second half. The Challenge Sets consist of 20-30 problems for which students should write full solutions and submit them to us for evaluation. This evaluation will include commentary both on the accuracy of the solution, and on the student's ability to write mathematics effectively.
DPatrick20:17:45
Here's an example of a trigonometry problem that we'll be doing in the course. The solution of this problem may use a couple of trig identities that you're not familiar with. Don't worry -- we'll be covering them in the class!
DPatrick20:17:52
DPatrick20:18:03
No calculators!!
DPatrick20:18:20
What should we do? Should we try to compute sin(20) and all the others?
simo1420:18:33
make tan=sin/cos and cot=cos/sin
emcat20:18:33
we could switch tan into sin/cos
emcat20:18:33
and cot into cos/sin
DPatrick20:18:51
We can write the tan and cot terms of sines and cosines (sines and cosines, sines and cosines; not always the way to the solution, but it's usually the best way to bet).
DPatrick20:18:56
DPatrick20:19:06
What about that sin 20 term?
simo1420:19:22
also make sin20=sin(10+10)=2sin10cos10
DPatrick20:19:48
Right. We also know that sin(20)=2sin(10)cos(10) by the double-angle formula (again, don't worry if you don't know this now, we'll be covering it in class).
DPatrick20:20:32
DPatrick20:20:47
How do we finish?
emcat20:20:59
2sin^2(10)+2cos^2(10)
Malorian_220:20:59
cancel out the sin10 and cos10
DPatrick20:21:06
We can cancel the denominators:
DPatrick20:21:17
DPatrick20:21:22
What's that?
emcat20:21:37
sin^2+cos^2=1
Twin Prime Conjecture20:21:37
2
simo1420:21:37
2
emcat20:21:37
2
DPatrick20:21:47
Right: sin^2(x) + cos^2(x)= 1 for any angle x, so this is just 2.
DPatrick20:22:00
This problem is on the easier end of the spectrum of problems we'll be doing in the class.
DPatrick20:22:06
Here are a few examples of more difficult trigonometry problems that we'll be covering in the class:
DPatrick20:22:14
(I won't discuss their solutions.)
DPatrick20:22:18
DPatrick20:22:24
DPatrick20:22:33
DPatrick20:22:43
Trigonometry and complex numbers are not only very interesting topics in their own right, they also are extremely useful problem solving tools, and can be used to solve many problems which, on the surface, don't appear to involve trigonometry or complex numbers at all.
DPatrick20:22:54
The following two problems are examples of such problems. The first one is solved using trigonometry, the second one is solved using complex numbers. They are very, very hard to solve without these tools. We won't work through the solutions here, but we'll cover them in the course:
DPatrick20:23:00
DPatrick20:23:07
DPatrick20:23:12
In the course, we will cover the definitions and properties of the trigonometric functions, various trigonometric identities, the Law of Sines and Law of Cosines, the definition and properties of complex numbers and the complex plane, the exponential form of complex numbers, De Moivre's Theorem, the geometry of complex numbers, and roots of unity.
DPatrick20:23:34
Are there any questions about this course?
qwertythecucumber20:23:43
is there a text for this class? if not, will one be available later?
DPatrick20:23:58
There is no textbook for this class. We're working on the trig book but it won't be published until later in 2009.
UnendingPi20:24:01
how long is this course?
DPatrick20:24:03
12 weeks.
tenisu_no_oujisama20:24:11
so what classes should we have taken before if we wish to take Trigonometry and Complex Numbers?
DPatrick20:24:19
The Intermediate Trigonometry/Complex Numbers course is for students who have mastered basic algebra and geometry, including: the Pythagorean Theorem, solving basic equations and systems of equations, graphing simple functions, the geometry of triangles, circles, and arcs, and the basic arithmetic of complex numbers. We do not assume any past experience working with trigonometry.
Math4620:24:39
Is it apropiate for a 8th grader?
DPatrick20:24:52
Only for an 8th grader who's mastered all of the above topics.
isiahbunshart1020:24:57
are you ready test
DPatrick20:25:20
Right: this class (like all of our subject classes) has an "Are you Ready?" diagnostic test in the Online Classes section of the website.
bwu20:25:28
which competitions are these classes aimed for?
DPatrick20:25:40
The material in the trig class will help on the AMC 12 and AIME contests.
tenisu_no_oujisama20:25:45
so if we've taken Algebra I, Algebra II, and Geometry in school, would this class be appropriate?
DPatrick20:25:48
It should be, yes.
DPatrick20:25:55
But look at the diagnostic test to be sure.
DPatrick20:26:13
At this point, are there any questions about any of the classes I've discussed tonight, or our online school in general?
Flying Radio20:26:19
are all classes aimed at a competition ... do they have value for students needing work on word problems, general SAT prep, etc
DPatrick20:26:29
The subject classes by and large are not aimed at any particular solutions.
DPatrick20:26:35
They are designed to teach problem-solving schools in general.
Math4620:26:42
Can you opt out of class after it has started? will there be a refund?
DPatrick20:26:51
Yes, you may drop any class prior to the 3rd class session for a full refund.
UnendingPi20:27:04
so we dont have to know about that much trig for intermediate trig/complex numbers?
DPatrick20:27:06
Correct.
bwu20:27:11
what if you can't make a few classes?
DPatrick20:27:30
All of our classes are transcripted; if you miss a class, you can log onto the website and view the transcript of the class you missed.
math22220:27:52
How do we find Areyou ready test for AMC8 class?
DPatrick20:28:23
We don't have them for our problem series classes (that is, our MATHCOUNTS or AMC/AIME classes); instead, read the "Who should take this class" description on each class's information page.
alexperkowski20:28:31
What will the amount of refund be after that 3rd class?
DPatrick20:28:37
There is no refund after the 3rd class.
simo1420:28:43
will it go beyond the trig covered in art of problem solving volume 2?
DPatrick20:28:51
Yes, there will be more detail and a wider variety of problems.
Kira Sekulow20:28:56
is the pace of the chat room during Intro Alg similar to the pace tonite, faster, or slower?
DPatrick20:29:03
The regular class pace is a bit slower than tonight's pace.
math=fun20:29:11
the geometry class, is it for student already taken geometry?
DPatrick20:29:30
It depends...our geometry class has more difficult problems than most "regular" geometry classes.
math22220:29:48
Do you have examples of past AMC8 class transcripts?
Godzilla!20:29:48
Are there some example class transcripts available?
DPatrick20:30:13
All of our Math Jam transcripts are available on the website, and in particular the transcript for tonight's Math Jam will be available shortly after we're done.
DPatrick20:30:29
Go to "Math Jams" under the Community link on the left side of the page, and then click the "Transcripts" button.
kelleyzhao20:30:52
do we need finish algebra II before start to trig?
DPatrick20:30:58
It is strongly recommend.
rikkilay20:31:12
my sister is currently taking Calculus BC in school... she has not yet taken the AMC 12.. will this course be too easy for her?
DPatrick20:31:26
Not sure...check the "Do You Need This?" diagnostic for the class.
davidd220:31:42
are these classes above the school classes? do they cover all the topics of regular class?
DPatrick20:32:09
They cover at least the same curriculum (and likely more) than a regular school class, and there is more of a focus on solving difficult, non-routine problems.
Godzilla!20:32:29
In the classes, if one reads the chapter and does the problems pre-class - what does the class do? different problems? Extra remarks?
DPatrick20:32:45
This question only applies to the Intro Algebra class (of the classes I discussed tonight)...it's the only one with a textbook.
DPatrick20:33:00
If you can self-study with the book and do all the problems, the class probably won't add too much.
DPatrick20:33:25
The problems in the class will be somewhat different than those in the book, and the biggest advantage of the class is the interaction with the instructor.
DPatrick20:33:42
Also, you'll get detailed written feedback on the solutions that you submit to the Challenge Sets.
123go20:33:47
who is going to teach?
DPatrick20:33:54
Richard Rusczyk is teaching Intro Algebra.
DPatrick20:34:02
Josh Zucker is teaching both MATHCOUNTS classes.
DPatrick20:34:07
Sean Markan is teaching Interm Trig.
DPatrick20:34:19
Their bios and the instructor listings for all our classes is on the website.
simo1420:34:26
do you get written feedback with the problem series?
DPatrick20:34:41
No. (As such, their enrollment fees are much lower.)
kelvin20:34:50
Do we still enter the same place?
DPatrick20:34:58
Yes, you enter the classroom the same way you entered the Math Jam tonight.
Campeona20:35:04
are most of the kids in this class in middle school?
DPatrick20:35:25
Intro Algebra is mostly grades 5-8; MATHCOUNTS is generally grades 6-8.
DPatrick20:35:30
Most of the Interm Trig students will be high school.
math22220:35:33
Would there ever be voice interactions with the instructor?
DPatrick20:35:41
No, we have no audio/video.
timwu22520:35:43
so you can get a full refund after 2 classes?
DPatrick20:35:45
Correct.
mofthree20:36:03
does each year's AMC class same? If you took it last year, do you need to take it this year?
DPatrick20:36:15
It's the same. (The only class that changes is the AIME class.)
math22220:36:30
The trig/complex number class has gone for 1 week already?
DPatrick20:36:36
No, it starts October 13.
Godzilla!20:36:58
Do the instructors get evaluated by the students? Can we see their ratings? Maybe some (like you) are really good, and some not so good?
DPatrick20:37:39
There are testimonials scattered through our Online Classes pages on the website. We don't have independent ratings (and, to be frank, as a business, there's no possible reason why we would publicize bad instructor ratings anyway).
UnendingPi20:37:47
I'm taking calculus, but I neve took trig. and we didn't really cover the stuff we discussed in pre-cal so should i take int. trig?
DPatrick20:38:22
Check the "Do You Need This?" test on the class's information page on the website.
DPatrick20:38:41
Our class will cover trig/complex numbers a lot more deeply than you've probably seen them before.
rikkilay20:38:44
what do you mean by 'no written feedback in the problem series'? Do you mean there is no classroom?
DPatrick20:39:43
The subject classes have periodic problem sets (called "Challenge Sets") for which students write up their solutions and turn them in (by mail, email, or fax) . We then read these solutions and provide detailed written feedback (essentially, we "grade" them, though we don't assign a grade, we instead give feedback).
DPatrick20:40:13
The problem series classes (that is, the MATHCOUNTS and AMC/AIME classes) don't offer this. (This is primarily why they are much less expensive, by the way.)
Campeona20:40:23
are there any high school students in the basic Mathcounts class?
DPatrick20:40:29
Probably not, but I don't know.
timwu22520:40:35
will there be transcripts for all of the classes
DPatrick20:40:37
Yes.
math22220:40:52
Even if you do not publicize bad instructor ratings, would you do independent ratings by students and remove ineffective instructors?
DPatrick20:41:12
We are very selective in our choice of instructors. We do solicit student feedback and the end of each course.
math22220:41:24
Did you say AMC8 class is for AMC 8 competition and trig/complex numbers is for AMC 12 competition. What class is good for AMC 10 competition?
DPatrick20:41:32
Any of our Intro-level classes would be helpful for AMC 10.
tenisu_no_oujisama20:41:41
how long will the transcripts be avaible online?
DPatrick20:41:54
They remain for about a month or so after the conclusion of each class.
DPatrick20:42:13
(meaning the entire course; so if the course ends in mid-April, all of the transcripts will stay up until mid-May.)
Campeona20:42:50
will today's transcript be available since it's not a class?
DPatrick20:42:58
Yes, all of our Math Jams are transcripted too.
DPatrick20:43:20
Go to the Math Jams page (by clicking Community->Math Jams on the left side of the site), and then click the transcript button.
DPatrick20:44:23
That's it for tonight, thanks for coming! If you have more questions, please post them on the "Classes Information" forum or send email to classes@artofproblemsolving.com
DPatrick20:44:38
We'll be having another Math Jam one week from tonight to discuss the remainder of our fall schedule.
DPatrick20:45:15
Good night!
