| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on Feb 19. |
| Math Jam hosted by rrusczyk
(Richard Rusczyk ). |
rrusczyk19:28:31
Hello, and welcome to the Introduction to Number Theory and Introduction to Geometry Classes Math Jam. Today we'll be discussing what these classes cover and how they work, and answering any questions you may have. We will also be doing a few problems from each class.
rrusczyk19:28:54
My name is Richard Rusczyk. I founded Art of Problem Solving and have written several Art of Problem Solving textbooks (including the Introduction to Geometry text, which is the text for one of the classes we will discuss tonight).
rrusczyk19:29:03
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.
rrusczyk19:29:18
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.
rrusczyk19:30:06
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.
rrusczyk19:30:36
Both of our classes tonight are Introduction level on specific subjects.
rrusczyk19:30:45
Each course consists of a series of 90-minute classes that are similar to the discussion we will have about a few problems later this evening. Each course has an accompanying textbook. (And we find that students who read the assigned chapters before each class do much better than those who don't!)
rrusczyk19:31:10
If a student cannot attend a specific class, the student can access a full transcript of the class on our website within 24 hours after class end.
rrusczyk19:31:34
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.
rrusczyk19:31:52
After each class, extra problems are given to the students on a course message board for full-class discussion. Each course also includes a Challenge Set every six weeks. 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.
rrusczyk19:32:41
Before I dive into talking about each of the specific classes, I'll address a few general questions you have asked about our courses:
howbeautiful9219:32:45
Is this an online class?
docpratshah19:32:45
Are all these classes on the internet??
rrusczyk19:32:52
Yes, the classes are all online.
Bindu19:33:04
do we fax, or submit online?
rrusczyk19:33:06
Homework can be faxed, emailed, or mailed.
stevie1719:33:27
is there any audio?
rrusczyk19:33:30
No, we have found that text-only classes are more effective.
rrusczyk19:33:38
Here's an explanation why we structure our classes as we do:
rrusczyk19:33:51
(This transcript will be online latter tonight, so you can read that later.)
mathtyro19:33:53
Can we still apply for these coming classes?
rrusczyk19:34:16
Yes, space is still available in both. Enrollment information is here:
Canton19:34:39
What if we don't get the book till after the class starts?
rrusczyk19:34:42
Don't worry about it too much; the first couple classes are more gradual than the later ones.
arkantosstevius19:35:01
there are two classes?
rrusczyk19:35:02
Yes, tonight we will discuss two different classes, Introduction to Number Theory, and Introduction to Geometry
stella7gem19:35:18
What grade level are these classes for?
rrusczyk19:35:20
Most students in the Introduction level classes are in grades 6-10.
vamathletes19:35:51
Do the problem series have the challenge sets too?
rrusczyk19:35:52
No. The "Problem Series" classes (i.e. those strictly for MATHCOUNTS or AMC prep) do not have challenge sets.
wesleyliu19:36:26
is there homework?
rrusczyk19:36:27
Yes. After each class, extra problems are given to the students on a course message board for full-class discussion. Each course also includes a Challenge Set every six weeks. 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.
howbeautiful9219:36:30
Are these classes at specific times and on specific days?
rrusczyk19:36:41
Yes. The schedule is available here:
rrusczyk19:37:26
Number Theory is on Thursday nights, 7:30 - 9 PM ET, and Geometry is 7:30 - 9 PM ET on Mondays. Number Theory starts Feb 28, Geometry starts Mar 3.
docpratshah19:37:43
I have a book with an intro to geometry and another on an intro to number theory from AoPS. Where can I check these are the ones I need for the courses???
rrusczyk19:38:01
If you have Introduction to Number Theory by Mathew Crawford, that's the right one for the Number Theory class.
rrusczyk19:38:24
If you have Introduction to Geometry by Richard Rusczyk, then you have the right book for the geometry class.
howbeautiful9219:39:03
Are these classes helpful for AMC preparation?
rrusczyk19:39:05
Very much so. These classes are taught to give students the problem solving skills for MATHCOUNTS and the AMC, as well as give them a good foundation for when they are ready to move beyond the AMC.
rrusczyk19:39:12
Both classes have many AMC problems in them.
mathtyro19:39:31
Are the solution manuals needed?
trivea19:39:31
How about MC?
rrusczyk19:39:53
You'll want them. Almost everyone who orders only the text comes back and gets the solutions manual.
rrusczyk19:40:01
Yes, there are many MATHCOUNTS problems in both classes.
stella7gem19:40:27
What is AMC?
rrusczyk19:40:29
American Mathematics Competitions - www.unl.edu/amc
howbeautiful9219:40:39
If I am unable to attend at the scheduled time is that a problem?
rrusczyk19:40:45
Not at all. If a student cannot attend a specific class, the student can access a full transcript of the class on our website within 24 hours after class end.
tennisstar11719:41:23
How long is h/w supposed to take?
rrusczyk19:41:25
The Challenge Sets should take around 10 hours to complete if you do most of the problems. That's why we give you 6 weeks to do it!
sunandiyer19:41:29
So is it ok if we miss a class becuase we have other things to do?
rrusczyk19:41:38
Yes; you can read the transcript when you have time.
arkantosstevius19:41:41
the transcript is only available to subscribers right
rrusczyk19:41:43
Correct.
Canton19:41:45
Who is the teacher/moderator for the Number Theory class?
Bindu19:41:45
what will we be coering this class??
rrusczyk19:42:20
I'll discuss these details when I get to each specific course (which will be very soon - about 5 minutes). We'll do some sample problems then, too.
triangleman19:43:01
How do i access the transcript?
adamw19:43:01
How do we suscribe to the transcript?
rrusczyk19:43:02
Each class has a homepage (if you are already enrolled, then you already have access to it: Click My Classes on the Forum). There will be links to class transcripts there when the class starts.
sunandiyer19:43:04
Can we leave in the middle of the class?
rrusczyk19:43:09
Yes.
wesleyliu19:43:22
what happens if you dont do homework?
rrusczyk19:43:23
Nothing. The work is optional -- you get out of this class what you put into it.
howbeautiful9219:43:40
How long are the courses?
rrusczyk19:43:41
Number theory: 12 weeks. Geometry: 24 weeks.
rrusczyk19:43:53
You have 6 weeks for the Challenge Set homeworks.
tennisstar11719:44:12
What ages is this class for?
rrusczyk19:44:14
Most of the students are in grades 6-10 in the Introduction classes.
Rocket9519:44:16
the lengths of each class are 1.5 hours right?
rrusczyk19:44:18
Yes.
howbeautiful9219:44:23
How do you enroll? And is their a price?
rrusczyk19:44:30
Here is the enrollment page, with pricing:
sunandiyer19:44:51
What happens if we can not solve the homework
rrusczyk19:44:52
We will provide solutions, and there is a class message board where you can ask questions after class time.
stevie1719:44:59
what are the challenge set homeworks? are they extra homework on the side of regular homework?
rrusczyk19:45:20
After each class, extra problems are given to the students on a course message board for full-class discussion. These are not turned in, and are mainly for extra practice. Each course also includes a Challenge Set every six weeks. 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.
adamw19:45:42
What is the latest enrollment date for Number Theory.
rrusczyk19:45:51
March 9
tennisstar11719:45:54
Does weather affect this online connection?
rrusczyk19:45:59
Not most people's.
sunandiyer19:46:27
Can we do this course without the book?
rrusczyk19:46:28
The texts are required: we have assigned reading, etc.
siddhant19:46:53
is there going to be a contest for art of problem solving
rrusczyk19:46:54
We don't have an Art of Problem Solving contest, but we do run the USAMTS (www.usamts.org). You can check that out later.
stevie1719:47:19
when is the homework due for each week?
rrusczyk19:47:21
The homework you turn in is only due every 6 weeks.
sunandiyer19:47:22
Do we have timed questions
rrusczyk19:47:25
No.
triangleman19:47:27
how do we access the course message board?
rrusczyk19:47:52
It will appear in the Forum in the Online Classes section. You will have access to it roughly 1 week before class starts.
adamw19:47:56
Can you acces the course fourum without enrolling?
rrusczyk19:47:58
No.
howbeautiful9219:48:38
Where can I find the transcript for this MathJam, if I need to leave now?
rrusczyk19:48:46
Click on Math Jams on the Forum, then on transcripts.
stevie1719:48:48
do we have homework often then? how long is the homework? do we get graded for it? how do we send in the homework?
rrusczyk19:48:52
After each class, extra problems are given to the students on a course message board for full-class discussion. Each course also includes a Challenge Set every six weeks. 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.
rrusczyk19:49:07
We don't give letter grades unless specifically requested to.
sunandiyer19:49:12
What happens if you forget your ID or password
rrusczyk19:49:21
You email webmaster@artofproblemsolving.com
pshweh19:49:23
I will need letter grade. What does it take to get an A?
rrusczyk19:49:42
Do all the Challenge Sets (not every problem, but as many as you can). Participate in class and on the message board.
MathCount12319:49:45
What is the style of the class? Is the commnication by sending message only?
rrusczyk19:50:01
Let's take a look at some sample problems for the Number Theory class so you can get a feel for the class style.
rrusczyk19:50:22
I'll take more questions in a bit.
rrusczyk19:50:25
Introduction to Number Theory
rrusczyk19:50:38
Many students who already know how to solve MATHCOUNTS level problems about divisibility, base numbers, and divisor counting might think they have little need for this class, but many of the students who have made the most of this class were participants at national MATHCOUNTS and the AIME and found many of the problems discussed in class very challenging.
rrusczyk19:51:45
The Introduction to Number Theory class covers divisibility problems, clever uses of prime factorization, base numbers, linear Diophantine equations, the Euclidean Algorithm, and covers the mechanics of modular arithmetic in thorough detail. There are also many other topics covered in this course that are rarely mentioned in math books at all, but these topics become increasingly important as you move to higher mathematics.
rrusczyk19:52:02
For example, the topics covered in this course are crucial to an understanding of such areas as cryptography and computer science.
rrusczyk19:52:30
The following mini-lesson is excerpted from one of the Number Theory classes.
rrusczyk19:52:35
COUNTING DIVISORS
rrusczyk19:53:02
Once we know how to find the prime factorization of numbers, we can begin to use this tool to solve other problems.
rrusczyk19:53:10
One such problem is answering the question 'how many positive divisors does a particular integer have?' This kind of counting problem is common in number theory.
rrusczyk19:53:18
rrusczyk19:53:51
How can we describe any divisor of 200?
imuh319:54:19
have prime factors with 2 or 5
Aquila_219:54:19
It consists of 2s and 5s
rrusczyk19:54:34
(I see a lot of you citing a formula to me. See if you can prove the formula!)
rrusczyk19:54:47
How many 2s and how many 5s can we have?
imuh319:55:14
3 twos and two fives
Jerry_Zhang19:55:14
3 2s 2 5s
trivea19:55:14
3 2s and 2 5s
womath19:55:14
three twos and two fives
rrusczyk19:55:21
trivea19:55:52
there are only 3 2s in 200
desperado19:55:52
because there are only 3 2s
Rocket9519:55:52
too high
adamw19:55:52
There is not enough two's in 200
mathtyro19:55:57
because there can only be 3 2's. 2^4 has 4 twos
rrusczyk19:55:59
rrusczyk19:56:08
Now, we're ready to solve the problem.
rrusczyk19:56:10
rrusczyk19:56:22
desperado19:57:01
a=0, 1, 2, 3 b= 0, 1, 2
trivea19:57:01
0,1,2,3 and 0,1,2 (these are the powers)
Rocket9519:57:01
a=0,1,2,3
b=0,1,2
rrusczyk19:57:13
rrusczyk19:57:28
Now that you know what the possible values for a and b are, how can you use this to count the total number of divisors of 200?
trivea19:57:54
multiply 4 and 3
desperado19:57:54
4*3
vamathletes19:57:54
4 times 3 is 12 possibilities
Aquila_219:57:54
3*4=12
rrusczyk19:58:03
Let's see why:
rrusczyk19:58:04
Consider this tree diagram:
rrusczyk19:58:10
Gevalt19:58:14
for eaCH A THERE ARE 3 BS
rrusczyk19:58:17
Exactly.
rrusczyk19:58:21
rrusczyk19:58:43
Indeed we can see that this is true by listing the divisors:
1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200 are all the positive divisors of 200.
rrusczyk19:59:10
The reason we simply multiply the numbers of values for the exponents together is because we can select the values for each exponent independently from the values of the other exponent(s).
rrusczyk19:59:23
rrusczyk19:59:41
rrusczyk20:00:14
In general what can we say about the number of positive divisors of an integer n with a prime factorization
rrusczyk20:00:17
Bindu20:00:46
you add 1 to both exponents and add the resulting sums????
desperado20:00:46
(e_1+1)(e_2+1)...(e_m+1)
arkantosstevius20:00:46
(e1 +1) (e2 +1)......(em + 1)
as366520:00:52
(e1+1) *(e2+1) ....(em+1)
rrusczyk20:00:56
rrusczyk20:01:28
rrusczyk20:02:15
rrusczyk20:02:35
To make this method more clear, we will now work through a couple exercises.
rrusczyk20:02:42
Exercise: How many positive divisors does 60 have?
rrusczyk20:02:58
First, what is the prime factorization of 60?
desperado20:03:16
2^2 * 3 * 5
trivea20:03:16
2^2 * 3*5
vamathletes20:03:16
2^2 *3*5
arkantosstevius20:03:16
2^2 * 3 * 5
rrusczyk20:03:21
rrusczyk20:03:32
rrusczyk20:03:40
trivea20:04:05
3,2,2
mathtyro20:04:05
3,2,2
natiator20:04:05
3,2,2
Bindu20:04:05
3,2,2
Gevalt20:04:05
3,2,2
rrusczyk20:04:26
rrusczyk20:04:36
Given the number of possibilities for each exponent, how do we use this to determine the total number of divisors of 60?
Aquila_220:04:57
there are 3*2*2 posibilities = 12
Aquila_220:04:57
(e_1+1)*(e_2+1)*(e_3+1)
mathtyro20:04:57
3x2x2
vamathletes20:04:57
3*2*2=12
desperado20:04:57
3*2*2 =12
as366520:04:57
3x2x2=12
Rocket9520:04:57
multiply the # of options together
trivea20:04:57
3*2*2 = 12
rrusczyk20:05:00
rrusczyk20:05:24
All right, we know how to crank with our formula now. Our next problem shows why it is important to understand *why* our formulas work!
rrusczyk20:05:32
rrusczyk20:06:32
First, how many positive integral divisors does 792 have?
ss518820:06:51
first find prime factorization
rrusczyk20:06:56
OK, what is the prime factorization?
rihim77.720:07:06
2^3 * 3^2 * 11
ayahska20:07:06
2^3x3^2x11
desperado20:07:06
2^3*3^2*11
rrusczyk20:07:13
womath20:08:09
24
adamw20:08:09
24
fbfan20:08:09
so there are 24 positive divisors.
rrusczyk20:08:19
Aquila_220:08:46
To count evens we need 3*3*2 = 18
desperado20:08:46
18
mathtyro20:08:46
18
rrusczyk20:08:57
Bold Claim! We have to prove it. Why is this the case?
as366520:09:22
3x3x2 = 18 because it has to include 2 to be even
vamathletes20:09:22
18, because there are only 3 options for the 2 in the prime factorization (can't be 0)
mathtyro20:09:22
It has to have at least 1 '2' to make it even
rrusczyk20:09:31
The even divisors have a positive exponent for the prime 2.
rrusczyk20:09:41
rrusczyk20:09:55
rrusczyk20:10:05
A couple of you showed a different way to do the problem:
arkantosstevius20:10:15
(3+1)(2+1)(1+1) - (2+1)(1+1)
desperado20:10:15
24 - 3!
rrusczyk20:10:43
These two take our count of all the divisors, and then subtract the odd divisors. To count the odd divisors, we simply ignore the 2's.
rrusczyk20:11:33
That's why there are (2+1)(1+1) odd divisors: 2+1 choices for the power of 3, and 1+1 choices for the power of 11.
rrusczyk20:12:03
So, those were a couple of sample problems for the number theory class.
rrusczyk20:12:31
The course builds through a study of primes and divisibility until it reaches the BIG IDEA of number theory: modular arithmetic.
rrusczyk20:12:40
If you want to know what that is, take the course or read the book!
Canton20:13:11
Can the NT kids try geometry problems?
rrusczyk20:13:25
Yes. We'll get to the geometry after taking a few more questions about the classes.
Bindu20:13:27
that stuff is soooooo fun!
rrusczyk20:13:30
Yes it is!
ayahska20:13:45
who are the NT kids
trivea20:13:45
What's NT
rrusczyk20:14:12
Number Theory. Canton refers to the students in the number theory class, and is asking if they can participate in the geometry part of the Math Jam (coming up).
MathIsGreat20:14:29
What happens if you forget to come to a lesson?
rrusczyk20:14:30
If a student cannot attend a specific class, the student can access a full transcript of the class on our website within 24 hours after class end.
rrusczyk20:14:40
Are there any questions about the Number Theory course?
sunandiyer20:14:45
How do you access the transcript
rrusczyk20:15:05
For this Math Jam, you can click on Math Jams on the Community, then click transcripts.
rrusczyk20:15:15
The transcript will be up about 30 minutes after the Math Jam ends.
Rocket9520:15:19
whos taking or thinking of thaking number theory say I
(this is just for fun)
sunandiyer20:15:41
I
Bindu20:15:41
I!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!:)
natiator20:15:41
I
Canton20:15:43
Me
mathtyro20:15:50
i
docpratshah20:15:50
me
adamw20:15:50
II
Kingman23020:15:50
I
sherman*quinnell20:15:54
I
Istanbul20:15:54
I am
stevie1720:15:56
me
Gevalt20:15:56
moi
Canton20:16:03
Sorry, I
srikarp_220:16:03
I
KevinSix20:16:03
I
womath20:16:03
YO
rrusczyk20:16:06
That was fun.
adamw20:16:09
Can you access a transcript when you are not enrolled in the course?
rrusczyk20:16:12
No.
Aquila_220:16:14
Why Number theory is so poorly covered in school. For example the "bases" are not covered but the problems are in contests
rrusczyk20:16:46
I suspect because many teachers have not yet been taught it. The curriculum will catch up eventually, then they'll teach the teachers, so they can teach the students.
MathIsGreat20:16:48
I dont understand about modules that much
rrusczyk20:16:57
The number theory class will help with that.
KevinSix20:17:22
Who is the moderator for the number theory class?
rrusczyk20:17:25
Good question.
rrusczyk20:17:26
The course will be taught by Ashley Ahlin. Ashley was the first female to win a medal at National MATHCOUNTS, placing 3rd in 1987. She also won 1st place in the Westinghouse Science Talent Search. Later, Ashley served on the MATHCOUNTS problem writing committee and spoke at the national awards banquet. She has taught at the high school and college levels, and in summer programs for all age levels. Ashley finished her Ph.D. in math at the University of Chicago in 2001.
Rocket9520:17:43
maybe the people think NT wont help as much in the long run
rrusczyk20:17:44
They're just behind the times. The curriculum hasn't caught up to the computer age.
sunandiyer20:17:49
Can we have after session help if we do not understand something covered in the session?
rrusczyk20:18:04
There is a course message board where you can post questions for the instructors at any time.
womath20:18:07
how do you write math notations
rrusczyk20:18:26
Click LaTeX on the sidebar of our site - that will teach you a ton about how to do it. You can practice on the message board.
stevie1720:18:28
so it starts this thursday at what time?
rrusczyk20:18:37
7:30 - 9:00 PM ET
Istanbul20:18:39
What happens if you live in a foreign country with a different time zone?
rrusczyk20:18:50
You'll have to convert that to your time. If you're not sure, email us.
siddhant20:18:54
WIll differnt instructors teach
rrusczyk20:19:00
We try to only have one instructor per course.
Canton20:19:02
Next Thursday
rrusczyk20:19:07
Yes, next Thursday.
adamw20:19:09
How come the enrol date is after the start date?
rrusczyk20:19:16
There are *always* people who sign up late.
MathIsGreat20:19:18
Where is the course message board
rrusczyk20:19:39
It will be on the Forum, in the Online Classes section. There will be a link in the Class homepage. You'll be able to access it Friday.
Canton20:19:43
If you turn in challenge problems late, will it be a mark off?
rrusczyk20:19:51
Not if it's just a couple days late.
srikarp_220:19:58
do we have homework
rrusczyk20:20:04
After each class, extra problems are given to the students on a course message board for full-class discussion. Each course also includes a Challenge Set every six weeks. 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.
sunandiyer20:20:27
Will be talking like this in the session (this type of board)
rrusczyk20:20:28
I'm not sure what you're asking, but class will flow a lot like the problem we just did (maybe not quite so fast, but close)
sunandiyer20:20:30
Can you fail this class?
rrusczyk20:20:48
We don't give formal grades for the courses unless your school requires it. So, no. Most students take this for fun.
triangleman20:20:51
how do we send the homework to you?
rrusczyk20:21:30
Fax, phone, or mail. There are instructions in the course information document on the class homepage. (Click on My Classes on the Forum when you are logged in if you are enrolled in a class)
siddhant20:21:33
does every problem affect your grade
srikarp_220:21:33
will we get a grade
rrusczyk20:21:50
No grade - we only give grades if a school requires it. This is rare.
adamw20:21:52
Do you copy you writing that is the third time you have said it?
rrusczyk20:21:56
Copy paste.
Bindu20:22:14
the challenge problems take 10 hrs, but you're allowed help, right?
rrusczyk20:22:24
Students can work together in groups, and often do.
sunandiyer20:22:51
Can we use parent help for the challenge problems
rrusczyk20:22:59
Yes, but please indicate this when you turn in the assignments.
ss518820:23:04
how do u work in groups?
rrusczyk20:23:11
Students have access to this classroom at certain times.
sherman*quinnell20:23:29
can you use a calculator for homework?
rrusczyk20:23:30
Yes, but we recommend not doing so, so you develop good number sense.
MathIsGreat20:23:36
When is that
rrusczyk20:23:45
It is posted in the Online Classes section.
stevie1720:23:50
what are you teaching?
greenfrog20:23:50
when are we going to start the geometry portion of this class?
rrusczyk20:23:59
Let's get to what I'm teaching: geometry.
rrusczyk20:24:08
I'll take more questions after talking about geometry.
rrusczyk20:24:44
Introduction to Geometry
rrusczyk20:24:49
In the Introduction to Geometry class we cover all the fundamentals of geometry. We will start with a few days covering the basic tools such as triangle congruence, similarity, power of a point, relationships between angles and circles, etc., then dive into using those tools and more to solve increasingly difficult problems.
rrusczyk20:24:52
Most of the problems in the course will be at the MATHCOUNTS and AMC-10 level of difficulty, but we will be throwing in a few harder problems occasionally to show how to use very basic ideas to solve very challenging problems.
rrusczyk20:24:59
I am teaching the Introduction to Geometry course.
rrusczyk20:25:24
For those of you who don't know me (I'm guessing that's most of you), here's my bio:
rrusczyk20:25:47
Richard Rusczyk is one of the co-authors of the Art of Problem Solving textbooks, and author of Art of Problem Solving's Introduction to Algebra and Introduction to Geometry textbooks (Intermediate Algebra due to be published in early 2008). He is also one of the co-creators of the Mandelbrot Competition, and the Director of the USA Mathematical Talent Search. 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). He graduated first in his major, top 5 in his class, with a BSE from Princeton University in 1993. Richard worked as a bond trader for D.E. Shaw & Company for four years. AoPS marks Richard's return to his vocation - educating motivated students.
rrusczyk20:26:36
Enough of all that, let's do a couple problems. In the actual class, we'll teach all the concepts needed to do the problems as we go. We won't assume knowledge of geometry in the course.
rrusczyk20:26:51
For these practice problems, however, we'll use some basic concepts that will be taught in the class.
rrusczyk20:27:05
rrusczyk20:27:13
rrusczyk20:27:38
In order to find AC, what will we need?
desperado20:27:54
OC
vamathletes20:27:54
length of OC
Aquila_220:27:54
We need to find OC
rrusczyk20:27:57
We need AO, which we already have, and OC. How can we find OC?
wesleyliu20:28:07
the side of the square
rrusczyk20:28:12
OC is a side length of a square. Do we know anything else about this square that will give us information?
rrusczyk20:28:25
desperado20:28:41
OD is the radius
mathtyro20:28:41
OD= radius, which is 6. The diagonal of the square is 6
trivea20:28:43
diagonal OD
lifeisacircle20:28:43
The diagonal is the radius also.
rrusczyk20:28:47
We know the 6 inch radius of the circle is also the diagonal of the square. Now what is the side length?
Rocket9520:29:12
vamathletes20:29:12
sqrt{18}
trivea20:29:12
so oc is 3(sqrt2)
as366520:29:16
its 45 45 triangle so side of square should be 3root 2
rrusczyk20:29:19
Because the diagonal has length 6 and OC is a side, we have:
rrusczyk20:29:22
rrusczyk20:29:28
rrusczyk20:29:34
How do we find AC?
desperado20:29:53
pythagorean theorm
MathIsGreat20:29:53
pythagorean theorem
vamathletes20:29:53
Pythagorean Theorem
trivea20:29:53
so then AC = 3(sqrt 6)
desperado20:29:59
so 3 sqrt 6
rrusczyk20:30:01
We have a right triangle, so we use the Pythagorean Theorem:
rrusczyk20:30:13
rrusczyk20:30:17
rrusczyk20:30:51
That was on the easier side of the problems we'll do in the course (if you didn't know the geometry facts we used, don't worry: they are taught in the course).
rrusczyk20:30:56
Here's one more problem:
rrusczyk20:30:58
Sector OAB is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form.
rrusczyk20:31:06
rrusczyk20:31:37
What do we need to find?
mathtyro20:32:03
Radius of smaller circle
Jerry_Zhang20:32:03
radius of inscribed circle
Rocket9520:32:03
radius of the circle
rrusczyk20:32:14
We need the radius of the small circle, so what should we do with our diagram?
vamathletes20:32:33
draw radius
rrusczyk20:32:35
Let's draw the radius. Where should we draw it?
trivea20:33:15
perpendicular to both sides of sector
vamathletes20:33:15
Perpendicular to BO or AO?
mathtyro20:33:17
From center to where it touches AB?
Rocket9520:33:22
center to arc tangent point?
rrusczyk20:33:24
Draw a radius to each point of tangency and label each with length r. We like to draw radii to points of tangency because we get right angles.
rrusczyk20:33:29
rrusczyk20:33:47
(We prove that the radius is perpendicular to the tangent in the course.)
rrusczyk20:33:52
Now what?
rrusczyk20:33:59
Anyone see anything interesting?
Rocket9520:34:11
we have a square
mathtyro20:34:17
we have a square.
greenfrog20:34:17
a square
rrusczyk20:34:22
We notice a square. How will the square help us?
rrusczyk20:34:33
natiator20:35:00
its sides are the radius
docpratshah20:35:05
Its sides are the radius of the small circle
rrusczyk20:35:12
True. And how can we take advantage of that?
Aquila_220:35:15
FP + PO = 3
rrusczyk20:35:23
And what do we know about PO?
greenfrog20:35:32
divide the square in half
mathtyro20:35:33
Diagonal of the square
rrusczyk20:35:47
Drawing a diagonal in the square will give us a radius of the large circle (FO) in terms of r. What is the length of the diagonal of the square?
vamathletes20:36:16
sqrt{2}*r
Aquila_220:36:16
r*sqrt(2)
rrusczyk20:36:22
Because the side length of the square is r, the diagonal is r*sqrt2.
rrusczyk20:36:27
rrusczyk20:36:29
Now what?
desperado20:37:01
r*sqrt2 = 3-r
vamathletes20:37:01
r+r(sqrt{2}=3
mathtyro20:37:01
r+r* sqrt2=3
Aquila_220:37:08
Now we have equation: r*(1+sqrt(2))=3
rrusczyk20:37:15
We set up an equation and solve it. We know that OF is a radius, so OF = 3. However, we can write OF = OP + FP = r*sqrt(2) + r.
rrusczyk20:37:21
Now we have an equation and we can solve for r:
rrusczyk20:37:25
rrusczyk20:37:29
Is this the simplest form we can express the answer in?
adamw20:38:06
no rationalise denominator
trivea20:38:10
conjugate!
desperado20:38:10
3 sqrt2 - 3
Aquila_220:38:10
r=3*(sqrt(2)-1)
vamathletes20:38:10
multiply by 1-sqrt{2}
rrusczyk20:38:13
We multiply both the top and bottom by 1-sqrt2.
rrusczyk20:38:18
rrusczyk20:38:42
(That last bit is algebra, not just geometry; don't worry if you don't get that right away!)
rrusczyk20:38:46
Notice that we don't just sit and stare at the problem and wait for it to solve itself. We have to add lines and variables so we can build equations.
rrusczyk20:38:55
The first of those two problems is on the easy end of problems we will discuss. The second is a bit easier than average. All the geometric tools we use to solve problems, such as all the special relationships we used to solve these two problems today, will be taught in the class. We don't expect students to have any background knowledge in geometry.
rrusczyk20:39:06
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
rrusczyk20:39:35
The course will meet for 24 weeks on Mondays, starting March 3, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes.
rrusczyk20:39:39
This course will use a textbook in conjunction with the course: our own Introduction to Geometry 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:
rrusczyk20:40:04
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 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. We also expect to spend some class time answering students' questions about problems from the textbook.
MathIsGreat20:40:26
What happens if you have vacation and there is no internet
rrusczyk20:40:33
Then you can review the transcript when you return.
rrusczyk20:40:58
Many of our summer students (and the geometry class runs into the summer) miss a few weeks, but keep up or catch up with the book and the transcripts.
rrusczyk20:41:08
You can ask questions on the message board if you have any.
rihim77.720:41:10
This is a good class
rrusczyk20:41:13
I think so :)
natiator20:41:15
will the transcript go away
rrusczyk20:41:25
Not until about 4 weeks after the whole course is over.
rrusczyk20:41:41
Are there any more questions about the courses?
greenfrog20:41:54
are we done with the math part of this program
rrusczyk20:42:04
Yes. I will just be taking questions from here on.
srikarp_220:42:07
What is a transcript
rrusczyk20:42:17
It is a webpage that contains *everything* that went on in class.
Aquila_220:42:21
Are you going to add more courses to AoPS?
rrusczyk20:42:31
Over time, yes, particularly as we write more books.
siddhant20:42:32
CAn you skip questions if you don't get them
rrusczyk20:42:35
Yes.
desperado20:42:36
when do we receive our homework?
rrusczyk20:43:00
The first challenge set is given out after week 2, the second after week 8.
docpratshah20:43:02
I nees to leave. will I miss anything except questions???
rrusczyk20:43:03
no
wesleyliu20:43:24
what if there are technical difficulties with the website and/or anything else?
rrusczyk20:43:26
If something terrible happens, like our website goes down, we will add a week to the end of the course.
MathIsGreat20:43:43
So there is going to be a transcript about this Jam?
rrusczyk20:43:55
Yes. In about 30 minutes, click Math Jams on the Forum, then transcripts.
sherman*quinnell20:44:00
do we have to finish the homework in a week?
rrusczyk20:44:15
No. The message board problems you can do whenever you like. You have 6 weeks for the Challenge Sets.
srikarp_220:44:18
how long should the homework take
rrusczyk20:44:31
The Challenge Sets should take you a good 10 hours (over the 6 weeks).
rrusczyk20:44:42
To get the most out of the course, you should expect to spend 4-6 hours a week.
desperado20:44:45
Do we have to enroll to receive our homework?
rrusczyk20:44:46
Yes
siddhant20:45:05
if the transcript is old will it be deleted
rrusczyk20:45:06
The transcripts for the classes stay up until 4 weeks after the class ends.
MathIsGreat20:45:40
Each class is how long?
rrusczyk20:45:42
90 minutes
rrusczyk20:46:21
Are there any more questions about the courses?
sherman*quinnell20:46:23
will the transcript of this class stay for four weeks as well?
rrusczyk20:46:38
The Math Jams transcripts will stay up indefinitely (i.e., for a long time)
boarder392220:46:52
About how many kids are in each class?
rrusczyk20:47:18
It varies a great deal. I expect the number theory class to be a touch bigger than class tonight, and the geometry class to be a touch smaller.
srikarp_220:47:21
I have to leave now will I miss anything
rrusczyk20:47:35
no. We're basically finished except for questions.
sherman*quinnell20:47:37
is therre a limited amount of people for each class?
rrusczyk20:47:51
Not really. If we have a ton of enrollments, we'll split the class into two sections.
rrusczyk20:48:36
That's it for class tonight. If you have any questions later, email us at classes@artofproblemsolving.com.
