| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on Sep 24. |
| Math Jam hosted by rrusczyk
(Richard Rusczyk ). |
rrusczyk19:32:30
Hello, and welcome to an Art of Problem Solving Math Jam. Today we'll be discussing the following four classes: Algebra 1, Algebra 2, Introduction to Counting & Probability and Introduction to Geometry.
rrusczyk19:32:45
My name is Richard Rusczyk. I founded Art of Problem Solving and have written several Art of Problem Solving textbooks.
rrusczyk19:32:51
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.
rrusczyk19:32:58
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. 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.
rrusczyk19:33:26
In general in our classes, we have assistant instructors in all of our classes, and all math questions get answered by the primary instructor of the assistants. Tonight, since there are so many of you, we might not be able to answer *every single question*, but we get them all in the classes.
rrusczyk19:33:52
As for questions about the classes, we will try to answer all of those tonight. I will let you know when to start asking questions about specific classes.
rrusczyk19:34:02
A couple of you have asked about sound. There is no sound in the classroom. This webpage explains why:
rrusczyk19:34:12
You can read that later; you don't have to check it out now. There will be a full transcript of this class available on the Math Jam pages about an hour after we finish.
rrusczyk19:34:33
In this Math Jam, I will briefly describe the course, then go through an example problem. Then, I will hold a question-and-answer session about the class.
rrusczyk19:34:50
Before we get started, I'd like to note that the mathematics we will discuss today cover a *very* wide range of difficulty. Moreover, I know that many of you are here just to check out the classroom before your classes start this fall.
rrusczyk19:35:10
Please understand that if you are enrolled in one of our introductory classes, or haven't much experience yet with advanced problem solving, then some of the problems we discuss tonight may be very hard for you. We won't be able to teach you all the math you need to understand this material in one night! So, don't be frustrated if you don't understand the problems we discuss for those classes -- your time will come!
rrusczyk19:35:48
My assistant tonight is Vincent Le, whose username is chesspro. He's in his first year at MIT. When he's not doing homework, he's solving the Rubik's cube faster than humans should be allowed to, which is easy for him because he is also a magician!
rrusczyk19:36:09
He will sometimes answer your questions privately, by whispering to you, or opening a 1-on-1 chat window with you.
rrusczyk19:36:20
Let's get to the classes!
rrusczyk19:36:27
Algebra 1 + Algebra 2
rrusczyk19:36:36
The Art of Problem Solving Algebra 1 covers the fundamental concepts of algebra, including exponents and radicals, linear equations and inequalities, ratio and proportion, systems of linear equations, factoring quadratics, complex numbers, completing the square, and the quadratic formula. (Note: This class is equivalent to the first half of the Introduction to Algebra course we used to offer.)
rrusczyk19:37:08
Algebra 2 covers quadratic equations, graphing , complex numbers, graphing, functions, sequences and series, and exponents and logarithms. Problem solving skills are emphasized throughout, and time is devoted to advanced topics like telescoping sums and piecewise functions. (Note: This class is equivalent to the second half of the Introduction to Algebra course we used to offer.)
rrusczyk19:37:31
Our algebra classes do not align precisely with the standard curriculum. We have divided them up and named them the way we have in order to provide some guidance for choosing which of our classes to take.
rrusczyk19:37:42
If you have never taken an algebra course, then our Algebra 1 class is probably the right place to start.
rrusczyk19:37:55
If you have taken an algebra 1 class in school, then much of our Algebra 1 course would be review (although there are some more challenging problems in our Algebra 1 class than you would see in school). You may wish to skip our Algebra 1 and move on to Introduction to Counting & Probability, Introduction to Number Theory, and/or Algebra 2.
rrusczyk19:38:14
If you have taken Algebra 2 in school, then Algebra 3 is probably the correct algebra class to take with us (though other non-algebra classes will likely be appropriate as well).
rrusczyk19:38:46
Our algebra classes emphasize 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.
rrusczyk19:39:17
There are a couple major goals of our algebra courses:
rrusczyk19:39:25
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.
rrusczyk19:39:46
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.
rrusczyk19:40:23
I'm now going to run through an algebra problem to give you a taste of the mechanics of the classes. After that I'll spend some time taking questions about the courses.
rrusczyk19:40:32
Here's a problem which involves some fairly sophisticated thinking.
rrusczyk19:40:36
What's the solution to the following equation?
rrusczyk19:40:39
rrusczyk19:40:50
Where do we start?
xxrxxhxx19:41:21
removing the square root of x
spazmaster10019:41:22
Get that sqare out of the squre root sign!
annie951219:41:22
square root
rrusczyk19:41:31
We want to deal with complication, which is the square root.
rrusczyk19:41:34
People have suggested this:
rrusczyk19:41:37
rrusczyk19:41:41
After all, the square root of the square of a number seems to be the original number. For example,
rrusczyk19:41:46
rrusczyk19:41:50
Let's try this and see where it goes. What's our new equation?
spazmaster10019:42:05
x-2x=1
sparkles25719:42:05
x-2x=1
stargazer41819:42:05
x - 2x = 1
Omnigeek619:42:05
X-2X=1
xxrxxhxx19:42:05
x-2x=1
rrusczyk19:42:09
rrusczyk19:42:14
And what do we get here?
mathISuber19:42:29
-x=1
xxrxxhxx19:42:29
-x=1
spazmaster10019:42:29
-x=1
stargazer41819:42:29
-x = 1
Iggy Iguana19:42:29
-1x=1
annie951219:42:29
-x=1
Bob Bobason19:42:29
countmath19:42:29
-X=1
mathworms19:42:29
-x =1?
rrusczyk19:42:31
We can combine like terms.
rrusczyk19:42:34
rrusczyk19:42:36
So what's x?
Jimmery19:42:45
x= -1
sakkorig19:42:45
x=-1
spazmaster10019:42:45
x=-1
Survivor8219:42:45
x=-1
Omnigeek619:42:45
X=-1
sparkles25719:42:45
x=-1
rrusczyk19:42:48
x = -1
rrusczyk19:43:01
Alright, now, it's always a good idea to check your work by plugging your solution into the original equation. Let's do that. Does the equation work when we put x = -1 in?
Iggy Iguana19:43:36
no
phanwort19:43:36
no
sparkles25719:43:36
no
Jimmery19:43:36
no?
rrusczyk19:43:46
What do we get on the left side when we put in x = -1?
Canis Lupus19:44:10
3
sparkles25719:44:10
3
countmath19:44:10
3
annie951219:44:10
3
thekingofhero19:44:10
3=1
mathworms19:44:10
3=1
Jimmery19:44:10
1+2=1 which is false...
Bob Bobason19:44:10
rrusczyk19:44:15
If we plug x = -1 into
rrusczyk19:44:18
rrusczyk19:44:26
the left side becomes:
rrusczyk19:44:31
rrusczyk19:44:48
Uhoh! Is x = -1 a solution?
stargazer41819:44:57
no
cding19:44:57
no
Iggy Iguana19:44:57
no
rrusczyk19:45:13
No! It gives us 3 on the left side, but we have 1 on the right side.
rrusczyk19:45:22
Hmm. . . . .
mathepic19:45:36
rrusczyk19:45:43
Correct.
rrusczyk19:45:49
So, where did we go wrong in our solution?
sparkles25719:46:22
+ or -x is sqrt(x^2)
mathepic19:46:22
We assumed that x was positive
cding19:46:22
you cant cancel the root and square root sign
Survivor8219:46:22
sqrt(x^2)DOES NOT EQUAL x
rrusczyk19:46:27
It was wrong to assume that
rrusczyk19:46:30
rrusczyk19:46:33
We can only make this assumption if we know that x is positive. If x is negative, what does
rrusczyk19:46:36
rrusczyk19:46:41
come out to?
thekingofhero19:46:57
-x
Iggy Iguana19:46:57
-x
tennis12319:46:57
-x?
math1719:46:57
-x
Omnigeek619:46:57
-x
rrusczyk19:47:01
rrusczyk19:47:08
Let's try solving the equation again, this time assuming x is negative. What does the equation become if x is negative and we simplify the square root?
thekingofhero19:47:31
-x-2x=1
stargazer41819:47:31
-3x = 1
Iggy Iguana19:47:31
-3x=1
Omnigeek619:47:31
-3X=1
DreamBird19:47:31
-x-2x=1
rrusczyk19:47:37
rrusczyk19:47:40
What do we get for x?
Survivor8219:48:02
-3x=1 and x=-1/3
thekingofhero19:48:02
-1/3
Iggy Iguana19:48:02
-1/3
math1719:48:02
-1/3
stargazer41819:48:02
x = 1/-3
mathISuber19:48:02
-1/3
rrusczyk19:48:14
The equation reduces to -3x = 1.
rrusczyk19:48:18
rrusczyk19:48:44
Now we've found the solution -- try it for yourself and see. When you substitute x = -1/3 into the solution, it works!
rrusczyk19:48:49
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.
rrusczyk19:49:01
You can find more questions like those we cover in the courses by checking out the Post Tests. The Post Test for Algebra 1 is here:
rrusczyk19:49:08
The Post Test for Algebra 2 is here:
rrusczyk19:49:27
The Algebra 1 course will meet for 15 weeks on Tuesdays, starting October 6, 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). The last day of the class is February 2.
rrusczyk19:49:36
The instructor for the course is Paul Ryu. Paul was a third-place winner at National MATHCOUNTS, a two-time perfect scorer on the American Mathematics Contest (AMC) 12, a participant at the Research Science Institute, and a prize winner at the Intel International Science and Engineering Fair. He has also helped students achieve success at Intel Science Talent Search and Siemens Westinghouse Science and Technology Competition. He attended Harvard.
rrusczyk19:49:53
The Algebra 2 course will meet for 15 weeks on Wednesdays, starting Sept 30, 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). The last day of the class is January 27.
rrusczyk19:50:04
The instructor for this course is Jeremy Copeland. Jeremy earned his Ph.D. in mathematics from the University of Chicago in 2006 and was on the math faculty at MIT from 2006 to 2009. He specializes in turning hard problems in geometry, algebra, and mathematical physics into easy problems in combinatorics and graph theory. He has spent most of this decade teaching gifted students at top tier colleges and brings this perspective to the AoPS staff. He secretly believes that every problem somehow reduces to the Chinese remainder theorem.
rrusczyk19:50:34
Both courses will use our Introduction to Algebra book. The material covered in the textbook is roughly equivalent to the material covered in the courses. You can see the table of contents and some excerpts from the book here:
rrusczyk19:51:01
The book is required for the courses. 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.
rrusczyk19:51:35
The homework for each course 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. Each course also has 5 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.
rrusczyk19:51:59
Are there any questions about this class or the textbook?
DreamBird19:52:35
the real classroom has no sound, is that correct?
rrusczyk19:52:37
That's correct; this is the room classes are in.
Kayak9719:52:56
is there are bokk past "introduction"
rrusczyk19:52:57
Yes, there is an Intermediate Algebra book.
Iggy Iguana19:53:08
are we going to do any more problems in this math jam?
rrusczyk19:53:10
Yes, counting and geometry problems.
Bob Bobason19:53:38
how is it compared to regular al1 and 2 classes in school?
rrusczyk19:53:41
Our classes are much more challenging than regular school.
Mycroft19:53:59
Is there any visual?
rrusczyk19:54:00
Yes, we can post images in the classroom, as you'll see when we get to geometry.
cding19:54:35
if we previously took algebra 1, shouldnt we do introduction to number theory and probiblity before algebra 2?
rrusczyk19:54:37
You can do those three classes in any order. We recommend students do the number theory or counting after algebra 1, but it's OK to plow ahead with Algebra 2.
DreamBird19:54:53
is it required for a student to know how the use LaTek?
rrusczyk19:54:54
Not at all; many of our students never bother learning it.
countmath19:55:14
How long does "homework" usually take in a week?
rrusczyk19:55:16
We expect students to take 5-8 hours a week, including class time, on all the classes we discuss tonight.
clemo19:56:04
Would the Alg. 1 / Alg 2. equate to Honors Alg. 1 / Honors Alg. 2 in school?
rrusczyk19:56:07
Our Algebra 1 and 2 together (which together are 30 weeks) cover all the curriculum of a typical honors algebra 1 class and most of a typical algebra 2 class (plus several topics that aren't usually taught in those classes).
math1719:56:30
What helps the most for amc 8 and mathcounts
rrusczyk19:56:31
All the classes we discuss tonight are good for AMC 8 and MATHCOUNTS.
DreamBird19:57:04
is there a money refund if i am not satisfied?
rrusczyk19:57:05
You can drop any course before the 3rd class and receive a full refund of the class fee. After the third class starts, you cannot drop anymore.
rrusczyk19:57:48
Of course, as chesspro notes, that's unlikely that you'll need to drop. You'll be satisfied :)
thekingofhero19:58:08
Is there a report card in these classes?
rrusczyk19:58:10
If you need a grade for school, we can assign one. Most students don't take our classes for grades, but some do.
countmath19:58:57
My daughter is taking Algebra II right now in school, do you think that it would be best if she takes the Algebra II course?
rrusczyk19:58:58
I recommend using the diagnostic tests as a guide. The counting or number theory classes might be good choices, too, as these will cover important areas of math that your school classes won't cover to much depth.
clemo19:59:20
Should you take Alg. 2 before the Geometry Class?
rrusczyk19:59:22
We recommend that you do so. The geometry class is the hardest of our Intro-level courses.
DreamBird19:59:48
are there tests/grading/quizes?
rrusczyk19:59:49
There are no tests; we use the Challenge Sets and work on the class message board to determine grades when students need grades.
cding20:00:27
so we can take these classes for credit at our schools?
pytheagle20:00:27
So taking Algebra I, Geometry, etc. earns you a credit in those areas?
Arcenas20:00:27
Do we receive High School credit for these courses, or are they just for learning ?
rrusczyk20:00:30
You have to arrange this with your schools. Some schools allow it, others don't. Many homeschool students use our classes as their "official" math classes.
rrusczyk20:00:47
We are happy to talk to school officials if they would like to learn more about our classes before making a decision.
rrusczyk20:00:54
Are there any more questions about the algebra classes?
DreamBird20:01:10
How long is the class period?
rrusczyk20:01:11
90 minutes
kevint20:01:44
How do challenge sets work - and how do I submit my work for them? Is it through email
rrusczyk20:01:46
You can email, fax, or snail-mail them to us. You will access your feedback on a webpage (and your work will be there, too).
pytheagle20:02:15
Are the challenge problems expected of us?
pytheagle20:02:15
I mean, are we SUPPOSED to be able to do all of them?
rrusczyk20:02:22
I won't send the ninja squad to your house if you don't do them, but you won't learn nearly as much if you don't.
DreamBird20:02:31
can i have another class example for algebra 1?
rrusczyk20:02:51
That is the posttest of Algebra 1. It gives examples of some of the more challenging problems we do in the course.
countmath20:03:03
Beyond the time spent in class room, if student has questions, do they contact teacher or post their queries on the bulletin board?
rrusczyk20:03:06
Post on the class message board. You can do so at any time.
Mycroft20:03:17
Do we write them on paper and then scan or do we type?
rrusczyk20:03:19
Either is fine. Or just mail the paper to us.
annie951220:04:02
i am taking geometry in school and also learning alg2 by my self, which class will be helpful?
rrusczyk20:04:04
You probably don't need the Algebra 1 class. I recommend looking into the counting or number theory class we offer this fall. You can use the diagnostic tests as a guide.
kevint20:04:27
My challenge set answers must be received by you before the next weekly class?
rrusczyk20:04:29
The Challenge Sets are every 3 weeks. The due dates are the dates they must be mailed by.
phanwort20:04:56
To take this course did we have to have learnt negatives in school
rrusczyk20:04:58
You should have learned negatives, yes. But you don't have to have learned them in school! There are plenty of other places to learn -- like AoPS!
DreamBird20:05:29
How many students are in a class?
rrusczyk20:05:31
There's a lot of variation -- 30-70, depending on the course.
DreamBird20:06:05
how many students in algebra 1?
rrusczyk20:06:18
We won't know until the class starts. Most people wait until the last minute to sign up.
rrusczyk20:06:30
We recommend signing up earlier than that, because our classes sometimes fill!
annie951220:06:35
if i miss class, what can i do??
rrusczyk20:06:38
There is a full transcript made of every class that all enrolled students have access to.
rrusczyk20:06:52
You can review that after class, and ask questions on the class message board if you have any.
xxrxxhxx20:07:38
For school, my teacher says that if I take an Alg 1 honors with him and take a geometry class, I can skip to Alg 2. Does your Geometry class count as a full course in geometry and include everything needed for tests in Geometry and the SAT?
rrusczyk20:07:40
Our geometry class is a full honors geometry course (plus some challenging topics not usually addressed in school). It will cover all you need for school and the SAT, and then some.
clemo20:08:17
I have signed my dd up for Alg 1 this week, at what point do I need to decide if I want a report/grade provided at the end of the course?
kevint20:08:17
How do I notifiy the teacher that I want to be graded for the class?
rrusczyk20:08:19
By the end of the course. You should let us know when you know for sure you need a grade. Email classes@artofproblemsolving.com. It is best to decide before the class starts, and to let us know.
DreamBird20:08:24
what is the transcript? is it exactly what is shown in a regular class?
rrusczyk20:08:25
Yes
phanwort20:08:45
what is the average age of the people in this corse
rrusczyk20:08:47
For the Algebra 1, the students are typically 10-14 years old.
Bob Bobason20:09:08
do any of them require you to have LaTeX or do they all dont need it?
rrusczyk20:09:10
You don't have to learn LaTeX for anything in our courses if you don't want to.
spazmaster10020:09:18
how can you type that fast?
rrusczyk20:09:29
I have 23 fingers. I have trouble when I have to type x, y, or z.
countmath20:10:27
If students have problems grasphing the math concepts, what recourse do they have? What kind of tools do you use to bring clarity?
rrusczyk20:10:28
Students must be willing to ask questions! We have extra assistant instructors in the room, like chesspro tonight, and they are there to help students 1-on-1 if they need it. Also, the class message board is always available, and you'll get good answers if you ask good questions.
k00lperson20:10:44
is counting and probability the easiest course?
rrusczyk20:10:46
Algebra 1 is probably easiest.
You-lost-the-game20:11:20
The sample algebra problem was a little difficult for me to follow. Mostly due to speed. Is this an indication that I may not be ready to take this class?
rrusczyk20:11:22
No -- that problem is in the middle-level difficulty for the Algebra 1 course. This is a sign that you're probably not ready for Algebra 2, but you'll probably be fine in the Algebra 1.
Mycroft20:11:50
Do slow typists have difficulty in these classes?
rrusczyk20:11:52
Not so much -- students are usually typing pretty short answers. And you'll get better over time. I sure have.
countmath20:12:12
Is it possible to explain abstract math concepts via message boards?
rrusczyk20:12:14
Absolutely. That's why there are over 1.5 MILLION posts on the Art of Problem Solving message board.
DreamBird20:12:23
but your answers are not so short
rrusczyk20:12:24
Because I have 23 fingers :)
annie951220:12:44
can i enroll in the class that has already started?
rrusczyk20:12:46
If the class has not filled, yes, you can enroll up to 2 weeks or so after the class starts.
rrusczyk20:13:10
Let's talk about the counting class,
rrusczyk20:13:16
and then I'll take more questions.
rrusczyk20:13:19
Introduction to Counting & Probability
rrusczyk20:13:58
In the Introduction to Counting and Probability class, we cover basic and intermediate counting concepts, including casework, multiplication, permutations, combinations, Pascal's triangle, probability, combinatorial identities, and the Binomial Theorem.
rrusczyk20:14:02
The main emphasis of this class is learning how to take an organized approach to counting, and understanding that nearly all of counting is learning when to use the basic arithmetic operations division, multiplication, addition, and subtraction (and of course why to use them when you use them).
rrusczyk20:14:25
Students completing this course, who work most of the problems, should come out of the course knowing how to tackle any MATHCOUNTS counting problems, most AMC 10 and AMC 12 counting problems, and even some AIME counting problems. The concepts in this course are also crucial to understanding computer science.
rrusczyk20:14:39
We'll now take a look at a couple of sample problems from the course, which highlight some of the tactics we'll investigate in this class.
rrusczyk20:14:53
Anamfija20:16:16
Does each girl count as a separate person, or is each girl just counted as a girl?
mathepic20:16:16
Is boy 1 different than boy 2 in the fact that if we swap them, we get a different way to seat them?
rrusczyk20:16:31
Are you the same as the girl you sit next to in your class?
rrusczyk20:16:39
I don't think that's a school I want to go to :)
rrusczyk20:16:54
Indeed, the girls are different, and the boys are different.
Iggy Iguana20:17:07
7!=5040
DreamBird20:17:07
7!
skylord581620:17:07
math1720:17:07
7!
rrusczyk20:17:24
Why?
spazmaster10020:17:56
bacuae for the first chair we have 7 choises for the second 6 third 5 etc.
math1720:17:56
because the first person has 7 way the next 6 and so on
DreamBird20:18:03
in the first seat, there are 7 choices(peaople)
Iggy Iguana20:18:03
there are 7 choices for who sits in the 1st chair, 6 choices for second chair...
rrusczyk20:18:06
This is a straightforward application of multiplication: there are 7 students who could sit in the first seat. For each of these choices we make for the first seat, there are six ways to choose a student for the next seat, so there are 7 x 6 ways to seat the first two students.
rrusczyk20:18:10
Continuing in this vein, for each of these 7 x 6 ways to seat the first two students, there are 5 ways to pick a student for the third seat. Thus, there are 7 x 6 x 5 ways to seat the first three students.
Anamfija20:18:12
There are 7 choices for seat #1, 6 choices for seat #2, 5 choices for seat #3, etc.
rrusczyk20:18:15
We keep going like this: there are 4 ways to seat the fourth student, 3 ways to seat the fifth, 2 ways to seat the sixth, and one way to seat the last student. This gives us 7 x 6 x 5 x 4 x 3 x 2 x 1 ways to seat all the students.
rrusczyk20:18:21
We run into products like 7 x 6 x 5 x 4 x 3 x 2 x 1 so much in mathematics that we have a symbol and a name for it. We write 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! and we call this 'seven factorial'.
rrusczyk20:18:31
Similarly,
rrusczyk20:18:35
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720.
Mycroft20:18:39
Then wjy do you bother mentioning gender?
DreamBird20:18:40
so the girl and boy thing is useles?
rrusczyk20:18:49
Ah, you taunt me. I'll get you for that:
rrusczyk20:18:53
That problem was pretty simple. Let's put a wrinkle in it. Suppose we must have a girl in the first chair and a girl in the last chair. Then how many seatings are there?
rrusczyk20:19:03
What's wrong with this answer:
rrusczyk20:19:08
There are 4 ways to choose the girl for the first chair. After that, we have 6 students left for the next chair, then 5 for the next, and so on, giving us a total of:
rrusczyk20:19:11
4 x 6 x 5 x 4 x 3 x 2 x 1 seatings.
rrusczyk20:19:17
What's wrong with that?
Iggy Iguana20:19:38
the last chair
skylord581620:19:38
Forgot the last girl.
xxrxxhxx20:19:38
the last chair can have a boy
Anamfija20:19:38
There is a girl in the last seat that you did not count.
Canis Lupus20:19:38
girl in last chair
Omnigeek620:19:41
that only has a girl in the first chair
rrusczyk20:19:45
The problem here is the last chair - we must have a girl in that last chair, but our approach above definitely does not guarantee this. We might end up with a boy left at the end, which would violate the problem.
rrusczyk20:19:59
How do we deal with that?
Iggy Iguana20:20:39
do them seperately
sparkles25720:20:39
the last chair should be three
rrusczyk20:20:47
We can think to ourselves 'How would we seat the kids according to these restrictions if we had to make up a seating ourselves?' Our answer is: we'd seat the girls at the ends first, so we make sure we satisfy that restriction. What do we find?
turak20:21:09
frst count the first chair then the last then the middle ones
rrusczyk20:21:13
And what do we get?
xxrxxhxx20:21:24
so 4*3 for the 2 restrictions, and then 5*4*3*2*1 because of the other seats. so 4*3*5*4*3*2
Mycroft20:21:24
4x5x4x3x2x1x3
skylord581620:21:24
Iggy Iguana20:21:24
4x5x4x3x2x1x3
Omnigeek620:21:24
4 ways for 1st, 3 for last, then the remaining 5 chairs as if they were a seperate group. 1440
rrusczyk20:21:37
As before, there are 4 ways to seat a girl in the first seat. Next we seat a second girl in the last seat - there are 3 girls left, so there are 3 choices. Now we have our restriction taken care of. We can then seat the rest of the students as before. There are 5 students left to choose one for the second chair, then 4 students for the third chair, and so on.
rrusczyk20:21:59
Thus, we have 4 x 3 ways to seat girls at each end, and for each of these seatings we have 5 x 4 x 3 x 2 x 1 ways to seat the rest of the students, for a total of:
rrusczyk20:22:02
4 x 3 x 5 x 4 x 3 x 2 x 1 = 1440
rrusczyk20:22:05
ways to seat the students such that there is a girl on either end.
rrusczyk20:22:26
This example brings up two important counting concepts.
rrusczyk20:22:30
First, when dealing with a counting problem that has restrictions, it often pays to think about how you would create one possible arrangement yourself. Here, we realize that if we seated the students ourselves, we'd start with the girls on the ends. This brings us to our second important counting concept:
rrusczyk20:22:39
When dealing with restrictions, it usually helps to deal with the restrictions first. Here, we took care of the girls on the ends first since that was our restriction.
rrusczyk20:23:00
However, there are other clever ways for dealing with restrictions. Let's check a couple others out:
rrusczyk20:23:05
We still have 7 students to seat in a row, but two of them, Ali and Brianna, refuse to sit next to each other. In how many ways can we seat the students now?
rrusczyk20:23:13
What's wrong with this solution:
rrusczyk20:23:16
There are 7 ways to seat Ali. We deal with the restriction first and realize that we can't seat Brianna in either of the seats next to Ali. Hence, Brianna has 4 choices. Then the next student has 5 choices, the one after that has 4 choices, and so on.
rrusczyk20:23:21
What's wrong?
turak20:23:59
if ali sits on the end
skylord581620:23:59
What if Ali sits on the end?
Omnigeek620:23:59
Ali could be one on end.
Anamfija20:23:59
Brianna might be sitting on the 1st or last seat.
hlee.mathcountscoach20:23:59
Brianna has 5 choices if Ali sits on the end.
spazmaster10020:23:59
What if he sits on the end?
rrusczyk20:24:03
The problem here is that there are not always 2 seats next to Ali - sometimes he may be put at the end. Hence, sometimes Brianna will have 5 choices for her seat.
rrusczyk20:24:09
We could deal with this by using casework (and we'll discuss very important casework strategies in the course - these tricky casework problems are often the difference in proceeding to the next level in MATHCOUNTS/AMC), but there is a slicker approach. What else could we do?
mathepic20:24:41
It assumes Ali is not on the end - A better method would be to get rid of the restrictions, and subtract invalid cases at the end
Iggy Iguana20:24:41
complementary
hlee.mathcountscoach20:24:41
Use complementary counting - count what you DON'T want.
xxrxxhxx20:24:41
complementary countin=count the ones that cant be first and subtract from total
rrusczyk20:24:55
What makes this problem hard is the restriction that Ali and Brianna are not adjacent. We know there are 7! ways to seat the students without any restrictions.
rrusczyk20:24:59
Instead of counting our desired seatings directly, we count what we don't want and subtract.
rrusczyk20:25:05
We know there are 7! ways without restrictions, so we will try to count those that violate our restriction that Ali and Brianna are separate. We'll then subtract these violators from our total.
rrusczyk20:25:15
In how many ways can we seat Ali and Brianna if they are together?
skylord581620:25:52
12 ways.
Iggy Iguana20:25:52
12
rrusczyk20:25:59
Why 12?
rrusczyk20:26:11
Again, we could seat Ali and then note that Brianna has . . . uh-oh. Brianna might have 1 or 2 choices. We don't want to do casework. What can we do with Ali and Brianna to easily count those cases in which they are together?
skylord581620:26:33
6 ways, but they could be swapped, so 12.
Kayak9720:26:33
6*2
mathepic20:26:33
6 different spots, 2 different orders.
rrusczyk20:27:05
There are 6 pairs of chairs they could sit in, but they could swap.
rrusczyk20:27:10
Here's another way to think about it:
hlee.mathcountscoach20:27:15
Consider Ali and Brianna as one "body" - then there are 6! ways to arrange the students, multiplied by 2 because Ali and brianna have 2 ways. This is 6!*2=1440.
rrusczyk20:27:30
We can pretend Ali and Brianna are one person: AliBrianna. Then, we have 6 students and we have no restrictions. Thus, we have 6! ways to seat these students.
rrusczyk20:27:42
But we have to remember that Ali and Brianna are not the same person. They could be AliBrianna or BriannaAli. Thus, for each of our 6! seatings, there are 2 orders in which we can seat Ali and Brianna in their slot. Hence, there are 2 x 6! ways to seat the students such that Ali and Brianna are together.
rrusczyk20:27:49
So, in how many ways can we seat them so that they are apart?
hlee.mathcountscoach20:29:19
So, the answer should be 7!-1440=5040-1440=3600.
Iggy Iguana20:29:19
3600
Iggy Iguana20:29:19
5040-1440=3600
Anamfija20:29:19
7! - 2 x 6!
rrusczyk20:29:37
There are 7! ways without restrictions, and 2 x 6! ways for them to be together. This leaves 7! - 2 x 6! ways for them to be apart.
rrusczyk20:29:49
So, the answer is 7! - 2 x 6! = 7 x 6! - 2 x 6! = 5 x 6! = 5 x 720 = 3600.
rrusczyk20:29:54
This example brings up a couple more important tactics.
rrusczyk20:30:00
First, when it looks hard to count something directly, try counting the opposite of what you're asked for. We call this approach complementary counting, since 'complement' in dealing with groups of objects in mathematics roughly means 'opposite'. I also call this 'counting what you don't want'.
rrusczyk20:30:35
Second, when your restriction is that some of your items must remain together when putting them in a row, a useful tactic is to consider the items all together as a single item, as we did AliBrianna above. Then you separately consider how many ways you can order the items within the group.
rrusczyk20:30:51
Here's another approach to the problem that uses organized casework:
Omnigeek620:30:53
There are 7 ways to seat Ali. For two, there are 5 ways to seat Brianna, for the other 5, there are 4 each. So, 30 times 5!
rrusczyk20:30:56
Very nice.
rrusczyk20:31:05
These three basic examples show why it is pointless to memorize your way through counting - I can ask zillions of variations of the above questions. Instead of memorizing your way through each variation, you should learn when to add, when to subtract, when to multiply, and when to divide. Since you already know how to perform these operations, once you know when to do them, you know how to count!
rrusczyk20:31:36
The first of these three problems was considerably easier than most of the problems we will do in the course. The second and third are closer to the middle in difficulty, though they are still a good bit easier than the average problem.
rrusczyk20:31:55
In general in the course we will go through the ideas more gradually than we did here - each idea will be explored with gradually more difficult examples. Thus, the pace at which new ideas are introduced is slower than we did here (in which we introduced 4 general tactics in two problems!)
rrusczyk20:32:03
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
rrusczyk20:32:12
We are offering the course at two different times this fall.
rrusczyk20:32:23
One offering of the course starts Monday, October 5, and is on Mondays until January 11. The timing of this class is designed specifically for homeschooled and European students, as the class is 1 - 2:30 PM ET (10 - 11:30 AM PT).
rrusczyk20:32:36
The other offering of the course is Thursday nights, from October 8 through January 14, 7:30 PM - 9 PM Eastern (4:30 - 6 PM PT).
rrusczyk20:32:50
Both offerings will be taught by Jeremy Copeland. Jeremy earned his Ph.D. in mathematics from the University of Chicago in 2006 and was on the math faculty at MIT from 2006 to 2009. He specializes in turning hard problems in geometry, algebra, and mathematical physics into easy problems in combinatorics and graph theory. He has spent most of this decade teaching gifted students at top tier colleges and brings this perspective to the AoPS staff.
rrusczyk20:33:04
This course will use a textbook in conjunction with the course: our own Introduction to Counting & Probability 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:
rrusczyk20:33:14
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.
rrusczyk20:33:55
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, however 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. These solutions will be evaluated by AoPS graders, and you will receive detailed feedback.
rrusczyk20:34:30
This course is also supplemented by our revolutionary online learning system, Alcumus. Alcumus consists of a database of problems and video lessons, and is backed by an algorithm that delivers problems to students based on their performance on previous problems. Throughout the course, students will be given guidelines about what they should be doing with Alcumus, which will provide much of the basic practice required to master the material. Because students will have Alcumus to reinforce the basics, most of the message board problems and Challenge Sets will focus on harder material.
rrusczyk20:34:54
And through Alcumus you'll get to see videos with a particularly handsome and dashing instructor.
rrusczyk20:35:07
Are there any questions about this class or the textbook?
Kayak9720:35:44
Are you the instructor?
rrusczyk20:35:46
Watch the videos and see!
k00lperson20:36:13
wat is the wesite for those videos
rrusczyk20:36:18
Click Alcumus on our site.
rrusczyk20:36:24
I also do videos for MATHCOUNTS
rrusczyk20:36:31
You can find those on their website.
rrusczyk20:36:49
That's the Alcumus page.
clemo20:37:06
Is Alcumus useful if you are taking the Alg.1 class?
rrusczyk20:37:07
Right now it only has counting problems. We will add algebra around year-end.
countmath20:38:02
Is this course suitable some one taking algebra I in a middle school?
rrusczyk20:38:04
If the student is eager about math and has experience with basic algebra, yes.
Mycroft20:38:22
Will it be getting Calculus or Number THeory problems?
rrusczyk20:38:23
Alcumus will get number theory and calculus problems some day, but it might be a while from now.
testguy10320:38:36
is there a video component to this virtual classroom for normal classes?
rrusczyk20:38:43
No; this page explains why: http://www.artofproblemsolving.com/Classes/classroom.php
rrusczyk20:39:01
Let's do some geometry, and then I'll take some more questions.
rrusczyk20:39:04
Introduction to Geometry
rrusczyk20:39:08
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
rrusczyk20:39:11
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.
rrusczyk20:39:21
In the diagram shown, DEOC is a square. The radius of circle O is 6 in. What is the number of inches in AC? Express your answer in simplest radical form.
rrusczyk20:39:25
rrusczyk20:39:47
You can use that link to open the diagram in another window if you want.
rrusczyk20:39:51
In order to find AC, what will we need?
skylord581620:40:27
CO
xxrxxhxx20:40:27
the length of co
testguy10320:40:27
CO
annie951220:40:27
length of oc
mathepic20:40:27
OC, which requires us to find OD
rrusczyk20:40:32
We need AO, which we already have, and OC. How can we find OC?
skylord581620:41:06
Luckily, we know OD!
pytheagle20:41:06
We have OD=6 since it's the radius.
rrusczyk20:41:12
And what will we do with that?
testguy10320:42:04
which is radius / sqrt(2)
bobcat12020:42:04
divide it by sqrt(2)
Kayak9720:42:04
divide it by sqrt2
Iggy Iguana20:42:04
OE = 3 sqrt2
rrusczyk20:42:08
OC is a side length of a square
rrusczyk20:42:16
We know the 6 inch radius of the circle is also the diagonal of the square.
rrusczyk20:42:21
Because the diagonal of the square has length 6 and OC is a side, we have:
rrusczyk20:42:25
rrusczyk20:42:30
How do we find AC?
bobcat12020:43:32
pythagorean theorem
stargazer41820:43:32
pythagorean theorem
annie951220:43:32
pythagorian theorm
skylord581620:43:32
Iggy Iguana20:43:32
3 sqrt 2 squared + 6^2 then take sqrt
mathepic20:43:34
Phythagorean Theorem
rrusczyk20:43:39
And what do we find for AC?
Iggy Iguana20:44:41
3 sqrt 6
pytheagle20:44:41
3\sqrt{6}
pytheagle20:44:41
bobcat12020:44:41
3sqrt(6)
annie951220:44:41
3sqrt 6
rrusczyk20:44:45
rrusczyk20:44:57
Let's try a harder one.
rrusczyk20:45:01
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.
rrusczyk20:45:05
rrusczyk20:45:23
We need the radius of the small circle, so what should we do with our diagram?
bobcat12020:45:54
set the radius to r
rrusczyk20:45:58
OK, then what?
mathISuber20:46:05
draw in a radius
rrusczyk20:46:10
Where should we draw it?
pytheagle20:46:42
From the center to the tangent
bulutcocuk20:46:42
to the tangent points
bobcat12020:46:42
tangent to arc AB
skylord581620:46:42
Also, draw a radius of the small circle touching the large circle
rrusczyk20:46:48
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.
rrusczyk20:46:53
pytheagle20:46:59
Make a square from the center to the 2 tangents made by the straight lines.
skylord581620:47:02
Make a square such that the upper-left point is the center of the small circle, and is perpendicular/parallel to the walls of the sector.
rrusczyk20:47:08
We notice that HOGP is a square. How will the square help us?
bobcat12020:47:41
find the lenght of PO
turak20:47:41
PO is r\sqrt2
bulutcocuk20:47:41
now set up a relationship with the diagnol of the square + r = 3
rrusczyk20:47:46
Drawing a diagonal in the square will give us a radius of the large circle (FO) in terms of r.
rrusczyk20:47:50
Because the side length of the square is r, the diagonal is r*sqrt2.
rrusczyk20:47:54
rrusczyk20:48:11
Now what?
Anamfija20:48:34
The length of PO is equal to sqrt(2)*r, so r + sqrt(2)*r equals 3 cm, the radius of the big circle.
Iggy Iguana20:48:34
r + r sqrt2=3
Canis Lupus20:48:34
r+rsqrt2=3
bobcat12020:48:34
now r(1+sqrt(2))=3
rrusczyk20:48:43
We set up an equation and solve it. We know that OF is a radius of the quarter circle, so OF = 3. However, we can write OF = OP + FP = r*sqrt(2) + r.
rrusczyk20:48:51
So, how do we find r?
Iggy Iguana20:49:12
factor the left side to r(1+sqrt2) =3
rrusczyk20:49:17
Good. And then?
Iggy Iguana20:49:36
divide by 1+sqrt2
skylord581620:49:42
rrusczyk20:49:44
rrusczyk20:49:48
Can we simplify this?
bulutcocuk20:50:00
and conjugate it
MuffinMan0920:50:00
conjugates
Iggy Iguana20:50:00
multiply each side by 1-sqrt2
pytheagle20:50:00
Bob Bobason20:50:00
r=-3+3sqrt2
rrusczyk20:50:03
We multiply both the top and bottom by 1-sqrt2
rrusczyk20:50:07
rrusczyk20:50:14
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.
rrusczyk20:50:22
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.
rrusczyk20:50:28
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
rrusczyk20:50:49
This course is a full geometry course. The equivalent (and a bit beyond) of a typical honors geometry course.
rrusczyk20:50:53
The course will meet for 24 weeks on Fridays, starting October 9, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and the course ends on April 9.
rrusczyk20:50:58
I will be teaching the course. I wrote the text for the course, and have written a number of other AoPS texts. You can read the rest of my bio here: http://www.artofproblemsolving.com/Classes/instructors.php.
rrusczyk20:51:03
Speaking of the text, the required text for the course is our 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:
rrusczyk20:51:13
Students will be able to read additional material in the text 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.
rrusczyk20:51:48
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, however you may post them to the message board if you like. The class also has 8 longer problem sets for which you should write up your full solutions and submit them. These solutions will be evaluated by AoPS graders, and you will receive detailed feedback.
rrusczyk20:51:52
Are there any questions about this class or the textbook?
bulutcocuk20:52:32
is there a intermediate geometry on the queue?
rrusczyk20:52:45
We have an "Olympiad Geometry" class that we offer every 12-24 months.
countmath20:52:57
Between Introduction to counting & probability and Introduction to Geometry, which one would you recommend a student currently studying algebra I?
rrusczyk20:52:58
Definitely the counting. The geometry is much harder.
pytheagle20:53:14
When is the next "Olympiad Geometry"?
rrusczyk20:53:16
We haven't decided yet.
rrusczyk20:53:52
Looks like I've exhausted all of your questions! Thanks for coming tonight. That's it for the Math Jam tonight. If you have any more questions about the classes, you can write me at classes@artofproblemsolving.com.
