New AOPS Precalculus Book!

Richard Rusczyk’s long awaited Precalculus Book is now available!  Here is the description:

Precalculus is part of the acclaimed Art of Problem Solving curriculum designed to challenge high-performing middle and high school students. Precalculus covers trigonometry, complex numbers, vectors, and matrices. It includes nearly 1000 problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics competitions such as the American Invitational Mathematics Exam and the USA Mathematical Olympiad. Almost half of the problems have full, detailed solutions in the text, and the rest have full solutions in the accompanying Solutions Manual.

As with all of the books in Art of Problem Solving’s Introduction and Intermediate series, Precalculus is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which new techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text.

About the authors: Richard Rusczyk is the founder of www.artofproblemsolving.com. He is co-author of the Art of Problem Solving, Volumes 1 and 2 and Intermediate Algebra, and author of Introduction to Algebra and Introduction to Geometry. He was a national MATHCOUNTS participant, a three-time participant in the Math Olympiad Summer Program, a perfect scorer on the AIME, and a USA Math Olympiad Winner. The solutions are co-authored by Naoki Sato. He is a curriculum developer and the director of WOOT at Art of Problem Solving. He won first place in the 1993 Canadian Mathematical Olympiad and is a 2-time medalist at the International Mathematical Olympiad. He has been Deputy Leader of the Canadian IMO team three times.

ISBN: 978-1-934124-16-1

Text: 528 pages. Solutions: 272 pages.

Paperback. 10 7/8 x 8 3/8 x 1 1/16 inches.

Telescoping Sums and Products (Problem 1)

While we wait for the Metroplex Math Circle to continue again in mid January, I will be posting a series of problems and solutions from our last session. Those who were fortunate enough to attend learned useful techniques for solving problems common in math contests.

Telescoping Sums and Products

The telescoping sums and products idea is used to solve many problems involving sums or products in algebra. For problems involving sums, the idea is to use identities, to write the sum in the form

and then cancel out terms to get F(n+1)-F(1). Sometimes the desired identity is hard to find, but basically you are searching for it in the recursive form of the sequence, or you can look foor the “conjugates” for the terms you have. The first example is classical. You Certainly know these formulas.

What about ?

Example 1.  Prove that

December 5 – Dr. Andreescu – Sums and Products

Don’t miss our last Math Circle of the 2009 Fall season as Dr. Titu Andreescu presents a lecture and problems on “Sums and Products.”

November 20 – AOPS Linguistics Math Jam

The Art of Problem Solving is hosting a free for Math Jam those interested in linguistics:

 Date: Nov 20 (Fri)
Time: 7:30 PM Eastern
Instructor: Christina Skelton
Two linguists discuss the the Comparative Method, a linguistic theory which describes how languages change over time, and use it to demonstrate that English is related to some exotic languages you never would have imagined, like Sanskrit and Hittite.

 

MITnews – The Math Gap

Here is an interesting article from MIT on patterns in high math achievement based on a study of AMC data.  This research follows on the heels of the paper written by Dr. Andreescu and his colleagues last year.  The study seems to conclude that girls (and presumably also boys) thrive when they are able to study math in a community that reinforces their interests and encourages their talents.

Here are some key excerpts from the article:

Ellison and Swanson arrived at their findings by using a novel source of data: the American Mathematics Competitions (AMC), a 60-year-old annual contest involving 125,000 exceptional high-school students. A select group of students who do especially well on the AMC compete in a series of annual competitions, the U.S. Mathematical Olympiad and the International Mathematical Olympiad. This focus on standout students differs from most studies about math and gender in schools…

The numbers Ellison and Swanson scrutinized indicate that the gender disparity among star math students widens as performance levels increase. In 2007, about 800,000 girls took the math SAT, compared to about 700,000 boys. Yet at the 99th percentile of the math SATs, boys outnumber girls two to one. In their research, Ellison and Swanson divide that upper tier into even smaller segments, using AMC results. Among students in the 94th percentile of the AMC tests, they found, the top boys outnumbered the top girls four to one; at the 99th AMC percentile, six to one; and at the 99.9th AMC percentile, 12 to one…

Ellison thinks this huge gender disparity is linked to another fact: Among those students scoring so highly on the AMC and participating in the math Olympiads, the range of high schools represented is much greater for boys than for girls. “The top boys in the Olympiads come from all over the United States,” says Ellison. “Some of them are from big powerhouse schools, and some are from schools where they’re the only student who’s really good at math. But it’s these 20 high schools where the majority of the girls are coming from.” Those institutions range from Phillips Exeter Academy, an elite New England prep school, to a fistful of public high schools in Northern California, from Palo Alto to San Jose. By contrast, Ellison and Swanson note, half of the boys in the Olympiads come from about 200 high schools…

O’Keeffe, who has a daughter who competed in the math Olympiad, is inclined to agree. “Anecdotally, I do think the difference a community makes is enormous,” she says. “If you’re lucky enough to be at a school with a math club, you might be the only girl in it. At Exeter or Stuyvesant [a prominent Manhattan public high school], you might be in a minority, but you won’t be alone.” To be more rigorous, though, Ellison wants to track many individual students over time…

Nov 14 – Ivan Matic – Combinatorial Games

Ivan Matic, assistant director of the Berkeley Math Circle and a former IMO medalist will teach our students a variety of pricinples related to Combinatorial Games.

Winning strategy for a particular game is a procedure that will ensure a victory no matter how the opponent is playing.   We will discuss some of the two player games that have winning strategies, and try to recognize the patterns for finding the strategies.   After that we will talk about multi-player games and probabilistic games.

Nov 7 – Dr. Harold Reiter – Exploding Dots

This weekend should be a very interesting and accessible lecture by Dr. Harold Reiter:

Exploding Dots is an alternative method for examining place value. We’ll look at some exotic methods of representing numbers and see a few applications. Some methods make use of antidots and some require black holes.

Dr. Reiter is a professor at the University of North Carolina.  Following are some of his many accomplishments:

• Member of the College Board’s College-Level Examination Program (CLEP) Pre Calculus Mathematics Committee
• Chair of the MAA’s Edith May Sliffe Award Committee.
• Former chair of the MATHCOUNTS Question Writing Committee
• Former member of AIME and USAMO committees
• Former chair of SAT II Test Development Committee with ETS. advanced mathematics

Edutopia – “Math test scores soar if students are given the chance to struggle”

Richard Rusczyk linked to a great article based on some recent research suggesting that students benefit from struggling with Hard Problems.

“We’ve found there is a healthy amount of frustration that’s productive; there is a satisfaction after having struggled with it,” says Roberta Schorr, associate professor in Rutgers University at Newark’s Urban Education Department. Her group has also found that, though conventional wisdom says certain abilities are innate, a lot of kids’ talents and capabilities go unnoticed unless they are effectively challenged; the key is to do it in a nurturing environment.

“Most of the literature describes student engagement and motivation as having to do with their attitudes about math — whether they like it or not,” Schorr says. “That’s different from the engagement we’ve found. When students are working on conceptually complex problems in a supportive environment, they do better. They report feeling frustrated, but also satisfaction, pride and a willingness to work harder next time.”

“Motivation is a key aspect of achievement that we often ignore in math; it’s the missing link,” Schorr says. “We need to provide kids with conceptually challenging math problems in an emotionally safe environment, and the teacher plays a critical role in that. Kids can view frustration as an opportunity for success instead of an indication of failure, but that won’t happen without teachers letting the students experience productive struggles.”

October 31 – More of the Triangle Inequality

This Saturday Dr. Dorin Andrica will continue exploring the Triangle Inequality.  For students who may have missed last week’s session he will begin with a brief review.

Arthur Benjamin’s Formula for Changing Math

A good friend of the Metroplex Math Circle, Dr. Arthur Benjamin, is a popular speaker at the world famous TED conference.  Recently, he offered his own idea for fundamentally changing math education in our country.  Like Richard Rusczyk, he sees the singular focus on Calculus as insufficient and distracting from a full math education.