Art of Problem Solving: The Basics and Beyond

February 7, 2011
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Art of Problem Solving

Content: Math
Grade(s): 7th – 10th
Perspective: Secular
Prep Time: Minimal
Teacher Manual: No
Teacher Involvement: Essential
Cost: $$ || ?
Pages: 272
Publication Date: 2011
Updated Review: May 26, 2015
Reviewer: Alessa Giampaolo Keener, M.Ed.

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Back in the 1990s, a group of 3 college friends, who had met during the Math Olympiad Summer Program, created a new math contest which they called the Mandelbrot. The first year of the competition didn’t go so well, so the trio decided to write their own math book to help students think deeper in math. That book is what we now call Volume 1 in the Art of Problem Solving series.

The Art of Problem Solving is the classic problem solving textbook used by many successful MATHCOUNTS programs. It is not designed to be a curriculum that you can say, “Hey, let me substitute this out for 7th grade math.” Instead, AoPS Volumes 1 & 2: The Basics and Beyond are in-depth workbooks for students in grades 7 – 10 who are preparing to compete in math contests. Volume 1 is great for those just beginning to the competition adventure and Volume 2 is geared for high school kids ready for more of a challenge.

On the surface, it might seem like the books are just a collection of math tricks. But, once you get into reading them and doing the problems, you realize the emphasis focuses on learning and understanding methods rather than memorizing formulas. The result is that students come away with skills to be able to solve large classes of problems beyond what is presented in the book.

AoPS is extremely text-heavy, which could be a problem for kids with visual processing or tracking weaknesses. Margins are thin and there just isn’t a whole lot of white space on the pages. The book is black and white and the mathematical images are compact and efficiently sized.

The book’s conversational tone and thorough explanations, however, make up for the lack of visual appeal. You don’t have to be a math whiz to understand what the authors are trying to explain. They introduce a concept, like prime numbers, and then walk you through step-by-step in breaking down how it relates to prime factorization. They also use 3 icons to help readers be aware of tricky and confusing concepts.

If you do decide to use AOPS The Basics and Beyond as a math curriculum, it’s not necessarily the kind of book you just hand off to your kid and say, “Do pages 36 and 37 today.” Instead, you may actually find yourself keeping a notebook as you read through the book out-loud – rewriting the conversational explanations in a math friendly format.

Table of Contents

Volume 1: The Basics

Exponents and Logarithms
Complex Numbers
Linear Equations
Using the Integers
Quadratic Equations
Special Factorizations and Clever Manipulations
What Numbers Really Are
An Introduction to Circles
Triangles, a.k.a. Geometry
Angle Chasing
The Power of Coordinates
Power of a Point
Three Dimensional Geometry
Shifts, Turns, Flips, Stretches, and Squeezes
A Potpourri of Geometry
Operations and Relations
Sequences and Series
Learning to Count
Statistics and Probability
Prove It

Volume 2: And Beyond

Not Just For Right Triangles
More Triangles!
Cyclic Quadrilaterals
Conics and Polar Coordinates
Taking it to the Limit
Complex Numbers
Vectors and Matrices
Cross Products and Determinants
Analytic Geometry
Equations and Expressions
Sequences and Series
Counting in the Twilight Zone
Again and Again
Probability Find It and Make It
Collinearity and Concurrency
Geometry Tidbits
Number Theory
Diophantine Equations
Graph Theory

A Solutions Manual demonstrates the full process for deriving each answer for every problem.

About the Authors:
Richard Rusczyk is the founder of the Art of Problem Solving website. He was a national MATHCOUNTS participant in 1985, a three-time participant in the Math Olympiad Summer Program, a perfect AIME scorer in 1989, and a USA Mathematical Olympiad winner. He is author or co-author of 6 Art of Problem Solving textbooks. Sandor Lehoczky participated in the Math Olympiad Summer Program in 1989, and in 1990 earned the sole perfect AIME score and led the national first place team on the AHSME (now AMC 12). Lehoczky and Rusczyk were co-founders of the Mandelbrot Competition.

Because one opinion is never enough! Have you ever used Art of Problem Solving? How did it work for your family? Share your review below.

One Comment

  • Julie May 28, 2015 at 9:20 pm

    I have loved these books for my math-loving son. They cater to students that are math-inclined. The teaching is great and the problems progress to very difficult problems. The answer book has all the solutions worked out.

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