Art of Problem Solving, High School

February 7, 2011
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Art of Problem Solving

Content: Math
Grade(s): 9th and up
Perspective: Secular
Prep Time: Minimal
Solutions Manual: Yes
Teacher Involvement: Essential
Cost: $$$ || ?
Pages: Varies
Publication Date: 2011
Updated Review: May 26, 2015
Reviewer: Alessa Giampaolo Keener, M.Ed.

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We’re just going to say it like it is: Art of Problem Solving is a math curriculum designed for kids who LOVE math and are really, really good at it. We’re talking about the kiddos who may be starting Algebra 1 at the age of 10.

That’s not to say that other students would not benefit from AoPS. But, if your teen enjoys math textbooks with the visual appeal of lots of bright pictures and sideboxes filled full of random fun facts and real world applications of math skills, then AoPS will probably not be a good fit.

AoPS textbooks are highly regarded for their rigor and depth, but they are dense and intense – which is not a bad thing, if you love math. Written in a conversational style, it’s like reading the transcript from a math lecture. Concepts are introduced and sample problems are explained step-by-step in text and number format.

Students then get the chance to try problems on their own: Exercise problems, review problems, and challenge problems. Don’t worry, though. The Solutions Manual not only gives you the answer, but it also gives you explanations on how to solve the harder problems.

Not sure what level you should start with? Visit the AoPS website’s Recommendation page and look for the Are You Ready? link to find a placement test for each textbook in the series.

Also, be sure to check out the Free Alcumus site on AoPS. You’ll find thousands of problems to supplement some of the textbooks. And, you can even watch video lessons that go with the Introduction to Counting & Probability book.

1. Introduction to Counting & Probability

  • Casework
  • Multiplication
  • Permutations
  • Combinations
  • Pascal’s triangle
  • Probability
  • Combinatorial identities
  • Binomial Theorem
2. Introduction to Geometry

  • Similar triangles
  • Congruent triangles
  • Quadrilaterals
  • Polygons
  • Circles
  • Funky areas
  • Power of a point
  • Three-dimensional geometry
  • Transformations
  • and much more
3. Introduction to Number Theory

  • Primes and composites
  • Divisors and multiples
  • Prime factorization
  • Divisibility rules
  • Remainders
  • Modular arithmetic
  • Number bases
  • Linear congruences
  • How to develop number sense
  • and more
4. Intermediate Algebra

  • Review of basic algebra topics
  • Complex numbers
  • Quadratics and conic sections
  • Polynomials
  • Multivariable expressions
  • Sequences and series
  • Identities
  • Inequalities
  • Exponents and logarithms
  • Piecewise-defined functions
  • Functional equations
  • and more
5. Intermediate Counting & Probability

  • Inclusion-exclusion
  • 1-1 correspondences
  • Pigeonhole Principle
  • Constructive expectation
  • Fibonacci and Catalan numbers
  • Recursion
  • Conditional probability
  • Generating functions
  • Graph theory
  • and more
6. Pre-Calculus

  • Trigonometry
  • Complex numbers
  • Vectors
  • Matrices


7. Calculus

  • Limits
  • Continuity
  • Derivatives
  • Integrals
  • Power series
  • Plane curves
  • Differential equations

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.


  • Liz W. May 6, 2014 at 9:18 pm

    My daughter is not mathy, but she did really enjoy Counting and Probability. AoPS was not for her at higher levels, but this book was a hit. It’s a shorter book than your average math text and she went through it pretty fast. I wouldn’t consider it a year long math curriculum.

  • Cindy H February 14, 2014 at 9:20 pm

    Art of Problem Solving was very good for us. My math-loving son took a class in Intermediate Counting and Probability and one in Trig and Imaginary Numbers. It was quite a while ago, and things have probably changed. At the time, the classes were enrichment, offering a once-a-week online classroom but no grades. The online classroom was well-regulated, and somehow they managed to keep it on-topic as well as fun.

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