AoPS Intro to Algebra Review – An Active Approach

For a middle or high school student who is motivated to take a more active role in learning Algebra, check out Art of Problem Solving Introduction to Algebra, 2nd Edition, by Richard Rusczyk.  Intended for students in grades 6-10, I’ve found it to be effective in not only teaching Algebra, but also in developing general problem solving skills.


The pages are uncluttered with mainly black and white print and a few boxes highlighted in a peaceful blue – making it easy to follow and not distracting.

Jump Right In

Building concepts in a logical, step-by-step manner, the textbook is thoughtfully written to the student. The method appears to be this: the author carefully introduces new topics and engages the student by posing simple, related example problems, and then gives her a chance to jump in and develop her own solutions before presenting the answer to her.  For example, the chapter on quadratic equations begins by defining quadratic terms and expressions.  Then it leads the student by posing five such questions – gradually increasing in difficulty – related to quadratics.  This starts the student in the process of thinking about how she will answer the questions.  But the solutions are not revealed just yet.  The idea seems to be to motivate the student to reach her own solution before the book explains it to her.  Eventually the solution is explained, but by that time the student will be more aware of it, since she should have at least begun to construct her own.  Contrary to many textbooks I’ve seen, more up front thinking is expected of the student rather than being spoon-fed the material.  This active involvement makes learning math concepts less dull, more enjoyable and easier to remember.

Breaking It Down

Throughout the book there is a parallel with how new material is presented and the fact that complicated problems can often be solved by breaking them into smaller parts or steps.  For a quick example, when imaginary numbers are introducedand defined, the student is asked to evaluate a few simple cases on his own (which he should be able to do by applying the definition).  Next he is asked to solve aquestion that is slightly more difficult, but building off the previous question.  Then another – more difficult, but similar.  Finally, he is to simplify a set of more complicated imaginary number questions.  But if he has been solving the little problems along the way, the difficult questions are more easily diffused because he can break the difficult problem into smaller, simpler parts.  It no longer looks scary or confusing, and he will probably be able to quickly calculate the answer.  As I said above, only after he has had time to develop his own solution are the explanations presented in the text.  This approach makes sense, and is a helpful model for problem solving in general.

Following the above-mentioned example problems, there are Exercises that evaluate the concepts for each section.

At the end of each chapter is a thorough Summary section which highlights definitions, concepts, and/or other important information.  Many of the these sections also contain Problem Solving Strategies, and it is worth your time to read, absorb and apply these.  Next are Review Problems which seem to help measure how well a student understands the chapter.  And finally there is a Challenge Problems section in which will be found the more difficult problems that are likely to really stretch the student’s understanding and help her master the material.


Problems previously seen in MathCounts, the American Mathematics Competitions (AMC8, AMC10, and AMC12,) the American Invitational Mathematics Examination (AIME), the USA Mathematical Olympiad (USAMO) and others are included in some of the exercise, review, and challenge problems.  Working these problems not only helps with understanding the idea of the chapter, but also in preparing for future contests.  In our situation, this book greatly contributed to the preparation of my daughter in past competitions.

Additional Helps

AoPS’s website also provides Intro to Algebra Videos which give the student another visual aid if desired.  Though my daughter did not use these, they are there if you would like them – which I appreciate.

There is a Hints section located in the back of the book for selected problems.  Any problem with a hint will note that at the end of the problem.

I strongly recommend the Solutions Manual.  It provides well written explanations with step-by-step solutions.


It is important to read the How to Use This Book section at the beginning of the book for clarification.  A student should attempt solving a tough problem several times before looking at any hints in the back of the book.  For example, a student may work on a difficult problem for half of an hour or more, but not solve it.  She may feel frustrated and that she is getting nowhere.  But she is learning where her weaknesses and strengths are.  At some point she may want to look at a hint or review the concept notes of the chapter.

If you find that the lessons are taking too long, another option is to pick and choose challenge problems.

Highly Recommended

Overall, we thought highly of this book because there were challenging problems that made you think. 

The book seems creatively written with the intention of not only teaching Algebra, but also developing robust problem solving skills in its reader.  From the perspective of a teacher and especially if a student uses the materials as intended, I think it accomplishes this goal.