**Companion Guide to Understanding Mathematics: From Counting to Calculus**

*Over 10,000 problems and solutions, all written to complement each section of explanation provided in Understanding Mathematics: From Counting to Calculus. Includes end-of-chapter tests, helpful hints to problem solving, and expanded explanations of key topics.*

**Why Anyone Needs This Math Book**

She sees the wave coming. It's a big one. She starts paddling. She draws her breath in as the wave swells up beneath her. She is plunging down the face of the wave. She cannot see through the spray in her eyes, can't hear over the roar of the crashing water. It's a ten foot fall, face first, and then the wave will crash down on top of her. As her stomach drops, her arms push up on the board and her legs are under her. She is on her board going hard to the right. The ripples from takeoff are drops in the explosion of the crashing wave.

No surf video or book can prepare someone to pop up on a big day at Blacks beach. The only way to learn is through practice. It is equally naive to try to learn math without practice problems. From addition in grade school to functional analysis in graduate school, no one can master mathematical material without practice problems.

**What This Book Provides**

** ** The beauty and elegance of mathematics can lull even the most alert of students into a sense of false confidence. After reading a mathematical idea the student thinks, "That makes sense, I get it" only to find himself lost when asked to solve a specific problem. The subtle gaps in the student's understanding are like black holes they can not be seen directly they appear only against the background of problems he can't solve.

This companion text provides hints, examples, explanations, exercises, problems and solutions organized to complement a student's reading of "Understanding Mathematics."

**The Layout of This Text**

Throughout this book we refer to "Understanding Mathematics: From Counting to Calculus" as the "main text" and refer to this book as the "companion text."

The chapters and sections of this companion text correspond to the chapters and sections of the main text. Each section of this text should be completed after the student has read through the corresponding section of the main text.

The principal purpose of each section is to provide problems for the student to practice the newly learned material. As suits the subject, some sections supply examples and explanations preceding the exercises. Occasionally, additional topics and techniques are introduced.

The problems in this text follow a progression from section to section and chapter to chapter. Each section builds on the earlier sections. For example, solving algebra problems will require understanding how to add, multiply, use fractions and exponents. The initial exercises in a given section are straightforward applications of the material in the main text. The latter problems in the section will combine the new material of the section with material from previous sections and may be more challenging. The final problems in the section will combine the new material with topics from previous sections and previous chapters. Each chapter summary contains a 20 question "End of chapter review."

The second part of this book consists of the solutions to the odd-numbered problems. The solutions provide an opportunity for the student to check her work. More importantly the solutions demonstrate how to solve the problems. A student could learn a great deal from studying the solutions to all of the odd numbered problems in a section before even attempting the even-numbered problems.

**Who Would Benefit from This Book**

Parents teaching their children will find the problems an invaluable source of assignments to help their kids master the material. A motivated student reading the main text to get ahead can solidify her knowledge by working through the problems and learn to solve more difficult problems by examining the solutions. A college graduate looking to review for economics or business school will have a full range of problems to brush up on. A high school student studying for a standardized test can practice the relevant exercises and end of chapter tests.

**Why Buy One Book that Contains 12 Years of Mathematics?**

Standard math texts include formal explanations and proofs. Though proofs are essential for mathematicians they can intimidate and confuse some students. The danger of giving up too much rigor is that one can inadvertently foster misconceptions and engender false assumptions. We try to maintain a balance by only making true, mathematically rigorous statements (or at least including disclaimers when we over simplify) and consistently warning the reader of common mistakes to avoid. Trading formality for readability, this companion text fits 12 years of math problems into one volume.

**This Math is Really Used in Practice**

** **Saving up for gas for a road trip, figuring out the interest an investment will return, and calculating the acceleration of a falling object are some of the problems in this text. The application of everything from addition to the Pythagorean Theorem in day-to-day life is emphasized throughout this companion text. Most people, from the shores of San Diego to the research labs of the University of Pennsylvania in Philadelphia, learn math to solve the problems that they encounter.

**What the Reader Should Already Know **

Each section of this companion text assumes the student has completed the previous sections and the corresponding section in the main text. These sections certainly don't assume the student has a perfect understanding of the material from the main text. If she did, these problems wouldn't be necessary.