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| Introduction: A brief travel guide to the journey ahead | |
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| To the student: A great adventure | |
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| To the instructor: The mathematics to come | |
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| Puzzles and patterns: A precursor to proofs | |
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| Senators and snakes | |
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| Checkerboards undercover | |
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| Flipping cards and coins | |
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| Juggling balls | |
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| Counting on a good hand and a good grade | |
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| Stepping back | |
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| Bringing theorems to justice: Exposing the truth through rigorous proof | |
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| Making a statement | |
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| The truth, the whole truth, and other stuff besides the truth | |
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| Arguing the case through proof | |
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| The domino effect of bringing everyone down through guilt by induction | |
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| Stepping back | |
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| Delving into the dependable digits: Counting on counting numbers | |
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| Divide and conquer | |
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| Solving equations by simple division | |
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| Now ready for prime time | |
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| Stepping back | |
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| Going around in circles: The art of modular arithmetic | |
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| The mod side of mathematics | |
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| Old-school algebra gets an extreme makeover | |
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| The power of congruences | |
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| Public secret codes | |
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| Stepping back | |
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| The irrational side of numbers: A world of nonrepeating digits | |
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| Divisible and indivisible | |
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| Forgoing fractions | |
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| Irrationalit-e | |
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| Stepping back | |
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| Discovering how to function in mathematics: Moving beyond ordinary relations | |
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| Get ready, get set | |
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| Fun with functions | |
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| Moving onto an intimate one-to-one relationship with functions | |
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| An intuitively-believable-yet-challenging-to-verify correspondence | |
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| Stepping back | |
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| Infinity: Understanding the unending | |
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| Comparing cardinality | |
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| Countable collections | |
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| The return of power sets | |
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| A pair of paradoxes | |
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| Stepping back | |
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| Recursively defined functions: The next generation | |
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| Moving ahead by looking back | |
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| Forgoing convergence-A formal look at power series | |
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| Generating closed formulas through generating functions | |
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| Stepping back | |
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| Discrete thoughts of counting: Quantifying complicated quantities | |
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| Pigeons without a home | |
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| PERMUTATION = UP TO A REMINT | |
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| A combination that unlocks counting | |
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| Stepping back | |
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| Quantifying uncertainty with probability: A likely story? | |
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| What are the chances? | |
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| A coin toss? | |
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| What's the deal? | |
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| Stepping back | |
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| The subtle art of connecting the dots: Edging up to graphs | |
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| Bridging the graph | |
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| Making the circuit | |
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| Branching out to trees | |
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| Completely connected graphs | |
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| Stepping back | |
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| Just plane graphs: Drawing without being cross | |
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| Drawing without crossing | |
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| Seemingly similar graphs | |
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| Sensing symmetry | |
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| Stepping back | |
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| Visible and invisible universes: Geometric vignettes | |
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| Sizing up symmetry | |
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| Keeping an artful eye on art galleries | |
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| Up, up, and away-The fourth dimension | |
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| Stepping back | |
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| A synergy between geometry and numbers: Circles and Pythagorean triples | |
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| Rightful triangles | |
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| Determining which triangles are allright | |
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| A rational look at the circle | |
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| Stepping back | |
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| The mathematical mysteries within a sheet of paper: Unfolding pattern and structure | |
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| Getting into the fold | |
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| The story unfolds | |
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| Adding arithmetic to generate a wild parity | |
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| Stepping back | |
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| Take it to the limit: An initial approach to analysis | |
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| Getting really close | |
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| Grasping the cloudlike Cantor set | |
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| Turning to ternary | |
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| Stepping back | |
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| Uninterrupted thoughts of continuity: A jump-free journey | |
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| Functions without lifting a finger | |
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| Valuing intermediate values | |
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| A devilish staircase | |
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| Stepping back | |
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| An abstract world of algebra: Reconciling with your X | |
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| Grasping at groups | |
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| A group mentality | |
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| Building a group table | |
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| Stepping back | |
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| Cycles and curves: Algebraic structure in numbers and geometry | |
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| The subculture of subgroups | |
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| Cyclic groups | |
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| A circular group | |
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| Stepping back | |
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| Further frontiers | |
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| Hints, remarks, and leading questions | |
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| A proof primer: The proof is in the pudding | |
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| Preproof prep | |
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| Penning the proof | |
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| Post proof | |
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| Commentary for instructors | |
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| How I used this material in my classroom | |
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| A math personality questionnaire | |
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| Acknowledgments | |
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| Index | |