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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 | |