Content Outline
Contents
Introduction
Part l: Algebra 1
Chapter 1: The Power of Abstraction A). Defining whole numbers B). Pattern recognition and how generalization leads to axiom systems C). The inductive/deductive package 1). Plan making: Upper-Paleolithic survival strategies 2). The power of being numerate Chapter 2: The Creative Mind A). Defining the rational numbers B). Introduction to Zeno’s paradox C). Discovering the laws of exponents Chapter 3: Discovery A). Gaining a sense of the cosmos 1). Scientific notation and the speed of light 2). The light-year, space-time, and the size of the universe B). The distributive pattern, negative numbers, and simplifying polynomials Chapter 4: Balancing the World A). Solving for an unknown B). Variables as second-order abstractions C). Equations as models of the real world
Chapter 5: Functions A). Introduction to SETI and the Drake equation B). The spring function C). The pendulum and simple harmonic motion D). Defining the scientific method 1). Descartes’ leap of imagination; mathematics as the language of science 2). Archimedes and the law of levers 3). Longing for the harmonies and dealing with the “irrational” 4). Bucky Fuller and the concept of synergy
Chapter 6: Interdisciplinary Connections to Algebra l A). English: The power and creation of myth B). Social studies 1). Plato’s allegory of the cave 2). Greek culture and the rise of democracy C). Science: Greek astronomy, phases of the moon, and Ptolomy’s model of the heavens D). The Murth Game
Part ll: Geometry
Chapter 7: The Greek Mind A). The power of rational thinking B). The demands of space 1). The Pythagorean theorem 2). Symmetry and the nature of crystals C). From Thales to Euclid and the advent of Aristotelian logic Chapter 8: Essential Proofs A). The axioms of Euclidian geometry B). From axioms to theorems; deductive power 1). The supplementary angle theorem 2). The vertical angle theorem 3). If lines are parallel . . . 4). Flat space and the sum of the angles of a triangle Chapter 9: A Journey into Curved Space A). Introduction to “Lineland, Flatland and Spaceland; analogy and mathematical models B). The fourth dimension 1). Building the tesseract 2). A visit from the oversphere C). Finite but unbounded space 1). From lineland to circleland and from flatland to sphereland 2). Geodesics and the absence of parallels 3). Curved space and the sum of the angles of a triangle 4). Einstein’s conception of a curved, 4-D space Chapter 10: Can’t Be but Must Be A). Indirect proofs 1). Irrational numbers and a return to Zeno’s paradox 2). Cantor’s proof of larger orders of infinity 3). Non-Euclidian geometries B). Unresolved paradoxes and self-referential statements 1). Epimenides’ paradox 2). Gödel’s theorem Chapter 11: Deductive Power A). The amazing nine-point circle B). Limits and the derivation of formulas for area and volume of cones and spheres C). Ratios in similar objects and the connection to biology Chapter 12: Think Deeply A). Changing assumptions; the nature of revolution B). Unifying the learning experience Chapter 13: Student Geometry Work
Part lll: Algebra 2
Chapter 14: Planning Ahead A). The forebrain and cultural evolution B). Introduction to mathematics and the human condition Chapter 15: Let the Games Begin A). Analysis and synthesis B). The “real” games 1). Tic-Tac-Toe vs. Rock, Paper, Scissors 2). Checkers vs. Undercut 3). The Prisoner’s Dilemma and non-zero-sum games C). The dollar auction and the cold war
Chapter 16: The Inductive Leap A). Generalizing sums of sequences B). Resolving Zeno’s paradox; limits are real C). Deriving the equations of the conic sections D). Galileo’s confirmation of the Copernican model E). Newton’s examination of instantaneous change Chapter 17: Challenging the Axioms A). Gravitation and curved space B). Relative time Chapter 18: Models of Change A). Synergy vs. entropy; Bolzmann’s great idea B). The exponential function 1). Population and compound growth 2). Radioactive decay C). Darwin and Mendel; the how and why of physical evolution Chapter 19: Imagination Unleashed A). Elements, alloys, and compounds; an essential look at analysis and synthesis B). Mendeleev’s chart, a classic example of the scientific method C). Imaginary numbers and modeling vectors D). The Gaussian curve and building a model of the atom E). The paradoxical quantum Chapter 20: Mathematics and Morality A). Cooperation and defection as part of the human condition B). Humans as social critters; the ego dilemma C). Creating grand survival strategies; the role of education D). The final project Chapter 21: The Handout Chapter 22: Student Algebra 2 Work
Part lV: A Model for Integration: Creating a department of interdisciplinary studies
Recommended Reading Bibliography
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