Table of Contents. Montessori Algebra for the Adolescent Michael J. Waski"

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1 Table of Contents I. Introduction II. Chapter of Signed Numbers B. Introduction and Zero Sum Game C. Adding Signed Numbers D. Subtracting Signed Numbers 1. Subtracting Signed Numbers 2. Rewriting as Addition 3. Addition and Subtraction on a Number Line E. Dividing Signed Numbers F. Multiplication of Signed Numbers G. Games: Integer Bingo III. Chapter of Graphing B. The Coordinate Plane C. Midpoint and Distance 1. Midpoint Visually 2. Midpoint Formula 3. Distance Between Points Visually 4. Distance Formula (Abstractly) D. Slope 1. Slope (Visual) 2. Slope Definition and Formula (Abstraction) E. Parallel and Perpendicular Lines 1. Parallel Lines Without Slope 2. Parallel Lines With Slope 3. Perpendicular Lines (Visual and Verified) 4. Perpendicular Lines: Towards Abstraction F. x-y Charts G. x and y Intercepts H. Symmetry 1. Symmetry of Graphs 2. Symmetry in Functional Notation IV. Chapter of Lines B. The Babysitting Problem C. Graphing Lines in Slope-Intercept Form D. Jersey s Pizza Project E. Graphing Types of Lines 1. Graphing Any Linear Equation v

2 2. Graphs and Equations of Horizontal Lines 3. Graphs and Equations of Vertical Lines F. Parallel and Perpendicular Lines G. Equations Given a Point and Slope H. Summary of Forms I. Combining Parameters J. Intersecting Lines 1. Intersection of Lines: Visual Guess and Verification 2. Substitution Method 3. Method of Elimination 4. Three by Three Systems of Equations V. Chapter of Inverse Operations B. Order of Operations 1. The Order of Operations 2. Evaluating Expressions by Using Order of Operations C. Solving Linear Equations 1. Solving One-Step Equations With Positive Numbers 2. Solving One-Step Equations With Negative Numbers 3. Solving One-Step Equations With Integer Multiples of x 4. Solving One-Step Equations With Fractional Multiples of x 5. Solving One-Step Equations in Abstraction 6. Solving Two-Step Equations 7. Solving Two-Step Linear Equations With Several Fractions 8. Variables on Both Sides of the Equation 9. Solving Multi-Step Linear Equations D. Solving Non-Linear Equations 1. Solving Simple Quadratic Equations 2. Solving Simple Equations of Any Power 3. Solving Equations Within Parentheses to Powers E. Solving for Variables VI. Chapter of Inequalities B. Inequalities and the Number Line C. Three Forms of Inequality Notation D. Linear Inequalities 1. Solving Linear Inequalities 2. Double Inequalities E. Non-Linear Inequalities 1. Solving Polynomial Inequalities From a Graph 2. Solving Polynomial Inequalities 3. Inequalities With Rational Expressions F. Graphing Inequalities in Two Variables vi" Montessori Algebra for the Adolescent Michael J. Waski

3 VII. Chapter of Exponents B. Exponents 1. Exponents as Repeated Multiplication 2. Multiplying Exponential Expressions 3. Dividing Exponential Expressions 4. Exponential Expressions to Powers 5. Non-Positive Integer Exponents With Materials 6. Fractional Exponents C. Radicals 1. Converting Fractional Exponents to Radicals 2. Operations With Radicals 3. Irrational Numbers 4. Simplification of Radicals 5. Simplification of Radicals With Exponent Laws 6. Rationalizing the Denominator VIII. Chapter of Combining Like Terms B. Combining Like Objects C. Combining Like Terms With Algebra Tiles D. Combining Like Terms in Abstraction E. Same Name, Different Look IX. Chapter of Factoring B. Distributive Property 1. Elementary Lessons 2. Review of the Distributive Property 3. Distributive Property With Numbers 4. Distributive Property With Variables 5. Introduction to Algebra Tiles 6. Distributive Property as Area 7. Factoring as Area 8. Removing Common Factors 9. Distributive Property and Factoring Higher Powers in Abstraction 10. Review of Multiple Term Distribution 11. Multiplying a Polynomial by a Polynomial 12. Binomial Multiplication With Algebra Tiles C. Factoring Trinomials With Algebra Tiles 1. Trinomial Factoring With the Variation of Set A 2. Trinomial Factoring With the Variation of Set B 3. Transition to Abstraction Through Symbolism 4. Trinomial Factoring With the Variation of Set C 5. Trinomial Factoring With the Variation of Set D vii

4 6. Trinomial Factoring With the Variation of Set E 7. Trinomial Factoring With the Variation of Set F 8. Trinomial Factoring With the Variation of Set G D. Factoring by Grouping E. Matching Card Sorting F. Perfect Square Trinomials 1. Perfect Square Trinomials With Algebra Tiles 2. Perfect Square Trinomials With the Binomial Square G. Difference of Squares 1. Difference of Squares Numerically With the Pegboard 2. Difference of Squares Algebraically 3. Difference of Squares With Algebra Tiles H. Sum and Difference of Cubes 1. Difference of Cubes Numerically 2. Difference of Cubes Algebraically 3. Sums of Cubes Numerically 4. Sums of Cubes Algebraically I. Follow-Up Work in Abstraction 1. Complete Factorization 2. Simplifying Rational Expressions 3. Simplifying Opposite Factors 4. Multiplying and Dividing Rational Expressions 5. Adding and Subtracting Rational Expressions 6. Partial Fraction Decomposition X. Chapter of Absolute Value B. Absolute Value: Definition and Evaluation C. Linear Equations 1. Solving Linear Equations With Absolute Values 2. Linear Equations With Absolute Values: Special Cases D. Linear Inequalities 1. Solving Linear Inequalities With Absolute Value 2. Solving Linear Inequalities: Special Cases E. Graphing Equations With Absolute Values XI. Chapter of the Binomial Theorem B. Binomial Square C. Algebraic Binomial Cube D. Higher Powers 1. Fourth Powers of a Binomial 2. Further Work With the Fourth Powers 3. Fifth Powers of a Binomial E. The Binomial Theorem viii" Montessori Algebra for the Adolescent Michael J. Waski

5 1. Pascal s Triangle and Binomial Coefficients 2. Pascal s Triangle and the Binomial Theorem 3. Formalization of the Binomial Theorem XII. Chapter of Quadratics B. Zero Product Property C. Solving Quadratic Equations by Factoring D. Completing the Square 1. Completing the Square With Algebra Tiles 2. Completing the Square: Towards Abstraction 3. Completing the Square to Solve Equations 4. Completing the Square: Odd Linear Terms 5. Completing the Square With Leading Coefficient Greater Than One With Algebra Tiles 6. Completing the Square With Leading Coefficient Greater Than One in Abstraction E. The Quadratic Formula F. The Discriminant G. Graphing Quadratic Equations With x-y Charts H. Finding the Vertex From the x-intercepts I. Finding the Vertex Through Completing the Square XIII. Chapter of Transformations B. Parent Graphs C. Vertical Translations of Graphs D. Horizontal Translations of Graphs E. Transforming Graphs by Combining Translations F. Transforming Graphs by a Scaling Factor G. Combining all Three Types of Transformations XIV. Chapter of Sequences B. What Comes Next? Finding Patterns in Sequences C. Arithmetic Sequences 1. Exploring Arithmetic Sequences 2. Generating a Formula for Any Term of an Arithmetic Sequence 3. Arithmetic Sequences and Slope 4. Sum of Consecutive Natural Numbers 5. The Sum of an Arithmetic Sequence D. Summation Notation E. Geometric Sequences 1. Exploring Geometric Sequences 2. Generating a Formula for Any Term of a Geometric Sequence 3. The Sum of a Finite Geometric Sequence 4. The Sum of an Infinite Geometric Sequence ix

6 F. Further Series and Sequences 1. Fibonacci Numbers 2. Sums of Squares 3. Sums of Cubes 4. Sums of Rectangular Numbers XV. Chapter of Functions B. Function Properties 1. Function Machines 2. Evaluating Using Function Notation 3. Definition of a Function, Domain, and Range 4. The Vertical Line Test 5. One-to-One Functions C. Domain and Range 1. Finding the Domain and Range of a Function From a Graph 2. Finding the Domain of a Function From the Equation D. Composition of Functions E. Inverses 1. Composition of Inverses 2. Finding the Inverse of a Function 3. Graphs of Inverses XVI. Chapter of Exponential and Logarithmic Functions B. Exponential Equations 1. Doubling and Halving Pennies 2. Exponential Equations and Their Graphs 3. Formal Definition of Exponential Equations 4. Discovering the Number e 5. Compound Interest 6. Continuous Interest C. Logarithmic Functions 1. A History of Logarithms 2. Evaluating Logarithms 3. Change of Base Formula as a Ratio 4. Solving Simple Exponential and Logarithmic Equations 5. Logarithms as Functions 6. Three Properties of Logarithms 7. Change of Base Formula Proof 8. Solving Multi-Step Exponential Equations 9. Solving Multi-Step Logarithmic Equations 10. Story Problems With Exponential and Logarithmic Equations XVII. Chapter of Polynomials x" Montessori Algebra for the Adolescent Michael J. Waski

7 B. Defining a Polynomial Function C. Graphs of Polynomials D. Rational Functions 1. Graphs of Rational Functions: Horizontal Asymptotes 2. Graphs of Rational Functions: Vertical Asymptotes 3. Complete Graphing of Rational Functions 4. Graphing With Slant Asymptotes E. Division 1. Polynomial Division 2. Synthetic Division 3. The Division Algorithm and Remainder Theorem F. Roots and Zeros 1. Factorization of Polynomials Over the Rationals 2. The Rational Zero Test 3. Descartes Rule of Signs 4. Factorization of Polynomials Over the Reals 5. Factorization of Polynomials Over the Complex Numbers 6. The Fundamental Theorem of Algebra XVIII. Chapter of Trigonometry 1. Introduction 2. Geometric Trigonometry a) Trigonometry With Right Triangles (1) Discovering Tangent (2) Inverse Functions (3) Sine and Cosine (4) Reciprocal Functions (5) Etymology of Trigonometric Functions b) Trigonometry With Non-Right Triangles (1) Law of Sines and Area of Triangles (2) Geometric Proof of Law of Sines (3) Law of Cosines With Insets (4) Law of Cosines Algebraically (5) Brahmagupta s Theorem 3. Analytic Trigonometry a) Unit Circle (1) Radian Measure (2) Converting Radians and Degree Measure (3) Negative Angles (4) Coterminal Angles (5) Special Triangles (6) Discovering Coordinates of the Unit Circle (7) Definitions of Sine and Cosine (8) Tangent as Ratio and Slope xi

8 (9) Inverses With the Unit Circle 4. Graphing Trigonometric Functions a) Graphing Sine and Cosine b) Graphing Tangent c) Graphing Reciprocal Functions d) Addition of Trigonometric Functions 5. Solving Trigonometric Equations 6. Identities 7. Sum and Difference Formulas XIX. Chapter of Complex Numbers B. Imaginary Numbers C. Powers of i D. Complex Numbers E. Addition, Subtraction, and Multiplication of Complex Numbers F. Complex Conjugates G. Division of Complex Numbers H. Graphing Complex Numbers I. Extension Activities 1. Gaussian Primes 2. Complex Numbers as Vectors 3. Trigonometric Form of Complex Numbers 4. Fractional Powers of i 5. DeMoivre s Theorem 6. Euler s Formula XX. Chapter of Further Work B. Matrices 1. Systems of Equations 2. Multiplication of Matrices 3. Inverse and Identity Matrix 4. Determinants 5. Further Matrix Applications C. Vectors 1. Direction 2. Magnitude 3. Addition and Subtraction 4. Position Vectors 5. Multiplication 6. Vector Equations of Lines D. Conic Sections 1. Definitions 2. Circles xii" Montessori Algebra for the Adolescent Michael J. Waski

9 3. Parabolas 4. Ellipses 5. Hyperbolas XXI. Chapter of Calculus B. Branches of Calculus C. Limits D. Differential Calculus 1. Derivatives 2. Derivative Formulas 3. The Chain Rule 4. Complete Graphing 5. Notes E. Integral Calculus 1. Integration 2. The Fundamental Theorem of Calculus 3. Areas and Volumes 4. Notes F. Conclusion XXII. Appendix A: Geometry XXIII. Appendix B: Arithmetic XXIV. Appendix C: Sample Organizational Charts XXV. Appendix D: Materials A. Materials From the Book B. Further Materials C. Further Reading XXVI. Bibliography xiii

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