# Pre-Order Montessori Algebra for the Adolescent Vol 2: The Older Adolescent

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Table of Contents

0 Introduction

How to use the Lessons

How this Book is Organized

Notes for the Second Edition

I. Chapter of the Binomial Theorem

A. Introduction

B. Binomial Square

C. Algebraic Binomial Cube

D. Higher Powers

1. Fourth Powers of a Binomial

2. Fifth Powers of a Binomial

E. The Binomial Theorem

1. Pascal’s Triangle and Binomial Coefficients

2. Pascal’s Triangle and the Binomial Theorem

3. Formalization of the Binomial Theorem

4. Application of the Binomial Theorem

II. Chapter of Sequences

A. Introduction

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

F. Further Series and Sequences

1. Fibonacci Numbers

2. Sums of Squares

3. Sums of Cubes

4. Sums of Rectangular Numbers

III. Chapter of Functions

A. Introduction

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

IV. Chapter of Exponential and Logarithmic Functions

A. Introduction

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

V. Chapter of Polynomials

A. Introduction

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

VI. 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

(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

VII. Chapter of Complex Numbers

A. Introduction

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

VIII. Chapter of Further Work

A. Introduction

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

3. Parabolas

4. Ellipses

5. Hyperbolas

IX. Chapter of Calculus

A. Introduction

B. Branches of Calculus

C. Limits

D. Differential Calculus

1. Derivatives

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

X. Appendix A: Arithmetic

XI. Appendix B: Sample Organizational Charts

XII. Appendix C: Materials

A. Materials from the Book

B. Further Materials

C. Further Reading

XIII. Bibliography

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