Introduction

This text contains an overview of topics in liberal arts mathematics from the perspective of elementary education. The mathematics is college-level, but with an eye for how the deeper understanding of mathematics will help college students prepare to become elementary school teachers. Topics include: Numeration Systems and Bases, Alternate Methods of Computation, Topics in Number Theory, Problem Solving Methods, and Sets and Logic. Algebra extensions are included for students who need to review algebra topics. Each chapter includes a project to further understanding of the chapter topics.

Table of Contents

Chapter 1: Numeration Systems and Alternate Bases 

1.1: Introduction to Base Ten Additive, Multiplicative and Place Value Numeration Systems

1.2: The Mayan Number System, Place Value in Base 20

A 1.1: Introduction to Exponents and Rules for Exponents

A 1.2: Zero and Negative Exponents; Combining Negative Numbers and Exponents; Connections to Base Ten Decimals

1.3: Adding and Subtracting in the Egyptian Numeration System; Connection to “Combining Like Terms” in Algebra; Base Ten Blocks Blocks and Combining “Like Terms”

A 1.3: Adding and Subtracting Polynomials

1.4: Converting Between Base Ten and Other Bases

A 1.4: Scientific Notation and Relative Size

1.5: The Babylonian Number System, Review of Converting

A 1.5: The Order of Operations and Exponents

1.6: Adding and Subtracting in Different Bases

1.7: Adding the Numbers from 1 to 100, Alternate Methods of Subtraction

Chapter 2: Problem Solving and Number Theory

2.1: The Sieve of Eratosthenes, Prime Numbers, Goldbach’s Conjecture, Mersenne Primes

A 2.1: What is a Variable? Writing and Solving Equations

2.2: Multiplication as Area; Commutative and Distributive Properties; Slope

A 2.2: Operations on Polynomials: Adding, Subtracting, Multiplying and Factoring

2.3: Problem Solving Strategies for All Ages

A 2.3: Solving Equations, Literal Equations; Finding Solution Sets

2.4: Fibonacci Numbers; the Golden Ratio, Figurate Numbers; Non-Linear Equations; Using Subscripts to Find Patterns

A 2.4: The Meaning of Slope; Parallel Versus Intersecting Lines; Systems of Equations

2.5: Introduction to Fractions; Patterns in Pascal’s Triangle

A 2.5: Simple Algebraic Fractions; Simplifying, Adding and Subtracting

2.6: Babylonian Fractions; Writing Terminating and Repeating Decimals as Fractions; Multiplying Fractions Visually

A 2.6: Exponential Equations: Graphing and Applications

2.7: Irrational and Rational Numbers, Pi; Multiplying and Dividing Fractions Using Models

A 2.7: Solving Quadratic Equations by Factoring and by Graphing

A 2.8: Solving Exponential Equations using Logarithms and by Graphing

A 2.9: Optional Topic: Solving Systems of Equations by Elimination; Graphing and Writing Equations Given a Point and a Slope, or Given Two Points

Chapter 3: Sets and Logic

3.1: Venn Diagrams, Complement, Intersection and Union

A 3.1: Common Factors and Multiples in Algebraic Expressions; Using Sets to Find Common Factors

3.2: Set Operations with Two and Three Circle Venn Diagrams, Using Regions, DeMorgan’s Laws

A 3.2: Multiplying and Dividing Algebraic Fractions; Graphing Rational Equations

3.3: More Set Operations with Three Circle Venn Diagrams, Using Regions with Three Set Diagrams, More on DeMorgan’s Laws; Elementary Education – Guess My Rule; Common Factors and Multiples

3.4: Math Topics – Writing Sets Using Inequalities, Subsets; Elementary Education – One-to-One Correspondence in Children

A 3.3: Solving Inequalities

A 3.4: Complex Fractions; Solving Equations with Fractions

3.5: Euler Diagrams: “All”, “Some” and “None”; Elementary Education – Engaging Multiple Learning Modalities

A 3.5: Complex Numbers

3.6: Euler Diagrams and Logical Arguments, “If…then” Statements

A 3.6: The Pythagorean Theorem

3.7: The Contrapositive, Combining Argument Types