Math Foundations
Edition: 1
Copyright: 2025
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$55.00 USD
Mathematical Foundations is a developmental mathematics textbook designed for students entering college with limited mathematical preparation. Serving as a companion to Applied Algebra, it provides a one-semester course that develops number sense, quantitative reasoning, algebraic thinking, and spatial reasoning through a carefully sequenced progression from arithmetic to algebra and geometry.
Grounded in contemporary research on learning, the text emphasizes conceptual understanding alongside procedural fluency through visual models, active-learning explorations, and real-world applications. Rather than focusing on memorization alone, it helps students build meaningful mathematical understanding and confidence.
A central theme of the book is that mathematics should be learned from first principles. Concepts are developed through units, quantities, patterns, and visual representations before being expressed symbolically, allowing students to understand not only how mathematical procedures work, but why they work.
Designed to strengthen the foundational skills needed for college-level mathematics, the book helps students become better prepared for quantitative courses and academic programs that require mathematical reasoning. It also includes teaching notes and assessment guidance to support instructors. By combining research-based pedagogy with accessible mathematics, Mathematical Foundations offers a modern approach to developmental mathematics focused on college readiness, student success, and long-term learning.
Introduction
1 Integer Arithmetic Foundations
Launch Exploration: Grouping and Regrouping
1.1 Understanding Counting Numbers
1.1.1 Numeration
1.1.2 Rounding
1.1.3 Number Comparison
1.1.4 The Number Line
Exercises
1.2 The First-Order Operations
1.2.1 Addition
1.2.2 Subtraction
1.2.3 The Integers
1.2.4 Operations with Integers
Exercises
1.3 The Second-Order Operations
1.3.1 Multiplication
1.3.2 Division
1.3.3 Operations with Integers
1.3.4 Properties of Operations
1.3.5 Order of Operations
1.3.6 Numerical Expressions: Translations
Exercises
1.4 Structure and Properties of Integers
1.4.1 Divisors and Multiples
1.4.2 Prime Numbers
1.4.3 Divisibility Criteria
1.4.4 Division with Remainder
1.4.5 Mental Math
Exercises
2 Rational Number Arithmetic
Launch Exploration: Part of a Whole
2.1 Introduction to Fractions
2.1.1 The Fractional Unit and Mixed Numbers
2.1.2 The Area Model and Equivalent Fractions
2.1.3 Number Line Representations
2.1.4 Comparing Ordinary Fractions
Exercises
2.2 Fraction Operations
2.2.1 Adding and Subtracting Fractions
2.2.2 Multiplying and Dividing Fractions
2.2.3 Properties and Order of Operations
Exercises
2.3 Working with Decimal Fractions
2.3.1 Comparing and Rounding Decimal Fractions
2.3.2 Operations with Decimal Fractions
Exercises
2.4 Proportional Reasoning and Percent
2.4.1 Ratios and Proportions
2.4.2 Derived Proportions
2.4.3 Direct and Inverse Proportionality
2.4.4 Percent Representations
2.4.5 Solving Percent Problems
Exercises
3 Algebra Foundations
Launch Exploration: Mystery Number Games
3.1 Higher-Order Operations
3.1.1 Exponents
3.1.2 Radicals
3.1.3 Real and Imaginary Numbers
3.1.4 Logarithms
3.1.5 Order of Operations
Exercises
3.2 Building Blocks of Algebra
3.2.1 A Primer on the Language of Sets
3.2.2 Properties of Real Numbers
3.2.3 Introduction to Algebraic Expressions
Exercises
3.3 Algabraic Equations
3.3.1 Introduction to Algebraic Equations
3.3.2 Linear and Power Equations
3.3.3 The Mapping Diagram
3.3.4 Problem Solving with Linear Equations
Exercises
4 Geometry Foundations
Launch Exploration: Construct a Square from Two Squares
4.1 Fundamentals of Angles and Triangles
4.1.1 Primitive Notions
4.1.2 Angles and Circles
4.1.3 Parallel Lines
4.1.4 Triangles
Exercises
4.2 Area and Perimeter of 2D Shapes
4.2.1 Rectangles and Parallelograms
4.2.2 Perimeter versus Area
Exercises
4.3 Exploring Volume in 3D Geometry
4.3.1 Shapes Generated by Translation
4.3.2 Volumes of Prisms and Cylinders
Exercises
5 Teaching Notes
5.1 Units, Numeration, and the Foundations of Arithmetic
5.2 Understanding Fractions, Decimals, and Percentages
5.3 Integrating Geometry and Arithmetic: Key Concepts
5.4 Using Algebra to Simplify Arithmetic and Geometry
5.5 Assessment Based on Learning Objectives
Index