Understanding College Algebra: Through Inquiry and Logical Reasoning

College Algebra actually makes sense and is something that you can figure out and understand why it works. This text focuses on a conceptual understanding of the big ideas in algebraic thinking, engaging the student in authentic problem solving and exploring the logical reasoning that underlies the various techniques and procedures in college algebra.

An Inquiry-based Approach. Each section starts with a Class Activity to engage students in actually doing mathematics. Doing math is not just calculating or following a procedure. Doing math is figuring things out: investigating, making and testing conjectures, making arguments, and communicating your reasoning to others.

The class activities are designed to highlight big algebraic ideas and spark a discussion of algebraic habits of mind, as well as students’ alternate conceptions that lead to common algebra mistakes. Students are asked to analyze solutions, explore representations, explain why valid methods for simplifying expressions or solving equations work, and explain why invalid methods do not work.

This book is intended to be read. Often math textbooks do not end up being read, but instead are used merely as a reference for their step-by-step procedures. Each section of this text has a “Read and Study” section that discusses the mathematics raised by the Class Activity and focuses on the mathematical reasoning and proof needed to nurture longer-lasting understanding of the content. This is meant to be read slowly and carefully, with pencil in hand. We pose questions that you should think about and answer before reading on. When we do work out an example, we do so to discuss the big ideas and illustrate our reasoning, not with the intention of providing you with a model to copy.

Exercises vs. Problems. In the homework, we distinguish between “exercises” and “problems.” Exercises are more routine, intended to give you more practice thinking about the big concepts. In contrast, the “problems” are intended to be problematic, to take time to explore, develop and make connections, and often to extend your reasoning to develop new ideas. We do not include “answers” to these homework exercises and problems. Why struggle and persevere to figure something out and understand it when you can just look it up? Mathematics is not about getting the right answer; it’s about figuring things out. It’s about logical reasoning and being able to justify that what you claim is true.

This doesn’t mean that you are on your own. We will do our best in the Read and Study sections to discuss the big ideas, offer explanations, and show you some good examples of problem solving and making mathematical arguments.

This text addresses the topics of a standard course in College Algebra, with the following sections:

1.Features of Algebraic Thinking
2. Algebraic Symbols
3. Sequences of Operations
4. Properties of Operations
5. The Distributive Law
6. Additive and Multiplicative Inverses
7. Using Inverses
8. Exponents
9. Roots of Numbers
10. Irrational and Imaginary Numbers
11. Testing and Justifying Simplifications
12. Types of Equations
13. Properties of Equality and Solving Equations
14. Techniques for Solving Equations
15. The Distance Formula
16. Finding Equations for Graphs
17. Ellipses
18. Function Defintions
19. Functional Thinking
20. Function Forms
21. Linear Function Forms
22. Quadratic Expressions
23. Quadratic Functions
24. Transformations of Functions
25. Polynomials
26. Rational Functions
27. Exponential and Logarithm Functions
28. The Natural Exponent Base
29. Inverse Functions
30. Finding Inverse Function Formulas
31. Solving Equations Review