Topics and ideas discussed include: algebraic expressions, equations and identities, the idea of an extraneous solution, factoring, quadratic equations, absolute values, equations involving absolute values, square-roots, equations involving square-roots, fractions, and equations involving fractions. We believe that a sound understanding of these topics is essential for all students in all scientific disciplines. Furthermore, we are certain that if the teaching of these topics is done in the correct way we have shown, with the correct doses of rigor rooted in sound common sense, then the learner will benefit both in knowledge and in the acquisition of the kind of mathematical thinking so valuable in other fields of endeavor. For, mathematics is not a mere body of knowledge; rightly viewed, it is also a way of thinking that encourages creativity, and is free of all constraints but the requirement of appropriate rigor.