Teaching School Mathematics

Cover -- Title page -- Chapters in the Companion Volume -- Preface -- Suggestions on How to Read This Volume -- Chapter 1. Symbolic Expressions -- 1.1. Basic protocol in the use of symbols -- 1.2. Expressions and identities -- 1.3. Mersenne primes and finite geometric series -- 1.4. Polynomials and order of operations -- 1.5. Rational expressions -- Chapter 2. Translation of Verbal Information into Symbols -- 2.1. Equations and inequalities -- 2.2. Some examples of translation -- Chapter 3. Linear Equations in One Variable -- 3.1. Solving linear equations -- 3.2. Some word problems -- Chapter 4. Linear Equations in Two Variables and Their Graphs -- 4.1. Coordinate system in the plane -- 4.2. Linear equations in two variables -- 4.3. The concept of slope -- 4.4. Proof that graphs of linear equations are lines -- 4.5. Every line is the graph of a linear equation -- 4.6. Useful facts and examples -- Chapter 5. Simultaneous Linear Equations -- 5.1. Solutions of linear systems and the geometric interpretation -- 5.2. The algebraic method of solution -- 5.3. Characterization of parallel lines by slope -- 5.4. Algebraic criterion for solvability -- 5.5. Partial fractions and Pythagorean triples -- 5.6. Appendix -- Chapter 6. Functions and Their Graphs -- 6.1. The basic definitions -- 6.2. Why functions? -- 6.3. Some examples of graphs -- 6.4. Remarks on graphs and coordinate systems -- Chapter 7. Linear Functions and Proportional Reasoning -- 7.1. Constant rate and linear functions -- 7.2. Proportional reasoning -- Chapter 8. Linear Inequalities and Their Graphs -- 8.1. How do inequalities arise in real life? -- 8.2. The symbolic translation -- 8.3. Basic facts about inequalities and applications -- 8.4. Graphs of inequalities in the plane -- 8.5. Solution of the manufacturing problem -- 8.6. Behavior of linear functions in the plane

Chapter 9. Exponents -- 9.1. Positive-integer exponents -- 9.2. Rational exponents -- 9.3. Laws of exponents -- 9.4. Scientific notation -- 9.5. Three additional remarks on rational exponents -- Chapter 10. Quadratic Functions and Their Graphs -- 10.1. Quadratic equations -- 10.2. A special class of quadratic functions -- 10.3. Properties of quadratic functions -- 10.4. The graph and the parabola -- 10.5. Some applications -- Appendix: Facts from [Wu-PreAlg] -- Bibliography -- Back Cover

This is a systematic exposition of introductory school algebra written specifically for Common Core era teachers. The emphasis of the exposition is to give a mathematically correct treatment of introductory algebra. For example, it explains the proper use of symbols, why "variable" is not a mathematical concept, what an equation is, what equation-solving means, how to define the slope of a line correctly, why the graph of a linear equation in two variables is a straight line, why every straight line is the graph of a linear equation in two variables, how to use the shape of the graph of a quadratic function as a guide for the study of quadratic functions, how to define a parabola correctly, why the graph of a quadratic function is a parabola, why all parabolas are similar, etc. This exposition of algebra makes full use of the geometric concepts of congruence and similarity, and it justifies why the Common Core Standards on algebra are written the way they are