Teaching school mathematics : Algebra

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 Cover 1 Title page 4 Chapters in the Companion Volume 10 Preface 12 Suggestions on How to Read This Volume 20 Chapter 1. Symbolic Expressions 22 1.1. Basic protocol in the use of symbols 23 1.2. Expressions and identities 26 1.3. Mersenne primes and finite geometric series 32 1.4. Polynomials and order of operations 38 1.5. Rational expressions 45 Chapter 2. Translation of Verbal Information into Symbols 48 2.1. Equations and inequalities 48 2.2. Some examples of translation 51 Chapter 3. Linear Equations in One Variable 58 3.1. Solving linear equations 58 3.2. Some word problems 66 Chapter 4. Linear Equations in Two Variables and Their Graphs 74 4.1. Coordinate system in the plane 75 4.2. Linear equations in two variables 78 4.3. The concept of slope 82 4.4. Proof that graphs of linear equations are lines 93 4.5. Every line is the graph of a linear equation 97 4.6. Useful facts and examples 99 Chapter 5. Simultaneous Linear Equations 106 5.1. Solutions of linear systems and the geometric interpretation 106 5.2. The algebraic method of solution 108 5.3. Characterization of parallel lines by slope 114 5.4. Algebraic criterion for solvability 119 5.5. Partial fractions and Pythagorean triples 122 5.6. Appendix 130 Chapter 6. Functions and Their Graphs 138 6.1. The basic definitions 138 6.2. Why functions? 143 6.3. Some examples of graphs 147 6.4. Remarks on graphs and coordinate systems 154 Chapter 7. Linear Functions and Proportional Reasoning 158 7.1. Constant rate and linear functions 158 7.2. Proportional reasoning 165 Chapter 8. Linear Inequalities and Their Graphs 176 8.1. How do inequalities arise in real life? 176 8.2. The symbolic translation 178 8.3. Basic facts about inequalities and applications 181 8.4. Graphs of inequalities in the plane 184 8.5. Solution of the manufacturing problem 201 8.6. Behavior of linear functions in the plane 208 Chapter 9. Exponents 212 9.1. Positive-integer exponents 215 9.2. Rational exponents 221 9.3. Laws of exponents 226 9.4. Scientific notation 235 9.5. Three additional remarks on rational exponents 241 Chapter 10. Quadratic Functions and Their Graphs 244 10.1. Quadratic equations 245 10.2. A special class of quadratic functions 259 10.3. Properties of quadratic functions 267 10.4. The graph and the parabola 273 10.5. Some applications 281 Appendix: Facts from [Wu-PreAlg] 286 Bibliography 294 Back Cover 296