Rational Numbers to Linear Equations

This is the first of three volumes that, together, give an exposition of the mathematics of grades 9 12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K 12 as a totally transparent subject. The present volume begins with fractions, then rational numbers, then introductory geometry that can make sense of the slope of a line, then an explanation of the correct use of symbols that makes sense of variables , and finally a systematic treatment of linear equations that explains why the graph of a linear equation in two variables is a straight line and why the usual solution method for simultaneous linear equations by substitutions is correct. This book should be useful for current and future teachers of K 12 mathematics, as well as for some high school students and for education professionals. Cover 1 Title page 4 Contents of the Companion Volumes and Structure of the Chapters 10 Preface 12 To the Instructor 28 To the Pre-Service Teacher 42 Prerequisites 46 Some Conventions 48 Chapter 1. Fractions 50 Overview of Chapters 1 and 2 50 1.1. Definition of a fraction 53 1.2. Equivalent fractions 68 1.3. Adding and subtracting fractions 81 1.4. Multiplying fractions 92 1.5. Dividing fractions 103 1.6. Complex fractions 117 1.7. Percent, ratio, and rate problems 121 1.8. Appendix: The basic laws 135 Chapter 2. Rational Numbers 138 2.1. The rational numbers 139 2.2. Adding rational numbers 140 2.3. The vectorial representation of addition 148 2.4. Multiplying rational numbers 154 2.5. Dividing rational numbers 161 2.6. Comparing rational numbers 170 2.7. FASM 182 Chapter 3. The Euclidean Algorithm 186 3.1. The reduced form of a fraction 186 3.2. The fundamental theorem of arithmetic 196 Chapter 4. Basic Isometries and Congruence 206 Overview of Chapters 4 and 5 206 4.1. The basic vocabulary, Part 1 213 4.2. The basic vocabulary, Part 2 229 4.3. Transformations of the plane 248 4.4. The basic isometries: Rotations 265 4.5. The basic isometries: Reflections and translations 278 4.6. Congruence, SAS, and ASA 288 4.7. A brief pedagogical discussion of proofs 301 Chapter 5. Dilation and Similarity 304 5.1. The fundamental theorem of similarity 304 5.2. Dilation 317 5.3. Similarity 332 Chapter 6. Symbolic Notation and Linear Equations 346 6.1. Symbolic expressions 347 6.2. Solving linear equations in one variable 371 6.3. Setting up coordinate systems 380 6.4. Lines in the plane and their slopes 386 6.5. The graphs of linear equations in two variables 400 6.6. Parallelism and perpendicularity 412 6.7. Simultaneous linear equations 422 Glossary of Symbols 434 Bibliography 436 Index 440 Back Cover 452