Teaching school mathematics : from pre-algebra to algebra

"This is a systematic exposition of a major part of the mathematics of grades 5 to 8 (excluding statistics), written specifically for Common Core era teachers. It differs from other books for teachers in that the mathematics is correct, in the sense that all the concepts are clearly and correctly defined, and a grade-appropriate explanation (that is, proof) is given for every assertion. For example, it gives a precise definition of percent and explains how to use the definition to do all the standard problems about percent in an entirely routine manner. It also gives a leisurely explanation for “negative times negative is positive”. Another key feature is an intuitive introduction to plane geometry via rotations, translations, reflections, and dilations that, instead of treating these transformations as merely fun activities, shows how they make sense of the usual geometric topics in middle school, including congruence, similarity, length, area, and volume. In short, the readers will find in this volume a clear explanation of whatever was once puzzling to them in the mathematics of grades 5 to 8."--Publisher's description Cover 1 Title page 4 Chapters in the Companion Volume 10 Preface 12 Suggestions on How to Read This Volume 22 Chapter 1. Fractions 24 1.1. Definition of a fraction 26 1.2. Fractions as division 44 1.3. Equivalent fractions 50 1.4. Adding and subtracting fractions 66 1.5. Multiplying fractions 79 1.6. Dividing fractions 93 1.7. Complex fractions 111 1.8. FASM 118 1.9. Percent, ratio, and rate problems 122 1.10. Finite probability 144 1.11. Appendix 166 Chapter 2. Rational Numbers 168 2.1. The two-sided number line 168 2.2. Adding rational numbers 170 2.3. Subtracting rational numbers 183 2.4. Multiplying rational numbers 187 2.5. Dividing rational numbers 197 2.6. Comparing rational numbers 206 2.7. FASM, revisited 220 Chapter 3. The Euclidean Algorithm 226 3.1. The reduced form of a fraction 226 3.2. The Fundamental Theorem of Arithmetic 242 Chapter 4. Experimental Geometry 252 4.1. Overview 252 4.2. Freehand drawing 259 4.3. Constructions using tools 262 4.4. The basic isometries 275 4.5. Congruence 310 4.6. Dilation 324 4.7. Similarity 343 Chapter 5. Length, Area, and Volume 362 5.1. The concept of geometric measurement 363 5.2. Length 371 5.3. Area 379 5.4. Volumes of cylinders and cones 399 Bibliography 404 Back Cover 407 This two-volume set includes a systematic exposition of a major part of the mathematics of grades 5-9 (excluding statistics) written specifically for Common Core era teachers.