Teaching School Mathematics

Cover -- Title page -- Chapters in the Companion Volume -- Preface -- Suggestions on How to Read This Volume -- Chapter 1. Fractions -- 1.1. Definition of a fraction -- 1.2. Fractions as division -- 1.3. Equivalent fractions -- 1.4. Adding and subtracting fractions -- 1.5. Multiplying fractions -- 1.6. Dividing fractions -- 1.7. Complex fractions -- 1.8. FASM -- 1.9. Percent, ratio, and rate problems -- 1.10. Finite probability -- 1.11. Appendix -- Chapter 2. Rational Numbers -- 2.1. The two-sided number line -- 2.2. Adding rational numbers -- 2.3. Subtracting rational numbers -- 2.4. Multiplying rational numbers -- 2.5. Dividing rational numbers -- 2.6. Comparing rational numbers -- 2.7. FASM, revisited -- Chapter 3. The Euclidean Algorithm -- 3.1. The reduced form of a fraction -- 3.2. The Fundamental Theorem of Arithmetic -- Chapter 4. Experimental Geometry -- 4.1. Overview -- 4.2. Freehand drawing -- 4.3. Constructions using tools -- 4.4. The basic isometries -- 4.5. Congruence -- 4.6. Dilation -- 4.7. Similarity -- Chapter 5. Length, Area, and Volume -- 5.1. The concept of geometric measurement -- 5.2. Length -- 5.3. Area -- 5.4. Volumes of cylinders and cones -- Bibliography -- Back Cover

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