The Number Line through Guided Inquiry
- Length: 124 pages
- Edition: 1
- Language: English
- Publisher: American Mathematical Society
- Publication Date: 2021-12-10
- ISBN-10: 1470465043
- ISBN-13: 9781470465049
- Sales Rank: #0 (See Top 100 Books)
The Number Line through Guided Inquiry is designed to give future secondary teachers a deep understanding of the real numbers and functions on the reals. By presenting just that part of the subject that underlies the high school curriculum, this book offers an alternative to a standard real analysis sequence for advanced undergraduate or beginning graduate students. It will give any student a much deeper understanding of the mathematics that they were taught in high school. Written in a guided-inquiry format, this book consists of a carefully scaffolded sequence of definitions, problems, and theorems that guides students through each topic. Readers solve the problems and prove the theorems on their own and present their results to their peers with the instructor as a mentor and a guide. Students will learn not only the mathematics, but also how to help others learn mathematics. They will learn to think creatively and to make compelling arguments to justify their conclusions. They will learn to listen critically to others and give constructive feedback. Ultimately, they will learn to work as a team to answer the bigger questions and build a common understanding of the broader subject.
Title page Copyright Contents Preface To the Student Acknowledgement Chapter 1. Rational Numbers and Missing Numbers 1.1. Integers 1.2. Rational Numbers 1.3. Missing Numbers 1.4. Number Lines 1.5. Many Number Lines 1.6. Historical Note: Other Missing Numbers Chapter 2. Limit Points and Sequences 2.1. Intervals and Limit Points 2.2. Sequences and Convergence 2.3. Historical Note: Zeno’s Paradox Chapter 3. Decimal Representations of Numbers 3.1. Infinite Decimal Representations 3.2. Rational Numbers as Infinite Decimals 3.3. Existence and Uniqueness 3.4. Historical Note: The Hindu-Arabic Numerals Chapter 4. Complete Number Lines 4.1. The Completeness Axiom 4.2. The Square Root of 17 4.3. Historical Note: The Archimedean Dilemma Chapter 5. Continuity 5.1. Definitions and Examples 5.2. Generating Continuous Functions 5.3. Missing Numbers as Intermediate Values 5.4. Uniform Continuity 5.5. Historical Note: Continuity via Infinitesimals Chapter 6. Calculus 6.1. Integrals 6.2. Derivatives 6.3. The Fundamental Connection 6.4. Historical Note: Calculus via Infinitesimals Chapter 7. Log and Exponential Functions 7.1. Rational Exponents 7.2. Natural Logarithm 7.3. Exponents Reveal More Missing Numbers 7.4. Historical Note: The 19th-Century Transition Chapter 8. The Real Number Line 8.1. The Non-Negative Real Numbers PP 8.2. The Real Number Line R 8.3. All Number Lines 8.4. Historical Note: The Hyperreal Numbers Chapter 9. The Price of Completeness 9.1. Cantor’s Set Theory 9.2. Lebesgue’s Measure Theory 9.3. Historical Note: More Infinities! Appendix A. Historical Summary Appendix B. The Reals via Dedekind Cuts B.1. Rational and Ghost Downsets B.2. Construction of the Number Line B.3. Rational and Irrational Numbers Appendix C. Guidelines for the Instructor C.1. Chapter Options C.2. Teaching through Guided Inquiry Bibliography Index
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