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The Number Line through Guided Inquiry
 
David M. Clark SUNY, New Paltz, NY
Xiao Xiao Utica College, Utica, NY
The Number Line through Guided Inquiry
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6504-9
Product Code:  TEXT/69
List Price: $45.00
MAA Member Price: $33.75
AMS Member Price: $33.75
eBook ISBN:  978-1-4704-6799-9
Product Code:  TEXT/69.E
List Price: $45.00
MAA Member Price: $33.75
AMS Member Price: $33.75
Softcover ISBN:  978-1-4704-6504-9
eBook: ISBN:  978-1-4704-6799-9
Product Code:  TEXT/69.B
List Price: $90.00 $67.50
MAA Member Price: $67.50 $50.63
AMS Member Price: $67.50 $50.63
The Number Line through Guided Inquiry
Click above image for expanded view
The Number Line through Guided Inquiry
David M. Clark SUNY, New Paltz, NY
Xiao Xiao Utica College, Utica, NY
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6504-9
Product Code:  TEXT/69
List Price: $45.00
MAA Member Price: $33.75
AMS Member Price: $33.75
eBook ISBN:  978-1-4704-6799-9
Product Code:  TEXT/69.E
List Price: $45.00
MAA Member Price: $33.75
AMS Member Price: $33.75
Softcover ISBN:  978-1-4704-6504-9
eBook ISBN:  978-1-4704-6799-9
Product Code:  TEXT/69.B
List Price: $90.00 $67.50
MAA Member Price: $67.50 $50.63
AMS Member Price: $67.50 $50.63
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 692022; 124 pp
    MSC: Primary 97; 01

    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.

    Readership

    Graduate students interested in secondary school mathematics and general undergraduate mathematics majors.

  • Table of Contents
     
     
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 692022; 124 pp
MSC: Primary 97; 01

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.

Readership

Graduate students interested in secondary school mathematics and general undergraduate mathematics majors.

  • 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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