Softcover ISBN:  9781470465049 
Product Code:  TEXT/69 
List Price:  $45.00 
MAA Member Price:  $33.75 
AMS Member Price:  $33.75 
eBook ISBN:  9781470467999 
Product Code:  TEXT/69.E 
List Price:  $45.00 
MAA Member Price:  $33.75 
AMS Member Price:  $33.75 
Softcover ISBN:  9781470465049 
eBook: ISBN:  9781470467999 
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 
Softcover ISBN:  9781470465049 
Product Code:  TEXT/69 
List Price:  $45.00 
MAA Member Price:  $33.75 
AMS Member Price:  $33.75 
eBook ISBN:  9781470467999 
Product Code:  TEXT/69.E 
List Price:  $45.00 
MAA Member Price:  $33.75 
AMS Member Price:  $33.75 
Softcover ISBN:  9781470465049 
eBook ISBN:  9781470467999 
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 DetailsAMS/MAA TextbooksVolume: 69; 2022; 124 ppMSC: 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 guidedinquiry 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.
ReadershipGraduate 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 HinduArabic 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 19thCentury Transition

Chapter 8. The Real Number Line

8.1. The NonNegative 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


Additional Material

RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
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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 guidedinquiry 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.
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 HinduArabic 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 19thCentury Transition

Chapter 8. The Real Number Line

8.1. The NonNegative 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