Exploring Mathematics with Integrated Spreadsheets in Teacher Education

The following sections are included:

• Preface
• Contents

The following sections are included:

• Introduction
• Natural Numbers and a Spreadsheet
• Increment as a Variable Entity
• Slider-Controlled Variables in a Spreadsheet Environment
• Generating Arithmetic Sequences of a Variable Length
• Spreadsheets and the Agent-Consumer-Amplifier Framework
  • Spreadsheet as an agency for using mathematics
  • Spreadsheet as a consumer of mathematics
  • Spreadsheet as an amplifier of using mathematics
• Using Integrated Spreadsheets: An Illustration
• From Conjecturing to Formal Reasoning
• Generating Piece-Wise Monotonous Sequence with a Cyclic Behavior
• An Oscillating Sequence with a Cyclic Behavior

The following sections are included:

• Introduction
• Motivating the Construction of the Addition Table
  • Resolving brainteaser
  • Proof of formula (2.1)
• Motivating the Construction of the Multiplication Table
  • Proof of formula (2.3)
• Using a Spreadsheet in the Construction of the Addition and Multiplication Tables
• Constructing a Single Table of Variable Type and Size
• Divisibility Properties of Numbers in the Addition and Multiplication Tables
  • Counting the multiples of two in the addition and multiplication tables
    • Counting in the addition tables
    • Counting in the multiplication tables
  • Finding the sum of multiples of two in an addition table of even size
  • Proof of formula (2.5)
  • Finding the sum of multiples of two in an addition table of odd size
  • Proof of formula (2.7)
• The Sum of the Multiples of Three in the n × n Addition Table
  • Finding formulas for α(n, 3)
  • Proving formulas (2.8)-(2.10)
    • Proof of formula (2.8)
    • Proof of formula (2.9)
    • Proof of formula (2.10)
• The Sum of the Multiples of Two in the n × n Multiplication Table
  • The case n = 2k
  • The case n = 2k − 1
  • Proof of formula (2.11)
  • Proof of formula (2.12)
• The Sum of the Multiples of Three in the n × n Multiplication Table
  • The case n = 3k
  • The case n = 3k + 1
  • The case n = 3k − 1
  • Proof of formula (2.13)

The following sections are included:

• Introduction
• Spreadsheet Modeling of a Diophantine Equation with Three Unknowns
• 3-D Spreadsheet Modeling Involving Parameters
• Using 3-D Spreadsheet Modeling in Problem Posing
  • A notion of problem coherence
  • Modeling ambiguous problems using a spreadsheet
  • Using spreadsheets to pose problems
  • Formulating TITE problems through parameterization
• Extension into 4-D Modeling

The following sections are included:

• Introduction
• Generating Prime Numbers Within a Spreadsheet
• Construction of the Spreadsheet Sieve
• Twin Primes
• Exploring Gaps Between Consecutive Primes
• Testing Prime-Producing Polynomials Using a Spreadsheet
• Fermat Primes and Euler's Factorization Method
  • A numeric example
  • Generalization
  • Spreadsheet implementation of Euler's factorization method

The following sections are included:

• Introduction
• Arithmetical Properties of Rectangular Grids
• Exploiting Syntactic Versatility of a Spreadsheet
• From Data-Driven Conjecturing to Proving
• Emergence of a New Concept Through De-Contextualization of Meditational Means
  • Subgrids of the second order
  • “Real” and “hypothetical” subgrids
• The Sum of the Greatest Common Divisors as a Problem-Solving Tool
• Revisiting Familiar Concepts in New Environments
  • Improving computing complexity
  • Linking the Euclidean algorithm to Fibonacci numbers
  • Generating prime numbers through the Euclidean algorithm

The following sections are included:

• Introduction
• Towards Finding the General Solution to the Pythagorean Equation
• Pythagorean Triples Emerge
• Computer-Mediated Discourse on the Pythagorean Equation
• Achieving the Goal and Going Beyond It
• Discovering Arithmetical Properties of Pythagorean Triangles
• Visualizing the Meaning of Fermat's Last Theorem
• Pythagorean Triples Meet Euler Phi Function
• Generalizing Pythagorean Triples via the Law of Cosines
• Technology-Motivated Transition to γ–Triples

The following sections are included:

• Introduction
• A Real-Life Context for Recreational Mathematics
• Measurement Model for Division as a Tool in Computing Applications
• A Numeric Example
• The Case of an n-Digit Number
• Computational Separation of Digits Within a Spreadsheet
• Illustration 1: Resolving the Odometer Challenge
• Illustration 2: Palindromic Number Conjecture
• Converging Differences
• Exploring the Growth of the Number of Digits in Integer Exponents

The following sections are included:

• Introduction
• Modeling as Problem Solving in a Historical Context
• Spreadsheet as an Agent of a Mathematical Activity Leading to Inequalities
• Developing Skills in Conjecturing, Proving, and Improving Inequalities
• Technology-Motivated Conjecturing
• Spreadsheet Modeling vs. Using AM-GM Inequality
• Making Connections as a Good Teaching Practice

The following sections are included:

• Introduction
• Establishing Context for Triangular Numbers
• Using a Spreadsheet to Investigate Brain Teaser 9.1
• Developing Formulas for Triangular Numbers
• Visualizing Mathematical Induction Proof of Formula (9.4)
• Square Test for Triangular Numbers and Its Spreadsheet Implementation
• Developing Square Numbers in Context
• Investigation of Brain Teaser 9.2 Using a Spreadsheet
• Developing Formulas for Square Numbers
• Using Triangular Numbers as Problem-Solving Tools
• Two General Formulas for Polygonal Numbers
• Square Test for Polygonal Numbers
• Triangular-Like Numbers
• Inequalities and Search for Triangular-Like Numbers Using a Spreadsheet
• Square-Like Numbers
• Two Types of Ghosts on the Same Floor
• Exploring m-Gonal-Like Numbers
• Two Types of Ghosts Meeting Each Other

The following sections are included:

• Introduction
• Extending Fibonacci Numbers to New Contexts
• Fibonacci Sieve of Order k
• From the Golden Ratio to Its Generalization
• Generalized Golden Ratios
• Computational Experiments with Fibonacci-Like Sequences

The following sections are included:

• Appendix: One Hundred Problems
  • A.1: Introduction
  • A.2: Problems

The following sections are included:

• Bibliography
• Index