Chapter
6.6. Comparing the Rate of Growth of Squares and Rectangles with Special Properties
Chapter 2 ALGEBRAIC EQUATIONS WITH PARAMETERS
2. A LOCUS APPROACH TO QUADRATIC EQUATIONS WITH PARAMETERS
3. QUADRATIC EQUATIONS WITH A HYPERBOLA-LIKE LOCUS
4. QUADRATIC EQUATIONS WITH TWO PARAMETERS
5. CUBIC EQUATIONS WITH TWO PARAMETERS
6. SYSTEMS OF SIMULTANEOUS EQUATIONS WITH PARAMETERS
Chapter 3 INEQUALITIES AND SPREADSHEET MODELING
2. SPREADSHEET MODELING OF LINEAR EQUATIONS
3. INEQUALITIES AS TOOLS IN MODELING NON-LINEAR PROBLEMS
4. GEOMETRIC CONTEXT AS A SPRINGBOARD INTO NEW USES OF INEQUALITIES
4.1. Finding Rectangle with the Largest Perimeter
4.2. Finding Rectangle with the Smallest Perimeter
4.3. Geometric Proof of the Arithmetic Mean—Geometric Mean Inequality
4.4. Alternative Approaches to Problem 3
5. TRANSITION TO THREE-DIMENSIONAL MODELING
6. DISCUSSION OF MODELING DATA AND ITS ALTERNATIVE INTERPRETATION
7. FORMAL AND INFORMAL APPROACHES TO SOLVING THREE-DIMENSIONAL PROBLEMS
7.1. Finding the Prism with the Smallest Surface Area
7.2. Finding the Prism with the Largest Surface Area Using a Combination of Formal Argument and Numeric Computation
7.3. A Three-Dimensional Geometric Problem in the Plane
Chapter 4 GEOMETRIC PROBABILITY
2. FORMAL DEFINITION OF GEOMETRIC PROBABILITY
3. SPREADSHEET AS A GEOMETRIC MEDIUM
4. COMPARING PROBABILITIES OF TWO EVENTS THROUGH GEOMETRIZATION
5. A SYSTEMATIC APPROACH TO GENERATING “JUMPING” FRACTIONS
6. EXPLAINING THE BEHAVIOR OF FRACTIONS THROUGH THE BEHAVIOR OF FUNCTIONS
7. FUNCTIONS WITH PARAMETERS AND GEOMETRIC PROBABILITY
8. CALCULATING GEOMETRIC PROBABILITY THROUGH A SPREADSHEET-BASED SIMULATION
9. LOCI OF TWO-VARIABLE INEQUALITIES AS IMAGES OF POINTS AND PICK’S FORMULA
10. ADVANCED EXPLORATIONS WITH GEOMETRIC PROBABILITIES
Chapter 5 COMBINATORIAL EXPLORATIONS
2. TWO WAYS OF DEFINING PERMUTATIONS
3. COUNTING PERMUTATIONS OF REPEATED OBJECTS
4. DEFINING COMBINATIONS THROUGH A PARTIAL DIFFERENCE EQUATION
5. NUMERICAL MODELING AS A WAY OF MAKING MATHEMATICAL CONNECTIONS
6. COMBINATIONS WITH REPETITIONS
7. COMBINATORIAL IDENTITIES AND MATHEMATICAL INDUCTION PROOF
7.1. Identities Involving C(n, r)
7. 2. Identities Involving
8. CONNECTING NUMBERS WITH DIFFERENT COMBINATORIAL MEANING
9. PARTITIONING PROBLEMS AND RECURSIVE REASONING
10. MODELING THE SUMS OF PERFECT POWERS
11. CONNECTING THE SUMS OF PERFECT POWERS TO COMBINATIONS
13. CLOSED FORMULAS FOR THE SUMS OF PERFECT POWERS
Chapter 6 HISTORICAL PERSPECTIVES
2. THE SPIRAL OF THEODORUS MOTIVATES CONCEPT LEARNING WITH TECHNOLOGY
3. THE CONSTRUCTION OF THE SPIRAL OF THEODORUS BY USING A SPREADSHEET
4. PARAMETERIZATION OF RECURRENCE RELATION (2)
5. GENERATING POLYGONAL NUMBERS THROUGH A SIEVE-LIKE PROCESS
6. DEVELOPING CLOSED AND RECURSIVE FORMULAS FOR POLYGONAL NUMBERS
7. INTERPRETING SUMMATION FORMULAS FOR POLYGONAL NUMBERS IN THE LANGUAGE OF SULVASUTRAS
8. THE SPIRAL MOTIVATES TRANSITION FROM SUMMATION TO ESTIMATION
9. PROOF OF PROPOSITION 6 AS AN AGENCY FOR PROBLEM POSING
9.1. Revisiting Geometry through Verifying Base Clause
9.2. Posing Problems in the Context of Inductive Transfer
10. THE HARMONIC SERIES REVISITED
Chapter 7 COMPUTATIONAL EXPERIMENTS AND FORMAL DEMONSTRATION IN TRIGONOMETRY
2. ONE EQUATION—FOUR SOLUTIONS
3. A COMPUTER-SUPPORTED GRAPHICAL DEMONSTRATION
4. A FORMAL GEOMETRIC DEMONSTRATION
5. INTRODUCING A PARAMETER IN EQUATION (8)
5.1. Exploring the Case a = 3
5.2. Exploring the Case a = 0
5.3. Exploring the Case a = 1
6. DOUBLE PARAMETERIZATION OF EQUATION (8)
7. TRIPLE PARAMETERIZATION OF EQUATION (8)
8. EXPLORING THE EQUATION cossin
8.1. First Solution to Equation (33)
8.2. Second Solution to Equation (33)
8.3. Demonstration of the Case a = 2, b = -2
8.4. Third Solution to Equation (33).
Chapter 8 DEVELOPING MODELS FOR COMPUTATIONAL PROBLEM SOLVING
2. SETTING A CONTEXT AND INTRODUCING MODELING TOOLS
4. POLYGONAL NUMBERS REVISITED
5. CHANGE OF MODEL AFFECTS CONTEXT
6. CHANGE OF CONTEXT REQUIRES NEW MODEL
7. INTRODUCING NEW MODELING TOOLS
8. QUADRATIC FUNCTIONS AS TOOLS OF SUMMATION
9. FINDING THE SUM OF ROOM NUMBERS THAT BELONG TO THE SAME FLOOR
10. REFINING OLD MODELS TO MATCH NEW CONTEXT
11. INTERPRETING THE RESULTS OF COMPUTATIONAL EXPERIMENTS
Chapter 9 PROGRAMMING DETAILS
2. SPREADSHEETS USED IN CHAPTER 1
2.1. Programming Details for Figure 1.5
2.2. Programming Details for Figure 1.8
2.3. Programming Details for Figure 1.9
2.4. Programming Details for Figures 1.11 and 1.12
2.5. Programming Details for Figures 1.10 and 1.13
3. SPREADSHEETS USED IN CHAPTER 3
3.1. Programming Details for Figures 3.1 and 3.2
3.2. Programming Details for Figures 3.3 and 3.4
3.3. Programming Details for Figure 3.5
3.4. Programming Details for Figure 3.6
3.5. Programming Details for Figure 3.9.
4. SPREADSHEETS USED IN CHAPTER 8
4.1. Programming Details for Figure 8.6
4.2. Programming Details for Figure 8.9
4.3. Programming Details for Figure 8.10
4.4. Programming Details for Figure 8.11
4.5. Spreadsheet Programming for Figure 8.12
4.6. Spreadsheet Programming for Figure 8.16
4.7. Spreadsheet Programming for Figure 8.21
4.8. Spreadsheet Programming for Figure 8.22
Computer-Enabled Mathematics: Integrating Experiment and Theory in Teacher Education
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