Change and Invariance A Textbook on Algebraic Insight into Numbers and Shapes /
"What is the connection between finding the amount of acid needed to reach the desired concentration of a chemical solution, checking divisibility by a two-digit prime number, and maintaining the perimeter of a polygon while reducing its area? The simple answer is the title of this book. The wo...
| Κύριοι συγγραφείς: | , |
|---|---|
| Συγγραφή απο Οργανισμό/Αρχή: | |
| Μορφή: | Ηλεκτρονική πηγή Ηλ. βιβλίο |
| Γλώσσα: | English |
| Έκδοση: |
Rotterdam :
SensePublishers : Imprint: SensePublishers,
2016.
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| Θέματα: | |
| Διαθέσιμο Online: | Full Text via HEAL-Link |
Πίνακας περιεχομένων:
- Preface
- Acknowledgements
- The Concept of Invariance and Change: Theoretical Background
- Understanding Phenomena from the Aspect of Invariance and Change
- The Concept of Invariance and Change in the Mathematical Knowledge of Students
- The Basic Interplay between Invariance and Change
- Some Introductory Activities in Invariance and Change
- References
- Invariant Quantities – What Is Invariant and What Changes?
- Introduction: Understanding the Invariance of Quantity as a Basis for Quantitative Thinking
- Activity 2.1: Dividing Dolls between Two Children
- Mathematic and Didactic Analysis of Activity 2.1: Partitioning a Set into Two Subsets: Posing Problems and Partition Methods
- Activity 2.2: How to Split a Fraction. Almost Like Ancient Egypt
- Mathematic and Didactic Analysis of Activity 2.2: Invariance of Quantity and Splitting of Unit Fractions
- Activity 2.3: They Are All Equal, But …
- Mathematic and Didactic Analysis of Activity 2.3: From Equal Addends to Consecutive Addends
- Activity 2.4: Expressing a Natural Number as Infinite Series
- Suggestions for Further Activities
- References
- The Influence of Change
- Introduction: Changes in Quantity and Comparing Amounts
- Activity 3.1: Less or More?
- Mathematical and Didactic Analysis of Activity 3.1: The influence That a Change in One Operand Has on the Value of an Arithmetical Expression
- Activity 3.2: Plus How Much or Times How Much?
- Mathematical and Didactic Analysis of Activity 3.2: Different Ways of Comparing
- Activity 3.3: Markups, Markdowns and the Order of Operations
- Mathematical and Didactic Analysis of Activity 3.3: Repeated Changes in Percentages
- Activity 3.4: Invariant or Not?
- Mathematical and Didactic Analysis of Activity 3.4: Products and Extremum Problems
- Activity 3.5: What Is the Connection between Mathematical Induction and Invariance and Change?
- Mathematical and Didactic Analysis of Activity 3.5: What Is the Connection between Mathematical Induction and Invariance and Change?
- Suggestions for Further Activities
- References
- Introducing Change for the Sake of Invariance
- Introduction: Algorithms – Introducing Change for the Sake of Invariance
- Activity 4.1: The “Compensation Rule”: What Is It?
- Mathematical and Didactic Analysis of Activity 4.1: Changes in the Components of Mathematical Operations That Ensure the Invariance of the Result
- Activity 4.2: Divisibility Tests
- Mathematical and Didactic Analysis of Activity 4.2: Invariance of Divisibility and Composing of Divisibility Tests
- Activity 4.3: Basket Configuration Problems
- Mathematical and Didactic Analysis of Activity 4.3: Diophantine Problems and Determining the Change and Invariance
- Activity 4.4: Product = Sum?
- Mathematical and Didactic Analysis for the Activities in 4.4: Invariance as a Constraint
- Suggestions for Further Activities
- References
- Discovering Hidden Invariance
- Introduction: Discovering Hidden Invariance as a Way of Understanding Various Phenomena
- Activity 5.1: How to Add Numerous Consecutive Numbers
- Mathematical and Didactic Analysis of Activity 5.1: The Arithmetic Series: Examples of Use of the Interplay between Change and Invariance in Calculations
- Activity 5.2: Solving Verbal Problems: Age, Speed, and Comparing the Concentrations of Chemical Solutions
- Mathematic and Didactic Analysis of Activity 5.2: Solving Verbal Problems by Discovering the Hidden Invariance
- Activity 5.3: Mathematical Magic – Guessing Numbers
- Mathematical and Didactic Analysis of Activity 5.3: Discovering the Invariant in Mathematical “Tricks”: “Guessing Numbers”
- Activity 5.4: “Why Can’t I Succeed?”
- Mathematical and Didactic Analysis of Activity 5.4: Discovering the Hidden Invariance in “Why Can’t I Succeed?”
- Suggestions for Further Activities
- References
- Change and Invariance in Geometric Shapes
- Introduction: Invariance and Change in the World of Geometry
- Activity 6.1: Halving in Geometry – Splitting Shapes
- Mathematical and Didactic Analysis of Activity 6.1: Invariance and Change When Dividing Polygons
- Activity 6.2: What Can One Assemble from Two Triangles?
- Mathematical and Didactic Analysis of Activity 6.2: Invariance and Change When Constructing Polygons from Triangles
- Activity 6.3: How Can a Parallelogram Change?
- Mathematical and Didactic Analysis of Activity 6.3: Invariance and Change of Dimensions in the Set of Parallelograms
- Activity 6.4: Identical Perimeters
- Mathematical and Didactic Analysis of Activity 6.4: Preserving the Perimeter
- Summary of the Roles of Invariance and Change in Geometrical Shapes
- Suggestions for Further Activities
- References.
