Change and Invariance : : A Textbook on Algebraic Insight into Numbers and Shapes /

Change and Invariance : : A Textbook on Algebraic Insight into Numbers and Shapes / / by Ilya Sinitsky.

"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...

Full description

Saved in:
Bibliographic Details
VerfasserIn:
Sinitsky, Ilya.
Place / Publishing House:Rotterdam : : SensePublishers :, Imprint: SensePublishers,, 2016.
Year of Publication:2016
Edition:1st ed. 2016.
Language:English
Physical Description:1 online resource (XIV, 378 p.)
Tags: Add Tag
No Tags, Be the first to tag this record!
Table of Contents:
  • 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.

Similar Items