Change and invariance : a textbook on algebraic insight into numbers and shapes / Ilya Sinitsky and Bat-Sheva Ilany.

Format:

Book

Author/Creator:

Sinitsky, Ilya, author.

Series:

Education Series

Language:

English

Subjects (All):

Symmetry (Mathematics).

Mathematics--Study and teaching.

Mathematics.

Mathematical models.

Algebra.

Geometry, Algebraic.

Mathematics--Philosophy.

Physical Description:

1 online resource (XIV, 378 p.)

Edition:

1st ed. 2016.

Place of Publication:

Rotterdam, The Netherlands ; Boston ; Taipei : Sense Publishers, [2016]

Summary:

"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 world is an interplay of variation and constancy – a medley of differences and similarities – and this change and invariance is, largely, a language of science and mathematics. This book proposes a unique approach for developing mathematical insight through the perspective of change and invariance as it applies to the properties of numbers and shapes. After a short introductory chapter, each of the following chapters presents a series of evolving activities for students that focus on a specific aspect of interplay between change and invariance. Each activity is accompanied by detailed mathematical explanations and a didactic discussion. The assignments start with tasks familiar from the school curriculum, but progress beyond the menial to lead to sophisticated generalizations. Further activities are suggested to augment the chapter’s theme. Some examples: “How to represent all the integers from zero to 1000 using ten fingers?”, “How to win at the game of Nim?”, “Why do different square lattice polygons with the same area often have the same perimeter?” This book can be used as a textbook for pre-service mathematics teachers and is primarily intended for their academic instructors. Essentially, students, teachers and anyone interested in elementary mathematics will enjoy the elegant solutions provided for the plethora of problems in elementary mathematics through the systematic approach of invariance and change.".

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

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 Invarianceand Change?

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

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?”

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 DidacticAnalysis 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

References.

Notes:

Includes bibliographical references.

Description based on online resource; title from title page (Ebook Central, viewed December 26, 2025).

ISBN:

9789463006989

9463006982

OCLC:

964657030