Discovering Hidden Invariance
Chapters 4 and 5 both present different aspects of relationships between invariance and change. Whereas in Chapter 4 we initiated some change for the express purpose of ensuring the preservation of some value or property, in Chapter 5 we present some prob
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Change and Invariance A Textbook on Algebraic Insight into Numbers and Shapes
Ilya Sinitsky Gordon Academic College of Education, Shaanan Religious College of Education, Israel and Bat-Sheva Ilany Hemdat Hadarom Academic College of Education, Israel
A C.I.P. record for this book is available from the Library of Congress.
ISBN: 978-94-6300-697-2 (paperback) ISBN: 978-94-6300-698-9 (hardback) ISBN: 978-94-6300-699-6 (e-book)
Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands https://www.sensepublishers.com/
All chapters in this book have undergone peer review.
Cover image by David Sinitsky
Printed on acid-free paper
All Rights Reserved © 2016 Sense Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
TABLE OF CONTENTS
Prefaceix Acknowledgementsxiii Chapter 1: The Concept of Invariance and Change: Theoretical Background
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Understanding Phenomena from the Aspect of Invariance and Change 2 The Concept of Invariance and Change in the Mathematical Knowledge of Students8 The Basic Interplay between Invariance and Change 18 Some Introductory Activities in Invariance and Change 27 References31 Chapter 2: Invariant Quantities – What Is Invariant and What Changes?
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Introduction: Understanding the Invariance of Quantity as a Basis for Quantitative Thinking 36 Activity 2.1: Dividing Dolls between Two Children 43 Mathematic and Didactic Analysis of Activity 2.1: Partitioning a Set into Two Subsets: Posing Problems and Partition Methods 45 Activity 2.2: How to Split a Fraction. Almost Like Ancient Egypt 55 Mathematic and Didactic Analysis of Activity 2.2: Invariance of Quantity and Splitting of Unit Fractions 58 Activity 2.3: They Are All Equal, But … 69 Mathematic and Didactic Analysis of Activity 2.3: From Equal Addends to Consecutive Addends 72 Activity 2.4: Expressing a Natural Number as Infinite Series 82 Suggestions for Further Activities 91 References96 Chapter 3: The Influence of Change
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Introduction: Changes in Quantity and Comparing Amounts 100 Activity 3.1: Less or More? 108 Mathematical and Didactic Analysis of Activity 3.1: The influence That a Change in One Operand Has on the Value of an Arithmetical Expression 109 Activity 3.2: Plus How Much or Times How Much? 120 Mathematical and Didactic Analysis of Activity 3.2: Different Ways of Comparing122 Activity 3.3: Markups, Markdowns and the Order of Operations 127
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TABLE OF CONTENTS
Mathematical and Didactic Analysis of Activity 3.3: Repeated Changes in Percentages130 Activity 3.4: Invariant or Not? 139 Mathematical and Didactic Analysis of Activity 3.4: Products and Extremum Problems 140 Activity 3.5: What Is the Connecti
