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
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.
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.
Cover image by David Sinitsky
SensePublishers
ISBN 978-94-6300-697-2
DIVS
Ilya Sinitsky and Bat-Sheva Ilany
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?”
Change and Invariance
Change and Invariance
Spine
20.498 mm
Change and Invariance
A Textbook on Algebraic
Insight into Numbers and
Shapes
Ilya Sinitsky and Bat-Sheva Ilany
Change and Invariance
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
1
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?
35
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
99
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
v
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 Connection between Mathematical Induction
and Invariance and Change?
146
Mathematical and Didactic Analysis of Activity 3.5: What Is the
Connection between Mathematical Induction and Invariance
and Change?
147
Suggestions for Further Activities
151
References157
Chapter 4: Introducing Change for the Sake of Invariance
159
Introduction: Algorithms – Introducing Change for the Sake of Invariance 160
Activity 4.1: The “Compensation Rule”: What Is It?
165
Mathematical and Didactic Analysis of Activity 4.1: Changes in the
Components of Mathematical Operations That Ensure the Invariance
of the Result
167
Activity 4.2: Divisibility Tests
178
Mathematical and Didactic Analysis of Activity 4.2: Invariance of
Divisibility and Composing of Divisibility Tests
182
Activity 4.3: Basket Configuration Problems
189
Mathematical and Didactic Analysis of Activity 4.3: Diophantine
Problems and Determining the Change and Invariance
191
Activity 4.4: Product = Sum?
202
Mathematical and Didactic Analysis for the Activities in 4.4: Invariance
as a Constraint
204
Suggestions for Further Activities
215
References222
Chapter 5: Discovering Hidden Invariance
225
Introduction: Discovering Hidden Invariance as a Way of Understanding
Various Phenomena
226
Activity 5.1: How to Add Numerous Consecutive Numbers
232
Mathematical and Didactic Analysis of Activity 5.1: The Arithmetic Series:
Examples of Use of the Interplay between Change and Invariance in
Calculations235
Activity 5.2: Solving Verbal Problems: Age, Speed, and Comparing the
Concentrations of Chemical Solutions
244
Mathematic and Didactic Analysis of Activity 5.2: Solving Verbal
Problems by Discovering the Hidden Invariance
246
vi
TABLE OF CONTENTS
Activity 5.3: Mathematical Magic – Guessing Numbers
254
Mathematical and Didactic Analysis of Activity 5.3: Discovering the
Invariant in Mathematical “Tricks”: “Guessing Numbers”
258
Activity 5.4: “Why Can’t I Succeed?”
266
Mathematical and Didactic Analysis of Activity 5.4: Discovering the
Hidden Invariance in “Why Can’t I Succeed?”
269
Suggestions for Further Activities
279
References284
Chapter 6: Change and Invariance in Geometric Shapes
285
Introduction: Invariance and Change in the World of Geometry
286
Activity 6.1: Halving in Geometry – Splitting Shapes
295
Mathematical and Didactic Analysis of Activity 6.1: Invariance and
Change When Dividing Polygons
299
Activity 6.2: What Can One Assemble from Two Triangles?
319
Mathematical and Didactic Analysis of Activity 6.2: Invariance and
Change When Constructing Polygons from Triangles
321
Activity 6.3: How Can a Parallelogram Change?
329
Mathematical and Didactic Analysis of Activity 6.3: Invariance and
Change of Dimensions in the Set of Parallelograms
331
Activity 6.4: Identical Perimeters
341
Mathematical and Didactic Analysis of Activity 6.4: Preserving the
Perimeter347
Summary of the Roles of Invariance and Change in Geometrical Shapes
364
Suggestions for Further Activities
365
References377
vii
PREFACE
The world – according to philosophers, scientists, and artists – is an amalgamation
of that which is static and that which varies: an ongoing interplay of change and
invariance. Already in ancient times, this interplay was noted: King Solomon stated
“… there is nothing new under the sun” (Ecclesiastes 1:9), while Greek philosopher
Heraclitus claimed that “you cannot step twice into the same river.”
In the early twentieth century, renowned Dutch artist M. C. Escher expressed
the concept of the connection between invariance and change through his series of
“Metamorphosis” drawings – the shapes in the drawings gradually changed into
other shapes that were entirely different (fish into birds, birds into buildings, and so
forth). Authors worldwide, including such names as the Russian Ivan Bunin, or Rosa
Montero from Spain, explore the themes of body and soul based on the scientific
fact that each and every cell in the human body is completely replaced at least every
few years.
Alongside exploring that which is constantly changing, there is a parallel search
for those quintessential components that remain permanent in a constantly fluctuating
world. This has intrigued thinkers since time immemorial when, for instance, they
contemplated the periodicity of the heavens and the constancy of the constellations.
Indeed, most of the basic scientific laws were formulated by observing what
remained invariant despite certain changes. Today, conservation laws in physics
describe our fundamental understandings of the world, and every conservation law
(conservation of energy, conservation of charge, to name but two) expresses the fact
that a particular property remains constant despite changes in the system. Einstein, in
fact, defined his theory of relativity as “a theory of space-time invariance.”
In mathematics, the idea of invariants is explored intensively. For example,
topology investigates which properties of a shape remain invariant when space
undergoes transformation (shrinking, stretching, expansion, and so forth). In
general, the notion of isomorphism expresses the invariance of properties during the
transformation of objects, and forms a basic concept of modern algebra.
In early mathematics education, the concept of invariance and change is present
from the very outset: A cardinal number, which defines the size of a set, is the final
ordinal number obtained in the process of counting the set’s elements. This number
represents the invariance of the counting process. In fact, varied and diverse interplay
between invariance and change exists in virtually every mathematical exercise,
algorithm, or theorem, but these relationships will remain obscure to the student if
learning focuses on technical skills and not on the profound mathematical ideas and
concepts that are inherent in them.
Today, there is a clear trend in mathematical education to foster algebraic
thinking early in elementary school. Emphasizing the “why?” instead of focusing
ix
PREFACE
on the “how?” will, hopefully, develop in learners the ability to see the intrinsic
relationships between objects, and to appreciate the different aspects and properties
of the phenomenon or object under study.
Many mathematics teachers, unfortunately, look at algebra – and perhaps
even mathematics in general – as merely a collection of “manipulations done on
letters,” and they are convinced that algebraic reasoning is a way of cognition that
develops only during the formal study of algebra. For this reason, elementary school
teachers do not usually give much thought to fostering algebraic reasoning as part
of their mathematical syllabus. We thus made it our goal to search for and develop
a framework to inculcate and expand elementary school teachers’ mathematical
and pedagogical understanding of algebraic thinking so as to provide an algebraic
perspective of the mathematical routines in the basic mathematics curriculum.
Educators agree that the ability to make generalizations is the most significant
component of algebraic reasoning, and the maxim “a lesson without a generalization is
not a lesson in mathematics” reflects this agenda. Every mathematical generalization
begins with specific numbers or shapes. The relationships between these elements in
different situation are investigated, and then a generalization is formed by comparing
those different situations and discovering which components remain invariant and
which change during the transition from one event to the next.
We are convinced that the ability to appreciate and visualize the invariance and
change will give students “algebra eyes and ears,” as dubbed by Blanton and Kaput.
The ability for such visualization should be nurtured in students from the onset of
the formation of mathematical knowledge. It transpires that the idea of invariance
and change (and their relationships) does provide profound algebraic understanding
for almost every issue in elementary mathematics. We believe that practicing with
a variety of mathematical situations from the viewpoint of invariance and change
will give teachers an algebraic perspective of the mathematical material studied in
elementary school – and they will be able to cultivate this algebraic perspective in
their students.
This book offers a unique way of enhancing mathematical knowledge through
the perspective of invariance and change – and the relations between them – in
mathematical entities. It is meant to help develop algebraic thinking on an informal
level. The topics presented in this book are neither of one kind nor of one particular
aspect of algebra but rather an assortment of subjects and exercises that will allow
discovering and exploring the interplay between invariance and change in many
intriguing ways. This approach allows “a fresh view” of the issues of elementary
secondary and high school mathematics and provides surprisingly simple strategies
to deal with and arrive at solutions for even well-known problems and tasks.
Because of the wide range of activities offered, instructors can choose those
that appeal to them and that are appropriate for the goals of their course and their
audience. The relevant connections between invariance and change are initially
presented using subject matter taught in elementary school, but then go on to offer a
wide range of ideas that span the subject, including issues associated with advanced
x
PREFACE
mathematics. Each activity begins with an assignment to be presented to the class,
and is then followed by detailed mathematical and didactical explanations.
The book is intended for math educators, and especially for educators of future
elementary and junior-high mathematics teachers as the content is applicable to the
learning processes in school. The authors are confident that the approach and the
material presented within may also be used for courses on algebra and pre-algebra
that aim to develop algebraic reasoning in students, and can be particularly useful
for deepening in-service teachers’ content knowledge (in professional development
courses). Students, teachers, and anyone interested in elementary mathematics will
enjoy the plethora of ideas offered for solving problems with a systematic approach
of invariance and change.
xi
ACKNOWLEDGEMENTS
The authors wish to extend special thanks to Prof. David Ben-Haim and Prof.
Avikam Gazit for their ideas, stimulating comments, and clarifications that
proved so useful; to Dr. Raisa Guberman, who so enthusiastically experimented
with some of the activities with her students; and to Prof. Nitsa Movshovitz-Hadar,
Tami Giron, Isaac Nativ, and Marina Sinitsky for their professional advice. We are
also thankful to the many teachers and students – too numerable to mention
individually – who participated in the professional development classes,
seminars, and studies that accompanied the writing of the book.
We would like to express our deep gratitude to Linda Yechiel, whose efforts and
dedication were so invaluable in the preparation of this manuscript. We are also
grateful to David Sinitsky for his generous permission to use his artwork for the
cover, and thank Vitaly Chernikov for the delightful caricatures that open each
chapter.
We are also indebted to the Mofet Institute for Research, Curriculum, and
Program Development for Teacher Educators; the Gordon College of Education,
Haifa; the Hemdat Hadarom Academic College of Education; and the Shaanan
Academic Religious Teachers’ College for their support and financial assistance in
the preparation of this book.
xiii
CHAPTER 1
THE CONCEPT OF INVARIANCE AND CHANGE:
THEORETICAL BACKGROUND
1
Chapter 1
A joke: Gödel-style “incompleteness theorems” in math education:
Theorem 1: For any person, there exists a mathematical proposition
that is unprovable by that person.
Theorem 2: For any statement, there exists a person who can’t prove
this statement.
UNDERSTANDING PHENOMENA FROM THE ASPECT OF
INVARIANCE AND CHANGE
The Concept of Invariance and Change
Our world is a constant stream of different phenomena. Human attempts to
understand the world and the various phenomena that it entails, often lead to a need
to determine which properties remain invariant despite changes in a situation. This
quest arises from the understanding that some properties and characteristics of an
object can change while others remain invariant. Obviously, if one state is exactly
the same as another, then no doubt exists that all aspects are exactly the same in both.
However, once any changes are observed, considerations arise regarding the source,
causes, and forces that lead to them.
The verse “The thing that hath been, it is that which shall be; and that which is
done is that which shall be done” (Ecclesiastes 1:9) does not imply that change never
occurs. A few passages previous, David said, “One generation passeth away, and
another generation cometh: but the earth abideth forever” (Ecclesiastes 1:4). This
is the deterministic approach that sees a relationship between cause and effect in
every process. The well-known maxim “There is nothing permanent except change,”
attributed to Greek philosopher Heraclitus, emphasizes the essence of the ongoing
changes that objects, phenomena, and processes undergo. They are the antithesis to
the underlying assumptions of the Milesian school, whose scholars believed in the
existence of some essential substances whose variations and combinations provide
the basic elements from which all world is formed.
Many philosophies maintain that the cornerstones of the physical and spiritual
worlds are fixed, invariable components. For example, the Kabbala and some Eastern
philosophies mention three or four basic elements. In classical Greek philosophy, the
four basic elements are earth, fire, water, and air, each of which is a combination of
two different states of matter: wet or dry, and hot or cold. In his Theory of Essence,
Aristotle added a fifth “heavenly” quintessential element to the four physical ones.
All together, these five elements provide all the ingredients necessary to form any
substances in the world. Elements, by definition, are unchangeable, and therefore
they provide a basic definition to the concept of invariance.
Another implication of invariance relates to the stability of the properties of an
object under certain conditions. In any process of change, some properties of an
item – whether a physical object or an abstract idea – change and some do not. In
this respect, invariance defines a property that remains fixed despite the application
2
THE CONCEPT OF INVARIANCE AND CHANGE: THEORETICAL BACKGROUND
of some influence, even though it may change as a result of another. If the property
remains invariant in some cases, this means that the invariance holds over a range
of changes.
Ontology, one of the main branches of philosophy, is concerned with the nature
of being and the essence of change in the world. From the ontological perspective,
every process entails some link between what remains invariant and what changes.
Every object can be defined according to a range of properties, some of which can
be directly influenced, and others that can be examined according to their effect on
other elements and attributes.
The functional (cybernetic) approach involves two terms: independent variables
and dependent variables. In this context, the term variable has a special meaning: it
does not necessarily change in all instances; however, there is the possibility that it
can or might change.
Change and Invariance from the Epistemological Perspective
In the field of social sciences, epistemological theories are used to try to understand
thought processes and the formation of knowledge. Invariance and change are
integral parts of thought processes. The process of abstraction that accompanies
any thinking process can be examined. During the abstraction of the most basic,
simple items, our senses help us to focus on specific forms or features that can be
selected, emphasized, and separated from the object’s other attributes. For example,
one might focus on the different colors of objects and ignore their sizes, or focus on
their shape but not their material, and so on. This abstraction might begin by using
terms related to concrete objects and phenomena, but later, some of the properties
of the objects are ignored and the focus continues to be only on properties that are
critical to defining the concept. That is to say, the formation of a concept deals with
various phenomena and objects that are different and that change, but, nevertheless,
there will be some common and invariant property that is recognized as an intrinsic
feature. This common property is what determines the terms used – the essential
invariance regarding it. For example, the term “green” might define an invariant
condition that characterizes a color common to a number of objects.
The transition from the concrete to the abstract occurs in all cognitive processes.
Higher order thinking skills are based on abstractions that involve representations
that are more general and that can ignore “random divergences.” According to Back
(2012), the facts and details that we sense are complex. A table is constructed from
a specific material, has a specific color, a specific shape, a specific height, and so
on. To understand a phenomenon perceived by the senses, the scientist must analyze
its quintessence. The process of forming an abstract term includes many stages: the
results may have many properties that differ from each other, yet they share the main
(intrinsic) characteristics. Thus, as a result of the process of abstraction, objects can
be recognized by some general characteristics that are invariant for that set and all
the other variations are irrelevant: An object can be any color, size, material, height
3
Chapter 1
trivial mathematical tasks. Hopefully, exposure to this aspect will allow teachers to
realize the mathematical and didactic potential that exists in such an approach.
As in any meaningful learning process using the constructivist approach, the
role of academic-level educators is to assist future teachers in the process of such
learning. A discussion of the concepts of invariance and change and the aspects of the
relationship between them will be found further on in this chapter, and will describe
a method that will help develop the algebraic thinking of pre-service teachers and
demonstrate how such practices can be integrated into mathematics instruction in
school.
THE BASIC INTERPLAY BETWEEN INVARIANCE AND CHANGE
Different Types Invariance
The wide variety of mathematical entities and processes exhibit many examples of
invariance and change. Below are a few examples relevant to elementary school.
Invariance of the object. As mentioned above, in order to understand the concept
of number, one must grasp that the quantity of some objects is not dependent on
the way they are arranged. Here, the object – that is to say, the set of elements that
represents the number – is invariant. Even if the given set of items is changed into
another set (for example, by systematically replacing each element in the original
set with a different element), the attributive property of the set – how many elements
are in the set, that is to say, its number – remains invariant. The concept of number
has a central role in all further mathematical studies, and invariance of the object is
expressed in arithmetic as invariance of the number.
Most of the calculation techniques taught in school are based on rules of
arithmetic operations. In fact, the technique may be considered correct if it can be
seen that the value of the number remains invariant during the transition from one
given expression to another (for example, 8+4 = (8+2) + 2). In fact, all the equalities
in a chain of calculations express the invariance of the value (which is unknown until
the conclusion of the chain) through its different presentations.
Invariance of the set of numbers. When the arithmetic operations are studied, we
focus on the properties of each operation for some specific set of numbers. This set,
as a whole, typically has some features of invariance. Another example concerns
the set of even numbers, which is closed with respect to addition, meaning that the
sum of any number of elements of the set will also be even, that is to say, the result
belongs to the same set (contrary, for example, to addition in the set of odd numbers).
Invariance of geometric shapes. During their first exposure to geometric shapes,
students are usually asked to find prototype shapes in their environment or to form
physical models of different shapes (for example, constructing a polygon from
18
THE CONCEPT OF INVARIANCE AND CHANGE: THEORETICAL BACKGROUND
strips or cutting a polygon out of paper). The ability to transfer a shape from one
medium to another demonstrates that the student recognizes the characteristic
properties of the shape despite the way in which it is represented. Later, students
formally learn geometric transformations that keep the shape invariant (isometric
transformations).
Invariance of relation. Invariance can also be observed in the comparison of
values: an equality or an inequality between two magnitudes remains invariant if
they simultaneously change in an identical way (for example, the same number
is added to both parts of an equality). Sometimes, the relation remains invariant
with respect to one change, but does not maintain the invariance with another. For
example, if both sides of inequality a > b are multiplied by a value c, the inequality
ac > bc holds for c ΠN. However, the equality is not invariant if c is a negative
number.
The above examples show that both mathematical objects and their properties
may be invariant under specific changes that occur in the object or with the object.
The Taxonomy of Invariance and Change
In this book, we restrict ourselves to those aspects of invariance and change that
can directly influence the development of algebraic thinking of teachers and young
learners.2
The activities presented in Chapters 2 through 5 focus on arithmetic issues, on
numbers and their properties, and on arithmetic operations. Accordingly, they yield
and lead to conclusions of an algebraic nature, since one of the facets of algebra
is generalized arithmetic. In Chapter 6, we use geometric objects (particularly
in plane geometry) to demonstrate the same concepts of invariance-and-change
relationships.
Invariance and change can have varied associations within the same process
or situation, and how we focus on a specific aspect will depend on both the
mathematical and didactical situation. We focus on four main aspects, not all of
which are exclusive to one another, but each of which focuses on a different facet.
These four are as follows:
a. Invariance that is predetermined by an external factor;
b. Change that is predetermined and its influence on other properties;
c. Change initiated for the purpose of invariance; and
d. Hidden Invariance that is present within the change.
The first two are related more to the act of the observation: the invariance
or change is prescribed in one of the components of some process (such as a
mathematical operation) or object (such a set of items or a geometric shape). The
latter two are more pronounced during the process of active intervention, where
either the change is decided upon in accordance with some invariance imposed
19
Chapter 1
on one of the components, or the hidden invariance is detected and/or used in the
process of making the change.
a) Invariance that is predetermined by an external factor. This situation
is characterized by a specific invariance that can be perceived directly and
immediately. In other words, the value or property of one of the parameters is
determined in advance as a condition of the problem. Students then observe if and
how the other parameters change, given the constraint.
Some examples are the following:
• Studying what the possible divisors of odd numbers can be (odd being the
imposed invariance).
• Comparing how long two different objects released from the same height remain
in freefall. Here, the height remains invariant, and the objects differ in one or
more properties (size, weight, material) over the series of experiments. (The
height may be changed between one series of experiments and another, but in
each series, the height remains invariant.)
• Comparing the areas or perimeters of shapes obtained by cutting same-size
pieces of material (sheets of paper, for example) into sections. Two pieces of
identical size (although they can be of different colors or materials) are used so
that one remains “intact” (to be used for comparison after the manipulations have
been done to the second). The invariant property in this case is area, and the
change involved during the manipulations is to the number of sides of the figures
obtained, their specific geometric properties, or their perimeters.
• Examining how many ways a specific set of identical objects can be arranged in
a rectangular pattern. The number of objects remains invariant. This task belongs
to a large family of combinatorial problems that examine how a given finite
set can be represented according to specific criteria. The arrangement (in our
example, a rectangular shape) and the number of objects are given (invariant).
This “tangible” problem is equivalent to finding all the ways that a number
can be expressed as the product of two factors that can be determined by prime
factorization. The value of the number (product) remains invariant, and the
different combinations of the prime factors into pairs are the changing ways that
the number can be presented as the product of two factors. Note that the more
general problem allows decomposition of the value by non-integer factors, but
often students restrict the range of change of factors: they suppose, for example
that the maximum value of both factors cannot exceed the given natural number
(ignoring the possibility that a proper fraction can be another factor).
• Searching for the relationship between step length and number of steps within an
invariant distance (inverse proportionality).
Invariance imposed on one property may or may not lead to changes in other
properties, and understanding the process means being able to answer the following
questions:
20
THE CONCEPT OF INVARIANCE AND CHANGE: THEORETICAL BACKGROUND
• If one of the properties of a process is invariant, what changes can be made to the
others?
• Does the application of one particular invariance lead to any other invariance? If
so, what is it?
All the aspects mentioned above are explored more in detail in Chapter 2, where
the focus is on the invariance of value. This is the primary form of invariance that
can be determined in advance in arithmetic.
b) Change that is predetermined and its influence on other properties. Many
processes can be described as a predefined change in a particular property, often
within predetermined limits. To investigate such situations, one examines how the
predetermined change affects the other properties during the process. For example,
does modifying a quantitative property in one direction (increasing or decreasing)
necessarily result in a similar change (increase or decrease) of another property? If
so, can the relationship between the two be quantified? Are there any properties that
remain unchanged throughout the range of defined changes?
Some examples follow:
• A bag containing a specific amount of sand is released from various heights.
Does the time it takes to reach the ground differ as the height changes? How
do the parameters that describe the motion of an object with a particular mass
(acceleration, speed at each second, time of the motion) change with the height?
Here, the change is made to only one parameter: the changing height at which the
objects are released (the mass of the object is kept constant).
• Examining the distance covered as a function of number of steps or length of step.
In either case, one parameter is kept invariant and the other one changes. In both
cases, the distance covered is directly proportional to the change in the parameter
observed.
• All the sides of a given triangle are doubled in length. How does the perimeter
of the triangle change? How does its area change? Do the angles change? In this
case, the perimeter and the area both increase. This is intuitive: “The longer the
sides of a triangle, the larger its perimeter and area.” However, the change to the
perimeter is not on the same scale as that to the area: the perimeter doubles its
original length (similar to the change in each of its sides) whereas the area is four
times as much as the initial. None of the angles change (they remain invariant).
In contrast, if just one side of a triangle is elongated (the lengths of the other two
remain invariant), then, the area of the new triangle (provided it exists!) might
actually be smaller than the initial triangle or even identical. (A triangle with sides
of 8, 5, and 5 units is identical in area to that of a 5-5-6-unit triangle whose one
side was increased by two units.)
• Changing the sides proportionally may be generalized into a situation in which
the lengths of all the sides of a triangle are changed in same manner. However,
what if all the sides are shortened (or elongated) by a specific length (and not
21
Chapter 2
2. For some natural numbers, applying the method yields a series that includes nonpositive integer addends. However, the expression will include a series with an
odd number of terms, with the middle term 0, and all the others forming pairs of
additive inverses. All of these may be cancelled out while keeping the (total) sum
invariant. As a result, the initial number is expressed as the sum of an even number
of natural numbers. Thus, expressing any natural number as the sum of a series of
natural numbers remains invariant for both an even or an odd number of terms.
3. There is only one way that any odd prime number, 2n + 1, can be expressed
as the sum of consecutive numbers and it is always using only two terms:
2n + 1 = n + (n + 1). This expression was also derived in the framework of the
developed procedure, expressing the value as the sum of 2n + 1 terms, all of which
are 1.
4. The invariance of the method for determining all the ways that a number can be
expressed leads to the conclusion regarding powers of 2. These numbers have no
odd divisors (other than 1) and therefore it is impossible to express them as the
sum of consecutive natural numbers.
ACTIVITY 2.4: EXPRESSING A NATURAL NUMBER AS INFINITE SERIES
Activity 2.4 deals with expressing a number as a sum of an infinite number of
its parts. The significance here is that every positive number, including natural
numbers, can be expressed as the result of an infinite process of adding (smaller
and smaller) portions to form the number. One of the goals of this activity is to
show different ways of expressing a number as the sum of invariant parts of variant
quantities. During the activity, students are invited to formulate conclusions and then
to generalize and apply the procedures they have discovered.
This activity is important for understanding the significance of the concept of
number. Every natural number can be expressed as an infinite series that converges
toward that specific number. In advanced mathematics, this approach is used to
define the concept of real numbers using the Cauchy sequence.5
In contrast to most of the activities in this book (where students are asked to first
investigate the topic on their own), this activity is presented as a series of questions
followed by comments and points of discussion. During the discussion, ensure that
the correct terms regarding infinity are applied since many students intuitively use
statements such as “there cannot be such a sum because the number of terms is
infinite” or “the sum must be infinite because it is an infinite series.”
1. Open the discussion by asking what the magnitude of the terms in the desired
series will be. Intuitively, it is clear that if we want to use an unlimited
number of addends to arrive at a finite sum, the magnitude of those numbers
must decrease to approach zero.
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Invariant quantities – What is Invariant and What ChangeS?
We can illustrate how to express the number 1 geometrically by using a model that
is familiar to students as it is used to demonstrate fractions. A square area whose
dimensions are 1 × 1 is chosen. First, we select half the area, then we add half of the
remaining half, and continue such: at each stage, we add half of the remaining area.
(see Figure 2.3).
Figure 2.3. Multi-stage addition of parts of a square
• How is the sum of the areas expressed after the first two stages? After three
stages? After n stages?
These sums are equal to
to
, and to
,
respectively.
We recall that one of the formal ways of calculating the sum of n elements of a
1
1
geometric progression Sn = 12 + 14 + 18 + 16
+ 32
+ ... + 1n is based on identifying the
2
invariant sum on both sides of the expression with further algebraic manipulations:
leading to
Sn −1 + 1n = 12 + 12 × Sn −1 ,
2
which implies that 12 Sn −1 = 12 − 1n . Thus we have arrived at: Sn = 1 − 1n .
2
2
• What happens as the process continues? How is the final sum expressed?
• What is the value of this sum?
83
Chapter 2
The process requires an intuitive understanding of the concept of limit. If the
students identify the sum as a decreasing geometric series, they can apply the
formula for such a series to find its value. Simple substitution of a value (the first
term is 12 , and the ratio of the series is also 12 ) results in the sum being equal to 1:
S = 12 + 14 + 18 + ... + 1n + ... = 1 .
2
Students can “rediscover” the results by using the relevant algebraic technique,
similar to the case of a finite sum:
Therefore, S = 1 .
In this case, it is important to point out that the existence of a value of the series S
(in other words, the convergence of the infinite series) is a basic assumption required
for the validity of the procedure. Usually students tend to write the “infinite sum” as
“ 12 + 14 + 18 + ... + 1n + ... ” and then add “and continues to infinity.” However, when
2
students are asked to interpret the results, they often state that the sum “just tends to
1 but in fact it is smaller than 1.”
During the geometric demonstration of the process with the model of a square, it is
important to emphasize that each part that is selected in advance will be “included in
the sum” at some specific point. This is a visualization of an abstract infinite process:
the number of terms increases indefinitely at the same time that each individual term
gets smaller at each stage, such that their partial sum is approaching a limit:
2. In the second stage, focus on the sum of the areas that were developed
above: 21 + 41 + 81 + ...+ 1n + ... . We have already established that this is a
2
way of expressing the number 1
Describe the three ways that the terms of the series can be interpreted and the next
addend obtained: as parts of the whole, as parts of the selected area, or as parts of the
area that has not yet been selected.
i) Each term is a specific part of the whole (the number 1), and at each stage, this
portion is half as large (as the previous one). The parts are powers of 12 :
84
Invariant quantities – What is Invariant and What ChangeS?
ii) Beginning with the second addend in the series, each term represents a specific
invariant portion of the preceding one, that is equal to its half. Thus, in the
second stage of the process, we have chosen a quarter of the initial square, which
is only half the size of the section chosen at the first stage. The 1 × 1 square is
thus assembled of rectangles each of which has an area half the value of the one
preceding it.
iii) Beginning with the second addend in the series, each term represents a specific
portion of the part that will complete the desired total. This portion is constant
and equal to half of what remains up to the point where the whole has been
completed. Thus, the 1 × 1 square can be represented as a collection of rectangles
the areas of each being half that required to complete the square.
This explanation is based on the fact that the series is convergent and is an existing
number. (See also item 3 in “Further Activities” in this chapter.)
3. In this question, we continue the process we began in question 1 and attempt
to generalize how the number 1 can be represented. Does the method
demonstrated work only for halves? What happens if we use powers of other
fractions that are smaller than 1, such as powers of one third?
We saw that it is possible to express the number 1 as the sum of powers of 12 . We will
now examine if this can be done in a similar manner using powers of 13 . In other
words, is the series of powers of 13 also equal to 1? Usually, students will respond
to this question affirmatively and assume that it is exactly “the same case as the
previous series.” But is it?
Similar to calculating the previous series, we assume that the sum does, indeed,
exist, denote it by S, and derive the following:
From here we see that the series of the powers of 13 does converge, but the
value to which it converges is S = 12 , which is not equal to 1! In fact, it is easy
1
to generalize this result: the series of the powers of any proper unit fraction m
converges to S = m1− 1 , which is smaller than 1 for any natural denominator beside
2. In general, a series of powers of proper positive fractions a ( 0 < a < b ) converges
b
to S = a . The sum will be less than 1 for any positive fraction less than 12 , but
b −a
greater than 1 for any series of powers for positive proper fractions greater than 12 .
85
Chapter 2
Now we will discuss the additional understanding we can get by expressing 1
as the sum of its parts. “Isolating” one third of the given 1× 1 square leaves an area
that is equal to two thirds of the entire square. Using the method that is analogous to
adding halves, we will be adding (only) a third of what remains of the given square
(see Figure 2.4).
We express this numerically as follows: In the first stage we select a rectangle
with area b1 = 13 , leaving us with a rectangle of area . The area of
the rectangle chosen in the second stage will be b2 = 13 × r 1 = 13 × 23 . At this point, the
“chosen” shape has an area of b1 + b2 = 13 + 13 × 23 , and the area of the “remaining”
rectangle is
. In the third stage, we take one third of
the remaining area:
, and after transferring that to the “chosen
area,” the remaining area will be
.
Figure 2.4. The stages in adding a third of the portion remaining at each stage
It can thus be seen that the areas that are removed at each stage ( b1 , b2 , b3 , ... )
form a geometric progression with first element b1 = 13 and a ratio q = 23 . The value
of the series can be calculated either by using the known formula for an infinite
decreasing geometric series or by applying the following technique, which is typically
used to derive this formula:
With this method, we have successfully expressed 1 as a sum of thirds. However,
in this case (as opposed to the previous instance of “thirds”), at each stage a third
86
Invariant quantities – What is Invariant and What ChangeS?
is removed from the area remaining to complete the whole. This method allows
k that is subtracted from the whole
generalization for any constant proper part m
sum obtained up to 1. In this case, too, calculating the accrued sum, S, leads to the
following expression:
From this, we can see that S = 1.
In addition to presenting the formal calculation, the qualitative explanation of
the process is crucial: Since at each stage we are adding a portion of the area to
the accumulating area (the sum), the final area (once all the additions have been
made) will have completed the area to form a square (that is to say, it will have
reached 1). At no stage will the total sum be greater than the whole. Geometrically,
the “accumulating” shape is gradually filling up the area of the square, and therefore
the sum of the areas will never exceed 1. Nevertheless, for any number that is less
than 1, the sum will, at some point, “reach” the given number, and may even exceed
it. In other words, the number 1 is the limit of a series.
The most surprising thing about the representations shown here is that it is
possible to select any constant portion of the remaining area at each stage, even a
very small one. For example, we could add to one-hundredth of the number a portion
equal to one-hundredth of the 0.99 remaining, and then we select another hundredth,
and so on. No matter what “at the end” of this infinite process, we will arrive at the
number itself.
• Is the representation of an infinite series only valid for 1?
For each of the stages arrived at for 1, we can multiply both sides of the expression
by the same value A to obtain an expression for A as the infinite series of its parts. In
order to demonstrate this geometrically, all one has to do is to denote the given initial
area by A (instead of 1). The fractions that represent the area of each portion describe
a portion of the entirety A – half of A, a quarter of A, and so on.
4. From Zeno to fractals
We demonstrated above how to split any number into “changing halves” – one
half of the value, and then one half of what remains, and then one half of what
remains from that, and so on – in an infinite process of fragmentation. This method
of expressing a number is the simplest case for a series of fractions: every portion is
87
Chapter 2
equal to one-half of what remains. In this particular case, each portion is also onehalf of the previous one.
The concept of expressing the whole as a series of “halves” goes far back into
history and brings to mind the famous paradox of “Achilles and the Tortoise”6
proposed by Greek philosopher Zeno (495–430 BC). His purpose was to show
the logical contradiction inherent in the concept of motion: in order to travel some
specific distance, one must first travel half of it, and then one half of the remaining
half, and then half of the quarter that remains, and so on to infinity. According to
Zeno, because one cannot travel an infinite number of “halves” in a finite amount
of time, it will be impossible ever to reach the endpoint. Zeno was not searching
for the actual value of an infinite series. The point of his paradox was to exemplify
the problem in logic that arises with mathematical concepts such as limits and
convergence: even though constructing an infinite geometric series is essentially an
infinite process, the sum of the geometric series is not infinite when the ratio of the
series is less than 1.
Expressing the whole as an infinite sum of its parts is also related to the branch
of modern mathematics and physics concerned with “fractals.” A fractal is a shape
that is made up of parts that are identical to the initial shape – any expansion of
any part reveals a pattern that is identical to any other expansion (Uribe, 1994).
In other words, presenting a number as the infinite sum of parts of a quantity that
has not yet been selected is, in essence, presenting it in the “fractal manner”: after
selecting some portion from the “uncolored” section, we next select a portion that
is equivalent to the one we selected in the previous stage, but from a “different
whole.”
We can demonstrate this fractal-like aspect by using a rectangle that, at the point
where the process is completed, is the sum of the halves of all its “remaining”
sections. For a more remarkable demonstration, we shall use a rectangle with a side
ratio of 1: 2 . In such a case, the ratio between the sides of the whole rectangle is
equal to that between the sides of the rectangle that remains after taking away half
the rectangle:
b 2
b:b 2 = 2 :b .
In this case, there is a similarity between half the rectangle (which has become the
“new” whole rectangle in the second stage) and the initial rectangle.
In other words, after the first stage of division, the situation remains identical and
the rectangle that remains can be presented as the sum of rectangles that are similar
to the two originals (whole and half) but each has half the corresponding area. This
similarity is the basis for the definitions of the various international sizes of standard
sheets of paper: cutting an A3-size sheet of paper into two equal halves results in
A4 size, the ratio of whose sides is the same as the previous (see Figure 2.5). This
property allows increasing or decreasing the size of a photograph because changing
the layout of the photograph on the different size papers still retains the proportions.
88
Invariant quantities – What is Invariant and What ChangeS?
Figure 2.5. “Folding” the rectangle into two similar rectangles without
changing the ratio between the sides
k
, of a whole while keeping the
m
proportions of the shapes invariant, the initial rectangle must have appropriate
dimensions expressed as follows: the ratio between the lengths of the sides must be
b = 1 − k . For example, if we wish to form a complete rectangle from portions that
m
a
In general, if one wants to select some part,
are 34 of the whole, and then to add three quarters of the portion that was not “added”
at each stage, the initial rectangle must have a side ratio of b : a = 1 − 34 = 1: 2 (see
Figure 2.6).
Figure 2.6. Demonstrating the first two stages of division for a rectangle where the ratio
between the lengths of its sides is 1:2. We add 3 4 of the missing portion at each stage
maintaining the side ratios for the remaining portions
5. Back to 1 – Another astounding demonstration
In this demonstration, you will show the students how the number 1 can be
expressed as the sum of nine-tenths, and then another nine-tenths of what remains,
and so on.
This application is important for understanding the properties of rational numbers.
If we express the number 1 as the sum of nine-tenths of it plus nine-tenths of all
remaining portions and so on, we arrive at the following expression:
1 = 0.9 + 0.9 × 0.1 + 0.9 × 0.1× 0.1 + ... = 0.9 + 0.09 + 0.009 + ... .
The sum on the right (with the “9” digit repeating infinitely) is termed the infinite
periodic fraction 0.9. In other words, 1 is exactly equal to this infinite periodic
89
Chapter 2
fraction. If we truncate the sum at some given point, we obtain a number that is
close to 1, yet different from it. It is extremely important to discuss and emphasize
that 0.9 is a different way of expressing exactly 1, and should not be considered a
“good approximation” of the number. We have found that pre-service teachers have
real difficulty in comprehending this idea. They can, perhaps, be convinced by the
following:
• Have them try to find a rational number greater than 0.9 and smaller than 1
(that is, to find a number that they can insert between the two). Because no such
number exists, the two numbers must be equal.
• Have them try to calculate the average value of 0.9 and 1. They will discover
that it is equal to one of the numbers, meaning that both numbers must coincide.
xpressing a number this way is valid for any whole number (for example,
E
5 = 4.9 ) as well as for any rational number:
0.2 = 0.9 × 0.2 + 0.9 × 0.2 × 0.1 + 0.9 × 0.2 × 0.1× 0.1 + ... =
0.2 = 0.18 + 0.018 + 0.0018 + ... = 0.1 + 0.09 + 0.009 + ... = 0.19.
In other words, every rational number that can be expressed as a finite decimal
can also be written as an infinite periodic decimal number. Therefore, there is a
uniform way to express all rational numbers as infinite periodic decimals.
Summary of the Roles of Invariance and Change in Activity 2.4: Expressing a
Natural Number as Infinite Sums
1. This activity deals with expressing a natural number as a geometric series:
the activity begins with expressing 1 as the sum of the natural powers of 12 .
Changing the base, 12 , to another fraction while keeping the method of expressing
it invariant (a series of powers of the initial value) does not keep the value of the
series invariant.
2. A method was developed for expressing 1 as the value of a geometric series that
is expressed as “an accumulation” of constant parts of the given number. This
process is characterized by the invariance in how the added portions are chosen:
at each stage, a consistent portion accumulates out of the part of the whole that
was not included in the previous stages (this part changes). We showed that there
is invariance with respect to the sum of the series (the sum is equal to 1) for each
portion that was determined beforehand.
3. Similarly, the way in which any positive number can be expressed is invariant
for all positive numbers. Any quantity can be expressed as a rectangular area.
For each portion chosen to be part of the “accumulation,” conditions were found
that assured that at each stage of the process, the ratio between the sides of the
90
Invariant quantities – What is Invariant and What ChangeS?
accumulating rectangles remained invariant. As a result, the area of the rectangle
can be expressed as a series of areas of smaller, accumulating rectangles, the sides
of which are proportional to those of the original.
4. One of the applications of this method is to demonstrate that the value of a
rational number remains invariant no matter whether it is presented as a finite
decimal number or as a geometric series. Therefore, an infinite periodic decimal
is a legitimate and uniform way to express any rational number.
SUGGESTIONS FOR FURTHER ACTIVITIES
1. Expressing a number using your fingers
The concept of different base counting methods offers one example of how a
value remains invariant even when a presentation of the number (its expression) is
changed. The value that 8 represents in base ten is represented by 12 in base six, yet
the quantity (value) has not changed. Among all the base counting methods, the one
with the most applications, especially in today’s technological world, is the binary
system. It is a remarkable fact that all digital technology is based on expressing any
number as a sequence based on only two digits: 0 and 1.
A simple (and actually quite ancient) application of the binary system is expressing
a natural number using the fingers of the hands. What is the largest natural number
that can be expressed this way? The astonishing answer is that all the numbers up to
1000, and even slightly beyond, can be represented by using the ten fingers.
To do this, we assign values to the fingers as follows: 1 to the first finger (let us
say, the thumb on the right hand), 2 to the right index finger, 4 (not 3!) to the right
middle finger, and so on up to the tenth finger (the thumb on the left hand) which will
have the value of the ninth power of 2, that is to say 512.
Any natural number less than 1024 can be expressed as the sum of the various
powers of 2, from 1 (2 to the power of 0) to 512 (2 to the power of nine). For
example, 200 can be expressed as 200 = 128 + 64 + 8, that is to say, the following:
200 = 0 × 29 + 0 × 28 + 1× 27 + 1× 26 + 0 × 25 +
200 = + 0 × 24 + 1× 23 + 0 × 22 + 0 × 21 + 0 × 20.
In order to indicate 200 with our fingers, we raise the right ring finger (2 to the
power of 3), the left ring finger (2 to the power of 6), and the left middle finger (2
to the power of 7).7
Note that numbers expressed in the binary base are easily multiplied by 2 or by
powers of 2. For example, multiplying by 4 simply requires “moving the fingers”
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Chapter 2
two places to the left. Addition is also easy using this method – simply replace any
two fingers “in the same place” by moving to the “next finger.”
2. In the footsteps of Egyptian fractions: the value of a telescopic series
Calculate the following sum:
1
+ 1
2× 3 3× 4
+ 4 ×1 5 + ... + 98 ×1 99 + 99 ×1100 .
In order to calculate this sum, we can express each of the terms as the difference
between two unit fractions. In activity 2.2., we showed that any fraction of the form
1
n can be expressed as the sum of a pair of unit fractions where the denominator of
one of the pair will be consecutive to that of the initial fraction: n1 = n 1+ 1 + n (n1+ 1)
(for example, 12 = 13 + 16 ). This also means that the value of any unit fraction whose
denominator is a multiple of two consecutive numbers is invariant when it is
expressed as a difference:
1
1
1 .
n (n + 1) = n − n + 1.
In other words, changing the operation between fractions n1 and n 1+ 1 from
multiplication to subtraction does not affect the results! This being the case, in the
expression given above, we can cancel out all the terms except for the first and the
last.
By changing how each element in the series is expressed, most of the elements
in the sum can be cancelled out. The main challenge in such a series is to find an
appropriate way to manipulate its terms so that it can be expressed as an algebraic
sum of fractions with identical numerators. In general, the method is applicable to
series where the denominators are products of the terms of an arithmetic progression
with a difference of d:
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Invariant quantities – What is Invariant and What ChangeS?
Because this method of calculation is reminiscent of a “telescopic” extension, it is
no wonder that such series are called telescoping series. The series presented above
(resulting in 0.49) is the most well-known example.8
3. Tiling using regular convex polygons
This activity is related to unit fractions and through it, some surprising connections
between various areas of geometry can be discovered. Convex regular polygons can
be used to tile a surface if, and only if, the angles of the vertices at the junctions add
up to 360°. For example, four regular quadrilaterals (squares) meet with angles of
90° and therefore comply with this condition: (90° × 4 = 360°) (see Figure 2.7).
Figure 2.7. Tiling using regular quadrilaterals
Based on the formula for the value of angles in regular polygons and based on
the conditions given above, what other convex regular polygons can be used
for tiling?
The sum of the internal angles of a regular polygon of n sides is Sn = 180° × (n – 2)
and therefore each internal angle of a regular polygon is:
Tiling is possible by using m polygons of this type if, and only if, m angles of size
an add up to 360°. Therefore, regular polygons with n sides can be used provided
that an × m = 360°, that is to say
. Dividing each side of
the expression by 2m yields the relationship between the number of sides (n) of a
polygon that can be used for tiling and the number of such polygons (m) that will
1
meet at the same vertex: 12 = n1 + m
.
Remarkably, this equation is identical to the one we obtained in Activity
2.2 where we searched for rectangles that have the same numerical value for
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Chapter 2
both perimeter and area. The solutions are also familiar to us from the ways we
discovered to express 12 as the sum of two unit fractions. Expressing 12 = 14 + 14
indicates that tiling is possible using regular quadrilaterals where four squares
meet at their vertices. The other expression for 12 ( 12 = 13 + 16 ) produces two other
ways that such tiling may be done: tiling with regular triangles (6 triangles meet at
each junction) and tiling with regular hexagons (three at each junction). This is an
example of invariance of a mathematical model when going from one problem to
another (see Figure 2.8).
Figure 2.8. All possible ways of tiling using congruent convex regular polygons
To conclude this activity, we offer a solution with a different approach that does
not require calculation and is based on the invariance of the sum of the angles of the
polygons that meet at each common vertex.
When tiling with regular four-sided polygons (squares), four polygons meet at
each junction. Any other possibility involves polygons with either fewer sides than
a square, or more. The only polygon with fewer sides is a triangle, and it is easily
demonstrated that six triangles will fit the bill (6 × 60° = 360°). And while there are
an infinite number of polygons with more sides than a square, more sides means
that the polygon will have a greater interior angle, meaning that we will need fewer
polygons. Since the only number possible less than four can be three (there cannot
be only two polygons at a junction!), the angles will have to be 120°, and we have
found the only other possible polygon that can be used: hexagons. Thus, without any
calculation, and using only the constraint that the sum of the angles around the vertex
must remain invariant, we have determined that there are only three possibilities
where regular polygons can be used for tiling: squares, triangles, and hexagons.
4. “Caution: series” – Sums with an infinite number of terms
In Activity 2.4, numbers were written as a series, that is to say the sum of an infinite
number of terms. Because the exercise was to express a particular number, we knew
that such a number existed. However, this does not mean that all infinite sums have
some significance. Here we offer the infinite sum of 1–1+1–1… to highlight some
peculiarities of “infinite sums.”
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Chapter 3
a composite number. For example, 899 is not prime because 899 = 900 − 1= 302 −1
can be factorized as 899 = 29 × 31.
Summary of the Role of the Concepts or Invariance and change in Activity 3.4:
Invariant or Not?
This activity compares the products of different pairs of positive numbers where the
sums of the factors of each pair are invariant.
1. Under these restrictions, both factors must change in the transition from one
product to another. To compare the products, we can form an additional expression
that keeps one of the factors from each of the products invariant.
2. When the sums of the factors remain equal, the transition from one product
to another always changes the value of the product. The higher the difference
between the factors, the lower the value of the product.
3. For a given sum of two factors, the largest product possible is when the two factors
coincide. The value of any other product with the same sum of the factors will be
less than this square number by the square of the change in factors. The decrease
in the value of the product compared to the maximal one depends only on the
value of the change and not on the value of the factors themselves.
4. The results obtained may be interpreted geometrically:
• In the transition from one rectangle to another, keeping a constant perimeter
necessarily means a change in area. It is impossible for two different isoperimetric
rectangles to have equal areas.
• Among isoperimetric rectangles, the square is the one with the greatest area.
ACTIVITY 3.5: WHAT IS THE CONNECTION BETWEEN MATHEMATICAL
INDUCTION AND INVARIANCE AND CHANGE?
In this activity, we analyze the role of invariance and change in proving by
mathematical induction.
a) Using mathematical induction, prove that the following relationship is true
for every natural number, n:
1
1× 2
+ 2 ×1 3 + ... +
1
(n − 1) × n
+
1
n × (n + 1)
= 1 − n 1+ 1 .
b)Prove by mathematical induction that the sums of the squares of the first n
n ( n + 1) ( n + 2)
natural numbers Sn(2) is equal to
.
6
c) Prove that the arithmetic series of odd numbers starting from 1 is a square number.
d)Using induction, prove that for each natural number a (a > 1), the number
an − 1 is divisible by (a − 1).
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The Influence of Change
MATHEMATICAL AND DIDACTIC ANALYSIS OF ACTIVITY 3.5:
WHAT IS THE CONNECTION BETWEEN MATHEMATICAL
INDUCTION AND INVARIANCE AND CHANGE?
In preparation for this activity, the principle of mathematical induction should be
studied or reviewed. Students should be familiar with the three stages of proof by
induction:
• testing the validity of the claim for an individual case k = n0;
• assuming that the claim is correct for k = n;
• correctly proving the claim for k = n + 1 using to the principle of induction. The
claim will hold for every natural k, k ≥ n0 .
a) Using mathematical induction, prove that the following relationship is true
for every natural number, n:
1
1× 2
+ 2 ×1 3 + ... +
1
(n − 1) × n
+
1
n × (n + 1)
= 1 − n 1+ 1 .
This activity is a generalization of question 2 in Further Activities in Chapter 2.
Similar to what was discussed there, the sum of the given fractions can be found
by expressing each one as the difference between two unit fractions. In the present
question, students are asked to prove the validity of a sum by induction.
• When n = 1, the total comprises only the fraction 1 ×1 2 = 12 . This fraction is equal
to the expression which appears on the right side of the equation: 1− 1 +1 1 = 12 .
To illustrate the use of the formula, we calculate the sum for n = 2:
1
+ 2 ×1 3 = 12 + 16 = 32 = 1 − 13 . The sum of the fractions is less than 1, and the
difference between that sum and 1 is a unit fraction
the denominator of which is
1× 2
the larger factor in the denominator of the last fraction.
• Induction hypothesis: For any natural value k not greater than n, the following
expression is valid:
1
1× 2
+ 2 ×1 3 + ... +
1
(k − 1) × k
= 1− 1 .
k
• Induction step: To calculate the sum for n = k + 1, we rewrite in explicit form the
penultimate term of the total under consideration:
1
1× 2
+ 2 ×1 3 + ... +
1
k × (k + 1)
= 1 ×1 2 + 2 ×1 3 + ... +
1
(k − 1) × k
+
1
k × (k + 1)
.
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Chapter 3
Based on the hypothesis, the total of all the fractions except for the last one is
-
and the increment in the new sum is just the last addend. Continuing with
simple algebra, we’ll derive:
The induction step is proved, and therefore the general formula is valid.
b)Prove by mathematical induction that the sums of the squares of the first n
n ( n + 1) ( n + 2)
natural numbers Sn(2) is equal to
.
6
Similar to the previous question, we need to prove the validity of the expression
above.5 The proof is also based on splitting a new expression into a sum that includes
an invariant portion (the value of which is assumed to be known from the induction
hypothesis) and an additional addend. To simplify the algebraic calculations, we
suggest students write the inductive hypothesis6 for n = k − 1 by performing the
inductive step for n = k.
For n = k, the sum is
Sk(2) = 12 + 22 + ... + (k − 1) 2 + k 2 =

Sk −1
(k − 1) k (2k − 1)
6
+ k2 .
Using simple algebra, we arrive at the desired equality:
S (2) =
k
k (2k 2 − 3k + 1 + 6k ) k (2k 2 + 3k + 1) k (k + 1) (2k + 1)
=
=
.
6
6
6
Similar to the case in a), the new total of the first k squares of natural numbers is
expressed as the sum of an invariant portion, known from inductive hypothesis, and
an additional addend that changes the value of the expression.
c)Prove that the arithmetic series of odd numbers starting from 1 is a square
number.
148
The Influence of Change
In contrast to the previous questions, here the value of the series is not given and
the students’ task is to first find an expression for it and then prove that it is valid
over all n.
The process begins with an “observation stage”: students should examine the
sums obtained for a few sets with a small number of odd addends. The data on the
first five such arithmetic series are shown below:
Number of odd addends
1
2
3
4
5
Arithmetic series
1
1+3
1+3+5
1+3+5+7
1+3+5+7+9
Largest addend
1
3
5
7
9
Total
1
4
9
16
25
By this stage, students will probably realize that the total of the first n addends
equals the square of n, i.e. Sn = n2. To prove this formula, students must first construct
the algebraic expression for the nth term in the series of as a function of n. Using
the first and third rows of the table above leads to an = (2n − 1), and the inductive
hypothesis for the total of the first odd n numbers can be written as follows:
Sn = 1 + 3 + …+ (2n – 1) = n2.
The induction step is now straightforward: the next addend is 2(n + 1) − 1 = 2n + 1,
and adding it to the sum of the first n addends brings about the required identity:
Sn + 1 = 1 + 3 + ... + (2n − 1) + (2n + 1) =
S n + 1 = Sn + (2n + 1) = n 2 + 2n + 1 = (n + 1) 2 .
A simple visualization of the formation of new square number from the previous
one is presented in Figure 3.10.
d)Using induction, prove that for each natural number a (a > 1), the number
an − 1 is divisible by (a − 1)
Understanding this statement is not always easy for students because it includes
two variables, a and n, each with very different roles. Therefore, it may be useful to
formulate the statement for a specific natural number a, and then expand the formula
for the general case. To derive any usefulness from the initial step, a must be a > 3
because the statement is trivial both for a = 2 (every natural number is divisible by 1)
149
Chapter 3
Figure 3.10. The transition from a square with side, a, to a square with side (a + 1).
and a = 3 (every power of 3 is an odd number and its difference with 1 is necessarily
an even number).
We start directly with the general case. The proposition is true for n = 1, because
a − 1 (that is to say a1 − 1) is divisible by itself. According to the inductive hypothesis,
ak − 1 is divisible by a − 1. To perform the inductive step, we separate ak − 1 from
the expression ak + 1 − 1:
150
The Influence of Change
Now, the expression includes a times a part that remains invariant [and which has
been assumed to be divisible by (a − 1)] and the additional addend, (a − 1), which
is, of course, divisible by itself.
The inductive step can also be carried out using a slightly different technique:
a
k +1
−1 = a
k +1
− a + a − 1 = a ( a k − 1) + (a − 1) .
In this case, too, the validity of the proof lies in that we can express a new
expression as the sum of an invariant portion and an “extra” addend, both of which
have the property we are seeking.
SUGGESTIONS FOR FURTHER ACTIVITIES
1. Changes in percentages without calculations
This activity presents a collection of questions that require understanding percentage
change. Calculation is not used to solve the problem.
The initial situation for all the problems is as follows: In Grade 6-a, the
number of girls is greater than the number of boys.
a) What do we know about the percentages of girls in Grade 6-a? What do we
know about the percentage of boys?
b)There is another Grade 6 class (6-b) in which the number of boys is less than
the number of boys in 6-a. What can we conclude regarding what percentage
of the entire grade is made up by the boys of 6-a?
We are asked to compare the percentages of some quantity that is a portion of two
different quantities. The number of boys in 6-a constitute a portion that is less than
half of all the students in 6-a. At the same time, this number is greater than half the
number of boys in the entire sixth grade. Therefore, while boys make up less than
50% of 6-a, the number is greater than 50% of the total number of boys in both
classes. It is possible to generalize by saying that the same quantity (invariance)
constitutes a greater percentage in a smaller population and a smaller percentage in
a larger population (change).
c)A few more boys will be joining grade 6-a. What will happen to the
percentage of boys in the class? What will happen to the percentage of girls
in the class?
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Chapter 3
Usually, the students’ immediate response is that the percentage of boys will increase
since the number of boys increases. Unfortunately, this argumentation is not enough:
keep in mind that when calculating the new percentage, we cannot not ignore the new
(higher) total in the class. In other words, both the numerator and denominator of the
estimated fraction change. However, we can also look at it from the perspective of
the percentage of girls: their number does not change, but because the total number
of students has increased, their percentage has decreased. Therefore, the percentage
of boys must increase correspondingly.
d)An equal number of boys and girls will join the class. How will the
percentage of boys in the class change? The percentage of girls?
Understanding these changes is based on determining which portion does not
change (remains invariant). In this case, the difference between the number of girls
and boys remains invariant. This difference constitutes a specific portion of the
total number of students. After the change, this difference (which remains the same)
constitutes a smaller portion of the total number of students (due to the increase in
the total number of pupils in the class). Therefore, the gap between the percentage of
boys and the percentage of girls is reduced. This means that the percentage of boys
increases while the percentage of girls decreases.
When discussing this problem, point out that all the results are qualitative and
do not depend on any specific numerical values. The answers are obtained by
understanding the principles of change and invariance within the problem. (Note:
The issues proposed in this activity are also suitable for discussion in Chapter 5,
which deals with discovering hidden invariance.)
2. Calculating the squares of numbers mentally
As a continuation to question 5 in Activity 3.4, we offer students a method for easily
calculating the squares of numbers close to 100.
What is the square of 98?
In order to calculate the square of 98, one can express 98 as the sum 90 + 8 = 98
and use the abridged multiplication formula to calculate the value of 982 = (90 + 8)2
= 902 + 2 × 90 × 8 +82. Another similar – and easier – method is to express 98 as a
difference: 100 − 2 = 98. Here too, we use the abridged formula for the square of the
difference: 982 = (100 − 2)2 = 1002 – 2 × 100 × 2 + 22.
152
The Influence of Change
However, we can also take advantage of the fact that 98 is “close” to 100, which
is a number whose square is easily calculated. This method relies on a controlled
change of the product that we wish to calculate (changes of this type are one of
main issues in Chapter 4). As has been pointed out in Activity 3.4, increasing “the
distance” between the factors of the product 982 = 98 × 98 (while keeping their
sum) leads to a change in the product. We can easily discover the magnitude of the
change: 97 × 99 will be less than the desired product by 1, and 96 × 100 will be less
by 4, which is easy to calculate and equal to 9600. Therefore, 982 = 9600 + 4 = 9604.
Similarly, the values for 972 = 94 × 100 + 32 = 9409, 932 = 86 × 100 + 72 = 8649 and
so on can be determined (for every product with a similar structure). Encourage the
students to arrive at this method on their own (to this purpose, we suggest beginning
this activity as a “teacher versus calculator competition”). Also, ask them to justify
their solution (in addition to the technical explanation of the method of calculation)
and apply it. For example, ask them to determine the squares of numbers that are
slightly greater than 100, or the squares of numbers close to 50, and so on.
3. The divisibility of sums of consecutive numbers.
In question 1 of Activity 2.1 we showed that the sum of (2n + 1) consecutive
addends is divisible by (2n + 1). This activity expands and refines the considerations
discussed there.
In that proof, we discussed the various ways that the statement can be proved. For
example, the sum of any three consecutive numbers is greater by 3 than the previous
series: 2 + 3 +4 is 3 greater than 1 + 2 + 3, and the next sum will be 3 more than the
previous one, and 2 × 3 greater than the first. Because the first total is divisible by 3
(1 + 2 + 3 = 6), all subsequent totals will also be divisible by 3.
Similarly, one can see that the transition from any sum of k consecutive addends
to the next sum increases the sum by k. This is a uniform change to the total, and
therefore, any sum of k consecutive addends is greater than the sum of the first k
consecutive addends (from 1 to k) by a value of k. This means that if the sum of
the first k addends (from 1 to k) is divisible by k, then all the sums of k consecutive
addends will be divisible by k.
This consideration is equivalent to the proof of divisibility by induction, and this
is one of the methods used to prove a statement for any odd value of k. But what
about the divisibility of the sum of k consecutive numbers for an even value of k?
In this case, it is divisible by the number that is the half of the quantity of addends
(see Chapter 2, note 3). For example, the sum of six consecutive numbers will be
3-divisible (or will produce a remainder of 3 when divided by 6).
The following example reminds us how cautious one must be when using
the interplay of invariance and change. We offer a “proof” made by a student –
153
Change and Invariance in Geometric Shapes
Is it always possible to know whether the sum or the initial perimeter will
be greater? Explain your answer. Write down a formula that expresses the
relationship between the sum of the perimeters of the polygons obtained and
that of the initial polygon.
c) In question 3 we discovered that some polygons cannot be divided into two
congruent polygons. Try to divide the polygon into two polygons with the
same perimeters or into two polygons with the same area. Which of the two
properties (area or circumferences) can be used to illustrate the concept of
“parts of the whole,” “fraction,” and “percentage”? Why?
d)Dividing a triangle into equal-area triangles:
• How can you divide any triangle into two triangles with equal areas?
• How many solutions are there to this problem for any given triangle?
e) Exploring some generalizations:
• Can a triangle be divided into a triangle and a quadrilateral with equal areas?
Extend a section line through any point on one of the sides of a triangle and
check different situations.
• Is it essential that the initial polygon be a triangle? What generalization can
you formulate?
MATHEMATICAL AND DIDACTIC ANALYSIS OF ACTIVITY 6.1:
INVARIANCE AND CHANGE WHEN DIVIDING POLYGONS
In Activity 6.1, a line segment is used to divide polygons into other polygons. This is
analogous dividing a given quantity into two quantities under different requirements,
as was discussed in Chapter 2. The activities present opportunities to discuss many
aspects of invariance and change.
In this activity, shapes are divided into two similar shapes (comparable to Activity
2.4, in which a schema of similar rectangles was used to show that a finite number
can be shown to be the sum of an infinite number of parts).
1. Introductory question
Given any geometric figure. Draw a line segment to divide it. How many figures
were obtained? On what does your answer depend? Are there any “unusual”
situations?
In the remainder of the activity, use only straight lines to divide convex polygons. In
our experience, most students usually assume that the line is a straight-line segment
299
Chapter 6
and that the shape is convex, so they almost immediately answer that a line will
divide a shape into two. However, this only relates to the this particular case: a
straight line segment will divide any convex shape into two, and if the given shape is
a polygon, then the shapes obtained will also be polygons (see Figure 6.4).
Figure 6.4. Dividing a convex shape into two with a straight line
However, if the shape is not convex, there is always at least one line that can
divide the shape into more than two parts (see Figure 6.5).
Figure 6.5. Dividing a concave polygon into two or more shapes with a straight line
Thus, even a straight section line will not necessarily divide a shape into two. If
the shape is not convex, the number of shapes obtained may be greater than two.
Nevertheless, examining divisions of this sort is quite complex, so we shall limit our
discussion to convex polygons.
2. Preserving the type or the property of a polygon – Can the original
shape remain the same after division?
Dividing a convex polygon results in two shapes, both of which are also
convex polygons. The act of division, therefore, keeps the shape’s basic
property – its convexity – invariant. The name of a polygon is determined
according to the number of sides (or the number of vertices or angles), and
this is also one of the basic properties of a shape.
We will attempt to determine if it is possible to divide a convex polygon
with k sides so that at least one of the resultant polygons (or even both) have the
same number, k, of sides. If the number of sides does change, we will examine
the rules involved in such a change.
300
Change and Invariance in Geometric Shapes
Here we observe whether dividing a polygon will give rise to new polygons of
the same type (according to the number of sides) and then examine the factors
that determine the number of sides the new polygons will have according to the
number of sides of the initial polygon and the position of the dividing line. The
results might surprise students who – based on their investigations with triangles
and quadrilaterals – believe that any polygon with k sides can be divided into two
polygons with k sides: in fact, it is more complicated.
a) Choose any triangle and divide it into two polygons using a straight line.
Which polygons did you obtain?
• Is the result dependent on the direction or placement of the section line?
• Is it possible that neither of the resulting polygons are triangles?
There are two principal ways to divide a triangle into two polygons: the section
line may either pass through one of the vertices, or it passes through two sides (see
Figure 6.6). The first case always produces two triangles; the second, one triangle
and one quadrilateral. In other words, at least one polygon will be a triangle, thus
keeping the type of the initial polygon– a triangle – invariant (for at least one of the
resultant shapes).
Coming back to question 1, point out that it is impossible to obtain more than two
polygons when dividing a triangle using one straight line because each triangle is
convex figure.
Figure 6.6. Dividing a triangle into two polygons – two possibilities
In the case where one of the shapes obtained is a quadrilateral, it is interesting to
note that although its area and perimeter have been reduced compared to those of the
initial triangle, the number of it sides has increased.
b)Choose any quadrilateral and divide it into two polygons using a straight line.
• Which polygons did you obtain?
• Is the result dependent on the direction or placement of the dividing line?
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Chapter 6
• Formulate the conditions required for obtaining two quadrilaterals when
dividing a quadrilateral.
• Is it possible that neither of the resulting polygons will be a quadrilateral?
• Formulate the condition for obtaining at least one quadrilateral when
dividing the quadrilateral.
It is useful to sort the ways that a section line may divide quadrilaterals:
i) Through two vertices. In this case, the section line will be a diagonal and will
divide the quadrilateral into two triangles (see Figure 6.7a);
ii) Through one vertex of the quadrilateral and one of its sides. In this case, the
quadrilateral will be divided into a triangle and a quadrilateral (see Figure 6.7b);
iii) Through two sides, see Figures 6.8.
Figure 6.7. Dividing a triangle from a quadrilateral – What other polygon is obtained?
In the third case, the position of the section line will be either through opposite
sides or adjacent sides of the quadrilateral. Extending the section line through
adjacent sides divides the quadrilateral into a triangle and a pentagon (see Figures
6.7c, 6.8a, 6.8b); extending the line through opposite sides divides it into two
quadrilaterals (Figures 6.8c and 6.8d). In the last case, both the shapes obtained keep
the initial shape invariant: a quadrilateral has been divided into two quadrilaterals.
Figure 6.8. Dividing a quadrilateral into two polygons through two sides
In general, one can state that in order to keep the type (quadrilateral) of at
least one of the obtained polygons invariant, the section line must pass through a
point (not the vertex!) on one side and any point (may be also a vertex) on the
opposite side.
302
Change and Invariance in Geometric Shapes
c) Choose any convex polygon with more than three sides and use a line segment
to divide it into two polygons. Vary the placement of the line segment a
number of times.
What are the three “principally” different ways that a line can divide the
polygon? Sort the results according to how the division was made. Suggestion:
try to predetermine which polygon you wish to obtain and then predict the
number of sides the other polygon will have depending on how the division is
made. For example, one possibility might be that each of the polygons obtained
after the division will have a number of sides different than the number or sides
in the original polygon.
Determine how to divide the polygon so that at least one polygon will have
the same number of sides as the initial one.
It appears that many different pairs of polygons can be obtained when dividing
a polygon, and therefore we suggest performing the exercise by deciding on the
type of one of the polygons to be obtained before making the division, and then
investigating how this influences what the other polygon will be. For example, we
can begin by using a quadrilateral and the task is to obtain at least one triangle
for each of the ways that it can be divided. If we divide from vertex to vertex (a
diagonal), the second polygon will necessarily be a triangle (Figure 6.7a). If we
divide from a vertex to a side, the second shape is a quadrilateral (Figure 6.7b). If
the section line cuts two adjacent sides of the quadrilateral, the second polygon will
be a pentagon (Figure 6.7c). However, if the section line cuts two opposite sides of
the quadrilateral, no triangle can be obtained – both polygons will be quadrilaterals
(Figures 6.8c and 6.8d).
To summarize, when a triangle is “cut off” from a quadrilateral, the number
of sides of the second polygon may be one less than the original polygon, equal
to it, or one greater, and this is determined by how the section line divides the
quadrilateral.
The table below shows all the polygons that can result from dividing a convex
pentagon, based on how it is divided. The type of the first polygon (defined in
advance) obtained from dividing the pentagon is shown in the leftmost column. The
Line divides from
vertex to vertex
Line divides from
vertex to a side
Line divides from
side to side
Triangle
5–1=4
5
5+1=6
Quadrilateral
5–2=3
5–1=4
5
Pentagon
Impossible
5–2=3
5–1=4
Hexagon
Impossible
Impossible
5–2=3
303