Unformatted text preview: Mathematics of the Secondary School
Curriculum, I (Math 151)
H. Wu January 18, 2012
c
HungHsi Wu, 2011 Contents
Chapters Not in This Book v General Introduction vii Suggestions on How to Read This Book ix Prerequisites xi Some Conventions xiii 1 Fractions 1 1.1 Deﬁnition of a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Equivalent fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3 Adding and subtracting fractions . . . . . . . . . . . . . . . . . . . . 34 1.4 Multiplying fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.5 Dividing fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.6 Complex fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.7 Percent, ratio, and rate problems . . . . . . . . . . . . . . . . . . . . 72 1.8 Appendix: The basic laws . . . . . . . . . . . . . . . . . . . . . . . . 85 2 Rational Numbers 89 2.1 The rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.2 Adding rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.3 The vectorial representation of addition . . . . . . . . . . . . . . . . . 102 2.4 Multiplying rational numbers . . . . . . . . . . . . . . . . . . . . . . 108 2.5 Dividing rational numbers . . . . . . . . . . . . . . . . . . . . . . . . 117
iii iv CONTENTS
2.6 Comparing rational numbers . . . . . . . . . . . . . . . . . . . . . . . 126 3 The Euclidean Algorithm
141
3.1 The reduced form of a fraction . . . . . . . . . . . . . . . . . . . . . . 141
3.2 The Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . 155
4 Basic Isometries and Congruence
4.1 The basic vocabulary . . . . . . .
4.2 Transformations of the plane . . .
4.3 The basic isometries . . . . . . .
4.4 Congruence, SAS, and ASA . . .
4.5 A brief pedagogical discussion . . .
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. 5 Dilation and Similarity
5.1 The Fundamental Theorem of Similarity
5.2 Dilation . . . . . . . . . . . . . . . . . .
5.3 Similarity . . . . . . . . . . . . . . . . .
5.4 Appendix: The symmetry of similarity . .
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. 6 Symbolic Notation and Linear Equations
6.1 Symbolic expressions . . . . . . . . . . . . . .
6.2 Solving linear equations in one variable . . . .
6.3 The graphs of linear equations in two variables
6.4 Parallelism and perpendicularity . . . . . . . .
6.5 Simultaneous linear equations . . . . . . . . .
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369 Chapters Not in This Book
Mathematics of the Secondary School Curriculum, II and III
Chapter 7: Linear Functions Chapter 8: Quadratic Functions and Equations Chapter 9: Polynomial and Rational Functions Chapter 10: Exponential and Logarithmic Functions Chapter 11: Polynomial Forms and Complex Numbers
Chapter 12: Basic Theorems of Plane Geometry
Chapter 13: Ruler and Compass Constructions
Chapter 14: Axiomatic Systems
Chapter 15: Trigonometry
Chapter 16: The Concept of Limit
Chapter 17: The Decimal Expansion of a Number
Chapter 18: Length and Area
Chapter 19: 3Dimensional Geometry and Volume
Chapter 20: Derivative and Integral
Chapter 21: Exponents and Logarithms, Revisited v Structure of the chapters
Chapter 1 ✟
✟✟
✟
✟ ✟
✟✟
✟
✟
Chapter 4 ❤
❤❤ ✡
✡ ✡ ✡ ✡ Chapter 12 ✡ ❝
❝
❝ ✡ Chapter 2 ❤❤❤❤
Chapter 5
Chapter 3 ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ Chapter 6
Chapter 7 ✡ ✡ ✡ ✡ ✡ Chapter 8
Chapter 9 ❛❛ ✡
✡
✡✂
❝✡
✡✂
✡
❝
Chapter
✡✂
✡❝
✡✂
❝
✡
❝
✡
✂
✡
✡
❝
✂Chapter 13
Chapter 14 ✂
Chapter
✂
✂
✂
Chapter
✂
✂
✂
✂
Chapter
✂
✂
✂
✂
Chapter
✂
✘✘✘
✘✘✘
✂
✘✘✘
✂
✘✘✘
✂
✘
✘✘ Chapter 19 ❛❛
❛❛ 15 ✦✦ ✦✦
✦✦ 16
✦✦
17 ✦ 18 Chapter 20
Chapter 21 ❛ Chapter 11 10 ✦✦ ✦ ✦
✦✦
✦✦
✦✦
✦✦
✦ General Introduction
This book is a textbook for the mathematics curriculum of grades 6–12 written for teachers. It covers the relevant mathematics outside of probability and statistics in a selfcontained manner and with particular emphasis on mathematical integrity. If there was
something you never understood as a student, you would likely ﬁnd the explanation here.
Formally, it assumes only a knowledge of whole numbers. Informally, it also assumes a
certain level of mathematical maturity in order to explain all the relevant mathematical
issues. A teacher has to know these explanations and the purpose of the book is to provide
this needed knowledge.
Because you will be a teacher, you have to approach this book with a diﬀerent mindset
from all other textbooks. Reading the latter, you would likely congratulate yourself if you
achieve mastery over 90% of the material. But would a math teacher who is correct
only 90% of the time be considered a good teacher? To be blunt, such a teacher is a
disaster. So your mission in reading this book should be nothing short of total mastery.
You are expected to know this material 100%. This is the standard you have to set
for yourself.
You have to approach this book diﬀerently in yet another respect. The standard assumption in a math course is that if you can do all the homework problems, most of your
work is done. Think back on your calculus courses and you will understand how true this
is. In this course sequence, which is designed speciﬁcally for prospective teachers, your
emphasis cannot be just on doing the homework assignments. When you stand in front
of a classroom, what you will be talking about, most of the time, is not the exercises
at the end of each section but the materials in the exposition proper. For example, very
likely you will have to convince a class on geometry why the Pythagorean Theorem is
true. There are two proofs of this theorem in this book, one in Chapter 5 and the other
vii viii GENERAL INTRODUCTION in Chapter 18. Yet on neither occasion is it possible to assign a problem that asks for a
proof of this theorem. There are problems that assess whether you know enough about
the theorem to apply it when the need arises, but how to assess whether you know how
to prove the theorem when the proofs have already been given in the text? It is therefore
up to you to achieve mastery of everything in the text itself. For example, pick a random
theorem; can you prove it without looking at the book? Can you explain its signiﬁcance?
Can you give an intuitive explanation of why it is believable? I may add that the most
taxing part of writing the text was in fact to do it in a way that allows you, as much as
possible, to adapt it for use in a school classroom with minimal change.
The order of presentation in these volumes follows the school curriculum as much as
possible.1 When you read Chapter 1 on fractions, for instance, be aware that you are in a
sixth grade classroom and therefore, no matter how much algebra or geometry you know,
you have to be able to give explanations in the language of sixth grade mathematics.
Similarly, when you come to Chapter 6, you are developing algebra from the beginning
so that even the use of symbols becomes an issue. Therefore, temper your explanations
accordingly. The main goal of Volume I is to provide the necessary background for the teaching of
algebra. Currently, school students are generally deﬁcient in their knowledge of the two
pillars that support algebra: rational numbers and similar triangles. These two topics are
the subject of Chapters 1–2 and Chapters 4–5, respectively. Algebra begins in earnest in
Chapter 6.
I wish to thank Emiliano Gomez, Ted Slaman, and Shari Lind Scott for their contributions to the early drafts. My indebtedness to Ole Hald for his many corrections and
suggestions is too great to be expressed in words.
1
There are one or two exceptions. For example, the conversion of fractions to inﬁnite decimals is
taught as a rote skill in middle school but is taken up in Volume III only after the concept of limit has
been introduced because no explanation can be given any time earlier. Another is the presentation of
geometry: Although it is fully consistent with the recent Common Core Standards ([CCSS]), it does
deviate slightly from the traditional school curriculum in geometry. See the discussions in Section 4.5 of
Chapter 4 (Volume I) as well as Chapter 14 (Volume II). Suggestions on How to Read This
Book
The major conclusions in this book, like all mathematics books, are summarized into
theorems; depending on the author’s (and other mathematicians’) whims, theorems are
sometimes called propositions, lemmas, or corollaries as a way of indicating which
theorems are deemed more important than others (a formula or an algorithm is just
a theorem). This idiosyncratic classiﬁcation of theorems started with Euclid around 300
B.C. and it is too late to do anything about it now. The main concepts of mathematics are
codiﬁed into deﬁnitions. Deﬁnitions are set in boldface in this book when they appear
for the ﬁrst time; a few truly basic ones are even individually displayed in a separate
paragraph, but most of the deﬁnitions are embedded in the text itself so that you should
watch out for them.
The statements of the theorems as well as their proofs depend on the deﬁnitions, and
the proofs are the guts of mathematics.
I would like you to know everything in this book, of course. Please note that
when mathematicians talk about knowing something, they have in mind knowing all the
deﬁnitions and theorems by heart, knowing why the deﬁnitions are what they are, knowing
the proofs of all the theorems and the main idea of each proof, knowing why the theorems
are needed, knowing what the implications of the theorems are, and ﬁnally, knowing how
to apply the theorems in new situations if possible. At the very least, I want you to
know by heart all the theorems and deﬁnitions as well as all the main ideas of the proofs
because, if you don’t, it is futile to talk about the other aspects of knowing. Therefore,
a preliminary suggestion to help you master the content of this book is for you to
copy out the statements of every deﬁnition, theorem, proposition, lemma, and
ix SUGGESTIONS ON HOW TO READ THIS BOOK x corollary, along with page references so that they can be examined in detail
if necessary,
and also to
summarize the main idea of each proof.
These are good study habits. When it is your turn to teach your students, be sure to pass
on these suggestions to them.
You should also be aware that reading a mathematics book is not the same as reading
a gossip magazine. You can probably ﬂip through such a magazine in an hour, if not
less. But in this book, there will be many passages that require slow reading and rereading, perhaps many times. I cannot single out those passages for you because they
will be diﬀerent for diﬀerent people. We don’t learn the same way. What is true under
all circumstances is that you should accept as a given that mathematics books make for
exceedingly slow reading. I learned this very early in my career. On my very ﬁrst day as
a graduate student many years ago, a professor, who was eventually to become my thesis
advisor, was lecturing on a particular theorem in a newly published volume. He mentioned
casually that in the proof he was going to present, there were two lines in that book that
took him fourteen hours to understand and he was going to tell us what he found out in
those long hours. That comment greatly emboldened me not to be afraid of spending a
lot of time on any passage in my own reading. If you too get stuck in any passage of this
book, take heart, because you are supposed to. Prerequisites
In terms of the mathematical development of Volume I, only a knowledge of whole
numbers, 0, 1, 2, 3, . . . is assumed: place value, the four arithmetic operations, their
standard algorithms, and the concept of divisionwithremainder and how it is related
to the long division algorithm. The divisionwithremainder assigns to each pair of whole
numbers b and d, where d = 0, another pair of whole numbers q (the quotient) and r
(the remainder), so that
b = qd + r where 0 ≤ r < d Some subtle points about the concept of division among whole numbers will be brieﬂy
recalled at the beginning of Section 1.5 in Chapter 1.
Note that 0 is included among the whole numbers.
A knowledge of negative numbers, in particular integers, is not assumed. Negative
numbers will be developed ab initio in Chapter 2. In terms of the undergraduate curriculum, Volume I assumes a certain comfort level
with mathematical reasoning, so that readers of Volume I should have taken the usual
two years of college calculus. For Volumes II and III, readers should be at least taking an
upper division course in abstract algebra concurrently. xi Some Conventions
• Each chapter is divided into sections. Titles of the sections are given at the beginning
of each section as well as in the Table of Contents.
• When a new concept is ﬁrst deﬁned, it appears in boldface but is not often accorded
a separate paragraph all by itself. For example:
k More generally, by m copies of , we mean . . .
You will have to look for many deﬁnitions in the text proper.
• When a new notation is ﬁrst introduced, it also appears in boldface. For example:
for any x, y on the number line, x < y means . . . .
• Equations are labeled with numbers inside parentheses, and the ﬁrst digit indicates
the chapter in which the equation can be found. For example, “Thus (1.17) implies
that K is . . . ” means the seventeenth labeled equation in Chapter 1 is . . . .
• Exercises are at the end of each section.
• Bibliographic citations are labeled with the name of the author(s) inside square
brackets, e.g., [Ginsburg]. The bibliography is on page 369. xiii Chapter 1
Fractions
Overview of Chapters 1 and 2
We are going to develop a theory of fractions (positive rational numbers and 0) and
rational numbers that is suitable for use in upper elementary and middle school.
This theory is equally important for prospective high school teachers because most
students come to high school with a very defective understanding of fractions. High
school teachers who do not possess a knowledge of fractions that is accessible to school
students would be tremendously handicapped when they try to help their students.
From a broader perspective, every teacher must have a ﬁrm grasp of fractions and
rational numbers, because school mathematics as a whole is about rational
numbers. Real numbers are strictly the purview of college mathematics. To illustrate
this point, consider the following simple operation with irrational numbers:
√
√√
2
5
(4 × 2) + ( 3 5)
√+
√
=
4
3
43
√ 2
In school mathematics, one does not explain what √ and 45 are, much less the
3
meaning of adding the numbers on the left. By the same token, the meaning of the
√√
product 3 5 on the right is even more of a mystery. In the tradition of school
mathematics, this diﬃculty has never been confronted honestly. Implicitly, however,
the way school mathematics deals with such arithmetic operations is to appeal to
what we call the Fundamental Assumption of School Mathematics (FASM),
which states that 1 2 CHAPTER 1. FRACTIONS
if an identity or an inequality “≤” among numbers is valid
for all fractions (respectively, all rational numbers), then it is
also valid for all nonnegative real numbers (respectively, all
real numbers.)1 FASM plays a pivotal role throughout this book. For example, the above equality is
clearly patterned after the formula for the addition of fractions:
ac
ad + bc
+=
bd
bd
where a, b, c, d are whole numbers (bd = 0). One way or another, one can eventually
prove that the same equality also holds for all rational numbers a, b, c, d (bd = 0),
even if this is rarely done in standard texts. (See Section 5 of Chapter 2.) Therefore,
by FASM, the same equality is valid for all real numbers a, b, c, d (bd = 0), rational
√
√
or irrational. This is how we could let a = 2, b = 3, c = 5, and d = 4 to get
√
√
the previous result, regardless of the fact that 3 and 5 are no longer rational.
Clearly, FASM makes it mandatory that every school teacher, regardless of grade
level, acquire a ﬁrm grasp of fractions and rational numbers.
From a mathematical perspective, the rational numbers are among the simplest
mathematical structures. Let us brieﬂy recall how it is handled in abstract algebra. Given the integers Z, introduce an equivalence relation ∼ among ordered
pairs of integers (a, b) with b = 0, so that (a, b) ∼ (a , b ) iﬀ ab = ba . Then the
equivalence class containing the ordered pair (a, b) is what we usually call the quoa
tient b . For example, 2 stands for the inﬁnite set of ordered pairs of integers:
3
{(±2, ±3), (±4, ±6), (±6, ±9), . . .}. The collection of all such quotients is called the
ﬁeld of rational numbers Q. In Q, we deﬁne addition + and multiplication × by
ac
ad + bc
+=
bd
bd
and ac
ac
×=
bd
bd 1
See [Wu2008]. We should call attention to the subtle point that the weak inequality “≤” is used
in the statement of FASM; see the end of Section 20.1 in Chapter 20 for an explanation. You will
not see any mention of FASM in the school mathematics literature, but we will prove it in Theorem
20.2 of Section 20.1 in Chapter 20. 3
Since we are deﬁning the addition and multiplication between two inﬁnite sets of
c
numbers, as each of a and d is an inﬁnite set of ordered pairs of integers, we need
b
to check that these deﬁnitions are welldeﬁned, in the following sense. Consider,
for example, the second deﬁnition: if (a, b) ∼ (a , b ) and (c, d) ∼ (c , d ), then by
c
c
deﬁnition a = a and d = d . Since the deﬁnition of multiplication would have
b
b
a
c
a c
× = ,
b
d
bd
we must verify that ac
a c
=
bd
bd in order for the preceding deﬁnition of multiplication to make sense. It goes without
saying that such veriﬁcations are routine and these deﬁnitions are welldeﬁned. Then
we proceed to further verify that + and × are associative, commutative, and distributive, and that Q is a ﬁeld under these operations. For the purpose of mathematics,
the fact that Q is a ﬁeld is all that matters.
Now there are many reasons why this way of dealing with the rational numbers is
not suitable for use in grades 5–7, which is where fractions and rational numbers are
systematically taught in schools. Children around the age of twelve have little or no
conception of why a ﬁeld is interesting or important, and they can’t care less whether
fractions can be added or multiplied or not. They come to the concept of a fraction
through the common usage of parts of a whole, such as twothirds of a glass of orange
juice. Furthermore, they come to the addition and multiplication of fractions through
their experience with whole numbers, so that + connotes “combining things” and ×
signiﬁes “repeated addition”. The need of the school classroom therefore dictates that
fractions be introduced in a way that respects students’ prior experience with whole
numbers rather than the (to them) abstruse concerns about ﬁelds or the distributive
law. We cannot allow mathematicians’ favorite deﬁnitions to derail children’s learning
process and contribute to their mental disorientation.
What the foregoing discussion hints at is that school mathematics education is
not the teaching of straightforward mathematics. If engineering is the customization
of abstract scientiﬁc principles to meet human needs, then mathematics education
is Mathematical Engineering.2 In this case, we have to customize the abstract
2 For an extended discussion, see [Wu2006]. 4 CHAPTER 1. FRACTIONS theory of rational numbers to meet the needs of students around the age of twelve,
as well as the needs of their teachers. At the same time, the customization must be
consonant with mathematics in terms of reasoning and precision.
The tension between the need to be faithful to the basic principles of mathematics and
the need to make the mathematics gradelevel appropriate will be a dominant theme
in this book. We must not forget that this is the basic charge of good engineering.
This chapter and the next give an exposition of rational numbers that is suitable
for use in a classroom of grades 5–7. In the process, we will also make occasional
references to the abstract deﬁnition of Q above to remind ourselves that what we are
doing is nothing but an “engineered version of Q”. We start in this chapter with the
rational numbers which are ≥ 0. In the tradition of school mathematics, these are
called fractions.
In reading these two chapters, please keep in mind that the emphasis here will not
be on the individual facts or skills. For example, it is taken for granted that you are
entirely comfortable with identifying the fractions and rational numbers as certain
points on the xaxis (called the “number line” in school mathematics), and are ﬂuent in
the four arithmetic operations with rational numbers. Rather, the emphasis is on the
reorganization of familiar facts and skills in a new way to form a logical and coherent
whole that is compatible with the learning processes of upper elementary and middle
school students. The hope is that, with the reorganization, you as a teacher will be
able to explain fractions to your students in a way that makes sense to them and to
you yourself. This is the ﬁrst step toward establishing mathematical communication
between you and your students. For example, it may come as a surprise to you that
it is possible to develop the concepts of adding and multiplying fractions from the
standpoint that a fraction such as 2 can be identiﬁed with the interval [0, 2 ], or that
3
3
although the invertandmultiply rule for the division of fractions is a deﬁnition of
division in the ﬁeld Q, it is a theorem that requires proof in the context of school
mathematics.
There is the danger that, because the facts are so familiar to you, you will sleepwalk through these two chapters. If you do, that would be a major mistake.
The teaching of fractions is the most problematic part of school mathematics because, 5
in the usual way it is done, there is hardly any valid deﬁnition oﬀered and almost
nothing is ever explained. The resulting nonlearning of fractions is not only a national
scandal about the state of mathematics education, but also a major stumbling block in
students’ learning of algebra. The importance of fractions to the learning of algebra is
beginning to be recognized. See, for example, Recommendation 4 on page xvii of the
report of the National Mathematics Panel ([NMP1]). See also the article [Wu2009].
For example, can you recall from your own K12 experience if you were ever told
3
what it means to multiply 5 × 3 and, in addition, why is it equal to 5×7 ? This should
4
7
4×
give you incentive to do better when it is your turn to teach. What you will learn
in these two chapters, and perhaps all through this book, will be a reorganization of
the bits and pieces that you learned haphazardly in K–12 into a coherent body of
knowledge, and your job is to make this knowledge accessible to your students. You
are being asked to become an advocate of a new way to teach school mathematics:
give precise deﬁnitions to all the concepts and explain every algorithm and every skill.
The purpose of this book is to optimize your chances of success in this undertaking. 6 1.1 CHAPTER 1. FRACTIONS Deﬁnition of a fraction Mathematics rests on precise deﬁnitions. We need a deﬁnition of a fraction, not
only because this is what mathematics demands, but also because children need a
precise mental image for fractions to replace the mental imageof their ﬁngers for
whole numbers.
We begin with the concept of a number line, which is the name in school mathematics for the real line, i.e., the xaxis. So on a line which is (usually chosen to be)
horizontal, we pick a point and designate it as 0. Assuming that we can compare
distances between points to see if they are the same, we choose another point to the
right of 0 and, by reproducing the distance between 0 and this point, we get an inﬁnite
sequence of equispaced points to the right of 0 (i.e., points on the line so that the
distance between any two consecutive points is invariably the same). Think of this as
an inﬁnite ruler. Next we denote all these points by the nonzero whole numbers 1, 2,
3, . . . in the usual manner. Thus all the whole numbers N = {0, 1, 2, 3, . . .} are now
displayed on the line as equispaced points increasing to the right of 0, as shown:
0 1 2 3 4 A horizontal line with an inﬁnite sequence of equispaced points identiﬁed with N
on its right side is called the number line. By deﬁnition, a number is just a point
on the number line. Note that except for the original sequence of equispaced points
which we have chosen to denote by 1, 2, 3, etc., most numbers have no names as yet.
The next order of business will be to name more numbers, namely, the fractions.
Now fractions are introduced to students in the upper elementary grades, and
their basic understanding of a fraction is that it is “parts of a whole”. The transition
from “parts of a whole” to “a point on the number line” has to be handled with care.
This is because, while the idea of placing a fraction such as 2 on the xaxis comes
3
naturally to you because you are familiar with coordinate systems and calculus, it is
anything but natural to sixth graders or beginning high school students. To them,
“parts of a whole” is an object, e.g., an area, a part of a pizza, an amount of water in
a glass, or a certain line segment, but not a point on a line. Therefore, the following
informal discussion is intended to smooth out this transition as well as prepare you 1.1. DEFINITION OF A FRACTION 7 for the contingency of having to convince your students to accept a fraction as a
certain point on the number line.
We begin the informal discussion by considering a special case: how the fractions
with denominator equal to 3, i.e., 1 , 2 , 3 , 4 , etc., come to be thought of as a certain
3333
collection of points on the number line. We take as our “whole” the unit segment
[0, 1] from 0 to 1. The fraction 1 is therefore onethird of the whole, i.e., if we divide
3
1
[0, 1] into 3 equal parts, 3 stands for one of the parts. One obvious example is the
thickened segment below:
0 1 2 3 Of course this particular thickened segment is not the only example of “a part
when the whole is divided into 3 equal parts”. Let us divide, not just [0, 1], but every
segment between two consecutive whole numbers—[0, 1], [1, 2], [2, 3], [3, 4], etc.—into
three equal parts. Then these division points, together with the whole numbers, form
an inﬁnite sequence of equispaced points, to be called the sequence of thirds:
0 1 2 3 For convenience, we call any segment between consecutive points in the sequence
of thirds a short segment. Then any of the following thickened short segments is
“one part when the whole is divided into 3 equal parts” and is therefore a legitimate
representation of 1 :
3
0 1 2 3 0 1 2 3 8 CHAPTER 1. FRACTIONS
0 1 2 3 The existence of these multiple representations of 1 complicates life and prompts
3
the introduction of the following standard representation of 1 , namely, the short
3
segment whose left endpoint is 0 (see the very ﬁrst example above of a thickened
short segment). With respect to the standard representation of 1 , we observe that this
3
short segment determines its right endpoint, and vice versa: knowing this segment
means knowing its right endpoint, and knowing the right endpoint means knowing
this segment. In other words, we may as well identify the standard representation of
1
with its right endpoint. It is then natural to denote this right endpoint by 1 :
3
3
0 1 2 3 1
3 In like manner, by referring to the sequence of thirds and its associated short
segments, the fraction 5 , being 5 of these short segments, can be represented by any
3
of the following collections of thickened short segments:
0 1 2 3 0 1 2 3 0 1 2 3 Again, our standard representation of 5 is the ﬁrst one, which consists of
3
5 adjoining short segments abutting 0. This standard representation is completely 1.1. DEFINITION OF A FRACTION 9 determined by its right endpoint, and vice versa. Thus to specify the standard representation of 5 is to specify its right endpoint. For this reason, we identify the standard
3
representation of 5 with its right endpoint, and proceed to denote the latter by 5 , as
3
3
shown.
0 1 2 3 5
3 In general then, a fraction m for some whole number m has the standard rep3
resentation consisting of m adjoining short segments abutting 0, where “short segment” refers to a segment between consecutive points in the sequence of thirds. Since
we may identify this standard representation of m with its right endpoint, we denote
3
the latter simply by m . The case of m = 10 is shown below:
3
0 1 2 3
10
3 We note that in case m = 0, 0 is just 0.
3
Having identiﬁed each standard representation of m with its right endpoint, each
3
point in the sequence of thirds now acquires a name, as shown below. These are
exactly the fractions with denominator equal to 3.
0
0
3 1
1
3 2
3 3
3 2
4
3 5
3 6
3 3
7
3 8
3 9
3 10
3 11
3 In terms of the sequence of thirds, each fraction m is easily located: the point m is
3
3
the mth point to the right of 0. Thus if we ignore the denominator, which is 3, then
the naming of the points in the sequence of thirds is no diﬀerent from the naming of
the whole numbers. 10 CHAPTER 1. FRACTIONS Of course the consideration of fractions with denominator equal to 3 extends to
fractions with other denominators. For example, replacing 3 by 5, then we get the sequence of ﬁfths, which is a sequence of equispaced points obtained by dividing each
of [0, 1], [1, 2], [2, 3], . . . , into 5 equal parts. The ﬁrst 11 fractions with denominator
equal to 5 are now displayed as shown:
0
0
5 1
1
5 2
5 3
5 4
5 5
5 2
6
5 7
5 8
5 9
5 10
5 11
5 Finally, if we consider all the fractions with denominator equal to n, then we
would be led to the sequence of nths, which is the sequence of equispaced points
resulting from dividing each of [0, 1], [1, 2], [2, 3], . . . , into n equal parts. The fraction
m
n is then the mth point to the right of 0 in this sequence.
This ends the informal discussion.
We now turn to the formal deﬁnition of a fraction.
We will begin by formalizing some terminology. In particular, we wish to make
precise the common notion of “equal parts”. Given two points a and b, a to the left
of b, the segment from a to b consists of all the points between a and b together with
the points a and b themselves; denote this segment by [a, b] as usual. (In calculus, a
segment would be called an “interval”.) The points a and b are called the endpoints
of [a, b]. Two segments [a, b] and [c, d] are said to be of the same length if, by sliding
one segment along the number line until its left endpoint coincides the left endpoint
of the other segment, their right endpoints coincide. A segment [a, b] is said to have
length k for a whole number k if the segments [a, b] and [0, k ] have the same length.3
In particular, the unit segment has length 1. We say a segment [a, b] is divided into
m equal parts if [a, b] is expressed as the union of m adjoining, nonoverlapping
segments of the same length. A sequence of points is said to be equispaced if the
segments between consecutive points in the sequence are all of the same length.
3
It will be observed that, insofar as the only the whole numbers have been attached to points on
the number line, the only segments that have lengths will have to be a those which have the same
length as a [0, k ] for some whole number k . But as we proceed to name the fractions, the collection
of segments that have lengths will be broadened. 1.1. DEFINITION OF A FRACTION 11 Divide each of the line segments [0, 1], [1, 2], [2, 3], [3, 4], . . . , into 3 equal parts.
The totality of division points, which include the whole numbers, form a sequence
of equispaced points, to be called the sequence of thirds. By deﬁnition, the
fraction 1 is the ﬁrst point in the sequence to the right of 0, 2 is the second point,
3
3
3
m
is the third point, and in general, 3 is the mth point in the sequence to the right
3
of 0, for any nonzero whole number m. By convention, we also write 0 for 0 . Note
3
m
that 3 coincides with 1, 6 coincides with 2, 9 coincides with 3, and in general, 33
3
3
3
coincides with m for any whole number m. Here is the picture:
0
1
2
3
etc.
0
3 1
3 2
3 3
3 4
3 5
3 6
3 7
3 8
3 9
3 10
3 The fraction m is called the mth multiple of 1 . Note that the way we have
3
3
just introduced the multiples of 1 on the number line is exactly the same way that
3
the multiples of 1 (i.e., the whole numbers) were introduced on the number line. In
other words, if we do to 1 exactly what we did to the number 1 in putting the whole
3
numbers on the number line, then we would also obtain all the m ’s for all m ∈ N.
3
In general, if a nonzero n ∈ N is given, we introduce a new collection of points on
the number line in the following way. Divide each of the line segments [0, 1], [1, 2],
[2, 3], [3, 4], . . . into n equal parts, then these division points (which include the whole
numbers) form an inﬁnite sequence of equispaced points on the number line, to be
called the sequence of nths. The ﬁrst point in the sequence to the right of 0 is
1
2
3
denoted by n , the second point by n , the third by n , etc., and the mth point in the
0
sequence to the right of 0 is denoted by m . By convention, n is 0.
n
Deﬁnition The collection of all the sequences of nths, as n runs through the
nonzero whole numbers 1, 2, 3, . . . , is called the fractions. The mth point to the
right of 0 in the sequence of nths is denoted by m . The number m is called the
n
numerator and n, the denominator of the symbol m . By the traditional abuse of
n
language, it is common to say that m and n are the numerator and denomina0
tor, respectively, of the fraction m . 4 By deﬁnition, 0 is denoted by n for any n.
n
4
The correct statement is of course that “m is the numerator of the symbol which denotes the
fraction that is the mth point of the sequence of nths, and n is the denominator of this symbol.”
Needless to say, it takes talent far above the norm to talk like this. 12 CHAPTER 1. FRACTIONS The thing to keep in mind is that we ﬁrst identify a subcollection of points as
fractions before giving them names. The symbol m , which denotes the mth point in
n
the sequence of nths, is merely one way to signify the location of this number. There
will be other ways; see below. For example, as is wellknown, both 2 and 1 refer to
4
2
the same point on the number line, a phenomenon that will be explained in detail in
the next section.
1
As before, we shall refer to m as the mth multiple of n . In the future, we will
n
relieve the tedium of always saying the denominator n of a fraction m is nonzero by
n
simply not mentioning it. Remarks.
(A) In general, if m is a multiple of n, say m = kn, then it is
n
n
n
selfevident that n = 1, 2n = 2, 3n = 3, 4n = 4, and in general,
n
kn
= k,
n
In particular, for all whole numbers k , n, where n > 0.
m
=m
1 and (1.1) m
=1
m for any whole number m.
(B) For the study of fractions, the unit is of extreme importance. On the number
line, it is impossible to say which point is what fraction until the number 1 is ﬁxed.
This means that we do not know what a fraction is, precisely, until we know what
the unit is. In a classroom situation, it is wise to always remind students that, before
they wave a fraction around, they had better make sure they know what it refers to:
does 1 mean a third of the volume of the liquid in a cup, or a third of the liquid
3
by height? Or, a fraction 5 could be ﬁvesevenths of a bucket of water by volume,
7
ﬁvesevenths of a pie by area when looked at from the top, or ﬁvesevenths in dollars
of your lifesavings. An example of a common error is to refer to a pizza as the unit
(“the whole”), and ask what fraction is represented by putting one of the four pieces
below,
✬✩
✫✪ 1.1. DEFINITION OF A FRACTION 13 ✬✩ together with one of the eight pieces as shown: ❅
❅
❅
❅
✫✪ When the answer of 3 is not forthcoming, there is the usual bemoaning of students
8
“not getting the concept of a fraction”. Students would “get it” if, instead of saying
that the pizza represents 1, we tell them that the area of the pizza represents 1.
Students don’t know how to put two shapes together to get a fraction, because they
have been misled by confusing instruction.
(C) We have been talking about the number line, but in a literal sense this is
wrong. A diﬀerent choice of the line or even a diﬀerent choice of the positions of the
number 0 and 1 would lead to a diﬀerent number line. What is true, however, is that
anything done on one number line can be done on any other in exactly the same way,5
and therefore we identify all of them. Now it makes sense to speak of the number
line.
(D) This way of approaching fractions implicitly assumes a foundational knowledge
of Euclidean geometry. Mathematically, this is not an issue as Euclidean geometry
can be developed independently of the concept of numbers. More important is the fact
that this approach is pedagogically sound because the amount of geometric knowledge
that is implicitly assumed of the students is nothing more than what every student
around the age of ten would naturally take for granted.
(E) Although a fraction is formally a point on the number line, the informal
discussion above makes it clear that on an intuitive level, a fraction m is just the
n
m
segment [0, n ]. So in the back of our minds, the segment image never goes away
completely, and this fact is reﬂected in the language we now introduce. First, a
deﬁnition.
(F) With the availability of fractions, we can now say a segment [a, b] has length
m
for some fraction m if the segments [a, b] and [0, m ] have the same length.
n
n
n
Deﬁnition The concatenation of two segments L1 and L2 on the number line
is the line segment obtained by putting L1 and L2 along the number line so that the
5
In technical language, all number lines are isomorphic, corresponding to the fact that all complete
ordered ﬁelds are isomorphic. 14 CHAPTER 1. FRACTIONS right endpoint of L1 coincides with the left endpoint of L2 .
L1
L2
Thus the segment [0, m ] is the concatenation of exactly m segments each of length
n
1
1
12
−
, to wit, [0, n ], [ n , n ], . . . , [ mn 1 , m ]. Because we identify [0, m ] with the point m ,
n
n
n
n
1
1
and [0, n ] with n , it is natural to adopt the following suggestive terminology. We say
m
1
is m copies of n to mean that the segment [0, m ] is the concatenation of
n
n
k 1
exactly m segments each of length n . More generally, by m copies of , we mean
the segment obtained by concatenating m segments each of length k .
(G) In school mathematics, the meaning of the equal sign is a subject that is
much discussed, mainly because the meaning of equality is never made clear. You
will ﬁnd later on the traditional use of the word equivalent for fractions when equal is
meant. This adds to the confusion. For this reason, we make explicit the fact that,
k
m
by deﬁnition, two fractions and n are equal if they are the same point on the
number line. We have already seen in equation (1.1), for example, that kn = k = k
n
1
for any n, k ∈ N.
(H) The deﬁnition of a fraction as a point on the number line allows us to make
precise the common concept of one fraction being bigger than another. First consider
the case of whole numbers. The way we put the whole numbers on the number line,
a whole number m is smaller than another whole number n (in symbols: m < n )
if m is to the left of n. We expand on this fact by deﬁning a fraction A to be smaller
than another fraction B , (in symbols: A < B ) if A is to the left of B on the number
line:
A
B
Note that in the standard education literature, the concept of A < B between fractions is never deﬁned, one reason being that if the concept of a fraction is not deﬁned,
it is diﬃcult to say one unknown object is smaller than another unknown object.
Sometimes the symbol B > A is used in place of A < B . Then we say B is
bigger than A.
(K) A ﬁnal remark has to do with the symbol used to denote a fraction: 2 , 14 , k ,
35
etc. Students are known to raise the issue of why use three symbols (k , , and the 1.1. DEFINITION OF A FRACTION 15 “fraction bar” —) to denote one concept. Remember that a fraction is a point on
the number line, no more and no less. The symbols employed are merely means to an
end: they serve to indicate where the points are. Thus the symbol 14 says precisely
5
that, if we look at the sequence of ﬁfths, then 14 is the 14th point in the sequence to
5
the right of 0. We clearly need every part of the symbol 14 , namely, the number 5,
5
the number 14, and the fraction bar in between, to describe the location of this point.
The need of 5 and 14 is obvious, and the role of the fraction bar — is to separate 5
from 14 so that one does not confuse 14 with 145, for example.
5
Thus the symbol of a fraction is no more than a particular representation of the
real thing, namely, a point on the number line. This piece of information about fractions should be clearly conveyed to students in grades 5–7.
There is a special class of fractions that deserves to be singled out at the outset:
those fractions whose denominators are all positive powers of 10, e.g.,
1489
,
102 24
,
105 58900
104 These are called decimal fractions, but they are better known in a more common
notation under a slightly diﬀerent name, to be described presently. Decimal fractions
were understood and used in China by about 400 A.D., but they were transmitted to
Europe as part of the socalled HinduArabic numeral system only around the twelfth
century. In 1593, the German Jesuit priest (and Vatican astronomer) C. Clavius
introduced the idea of writing a decimal fraction without the fraction symbol: just
use the numerator and then keep track of the number of zeros in the denominator (2
in the ﬁrst decimal fraction, 5 in the second, and 4 in the third of the above examples)
by the use of a dot, the socalled decimal point, thus:
14.89, 0.00024, 5.8900, respectively (see [Ginsburg]). The rationale of the notation is clear: the number of
digits to the right of the decimal point, the socalled decimal digits, keeps track of
the power of 10 in the respective denominators, 2 in 14.89, 5 in 0.00024, and 4 in
5.8900. In this notation, these numbers are called ﬁnite or terminating decimals.
In context, we usually omit any mention of “ﬁnite” or “terminating” and just say
24
decimals. Notice the convention that, in order to keep track of the power 5 in 105 , 16 CHAPTER 1. FRACTIONS three zeros are added to the left of 24 to make sure that there are 5 digits to the right
of the decimal point in 0.00024. Note also that the 0 in front of the decimal point is
only for the purpose of clarity, and is optional.
You may be struck by the odd looking number 5.8900, because you are probably
more used to seeing 5.89 and have assumed all along that it is ok to omit the zeros
to the right of the decimal point. Before going any further with this thought, just be
aware that what you have taken for granted is the fact that the following two fractions
are equal:
58900
589
and
4
10
102
This fact is correct, but your job is not ﬁnished until you can prove that it is correct.
We will do that in the next section.
We conclude this section by giving some examples on locating fractions on the
number line. Let us start with 4 , for example. As usual, this is the fourth point to
3
the right of 0 in the sequence of thirds:
0 4
3 1
✻ ✻ ✻ ✻ Question: Can you locate the fraction 20
15 ✻ ? How is it related to 4
3 ? Next we consider the problem of locating a fraction such as 84 , approximately,
17
on the number line, i.e., on the following line, where should 84 be placed, approxi17
mately?
0 1 2 3 4 5 6 7 8 The key idea will turn out to be the use of divisionwithremainder for whole
numbers.6 First, look at the multiples of 17: 0, 17, 34, 51, 68, 85, . . . Thus the 68th
6
“Divisionwithremainder” in school mathematics is what is usually called the “division algorithm” in university algebra texts. There is a good reason why the latter terminology is not used in
school mathematics: it would be too easily confused with the “long division algorithm”. 1.1. DEFINITION OF A FRACTION 17 1
1
multiple of 17 is 4 (because 68 = 4 × 17), and the 85th multiple of 17 is 5 (because
1
85 = 5 × 17). Therefore, 84 lies between 68 (= 4) and 85 (= 5), and is just 17 shy of
17
17
17
1
5, i.e., 84 is the point on the number line which is 17 to the left of 5. In terms of
17
divisionwithremainder, since 84 = (4 × 17) + 16, we have 84
(4 × 17) + 16
=
17
17
1
So if each step we take is of length 17 , going another 16 steps to the right of 4 will
get us to 84 . If we go 17 steps instead, we will get to 5. Therefore 84 should be quite
17
17
near 5, as shown: 0 1 2 3 4 5 6 7 8 ✻
84
17 In general, if m is a fraction and divisionwithremainder gives m = qn + k ,
n
where q and k are whole numbers and 0 ≤ k < n, then
m
qn + k
=
,
n
n
and the position of m on the number line would be between q (=
n
+
(= (q+1)n , which is qnn n ).
n qn
n ) and q + 1 Caution. Because the above reasoning gives 84 as 16 beyond 4, most school
17
17
textbooks would tell you that 84 = 4 16 . The latter is of course an example of a
17
17
m
k
mixed number. Similarly, the above n is supposed to be written as q n . As a teacher,
however, you should exercise selfcontrol not to introduce mixed numbers at this
point, because (for example) the symbol 4 16 is a fraction, and if fraction addition has
17
been deﬁned, we would be able to say that it is the fraction 4 + 16 . Because fraction
17
addition is usually not yet deﬁned when the symbol 4 16 is introduced, confusion is
17
the inevitable result.
Exercises 1.1.
In doing these and subsequent exercises, please observe the following basic rules: 18 CHAPTER 1. FRACTIONS (a) Show your work; the explanation is as important as the answer.
(b) Be clear. Get used to the idea that, as a teacher, everything you say has to be
understood.
1. Indicate the approximate position of each of the following on the number line,
77
and brieﬂy explain why. (a) 186 . (b) 457 . (c) 355 . (d) 14.127.
11
13
2. Suppose the unit 1 on the number line is the area of the following shaded region
obtained from a division of a given square into eight congruent rectangles (and
therefore eight parts of equal area).7
Write down the fraction of that unit representing the shaded area of each of
the following divisions of the same square and give a brief explanation of your
answer. (In the picture on the right, two copies of the same square share a
common side and the square on the right is divided into two parts of equal
area.)
❅
❅
❅
❅
❅
❅
3. With the unit as in problem 2 above, write down the fraction representing the
area of the following shaded region (assume that the top and bottom sides of
the square are each divided into three segments of equal length):
7
We will give a precise deﬁnition of congruence in Chapter 4, and will formally discuss area
in Chapter 18. In this chapter, we only make use of both concepts in the context of triangles
and rectangles, and then only in the most superﬁcial way. For the purpose of understanding this
chapter, you may therefore take both concepts in the intuitive sense. If anything more than intuitive
knowledge is needed, it will be supplied on the spot, e.g., in Section 1.4 of this chapter. 1.1. DEFINITION OF A FRACTION 19
4. A text on professional development claims that students’ conception of “equal
parts” is fragile and is prone to errors. As an example, it says that when a circle
is presented this way to students
✬✩
✑◗
✑◗
✫✪ they have no trouble shading 2 , but when these same students
3
are asked to construct their own picture of
them create pictures with unequal pieces: 2
,
3 we often see ✬✩
✫✪ (a) What kind of faulty mathematical instruction might have promoted this
kind of misunderstanding on the part of students? (b) What would you do to
correct this kind of mistake by students?
5. Ellen ate 1 of a large pizza with a 1foot diameter and Kate ate 1 of a small
3
2
pizza with a 8inch diameter. (Assume that all pizzas have the same thickness
and that the fractions of a pizza are measured in terms of area.) Ellen told
Kate that since she had eaten more pizza than Kate, 1 > 1 . (i) Did Ellen eat
3
2
more pizza than Kate? (ii) Is Ellen’s assertion correct? Why or why not?
6. Take a pair of opposite sides of a unit square and divide each side into 478 equal
parts. Join the corresponding points of division to obtain 478 thin rectangles
(we will assume that these are rectangles). For the remaining pair of opposite
sides, divide each into 2043 equal parts and also join the corresponding points of 20 CHAPTER 1. FRACTIONS
division; these lines are perpendicular to the other 479 lines. The intersections of
these 479 and 2044 lines create 478 × 2043 small rectangles which are congruent
to each other (we will assume that too). What is the area of each such small
rectangle, and why ? (This problem is important for Section 1.4 below.)
7. (Review the above remark (B ) on page 12 on the importance of the unit before
doing this problem. Also make sure that you do it by a careful use of the definition of a fraction rather than by some transcendental intuition you possess
which cannot be explained to your students.)
(a) After driving 148 miles, we have done only twothirds of the driving for the
day. How many miles did we plan to drive for the day? Explain.
(b) After reading 180 pages of a book, I am exactly fourﬁfths of the way
through. How many pages are in the book? Explain.
(c) Alexandra was three quarters of the way to school after having walked 0.78
mile from home. How far is her home from school? Explain.
8. Three segments (thickened) are on the number line, as shown: A
3 137
25 B
4 5 C
6 7 It is known that the length of the left segment is 11 , that of the middle segment
16
8
is 17 , and that of the right segment is 23 . What are the fractions A, B , and
25
C ? (Caution: Remember that you have to explain your answers, and that you
know nothing about “mixed numbers” until we come to this concept in Section
1.3.)
9. The following is found in a certain thirdgrade workbook:
Each of the following figures represents a fraction: 1.2. EQUIVALENT FRACTIONS
21
Point to two figures that have the same fractions shaded.
How would you change this problem to make it suitable for classroom use? 1.2 Equivalent fractions Recall that two fractions are equal if they are the same point on the number line. We
observed in equation (1.1) of Section 1.1 that nk = k , as both are equal to k . The
n
1
following generalizes this simple fact.
m
k
Theorem 1.1. Given two fractions
and
, suppose there is a nonzero whole
n
number c so that
k = cm and = cn.
Then m
k
=
n
Proof. First look at a special case: why is 4 equal to 5×4 ? We have as usual the
3
5× 3
following picture:
0
✻ 1
✻ ✻ ✻ 4
3 ✻ ✻ Now suppose we further divide each of the segments between consecutive points in
the sequence of thirds into 5 equal parts. Then each of the segments [0, 1], [1, 2],
[2, 3], . . . is now divided into 5 × 3 = 15 equal parts and, in an obvious way, we have
obtained the sequence of ﬁfteenths on the number line: 22 CHAPTER 1. FRACTIONS
0
✻ 1
✻ ✻ ✻ 4
3 ✻ ✻ The point 4 , being the 4th point in the sequence of thirds, is now the 20th point in
3
the sequence of ﬁfteenths because 20 = 5 × 4. The latter is by deﬁnition the fraction
20
5× 4
4
5×4
15 , i.e., 5×3 . Thus 3 = 5×3 .
The preceding reasoning is enough to prove the general case. Thus let k = cm
and = cn for whole numbers c, k , , m, and n. We will prove that m = k . In other
n
words, we will prove:
m
cm
=
n
cn for all fractions m and all whole numbers c = 0
n (1.2) The fraction m is the mth point in the sequence of nths. Now divide each of the
n
segments between consecutive points in the sequence of nths into c equal parts. Thus
each of [0, 1], [1, 2], [2, 3], . . . is now divided into cn equal parts. Thus the sequence of
nths together with the new division points become the sequence of cnths. A simple
reasoning shows that the mth point in the sequence of nths must be the cmth
point in the sequence of cnths. This is another way of saying m = cm . The proof
n
cn
is complete.
The content of Theorem 1.1 is generically referred to in school mathematics as
the theorem on equivalent fractions. As mentioned earlier, it is a tradition in
k
m
school mathematics to say that two fraction (symbols) and n are equivalent
if they are equal, i.e., k and m are the same point. Thus Theorem 1.1 gives a
n
suﬃcient condition for two fractions k and m to be equivalent: if we can ﬁnd a whole
n
number c so that k = cm and = cn. For brevity, Theorem 1.1 is usually stated as in
equation (1.2). In this form, Theorem 1.1 is called the cancellation law for fractions:
One “cancels” the c from numerator and denominator. This is the justiﬁcation for
the usual method of reducing fractions, i.e., canceling a common divisor of the
numerator and the denominator of a fraction. Thus, 51 = 3 because
34
2
51
17 × 3
3
=
=
34
17 × 2
2 1.2. EQUIVALENT FRACTIONS 23 The usual role played by Theorem 1.1 in the school mathematics curriculum is a
minimal one: it is strictly used to reduce fractions. This is wrong, because Theorem
1.1 is the fundamental fact about fractions, and the validity of this claim will be amply
veriﬁed in the rest of this chapter. One reason why Theorem 1.1 is fundamental is
that it gives a suﬃcient condition for two fractions m and k to be equal. There is
n
obvious interest in such a suﬃcient condition since each symbol represents a point on
the number line and one would like to be able to decide whether the two points are
the same or not. On the other hand, the condition in Theorem 1.1 is not a necessary
condition, in the sense that the equality m = k does not imply that k = cm and
n
= cn for some whole number c. For example, Theorem 1.1 shows that 3 = 21 (as
2
14
21 = 7 × 3 and 14 = 7 × 2), so that coupled with the preceding remark about 51 , we
34
have
21
51
=
14
34
However, there is clearly no whole number c so that c times 21 yields 51 and that
the same c times 14 yields 34. It turns out that, with a mild twist, Theorem 1.1
can be used to give a necessary and suﬃcient condition for two fractions to be equal.
Precisely:
Theorem 1.2 (CrossMultiplication Algorithm). A necessary and suﬃcient condition for two fractions k and m to be equal is that kn = m.
n
For later needs, we pause to note that there are several diﬀerent but equally valid
ways to state Theorem 1.2. One way is to say that
k
m
=
n if and only if kn = m. Another says
m
k
=
n
A more symbolic way is is equivalent to kn = m. k
m
=
⇐⇒ kn = m
n
No matter how the theorem is stated, all it says is that both of the following statements
are valid: 24 CHAPTER 1. FRACTIONS
First, k
= m
n implies kn = m. Second, kn = m implies k = m .
n
As is wellknown, each is said to be the converse of the other.
Proof of Theorem 1.2. (Part 1) We prove k = m
n implies kn = m. By Theorem 1.1, k = kn and m = m . Because we are assuming k = m , we
n
n
n
n
therefore have
kn
m
=
n
n
What this says is that the knth multiple of 1 is equal to the mth multiple of
n
This is possible only if kn = m. 1
.
n (Part 2) We next prove kn = m implies k = m .
n
The hypothesis implies that
kn
m
=
n
n
By Theorem 1.1, the left side is k while the right side is m . Thus we have k = m .
n
n
The proof of Theorem 1.2 is complete.
The education literature often mistakes Theorem 1.2 for a rotelearning skill without knowing that, once a fraction has been clearly deﬁned and the equality of two
fractions also clearly deﬁned, Theorem 1.2 is something capable of being proved precisely. As a result of this misunderstanding, many students have been taught to
avoid using this theorem or have not been taught this theorem. In this book, we
explicitly ask you to make ample use of this useful result whenever the equality of
two fractions is discussed, and there will be many opportunities for you to do just that.
Two remarks about Theorem 1.2 are relevant. One is that sometimes Theorem
1.2 provides the only easy way to decide if two fractions are equal, e.g., 551 and 203
247
91
are equal because 551 × 91 = 203 × 247. A second remark is that from the vantage
point of abstract algebra, the importance of Theorem 1.2 (and hence of Theorem
1.1) is manifest because the crossmultiplication algorithm is exactly the equivalence
relation between ordered pairs of integers when fractions are deﬁned as equivalence 1.2. EQUIVALENT FRACTIONS 25 classes of such ordered pairs: (a, b) ∼ (a , b ) if and only if ab = a b.
As an application of Theorem 1.1, we bring closure to the discussion in the last
section about the decimal 5.8900. Recall that we had, by deﬁnition,
58900
= 5.8900
104
We now show that 5.8900 = 5.89 and, more generally, one can add zeros to or delete
zeros from the right end of the decimal point without changing the decimal. Indeed,
5.8900 = 58900
589 × 102
589
=2
= 2 = 5.89,
104
10 × 102
10 where the middle equality makes use of the cancellation law (equation (1.2)). The
reasoning is of course valid in general, e.g.,
12.7 = 127
127 × 104
1270000
=
=
= 12.70000
4
10
10 × 10
105 From the proof of Theorem 1.2, we can extract a very useful statement about pairs
of fractions, which we call the Fundamental Fact of FractionPairs (FFFP):
Any two fractions may be symbolically represented as two fractions with
the same denominator.
The reason is simple: if the fractions are m and k , then because of Theorem 1.1, we
n
have
m
m
k
nk
=
and
=
n
n
n
In terms of the new fraction symbols, they now share the denominator n.
We can paraphrase FFFP this way: any two fractions can be put on equal footing,
in the following sense. Given k and m as above. Assume = n. Then k is k copies
n
1
1
of 1 , and m is m copies of n . Since 1 = n , it is diﬃcult to get a sense of k relative to
n
m
. It is like talking about 155 inches and 4 meters, one doesn’t have a sense which
n
is longer until one expresses both measurements in terms of the same unit, e.g., an
inch. Then since 1 inch is 2.54 cm, 4 meters is 400 ÷ 2.54 = 157.48 . . . inches. This
is how we can tell 4 meters is longer. In the same way, FFFP allows us to think of 26 CHAPTER 1. FRACTIONS k
and m as nk and m , respectively. Therefore they become, respectively, nk copies
n
n
n
and m copies of the same 1 . Now we can make sense of these two fractions by
n
comparing nk and m. For example, given 2 and 4 , we may replace them with 14
3
7
21
and 12 , respectively.
21
In some special cases, such as 3 and 9 , the fractions can be put on equal footing
2
8
without having to multiply their denominators because 3 = 12 .
2
8
There will be numerous applications of FFFP in subsequent discussions.
There is at present little awareness of the power of Theorem 1.1 on equivalent
fractions the school mathematics literature, and for that reason, this theorem has
been insuﬃciently exploited. We give one illustration of this power. First, we have to
give a precise meaning to a common expression, “twothirds of something”, or more
generally, “ m of something”. What is meant by, for example, “I ate twothirds of
n
a pie”? The truth is that we say this without thinking, but if pressed, we would
probably agree that this means if we look at the pie as a circular disk and ignore its
depth, and we cut it into 3 parts of equal area, then I ate 2 parts. Another example:
what is meant by “he gave threeﬁfths of a bag of rice to his roommate”? Most likely,
he measured his bag of rice by weight and, after dividing the bag of rice into 5 equal
parts by weight, he gave away 3 parts. In each case, the choice of the unit (area in the
ﬁrst and weight in the second) is implicit and depends on the reader’s common sense.
While the choice in each of these two cases is not controversial, one can imagine that
such good fortune may not hold out in general. Consider the statement: “I put away
three quarters of the ham”. This leaves a lot of room for diﬀerent interpretations.
I could have cut the ham, lengthwise, into four parts of equal length. Or I could
have cut it into four parts of equal weight. I could even have managed to cut it into
four parts so that each part had (approximately) the same volume of meat. There is
an even more pertinent example from real life.8 A man obtained a construction loan
from a bank for his house, and it stipulated that he should be at such a percentage
of completion of the project at a certain point. When that time came, the bank said
he had not met his obligation. Whereupon, he wrote to the bank: “Percentage of
completion by what measure?” He explained that if by, say, volume of materials used,
then the bank might have been correct, but if by sheer brute labor, then he was way
8 As related to me by my friend David Collins. 1.2. EQUIVALENT FRACTIONS 27 ahead of schedule. The bank was ﬂummoxed by his response.
These examples illustrate the fact that statements about “a fraction of something”
could be ambiguous and, for the purpose of doing mathematics, the choice of the unit
of measurement must be made explicit at the outset. With this in mind, we proceed
to give a formal deﬁnition of “a fraction of something”. If we ﬁx a unit of measurement, then we will use the informal language of quantity, understood to be relative
to the unit, to mean a number on the number line where the number 1 is the given unit.
Deﬁnition. Suppose a unit of measurement has been chosen. Then k of a
quantity means the totality (relative to this unit) of k parts when that quantity is
partitioned into equal parts according to this unit.
The simplest quantity in the present context is that of the length of a segment.
In this case, the unit of measurement will always be understood to be the length.
Consider for example the case of
1
3 of 24
7 This is then the length of 1 part when the segment [0, 24 ] is divided into 3 parts of
7
24
1
equal length. Now, 7 is 24 copies of 7 , and since 24 = 3 × 8, clearly [0, 24 ] can be
7
divided into 3 equal parts so that each part is the concatenation of 8 copies of 1 .
7
Thus 1 of 24 is 8 . The key point here is that the numerator of 24 is divisible by 3.
7
3
7
7
Next, suppose we want
2
5 of 8
7 Now we have to divide [0, 8 ] into 5 equal parts and then measure the length of 2 of
7
those parts. But ﬁrst things ﬁrst: we have to divide 8 into 5 equal parts. Noting that
7
8 is not divisible by 5, we make use of equivalent fractions to force the numerator of
8
to be divisible by 5 by writing 8 = 5×8 . The numerator 5 × 8 is now divisible by
7
7
5×7
5, and so by retracing the preceding steps, we conclude that if [0, 8 ] is divided into
7
8
1
1
5 equal parts, each part would be 8 copies of 5×7 = 35 , i.e., each part has length 35 .
×
Two of these parts then have length 2358 = 16 . Thus, 2 of 8 is 16 .
35
5
7
35
Pictorially, what we did was to subdivide the segments between consecutive points
of the sequence of sevenths, as shown, 28 CHAPTER 1. FRACTIONS
0 8
7 1 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻
✻ into 5 equal parts:
0 8
7 1 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻
✻ The unit segment is now divided into 5 × 7 = 35 equal parts, so that the new division
points furnish the sequence of 35ths. The segment [0, 8 ] is now divided into 40 equal
7
parts by this sequence of 35ths. Taking every 8th division point (in this sequence of
35ths) then gives a division of [0, 8 ] into 5 equal parts. So the length of a part in the
7
8
latter division is 35 . (Of course, what we have done is merely to reprove the theorem
on equivalent fractions in this particular case of 8 = 5×8 .)
7
5×7
This way of exploiting equivalent fractions will be seen to clarify many aspects
of fractions, such as the interpretation of a fraction as division, or the concept of
multiplication; see below. It also allows us to solve word problems of the following
type.
Example. Kate walked 2 of the distance from home to school, and there was
5
still 4 of a mile to go. How far is her home to school?
9
We can draw the distance from home to school on the number line, with 0 being
home, the unit 1 being a mile, and S being the distance of the school from home.
Then it is given that, when the segment from 0 to S is partitioned into 5 equal parts,
Kate was at the second division point after 0:
0 Kate
S
4
9
mi 1.2. EQUIVALENT FRACTIONS 29 If we can ﬁnd the length of one of these ﬁve segments, which for convenience will be
called the short segments for the moment, then the total distance from home to
school would be 5 times that length. We are given that the distance from where Kate
stands to S is 4 of a mile, and this distance comprises 3 short segments. If we can
9
ﬁnd out how long a third of 4 of a mile is, then we would know the length of a short
9
segment and the problem would be solved. By the theorem on equivalent fractions,
4
3×4
3×4
=
=
,
9
3×9
27
1
and this exhibits 4 as (3 × 4) copies of 27 . Therefore 4
9
length of one third of 4 . The total distance from 0 to S
9
20
which is 27 . The distance from Kate’s home to school is 1
4
copies of 27 (i.e., 27 ) is the
1
is thus (5 × 4) copies of 27 ,
20
27 miles. Remark. This is one of the standard problems on fractions which is usually given
after the multiplication of fractions has been introduced and the solution method is
given out as an algorithm (“ﬂip over (1 − 2 ) to multiply 4 ”). We can see that there
5
9
is in fact no need to use multiplication of fractions for the solution, and moreover, no
need to memorize any solution template. The present method of solution makes the
reasoning very clear.
Using the same reasoning, we now give a completely diﬀerent interpretation of a
fraction. We will prove:
m
= the length of one part when a segment of
n
length m is partitioned into n equal parts
1
Recall that the original deﬁnition of m is m copies of n , which means to locate
n
m
, it suﬃces to consider the unit segment [0, 1], divide it into n equal parts and
n
concatenate m of these parts. The above statement, on the contrary, says that to
locate m , one should divide, not [0, 1], but [0, m] into n equal parts and take the ﬁrst
n
division point to the right of 0. So the two are quite diﬀerent statements.
The proof is simplicity itself. To divide [0, m] into n equal parts, we express
m = m as
1
nm
n 30 CHAPTER 1. FRACTIONS 1
That is, [0, m] is nothing but nm copies of n . So one part out of the n equal parts
1
into which [0, m] has been divided is just m copies of n , i.e., m .
n The full signiﬁcance of this assertion will emerge only after we reexamine the
meaning of division among whole numbers. This we proceed to do.
Let m, n be whole numbers and let m be a multiple of n, let us say m = kn
for some whole number k . Then m ÷ n is the number of objects in a group when
m objects are partitioned into n equal groups; clearly, there are k objects in each
group, and therefore, m ÷ n = k . In other words, m ÷ n is the length of one part
when a segment of length m is partitioned into n equal parts. (This is the socalled
partitive meaning of division.) This deﬁnition of division requires that we assume
m is a multiple of n at the outset, because we are doing division among whole numbers
and must make sure that m ÷ n comes out to be a whole number (i.e., k ). Now if
we are allowed to use fractions, then m would no longer need to be a multiple of n,
and the preceding deﬁnition of m ÷ n (i.e., the length of one part when a segment of
length m is partitioned into n equal parts) makes sense for arbitrary whole numbers
m and n.9 With this in mind, we now deﬁne for two arbitrary whole numbers m and
n, the general concept of the division m ÷ n of two whole numbers m and n:
m ÷ n is the length of one part when a segment
of length m is partitioned10 into n equal parts.
The previous assertion on m
n can now be rephrased as theorem. Theorem 1.3. For any two whole numbers m and n, n = 0,
m
= m÷n
n
This is called the division interpretation of a fraction. In school texts and
education writing, the meaning of m ÷ n when m is not a multiple of n is not made
explicit and, worse, the equality m = m ÷ n is taken to be another meaning of a
n
fraction that students must memorize without beneﬁt of explanation. When you
9 In fact, m could even be a fraction.
To avoid the possibly confusing appearance of the word “divide” at this juncture, we have
intentionally used “partition” instead.
10 1.2. EQUIVALENT FRACTIONS 31 teach, please remember that there needs to be a clear deﬁnition for m ÷ n, and that
the statement about the equality of the two numbers m = m ÷ n is a theorem, i.e.,
n
something that can be explained logically.
As a result of the division interpretation of a fraction, we will retire the
division symbol “ ÷” from now on and use fractions to stand for the division among whole numbers. As a ﬁnal remark on the concept of “ k of m ”, we prove a theorem that will be
n
useful for the consideration of fraction multiplication in Section 1.4. As motivation,
recall the fact proven above that 21 = 51 . If m is a fraction, then it would stand to
14
34
n
reason that
21
14 of m = 51 of m
n
34
n But this fact clearly needs a proof because, according to the deﬁnition of “ k of m ”,
n
m
this says the concatenation of 21 parts when [0, n ] is divided into 14 equal parts has
the same length as the concatenation of 51 parts when the same [0, m ] is divided into
n
34 equal parts. This is certainly not obvious. Moreover, we would also expect that if
m
= M , then also
n
N
21
14 of m = 51 of M
n
34
N This too needs a proof. All this is taken care of by the following theorem.
Theorem 1.4. If k = K and m = M . Then,
L
n
N
k
of m = K of M
n
L
N Proof. We ﬁrst prove that
k
of m
km
=
n
n (1.3) Because m = m , we see that [0, m ] is m copies of 1 . Therefore if we divide
n
n
n
n
m
[0, n ] into equal parts, each part will be m copies of 1 . Therefore if we concatenate
n 32 CHAPTER 1. FRACTIONS k of these parts, we get km copies of 1 , i.e., we get km . By the deﬁnition of k of
n
n
m
n, we have proved that k of m = km .
n
n
In like manner, we have
K
L of M = KM
N
LN Hence, to prove the theorem, we must prove km = KM . According to Theorem
n
LN
1.2 (crossmultiplication algorithm), this would be the case if we can prove kmLN =
nKM . In other words, we have to prove:
(kL)(mN ) = (K )(nM )
By the assumption that k = K and by Theorem 1.2, we have kL = K . Similarly,
L
by the assumption that m = M and by Theorem 1.2, we also have mN = nM .
n
N
Therefore (kL)(mN ) = (K )(nM ), as claimed. The proof of Theorem 1.4 is complete.
Exercises 1.2.
[Reminder] In doing these and subsequent exercises, please observe the following basic
rules:
(a) Show your work; the explanation is as important as the answer.
(b) Be clear. Get used to the idea that, as a teacher, everything you say has to be
understood.
1. Reduce the following fractions to lowest terms,11 i.e., until the numerator and
denominator have no common divisor > 1. (You may use a fourfunction calculator to test the divisibility of the given numbers by various whole numbers.)
42
,
91 52
,
195 204
,
85 414
,
529 1197
.
1273 2. Explain each of the following to an eighthgrader, directly and without using
Theorem 1.1 or Theorem 1.2, by drawing pictures using the number line :
6
3
=,
14
7 28
7
=,
24
6 and 12
4
=.
27
9 11
Theorem 3.1 on page 142 of Chapter 3 will show that every fraction can be reduced to a unique
fraction in lowest terms. 1.2. EQUIVALENT FRACTIONS 33 3. School textbooks usually present the cancellation law for fractions as follows.
Given a fraction m . Suppose a nonzero whole number k divides both
n
÷k
m and n. Then m = m÷k .
n
n Explain to a seventh grader why this is true.
4. The following points on the number line have the property that the thickened
segments [A, 1], [B, 2.7], [3, C ], [D, 4], [ 13 , E ], all have the same length:
3
A
0 B
1 2 CD
✻ 3 E
✻ 4 2. 7 5 13
3 If A = 4 , what are the values of B , C , D, E ? Be careful with your explana7
tions: we don’t know how to add or subtract fractions yet. (Rest assured that
on the basis of what has been discussed in this section, you can do this problem.)
8
is 11 of which number? (b) I was on a hiking trail, and after walking
7
of a mile, I was 5 of the way to the end. How long is the trail? (c) After
10
9
driving 18.5 miles, I am exactly threeﬁfths of the way to my destination. How
far away is my destination? 5. (a) 7
3 6. Explain to a sixth grade student how to do the following problem: Nine students
chip in to buy a 50pound sack of rice. They are to share the rice equally by
weight. How many pounds should each person get? (If you just say, “divide
50 by 9”, that won’t be good enough. You must explain what is meant by “50
divided by 9”, and why the answer is 5 5 .)
9
7. (a) A wire 314 feet long is only fourﬁfths of the length between two posts. How
far apart are the posts? (b) Helena was three quarters of the way to school
after having walked 8 miles from home. How far is her home from school?
9
8. (a) 3 of a fraction is equal to 5 . What is this fraction? (b)
7
6
equal to k . What is this fraction?
m
n of a fraction is 34 CHAPTER 1. FRACTIONS
9. James gave a riddle to his friends: “I was on a hiking trail, and after walking
7
of a mile, I was 5 of the way to the end. How long is the trail?” Help his
12
9
friends solve the riddle. 10. Prove that the following three statements are equivalent for any four whole
numbers a, b, c, and d, with b = 0 and d = 0:
c
(a) a = d .
b a
c
(b) a+b = c+d . (c) a+b = c+d .
b
d
(One way is to prove that (a) implies (b) and (b) implies (a). Then prove (a)
implies (c) and (c) implies (a).)
11. Place the three fractions 13 , 11 , and 9 on the number line and explain how
6
5
4
they get to where they are.
+1
12. (a) For which fraction m is it true that m = m+1 ? (b) For which fraction
n
n
n
m
m
m+b
n is it true that n = n+b , where b is a positive whole number? 13. Prove that between any two fractions A and C , there is a fraction B , i.e.,
A < B < C. ...
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 Math, Fractions, Rational number

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