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Teaching School Mathematics: AlgebraTeaching School Mathematics: Algebra

Hung-Hsi Wu

Teaching School Mathematics: Algebra

https://doi.org/10.1090//mbk/099

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

Teaching School Mathematics: Algebra

Hung-Hsi Wu

Department of MathematicsUniversity of California, Berkeley

P r o v i d e n c e , R h o d e I s l a n d

2010 Mathematics Subject Classification. Primary 97-01, 00-01, 97H20, 97G70,97H30, 97F80.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-99

Library of Congress Cataloging-in-Publication Data

Names: Wu, Hongxi, 1940-Title: Teaching school mathematics. Algebra / Hung-Hsi Wu.Description: Providence, Rhode Island : American Mathematical Society, 2016. | Audience:

Grades 6 to 8.- | Includes bibliographical references.Identifiers: LCCN 2016000118 | ISBN 9781470427214 (alk. paper)Subjects: LCSH: Algebra–Textbooks. | Algebra–Study and teaching (Elementary) | Algebra–

Study and teaching (Middle school) | AMS: Mathematics education – Instructional exposition(textbooks, tutorial papers, etc.). msc | General – Instructional exposition (textbooks, tutorialpapers, etc.). msc | Mathematics education –Algebra –Elementary algebra. msc | Mathe-matics education –Geometry –Analytic geometry. Vector algebra. msc | Mathematics educa-tion –Algebra –Equations and inequalities. msc | Mathematics education –Arithmetic, numbertheory –Ratio and proportion, percentages. msc

Classification: LCC QA159 .W8 2016 | DDC 512.9071/2–dc23 LC record available at http://lccn.loc.gov/2016000118

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10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16

To Kuniko

Wir sind durch Not und Freudegegangen Hand in Hand;vom Wandern ruhen wir beidenun uberm stillen Land.

Im AbendrotJoseph von Eichendorff (1788–1857)

Contents

Chapters in the Companion Volume ix

Preface xi

Suggestions on How to Read This Volume xix

Chapter 1. Symbolic Expressions 11.1. Basic protocol in the use of symbols 21.2. Expressions and identities 51.3. Mersenne primes and finite geometric series 111.4. Polynomials and order of operations 171.5. Rational expressions 24

Chapter 2. Translation of Verbal Information into Symbols 272.1. Equations and inequalities 272.2. Some examples of translation 30

Chapter 3. Linear Equations in One Variable 373.1. Solving linear equations 373.2. Some word problems 45

Chapter 4. Linear Equations in Two Variables and Their Graphs 534.1. Coordinate system in the plane 544.2. Linear equations in two variables 574.3. The concept of slope 614.4. Proof that graphs of linear equations are lines 724.5. Every line is the graph of a linear equation 764.6. Useful facts and examples 78

Chapter 5. Simultaneous Linear Equations 855.1. Solutions of linear systems and the geometric interpretation 855.2. The algebraic method of solution 875.3. Characterization of parallel lines by slope 935.4. Algebraic criterion for solvability 985.5. Partial fractions and Pythagorean triples 1015.6. Appendix 109

Chapter 6. Functions and Their Graphs 1176.1. The basic definitions 1176.2. Why functions? 1226.3. Some examples of graphs 1266.4. Remarks on graphs and coordinate systems 133

vii

viii CONTENTS

Chapter 7. Linear Functions and Proportional Reasoning 1377.1. Constant rate and linear functions 1377.2. Proportional reasoning 144

Chapter 8. Linear Inequalities and Their Graphs 1558.1. How do inequalities arise in real life? 1558.2. The symbolic translation 1578.3. Basic facts about inequalities and applications 1608.4. Graphs of inequalities in the plane 1638.5. Solution of the manufacturing problem 1808.6. Behavior of linear functions in the plane 187

Chapter 9. Exponents 1919.1. Positive-integer exponents 1949.2. Rational exponents 2009.3. Laws of exponents 2059.4. Scientific notation 2149.5. Three additional remarks on rational exponents 220

Chapter 10. Quadratic Functions and Their Graphs 22310.1. Quadratic equations 22410.2. A special class of quadratic functions 23810.3. Properties of quadratic functions 24610.4. The graph and the parabola 25210.5. Some applications 260

Appendix: Facts from [Wu-PreAlg] 265

Bibliography 273

Chapters in the Companion Volume

Teaching School Mathematics: Pre-Algebra ([Wu-PreAlg])

Chapter 1: FractionsChapter 2: Rational NumbersChapter 3: The Euclidean AlgorithmChapter 4: Experimental GeometryChapter 5: Length, Area, and Volume

ix

x CHAPTERS IN THE COMPANION VOLUME

Structure of the chapters in [Wu-PreAlg] ([PA]) and this volume ([A])

[A]Chapter 10

[A]Chapter 7

[A]Chapter 5

[A]Chapter 6

[A]Chapter 4

[A]Chapter 3

[A]Chapter 2

[A]Chapter 1

[PA]Chapter 4

[PA]Chapter 2

[PA]Chapter 1

[PA]Chapter 3

[PA]Chapter 5

[A]Chapter 9 [A]Chapter 8

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Preface

A main obstacle in the learning ofschool mathematics has always

been how to cope with the steadyincrease in abstraction with

the passage of each school year.

This volume and its companion volume—Teaching School Mathematics: Pre-Algebra ([Wu-PreAlg])—are textbooks written for teachers, especially middleschool teachers. They address the mathematics that is generally taught in grades6–8. In this volume, we give a presentation of school algebra as a direct continuationof arithmetic—whole numbers, fractions, decimals, and negative numbers—and wealso assume a basic acquaintance with the geometry of congruence and similarity.For this reason, we must draw on the readers’ knowledge of these topics. In theAppendix (pages 265 ff.), one can find a brief summary of most of the relevantfacts from [Wu-PreAlg] that we need.

The topics to be taken up in this volume are those to be found in any mid-dle school or high school course on Algebra I: linear equations in one and twovariables, linear inequalities in one and two variables, simultaneous linear equa-tions, the concept of a function, polynomial functions and exponential functions,and a detailed study of linear and quadratic functions. These topics are entirelyunexceptional. Such being the case, one may well ask why this volume deservedto be written. In general terms, an answer to this question has been given in thePreface to [Wu-PreAlg]. What follows is a more focused answer in the context ofthe teaching and learning of introductory school algebra.

At the moment, Algebra for All is a national goal (see Chapter 3 of [NMP]), andthere are various theories as to why this goal seems to be out of reach. Could it bethat the appropriate classroom manipulatives have not been sufficiently exploited,that the latest advances in technology have not yet been fully integrated into theinstruction, or that the teaching has slighted so-called sense-making, conceptualunderstanding, and higher-order thinking skills? Perhaps. All these questions,however, ignore a fundamental issue: there is ample evidence that students can-not learn algebra, not because they don’t like the packaging of the product, butbecause they find the product itself to be incomprehensible. We will refer tothis product—the mathematics in almost all the standard school textbooks of thepast four decades—as Textbook School Mathematics ([TSM]).1 TSM fails, oftenin spectacular fashion, to explain to students, clearly and correctly, what they are

1See, for example, [Wu2013] or [Wu2015] for more details.

xi

xii PREFACE

supposed to learn. Education researchers who look into the nonlearning of al-gebra do not appear to have given much thought to the fact that the TSM thatresides in student textbooks or standard professional development materials isriddled with ambiguities and errors, big and small. In short, TSM is not learn-able. Until a mathematically correct version of school algebra is readily accessibleto one and all, it will be premature to draw any conclusions about why studentscannot learn algebra. With this in mind, the main justification for this volume’sexistence is that it gives a logical and coherent exposition of the standard math-ematical topics in Algebra I in a way that not only is grade-level appropriate foreighth and ninth graders, but also meets the requirements of the following fivefundamental principles of mathematics:

(I) Precise definitions are essential.(II) Every statement must be supported by mathematical reasoning.(III) Mathematical statements are precise.(IV) Mathematics is coherent.(V) Mathematics is purposeful.

We will refer the readers to the Preface of [Wu-PreAlg] for a fuller discussions ofthese fundamental principles.

The grade-level requirements we have imposed on this volume by no meansimply that this is a student textbook. This volume is unequivocally a book forteachers with a sharp focus on mathematics. What this requirement means isthat a conscientious attempt has been made to minimize the distance between thecontent in this volume and what teachers have to teach in middle school (see, forexample, [Wu2006]). Consequently, this volume will not touch on any advancedtopics such as vector spaces and linear transformations, groups, rings, fields, andespecially finite fields. It turns out that the need for such advanced considerationsis not critical at this stage and, in any case, there will be no advanced topics to befound in this volume. Instead, we will focus on probing the basic structure thatundergirds the standard topics of school algebra. In the course of this probe, how-ever, the need for advanced—and often quite subtle—considerations does surfacefrom time to time. On these occasions, we will not shy away from giving thefull explanation in order to bring mathematical closure to the discussion. All thesame, we will also be explicit in pointing out that these advanced considerationsare more for broadening the teachers’ knowledge base than for school classroompresentations.

The fundamental principles of mathematics are of critical importance in theteaching of school algebra because algebra is inherently an abstract subject com-pared to arithmetic, and TSM’s lack of precise definitions and logical reasoning inan abstract environment has rendered the subject unlearnable. In greater detail,let us consider the following specific manifestations of these flaws in the algebraportion of TSM:

1. TSM considers the concept of a “variable” to be basic in school algebra.For example:

Understanding the concept of variable is crucial to the study ofalgebra; a major problem in students’ efforts to understand anddo algebra results from their narrow interpretation of the term.([NCTM], page 102)

PREFACE xiii

Many in the education establishment may be surprised to learn that “variable” isnot a mathematically well-defined concept and is only used informally in mathe-matical discussions in order to remove excessive verbiage.2 One should not ex-pend scarce instructional time trying to teach a phantom concept, much less makeit the cornerstone of algebra learning. When textbooks follow suit and elaborateon a “variable” as a quantity that changes or varies, they block beginners at thegate of the gate-keeper course that is algebra.

2. Once the concept of “variable” has taken root, an equation will naturally bedefined in terms of a “variable”. Here is a typical example:

A variable is a symbol used to represent one or more num-bers. A variable expression is an expression that contains a vari-able. . . . An equation is a statement formed by placing an equalsign between two numerical or variable expressions. ([Dolciani],pages 724 and 731)

This then raises the question of what it means for two variable expressions to beequal: if a variable can represent more than one number, does the equality oftwo variable expressions mean the expressions are equal for all the numbers sorepresented? If so, isn’t that an identity? If not, then for which numbers are theyequal?

When basic questions like these cannot be answered, it is a foregone conclu-sion that the fundamental process of solving an equation, in the way it is taught inschool algebra, becomes a faith-based ritual divorced from mathematical reason-ing (see the discussion on pages 37 ff.).

3. TSM introduces students to the concept of the slope of a nonvertical linestrictly as a rote skill: fix two chosen points on the line and compute their “riseover run”. There is no mention of the fact that, if two other points are chosen,the resulting “rise over run” will still be the same. Some students even ignore the“rise over run” and simply expect every line to come equipped with an equationy = mx + b so that they can conveniently identify the slope of the line with theconstant “m”. Recently, the scope of the misconception about slope has been cap-tured quantitatively in [Postelnicu-Greenes], but the education research literaturestill seems oblivious to the fundamental mathematical error in TSM’s definition ofslope and the glaring absence of reasoning surrounding this concept. Educationresearch also appears to be unaware that, until this error is honestly confronted,it will be premature—not to say futile—to talk about students’ “conceptual un-derstanding” of slope.

4. A natural consequence of not having a correct definition of slope is theabsence of any explanations for the interplay between a linear equation in twovariables and its graph. For example, why is the graph of a linear equation intwo variables a straight line? And is every straight line necessarily the graphof some linear equation in two variables? TSM’s answer to the first question isthat when several points in the graph of the linear equation are plotted, “theylook straight”. Reasoning plays no role. Consequently, students can only learnhow to find the equation of a line satisfying certain geometric conditions (e.g.,passing through two given points, passing through a given point with a givenslope, etc.) as a rote skill. Since linear equations constitute a major part of thefirst half of Algebra I, this means that students’ first encounter with algebra will

2We have already done so above by referring to “linear equations of one and two variables”, etc.

xiv PREFACE

consist mainly of a deeper immersion in learning-by-rote. After years of bruisingbattles with fraction-as-a-piece-of-pizza, students become convinced by such anencounter that math is unlearnable except by brute force memorization.

5. The theorem that two lines being parallel is equivalent to the lines havingthe same slope is routinely offered in textbooks as a definition or as a key concept ofparallel lines. Likewise, the theorem that two lines being perpendicular is equiv-alent to the product of the slopes of the lines being equal to −1 is often given asa seemingly sophisticated definition of perpendicularity. Because students are al-ready familiar with the concepts of parallel and perpendicular lines from earliergrades, they are confused by this spectacular about-face. Does a mathematicalconcept have any permanence, or is it liable to change with each grade? Thelikely conclusion from such confusion is that algebra doesn’t make sense. This is onereason that the current discussion about “sense-making” in mathematics learninghas no real traction: until we have a curriculum that makes sense, we cannot askstudents to make sense of the mathematics.

6. In elementary and middle school, students have already used the conceptof constant rate (e.g., constant speed ) extensively, but there is no precise definition ofthis concept in TSM. What there is in TSM is an abstruse discussion of a conceptcalled proportional reasoning; the implicit assumption is that if students have aconceptual understanding of proportional reasoning, they will be able to handleconstant rate. An introductory algebra course is the first opportunity to bringclarity and closure to “constant rate” by pointing out what it means and why itcorresponds to the linearity of an appropriate function. Yet this is hardly everdone. This is a prime example of the fractured school curriculum: the intrinsiccoherence between the mathematics of grades 5–7 and the foundations of algebrais too often missing.

7. The concept of the graph of an equation is not precisely defined in TSM, andconsequently not emphasized. It follows that simple facts about graphs such asthe solution of simultaneous linear equations being the coordinates of the pointof intersection of the two linear graphs become articles of faith rather than simplelogical consequences of the definitions. Students do not learn mathematics if allthey do is memorize facts on faith alone. Not surprisingly, some students do losefaith, which then makes any kind of learning—by rote or otherwise—impossible.

8. In TSM, the graph of a linear inequality of two variables is almost never defined,and the concept of a half-plane is also left undefined. Consequently, the theoremthat the graph of a linear inequality is a half-plane becomes either a decree or adefinition, and it is impossible to decide which it is. In asking students to learnabout linear inequalities and linear programming, we are in effect asking them(once again) to wade through, and memorize by rote, a morass of disconnectedshadowy statements while making believe that we are teaching mathematics. Un-der these circumstances, how can any mathematics learning take place?

9. The concept of a rational 3 exponent of a positive number is a source ofimmense confusion. TSM makes believe that, for any positive number a, a0 = 1

is a theorem rather than a definition, and the same goes for a−n =1an (for any

positive integer n). Moreover, TSM does not explain that the reason we want a

3We are using the term of “rational numbers” in its correct mathematical sense: fractions andnegative fractions.

PREFACE xv

definition of ar for all rational numbers r is that these are special values of theexponential function x �→ ax when x is an arbitrary number. As a consequence,the laws of exponents become just another set of senseless rote skills about astrange notation rather than remarkable properties of the exponential function.

10. TSM’s presentation of quadratic equations and functions is chaotic: toomany facts to memorize while no conceptual framework is provided for their un-derstanding. For example, students learn how to factor quadratic polynomialswith leading coefficients other than 1, learn the quadratic formula, learn the for-mula for the axis of symmetry of the graph, learn the formula for the vertex ofthe graph of a quadratic function, etc. How are these related to each other?

If one goes through the algebra curriculum of TSM carefully, one will uncoverthese and many more serious mathematical issues. (Many of them will be pointedout in this volume in due course.) The prospect of a student learning algebra istherefore daunting: it may be likened to walking through a minefield where allthe mines were put there by human errors. The least we can do is to removethe mines (and some of students’ concomitant fears)—in other words, eradicateTSM—in order to give learning a chance. The modest goal of this volume is togive you the tools to do exactly that. Briefly, one will find in the following pagesways to deal with the preceding difficulties:

1a. What students should be learning is not what a “variable” is but theproper use of symbols; see pages 4 ff. The meaning of each symbol must bespecified before it is put to use. For example, the equality of two functions of onevariable, f (x) and g(x), may be a prototypical statement involving variables, butthe precise definition of the equality f = g is that, for each fixed number x in theircommon domain of definition, f (x) = g(x). Nothing varies.

2a. The solving of equations is strictly a matter of computations with numbers.No variables are involved, and therefore there is no reason to confuse the issueby using balance scales or algebra tiles to explain the solution process. See thediscussion in Section 3.1 on page 37.

3a. The concept of slope needs to be defined with far greater care than TSMhas let on. One has to explain what “slope” tries to measure, how to measure it,and, most importantly, why this way of measuring it is correct and useful. In Section4.3 on page 61, there is an extended discussion to this effect. In particular, this iswhere the discussion of congruent triangles and similar triangles in Chapter 4 of[Wu-PreAlg] becomes absolutely essential.

4a. In Sections 4.4 and 4.5 on pages 72 and 76, we will give a careful proofof why the graph of a linear equation of two variables is a line and why eachline is the graph of some linear equation of two variables. In the process, it willbecome obvious how to write down the equation of a line that satisfies any of thestandard geometric conditions. See Section 4.6 on page 78.

5a. Because perpendicularity and parallelism have been defined in Chapter 4of [Wu-PreAlg], and because slope has been defined in Section 4.3 on page 61, anyassertion about parallelism (or perpendicularity) and slope becomes a theorem tobe proved. We will do exactly that in Sections 5.3 and 5.6 on pages 93 and 109,respectively.

xvi PREFACE

6a. In Section 7.1 on page 137, we review the definition of constant rate, andthen prove that constant rate is equivalent to the existence of an appropriate linearfunction that represents work done over time. In Section 7.2, we closely examinethe possible meanings of proportional reasoning and point out how—by eliminatingit altogether—its purported applications in school mathematics can all be put ona firm mathematical foundation.

7a. In Section 5.1, we explain precisely why the solutions to a pair of equa-tions are the set of all the points of intersection of the graphs of the two equationsin question. Such an explanation is possible only because the graph of an equationhas been precisely defined and put to use in reasoning.

8a. In Section 8.4, we define the half-planes of a line and the graph of a linearinequality. Then in Theorem 8.4 on page 172, we prove that the graph of a linearinequality is a half-plane of the graph of the associated linear equation.

9a. Section 9.2 re-orients the discussion of rational exponents by assumingthe existence of exponential functions from the beginning. (This is analogous tothe discussion of solving polynomial equations by assuming—at the outset—theFundamental Theorem of Algebra. In school mathematics, sometimes a centraltheorem has to be taken on faith for pedagogical reasons.) Then we make useof the characteristic property of the exponential functions (i.e., ax · ay = ax+y)to prove that a0 = 1 and a−x = 1/ax. This makes it possible for the followingsection (Section 9.3) to present complete proofs of the other laws of exponents forrational exponents.

10a. Chapter 10 begins with a general discussion of the shape of the graph ofa quadratic function and then shows how the graph can provide a framework forthe understanding of quadratic functions in the same way that straight lines pro-vide a framework for the understanding of linear functions. The basic techniquehere is that of completing the square; it will be seen that this technique unifies thediverse skills related to quadratic functions.

It can be persuasively argued that any form of professional development formiddle school teachers that makes any claim to legitimacy must make the neededcorrections of these flagrant errors in TSM. The content of this volume—in its var-ious incarnations—has been used for both inservice and preservice professionaldevelopment since 2006. Nevertheless, I have come to realize that, as of the year2015, this offering comes with some liabilities. While it provides an opportunityfor teachers to learn correct school mathematics, perhaps for the first time, it alsoobligates them to put in a tremendous amount of work in order to teach this ma-terial in the school classroom. In addition, the amount of steely resolve that isneeded to teach it without the support of a compatible student textbook and aschool’s or a district’s pacing guide may well be beyond the normal call of duty.To give a somewhat extreme example, if a teacher teaches slope more or less ac-cording to Section 4.3 on page 61 (see 3a above), then inevitably he or she willhave to steal many hours from other topics in order to introduce students to thebasic facts about similar triangles.

The advent of the CCSSM ([CCSSM]) should mitigate some of the difficultiesteachers have in teaching correct algebra. If they wish to implement the content ofthis volume in their own classrooms, they can do so now with the assurance that,in the Common Core era, much of what used to be outlandish in this volume is

PREFACE xvii

now becoming the accepted norm. I can only hope that, in the forthcoming years,better student textbooks will be written so that the CCSSM will finally bring aboutbetter student learning in school algebra.

Acknowledgements. This volume and its companion volume [Wu-PreAlg]evolved from the lecture notes ([Wu2010a] and [Wu2010b]) for the Pre-Algebraand Algebra summer institutes that I used to teach to middle school mathemat-ics teachers from 2004 to 2013. My ideas on professional development for K–12mathematics teachers were derived from two sources: my understanding as aprofessional mathematician of the minimum requirements of mathematics (see thefive fundamental principles of mathematics in the Preface of [Wu-PreAlg]) andthe blatant corrosive effects of TSM on the teaching and learning of mathematics.Those summer institutes therefore placed a special emphasis on improving teach-ers’ content knowledge. I would not have had the opportunity to try out theseideas on teachers but for the generous financial support from 2004 to 2006 by theLos Angeles County Office of Education (LACOE), and from 2007 to 2013 by theS. D. Bechtel, Jr. Foundation. Because of the difficulty I have had with funding bygovernment agencies—they did not (and perhaps still do not) consider the kind ofcontent-based professional development I insist on to be worthy of support—mydebt to Henry Mothner and Tim Murphy of LACOE and Stephen D. Bechtel, Jr. isenormous.

Through the years, I have benefited from the help of many dedicated teachers;to Bob LeBoeuf, Monique Maynard, Marlene Wilson, and Betty Zamudio, I owethe corrections of a large number of linguistic infelicities and typos, among otherthings. Winnie Gilbert, Stefanie Hassan, and Sunil Koswatta were my assistants inthe professional development institutes, and their comments on the daily lecturesof the institutes could not help but leave their mark on these volumes. In addition,Sunil created some animations (referenced in Chapters 2 and 4) at my request.Phil Daro graciously shared with me his insight on how to communicate withteachers. Sergei Gelfand made editorial suggestions on these volumes—includingtheir titles—that left an indelible imprint on their looks as well as their user-friendliness. R. A. Askey read through a late draft with greater care than I hadimagined possible, and he suggested many improvements as well as corrections.I shudder to think what these volumes would have been like had he not caughtthose errors. Finally, Larry Francis helped me in multiple ways. He createdanimations for me that can be found in Chapter 4. He is also the only personwho has read almost as many drafts as I have written. (He claimed to have readtwenty-seven, but I think he overestimated it!) He met numerous last minuterequests with unfailing good humor, and he never ceased to be supportive; moreimportantly, he offered many fruitful corrections and suggestions.

To all of them, it gives me great pleasure to express my heartfelt thanks.Hung-Hsi Wu

Berkeley, CaliforniaApril 15, 2016

Suggestions on How to Read This Volume

The major conclusions in this book, as in all mathematics books, are sum-marized into theorems; depending on the author’s (and other mathematicians’)whims, theorems are sometimes called propositions, lemmas, or corollaries as away of indicating which theorems are deemed more important than others (notethat a formula or an algorithm is just a theorem). This idiosyncratic classificationof theorems started with Euclid around 300 B.C., and it is too late to change now.The main concepts of mathematics are codified into definitions. Definitions areset in boldface in this book when they appear for the first time. A few truly basicdefinitions are even individually displayed in a separate paragraph, but most ofthe definitions are embedded in the text itself. Be sure to watch out for them.

The statements of the theorems as well as their proofs depend on the defini-tions, and proofs (= reasoning) are the guts of mathematics.

A preliminary suggestion to help you master the content of this book is foryou to

copy out the statements of every definition, theorem, proposi-tion, lemma, and corollary, along with page references so thatthey can be examined in detail if necessary,

and also tosummarize the main idea of each proof.

These are good study habits. When it is your turn to teach your students, besure to pass on these suggestions to them. A further suggestion is that you mightconsider posting some of these theorems and definitions in your classroom.

You should also be aware that reading mathematics is not the same as readinga gossip magazine. You can probably flip through such a magazine in an hour, ifnot less. But in this book, there will be many passages that require careful readingand re-reading, perhaps many times. I cannot single out those passages for youbecause they will be different for different people. We do not all learn the sameway. What is true under all circumstances is that you should accept as a giventhat mathematics books make for exceedingly slow reading. I learned this veryearly 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 ona particular theorem in a newly published volume. He mentioned casually thatin the proof he was going to present, there were two lines in that book that tookhim fourteen hours to understand and he was going to tell us what he found outin those long hours. That comment greatly emboldened me not to be afraid tospend a lot of time on any passage in my own reading.

If you ever get stuck in any passage of this book, take heart, because that isnothing but par for the course.

xix

Appendix: Facts from [Wu-PreAlg]

There are three parts in this Appendix:

Part 1. Assumptions

Part 2. Definitions

Part 3. Theorems and Lemmas

The section in [Wu-PreAlg] where each item first appears is indicated parenthetically atthe end of that item.

Part 1. AssumptionsFundamental Assumption of School Mathematics (FASM). We can add and

multiply real numbers, and the laws of operations for both addition andmultiplication (associative, commutative, and distributive), the formu-las (a)–(d) for rational quotients (page 270), and the basic facts aboutinequalities (A)–(E) for rational numbers (page 269) continue to be validwhen the rational numbers are replaced by real numbers. (Section 2.7)

(Iso1). Translations, reflections, and rotations preserve lengths of segments anddegrees of angles. (Section 4.4)

(Iso2). Under a translation or reflection or rotation, the image of a line is a line,the image of a segment is a segment, and the image of a ray is a ray.(Section 4.4)

Part 2. Definitions

Alternate interior angles. Let two distinct lines L1, L2 be given. A transversal ofL1 and L2 is any line � that meets both lines in distinct points. Suppose� meets L1 and L2 at P1 and P2, respectively. Let Q1 and Q2 be pointson L1 and L2, respectively, so that they lie in opposite half-planes of �.Then ∠Q1P1P2 and ∠P1P2Q2 are said to be alternate interior angles ofthe transversal � with respect to L1 and L2. (Section 4.6)

Angle. An angle ∠AOB is by definition a region in the plane whose boundaryconsists of two rays ROA and ROB, with a common vertex O; each of ROAand ROB is called a side of the angle and O is called the vertex of theangle. Because of the inherent ambiguity in this definition, ∠AOB isusually taken to be the intersection of two closed half-planes: the closedhalf-plane of LOA that contains B, and the closed half-plane of LOB thatcontains A. (Section 4.4)

265

266 APPENDIX: FACTS FROM [Wu-PreAlg]

Average speed. For an object in motion, its average speed over the time intervalfrom t1 to t2, t1 < t2, is

distance traveled from t1 to t2

t2 − t1.

(Section 1.9)Basic isometry. In the plane, a basic isometry refers to a translation, a rotation,

or a reflection. (Section 4.4)Between. Given a line L in the plane, let P and Q be two points on L. A point S

is said to be between P and Q if S lies on L and if, when we make L intoa number line, either P < S < Q or Q < S < P holds. The fact that oneand only one of these inequalities holds is independent of the way L ismade into a number line. (Section 4.4)

Bilateral symmetry. A geometric figure S is said to have bilateral symmetry withrespect to a line L if the reflection Λ across L has the property thatΛ(S) = S. Equivalently, S is symmetric with respect to L if Λ maps everypoint of S to a point of S (this is because Λ ◦ Λ = identity transforma-tion). The line L is called the line of symmetry or the axis of symmetry.(Section 4.4)

Binomial coefficients. Let n and k be whole numbers . Then the binomial coeffi-cients (n

k) for k satisfying 0 ≤ k ≤ n is the whole number

(nk

)=

n!(n − k)! k!

.

(Exercises 1.4)Closed interval. Let a and b be two numbers so that a < b. Then the closed

interval [a, b] is the set of all numbers x satisfying a ≤ x ≤ b. (Section2.6)

Closed half-plane. It is the union of a half-plane of a line together with the lineitself. (Section 4.4)

Complex fraction. A complex fraction is a fraction obtained by a division AB of

two fractions A and B (B > 0). We continue to call A and B the numera-tor and denominator of A

B , respectively. (Section 1.7)Congruence. A congruence is a transformation of the plane that is the composi-

tion of a finite number of reflections, rotations, and translations. (Section4.5)

Congruent figures. A geometric figure S is congruent to another geometric fig-ure S′, in symbols, S ∼= S′, if there is a congruence ϕ so that ϕ(S) = S′.(Section 4.5)

Constant speed. An object in motion is said to have constant speed if the averagespeed of the motion over any time interval (see page 266) is equal to afixed constant. This fixed constant is then called the (constant) speed ofthe motion. (Section 1.9)

Corresponding angles of a transversal. A pair of angles formed when two paral-lel lines are intersected by a transversal are called corresponding anglesif they are obtained by replacing one angle in a pair of alternate interiorangles (relative to this transversal) by its opposite angle. (Section 4.6)

APPENDIX: FACTS FROM [Wu-PreAlg] 267

Dilation. A transformation D of the plane is a dilation with center O and scale factorr (r > 0) if

(1) D(O) = O.(2) If P �= O, the point D(P), to be denoted by P′, is the pointon the ray ROP so that |OP′ | = r|OP|. (Section 4.6)

Distance between parallel lines. Given two parallel lines L1 and L2, the lengthof the segment intercepted on a transversal that is perpendicular to bothL1 and L2 is a constant, and this length is the distance between L1 andL2. (Section 5.3)

Equal sets or equal geometric figures. Two geometric figures S and S′ are equal,in symbols S = S′, if

(i) every point P of S is also a point in S′, and(ii) every point Q of S′ is also a point in S .

(Section 3.1; Section 4.4)Exponent. Let b be a nonzero number and let n be a positive integer. Then bn

is by definition equal to b · b · · · b (n times). In this case, n is calledthe exponent of bn. [In Chapter 9, this concept of an exponent will beexpanded to include n as a rational number, and even as a real number.](Section 1.1)

Figure. See geometric figure.Fraction division. If m

n and k� are fractions ( k

� �= 0), then the division, or quo-tient, of m

n by k� , in symbols, m/n

k/� , is the fraction ab so that m

n = ab ×

k� . (Section 1.6)

Fraction multiplication. The multiplication of two fractions k� × m

n is by defini-tion the length of the concatenation of k parts when [0, m

n ] is partitionedinto � equal parts. (Section 1.5)

Fraction subtraction. If k� >

mn , then the subtraction k

� − mn is by definition the

length of the remaining segment when a segment of length mn is taken

from one end of a segment of length k� . (Section 1.4)

Geometric figure. A figure, or geometric figure, is just a subset of the plane. Oc-casionally, it also refers to a subset in 3-dimensonal space. (Section 1.5)

Intersection. The intersection of a collection of sets consists of all the points whichbelong to each and every set in the collection. (Section 3.1)

LCM. The LCM (least common multiple) of a finite collection of positive in-tegers is the smallest positive integer that is a multiple of each positiveinteger in the collection. (Exercises 3.2)

Multiplicative inverse. The multiplicative inverse of a number x is the numberx−1 so that x · x−1 = 1. (Section 1.6; Section 2.5)

Open interval. Let a and b be two numbers so that a < b. Then the open interval(a, b) is the set of all numbers x satisfying a < x < b. In other words, theopen interval (a, b) is the closed interval [a, b] without the two endpointsa and b. (Section 2.6)

Opposite signs. Two numbers are said to have opposite signs if one of them ispositive and the other is negative. (Section 2.6)

Perpendicular bisector. Given a segment AB, its perpendicular bisector is theline passing through the midpoint of AB and perpendicular to AB. (Sec-tion 4.2)


Polygon. A polygonal segment A1 A2 · · · An An+1 in the plane so that An+1 = A1and so that the segments Ai Ai+1 (for i = 1, 2, . . . , n) do not intersect eachother except at the points A1, . . . , An+1, called the corners or vertices ofthe polygon. Each of A1 A2, A2 A3, . . . , An A1 is called a side or an edgeof the polygon. When a polygon A1 A2 · · · An A1 is clearly understood,we denote it by the simpler notation A1 A2 · · · An. (Section 5.2)

Polygonal region. The union of a polygon together with the region inside thepolygon. (Section 5.2)

Polygonal segment. A polygonal segment A1 A2 · · · An is a sequence of segmentsA1A2, A2 A3, . . . An−1An which need not be collinear and which couldintersect among themselves. The points A1, A2, . . . , An are called thecorners or vertices of A1 A2 · · · An. (Section 5.2)

Product formula. For any two complex fractions AB and C

D ,

AB× C

D=

ACBD

.

(Section 1.5)Ratio. Given two fractions A and B. The ratio of A to B, usually denoted by

A : B, is the complex fraction AB . (Section 1.9)

Rational quotient. A number that is the quotient (or division) of one rationalnumber by another. For example, if x and y are rational numbers andy �= 0, then x

y is a rational quotient. (Section 2.5)Ray. A ray is a semi-infinite line with a beginning point called its vertex.

(Section 4.4)Reflection. Given a line � in the plane, the reflection across � is the transformation

Λ of the plane so that, for every point P in the plane, � is the perpendic-ular bisector of the segment joining P to Λ(P). (Section 4.4)

Relatively prime. Two positive integers are relatively prime if their only commondivisor is 1. (Section 3.1)

Removing parentheses. This refers to any one of the following three identitiesabout all rational numbers x and y:

−(x − y) = −x + y, −(−x + y) = x − y, and − (−x − y) = x + y.

(Section 2.3)Rotation. The rotation of the plane with center O and degree e (−360 ≤ e ≤ 360)

is the transformation R so that R(O) = O, and for a point P �= O, P andP′ = R(P) lie on the same circle around O, so that (i) the degree of theangle ∠POP′ is |e|, and (ii) P′ is in the counterclockwise direction of Pif e > 0 and P′ is in the clockwise direction of P if e < 0. (Section 4.4)

Same sign. Two numbers are said to have the same sign if they are either bothpositive or both negative. (Section 2.6)

Segment. Given a line L in the plane and two points P and Q on L, the segmentPQ consists of P and Q together with all the points S between P and Q(see between on page 266). (Section 4.4)

Similar figures. A geometric figure S is similar to another geometric figure S′,in symbols, S ∼ S′, if there is a dilation D so that D(S) is congruent toS′. (Section 4.7)

Similarity. A similarity is a transformation of the plane that is the compositionof a dilation followed by a congruence. (Section 4.7)


Square root. A square root t of a positive number x is a number so that t2 = x.(Section 3.1)

Transformation. A transformation F of the plane is a rule that assigns to eachpoint P of the plane a unique point F(P) (read: “F of P”) in the plane.(Section 4.4)

Translation. A translation along a vector−→AB is the transformation of the plane T

so that, if T maps a point P to Q, then Q has the following properties:(i) If P lies on the line LAB, then so does Q; if P does not lieon the line LAB, then the (line containing the) segment PQ isparallel to the (line containing the) segment AB,(ii) PQ has the same length as AB, and(iii) the two vectors

−→PQ and

−→AB point in the same direction.

(Section 4.4)Transversal. A transversal of two given lines L1 and L2 is a line that intersects

both L1 and L2. (Section 4.6)Two-sided number line. With the fractions already defined on the right side of 0

on the number line, given a fraction p, then −p is the point on the leftside of 0 which is equidistant from 0 as p.

0 p−p q−q

(Section 2.1)Union. The union of a collection of sets consists of all the points which belong

to at least one set in the collection. (Section 5.1)Vector. A vector

−→AB is a segment AB so that A is the starting point and B is the

endpoint. (Section 2.2)

Part 3. Theorems and Lemmas

AA criterion for similarity. If two triangles have two pairs of equal angles, theyare similar. (Section 4.7)

ASA. If two triangles have two pairs of equal angles and the common side ofthe angles in one triangle is equal to the corresponding side in the othertriangle, then the triangles are congruent. (Section 4.5)

Basic facts about inequalities in Section 2.6. If x, y, z, . . . are rational numbers,then:

(A) x < y ⇐⇒ −x > −y.(B) x < y ⇐⇒ x + z < y + z.(C) x < y ⇐⇒ x − y < 0.(D) If z > 0, then x < y ⇐⇒ xz < yz.(E) If z < 0, then x < y ⇐⇒ xz > yz.(Section 2.6)

Cancellation law for rational quotients. If x, y, and z are rational numbers, andy, z �= 0, then,

xy

=zxzy

.

(Section 2.5)


Cross-multiplication algorithm. (i) For rational numbers x, y, z, and w, withy �= 0 and w �= 0: x

y = zw if and only if xw = yz. (ii) For positive

numbers a, b, c, and d: ab < c

d if and only if ad < bc. (Section 2.5 andSection 1.7, respectively)

FFFP (Fundamental Fact of Fraction-Pairs). Any two fractions ab and c

d may be

regarded as two fractions with the same denominator, e.g., adbd and bc

bd .(Section 1.3)

Formulas for rational quotients in Section 2.5. Let x, y, z, w, . . . be rational num-bers so that they are nonzero where appropriate in the following.

(a) Cancellation law: xy = zx

zy for any nonzero z.

(b) Cross-multiplication algorithm: xy = z

w if and only if xw = yz.

(c) xy ± z

w = xw±yzyw .

(d) xy × z

w = xzyw .

(Section 2.5)Fundamental Theorem of Arithmetic. Every positive integer > 1 is a product of

a finite number of primes, and this collection of primes is unique (exceptpossibly for order). (Section 3.2)

Key Lemma. Suppose �, m, n are nonzero whole numbers, and � divides mn. If �and m are relatively prime, then � divides n. (Section 3.1)

Pythagorean Theorem. If the lengths of the legs of a right triangle are a and b,and the length of the hypotenuse is c, then a2 + b2 = c2. (Section 4.7)

SAS. If two triangles have a pair of equal angles (i.e., same degree) and thecorresponding sides of these angles in the two triangles are pairwiseequal (e.g., given ABC and A′B′C′, the following holds: |∠A| =|∠A′|, |AB| = |A′B′| and |AC| = |A′C′|), then the two triangles arecongruent. (Section 4.5)

SAS criterion for similarity. Given two triangles ABC and A′B′C′, if |∠A| =|∠A′| and

|AB||A′B′ | =

|AC||A′C′| ,

then ABC ∼ A′B′C′. (Section 4.7)SSS. If the three sides of a triangle and the three corresponding sides of an-

other triangle are pairwise equal, then the two triangles are congruent.(Section 4.5)

Theorem 1 in the Appendix of Chapter 1. For any finite collection of numbers,the sums obtained by adding them up in any order are all equal. (Section1.11)

Theorem 2 in the Appendix of Chapter 1. For any finite collection of numbers,the products obtained by multiplying them in any order are all equal.(Section 1.11)

Theorem 4.2. (a) An isosceles triangle has equal base angles. (b) In an isoscelestriangle, the perpendicular bisector of the base, the angle bisector of thetop angle, the median from the top vertex, and the altitude on the baseall coincide. (Section 4.5)


Theorem 4.4. If D is a dilation with center O and scale factor r, then for any twopoints P and Q in the plane, so that P′ = D(P) and Q′ = D(Q) are theirdilated images, we have

|P′Q′| = r |PQ|.(Section 4.6)

Theorem 4.5. Let D be a dilation with center O and scale factor r, and let P, Q betwo points not collinear with O. Further let P′ denote D(P). Then thedilated image Q′ of Q is the intersection of line LOQ and the line passingthrough P′ and parallel to LPQ. (Section 4.6)

Theorem 4.7. Alternate interior angles of a transversal with respect to a pair ofparallel lines are equal. The same is true of corresponding angles. (Sec-tion 4.6)

Theorem 4.9 If two lines have a pair of equal alternate interior angles or corre-sponding angles with respect to a transversal, they are parallel. (Section4.6)

Theorem 4.12. Given two triangles ABC and A′B′C′, their similarity, i.e., ABC ∼A′B′C′, implies the following equalities:

|∠A| = |∠A′|, |∠B| = |∠B′ |, |∠C| = |∠C′|,|AB||A′B′ | =

|AC||A′C′| =

|BC||B′C′| .

(Section 4.7)

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http://en.wikipedia.org/wiki/Conic_section

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�� cally for Common Core era teachers. The emphasis of the exposition is to give a mathematically correct treatment of introductory algebra. For example, it explains the proper use of symbols, why “variable” is not a mathematical ��slope of a line correctly, why the graph of a linear equation in two variables is a straight line, why every straight line is the graph of a linear equation in two variables, how to use the shape of the graph of a quadratic function as a guide ��graph of a quadratic function is a parabola, why all parabolas are similar, etc. �� !��are written the way they are.

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