Introduction to School Algebra [Draft]

[Pages:216]Introduction to School Algebra [Draft]

H. Wu July 24, 2010

Department of Mathematics, #3840 University of California Berkeley, CA 94720-3840 wu@math.berkeley.edu c Hung-Hsi Wu, 2010

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Contents

1 Symbolic Expressions

7

2 Transcription of Verbal Information into Symbolic Language

31

3 Linear Equations in One Variable

42

4 Linear Equations in Two Variables and Their Graphs

49

5 Some Word Problems

76

6 Simultaneous Linear Equations

89

7 Functions and Their Graphs

115

8 Linear functions and proportional reasoning

129

9 Linear Inequalities and Their Graphs

140

10 Exponents

177

11 Quadratic Functions and Their Graphs

195

12 The Quadratic Formula and Applications (outlined)

210

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Preface

This is a set of lecture notes on introductory school algebra written for middle school teachers. It assumes a knowledge of pre-algebra as given in the Pre-Algebra Institute of 2009:

(hereafter referred to as the Pre-Algebra notes)

There is nothing fancy about the content of these lecture notes. With minor exceptions, it covers only the standard topics of a first year algebra course in school mathematics (grade 8 or grade 9). These notes may therefore be called Introductory Algebra from a Somewhat Advanced Point of View. If there is any merit to be claimed for them, it may be the sequencing of the topics and the logical coherence of the presentation. The exposition is formally self-contained in the sense that the readers are not assumed to know anything about algebra. In practice, though, the readers are likely to have taught, or will be teaching beginning algebra so that they are already familiar with the more routine aspect of the subject. For this reason, while I try to shed light on the standard computations whenever appropriate, I have on the whole slighted the drills that usually accompany any such presentation.

Many of our algebra teachers are experiencing real difficulty in carrying out their duties, mainly because they have been told to emphasize certain ideas that they (like most mathematicians) cannot relate to, such as that of a quantity that "changes" or "varies" or that the equal sign is a "method that expresses equivalence". At the same time, they are denied the explanations of key facts which form the backbone of introductory algebra, such as the proper way to use symbols, why the graph of a linear equation is a straight line, why fractional exponents are defined the way they are and what they are good for, or why the significance of the technique of completing the square goes beyond the proof of the quadratic formula. The fact that many teachers do not even recognize that these are key facts in algebra speaks volumes about the present state of pre-service professional development in mathematics. The main impetus behind the writing of these notes is to propose a remedy. It gives an exposition of algebra ab initio, assuming only a knowledge of the rational number system and some elementary facts about similar triangles. I am quite aware that the latter is not standard fare in either the school curriculum or texts for professional

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development, but the Pre-Algebra notes give an adequate exposition of this topic and we will make frequent references to it. One of the stated goals of these notes is to make a strong case that this aspect of the school mathematics curriculum must change. Otherwise, the exposition of these notes is entirely unexceptional, and all it does is meet the minimum requirements of any exposition in mathematics, so that the unfolding of ideas is achieved not by appealing to any abstruse philosophical discussions but by use of clear and precise definitions and logical reasoning.

An integral part of the learning of algebra is learning how to use symbols precisely and fluently. This point of view is eloquently exposed in Chapter 3 of the National Mathematics Advisory Panel Report, Foundations for Success: Reports of the Task Groups and Subcommittees, U.S. Department of Education, 2008, available at



I believe if there is any meaning at all to the phrase "algebraic thinking" in school mathematics, it would be "the ability to use symbols precisely and fluently". In this regard, there is a need to single out the first two sections of these notes for some special comments. A principal object of study in introductory algebra is polynomials or, in the language of advanced mathematics, elements of the polynomial ring R[x]. Every exposition of school algebra must come to grips with the problem of how to properly introduce this abstract concept to beginners. The mathematical decision I made (which is of course not mentioned in the notes proper) is to exploit the theorem that R[x] is isomorphic to the ring of real-valued polynomial functions, so that in the context of introductory school algebra, the x in a polynomial anxn + ? ? ? + a1x + a0 may be simply taken to be a number. The purpose of Section 1 is to demonstrate how one can do algebra by taking x to be just a number, and school algebra then becomes generalized arithmetic, literally. Formal algebra in the sense of R[x] can be left to a later date, e.g., a second course in school algebra.

Section 1 is thus entirely elementary, and is nothing more than a direct extension of arithmetic. The exposition intentionally emphasizes its similarity to arithmetic. There is a danger, however, that precisely because of its elementary character, readers would consider it as something "they already know" and therefore not worthy of a second thought. I would like to explicitly ask algebra teachers to recognize that what is in Section 1 is genuine algebra, and that its simplicity is precisely the reason that

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algebra can be taught without any fanfare. At a time when Algebra for All is the

clarion call of the day in mathematics education, such a simple approach to algebra

merits serious consideration for implementation in the school classroom.

Section 2 addresses one of the main obstacles in the teaching of algebra: students'

apparent inability to solve word problems. It seems to me sensible to separate this dif-

ficulty into two stages: first teach how to transcribe verbal information into symbols,

and then teach the necessary symbolic manipulations to extract the solution from

the symbolic expressions. The need for such a separation does not seem to be well

recognized at present in mathematics education. Too many teachers have been con-

ditioned in the routine of plunging headlong into the solution of a problem by using

guess-and-check, and because guess-and-check is sometimes identified with conceptual

understanding, their students follow suit. It come to pass that students also fail to

recognize the critical need of the transcription process, and symbols are looked upon

with suspicion. The anti-mathematical practice of relying solely on guess-and-check

to achieve mathematical understanding gets recycled from generation to generation

and algebra students fail to learn the most fundamental aspect of algebra, namely,

the proper use of symbols. For the purpose of good education, I believe we should,

and must, promote the importance of transcription. The main purpose of Section 2

is to do exactly that.

Another perennial problem in the learning of introductory algebra is the absence

of reasoning even for the most basic facts of the subject, such as why the graph of a

linear equation in two variables is a line, why the graph of a linear inequality is a half-

plane, or why the maximum or minimum of a quadratic function f (x) = ax2 + bx + c

is achieved at the point

x

=

-

b 2a

.

As

a

result,

students

scramble

to

memorizing

all

four forms of the equation of a line, though not always with success, and come out of

the subject of quadratic functions and equations without learning the fundamental

skill of completing the square except as a rote skill to get the quadratic formula.

Concomitant with the absence of reasoning is the tendency to slight basic definitions.

Indeed, if one is not fully aware of the precise definition of the graph of an equation,

it would be difficult to prove what the graph of a linear equation must look like. Or,

if one doesn't say what the definition of the graph of a linear inequality is or what

the definition of a half-plane is, one cannot possibly prove that the graph of a linear

inequality is a half-plane. This monograph tries to compensate for these common

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deficiencies by addressing each of them in detail. There is at present an urgent need to emphasize precise definitions, coherence of

presentation, and logical reasoning in the mathematics education of teachers.1 This need may be greatest among algebra teachers because of the inherent abstraction of algebra. Without precise definitions and logical reasoning at each step of the mathematical processes, abstract mathematics is not learnable. The exposition in these notes fully reflects this sense of urgency.

1See the discussion in the article, The mathematics K-12 teachers need to know,

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1 Symbolic Expressions

Basic Protocol in the use of symbols Expressions and identities Mersenne primes and the finite geometric series Polynomials and order of operations Rational expressions

Basic Protocol in the use of symbols

It can be argued that the most basic aspect of algebra is the use of symbols.

Why symbols? When we try to assert that something is valid for a large collection

of numbers instead of just for a few specific numbers (e.g., for all positive integers,

or for all rational numbers), we have to resort to the use of symbols to express this

assertion correctly and succinctly. For example, suppose we observe that 2?3 = 3?2,

3?4

=

4 ? 3,

6 17

?

4 9

=

4 9

?

6 17

,

(-

8 3

)

?

82

=

82

?

(-

8 3

),

and

so

on,

for

any

two

numbers

that we care to multiply, and we want to say summarily that

for any two numbers, if we multiply them one way and, switching the order, we multiply them again, we get the same number.

Of course, what we wish to assert is what is known as the commutative law of multiplication. The question is how to say it completely, unambiguously, and succinctly. After many trials and errors through many centuries, starting with Diophantus around the third century2 to Ren?e Descartes (1596?1650) 3, people finally settled on the use of symbols as we know it today. For the problem at hand, the accepted way of enunciating the commutative law of multiplication is to say:

ab = ba for all numbers a and b

2Diophantus was a Greek mathematician who lived in Alexandria, Egypt (which was a Greek colony named after Alexander the Great). Unfortunately, his dates are unknown other than the fact that he probably lived in the third century A.D. His influence in the development of mathematics is quite profound, as evidenced by the fact that the terminology of Diophantine equations is standard in mathematics.

3A co-discoverer of analytic geometry with Pierre Fermat (1601-1665). He is also an important philosopher who is noted for the statement that, "I think, therefore I am".

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Compared with the preceding indented verbal statement, the brevity resulting from the use of symbols should be obvious.

It would seem that the fruits of some thirteen centuries of development of the symbolic notation have not filtered down to our school curriculum, and the use of symbols in standard textbooks is reckless at best. Major misconceptions ensue. A main theme throughout these notes is to give careful guidance on the etiquette of using symbols in order to undo these misconceptions.

One of the misconceptions that accompanies the abuse of the symbolic notation is the concept of a variable. At present, variable occupies a prominent position in school mathematics, especially in algebra. In standard algebra texts as well as in the mathematics education literature, there may be no explicit definition of what a variable is, but students are asked to understand this concept nevertheless because it is considered the gateway to algebra. When students are asked to understand something which is left unexplained, learning difficulties inevitably arise. Sometimes, a variable is described as a quantity that changes or varies. The mathematical meaning of the last statement is vague and obscure. At other times it is asserted that students' understanding of this concept should be beyond recognizing that letters can be used to stand for unknown numbers in equations, but nothing is said about what lies "beyond" this recognition. For example, in the National Research Council volume, Adding It Up (The National Academy Press, 2001), there is a statement that students emerging from elementary school often carry the "perception of letters as representing unknowns but not variables" (p. 270). The difference between "unknowns" and "variables" is unfortunately not clarified. All this deepens the mystery of what a variable really is.

These notes will not make the understanding of a variable a prerequisite to the learning of algebra. There is no need for that in mathematics.4 Instead, we will explain the correct way to use symbols, and once you understand that, you will feel

4In mathematics, a variable is an informal abbreviation for "an element in the domain of definition of a function", which is of course a perfectly well-defined concept (See Section 9). If, for example, a function is defined on a set of ordered pairs of numbers, it is referred to as "a function of two variables", and it must be said that, in that case, the emphasis is more on the word "`two" than on the word "variables". In the sciences and engineering, the word "variable" is bandied about with gusto. However, to the extent that mathematics is just a tool rather than the central object of study in such situations, scientists and engineers can afford to be cavalier about mathematical terminology. In these notes, we have to be more careful because we are trying to learn mathematics.

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