Pre-Algebra [DRAFT] - University of California, Berkeley

Pre-Algebra [DRAFT]

H. Wu April 21, 2010; revised, October 26, 2011

c Hung-Hsi Wu, 2010

General Introduction (p. 2) Suggestions on How to Read These Notes (p. 6) Chapter 1: Fractions (p. 8) Chapter 2: Rational Numbers (p. 112) Chapter 3: The Euclidean Algorithm (p. 169) Chapter 4: Experimental Geometry (p. 189) Chapter 5: Basic Isometries and Congruence (p. 233) Chapter 6: Dilation and Similarity (p. 296) Chapter 7: Length and Area (p. 325)

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General Introduction

The content of these notes is the mathematics that is generally taught in grades 6?8. This is a no frills, bare essentials course for helping you to teach mathematics in the middle school classroom, and is not designed to show you how mathematics, deep down, is just lots of fun. We review most of the standard topics of the middle school mathematics curriculum. However, the presentation of this material in the standard textbooks, be they traditional or reform, is riddled with mathematical errors. What is presented in this institute, while bearing superficial resemblances to what you normally find in textbooks, will likely be very different in terms of precision, sequencing, and reasoning. You will probably have to rethink some of this material even if you believe you already know them very well.

Let us look at the concept of congruence, a main point of emphasis in these notes. Most textbooks would have you believe that it means same size and same shape. As mathematics, this is totally unacceptable, because "same size" and "same shape" are words that mean different things to different people, whereas mathematics only deals with clear and unambiguous information. I will therefore suggest that you approach the teaching of this concept completely differently. First make sure that you know what reflections, translations and rotations are, then devise hands-on activities for your students to familiarize them with these concepts, and finally, teach them that two sets are congruent if one can carry one set onto the other by use of a finite number of reflections, translations and rotations.

You see right away that we will be doing standard middle school mathematics, but for a change, we will do it in a way that is consistent with how mathematics is supposed to be done. The hope is that by the time you are finished with these notes, you will begin to recognize school mathematics as a coherent subject with every concept and skill placed in a logically correct hierarchy. If I may express this idea by use of an analogy, it would be like bringing bookshelves to a roomful of books scattered all over the floor and trying to put the books on the shelves using a well-understood organizing principle. Once arranged this way, any book in the room can be easily accessed in the future. Likewise, if we can re-organize mathematical thoughts logically in our mind, we can much more easily access and make use of them.

But why? An obvious reason is that if we want students to see mathematics as a

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tool to help solve problems, the different parts of this tool must be freely accessible. This cannot happen if we as teachers do not have free access to these parts ourselves. A less obvious, but perhaps more compelling reason is that teaching mathematics merely as a jumbled collection of tasks has led our nation to a severe mathematics education crisis.1 It would be reasonable to attribute a good deal of students' nonlearning of mathematics to their being fed such jumbled information all the way from kindergarten to grade 12.2 These notes are dedicated to making improvements in mathematics instructions, one classroom at a time.

The main goal of these notes is to provide the necessary background for the teaching of algebra. Getting all students to take algebra around grade 8 is at present a national goal. For an in-depth discussion, see the National Mathematics Panel's Conceptual Knowledge and Skills Task Group Report:



Currently, most school students are deficient in their knowledge of the two pillars that support algebra: rational numbers and similar triangles; these two topics are the subject of five of the seven chapters in these notes. In the current school curriculum, one does not associate the learning of similar triangles as a pre-requisite to the learning of algebra. But it is, and this failure to give adequate support to our students' learning of algebra is one of the flaws in mathematics instructions that we set out to remedy.

Overall, these notes will strive to improve mathematics teaching by emphasizing, throughout, the following three principles:

(I) Precise definitions are essential. Definitions are looked upon with something close to disdain by most teachers because "they are nothing more than something to be memorized". Such an attitude stems from poor professional development that breeds such a misconception of mathematics. First of all, memorizing important facts

1See, for example, Rising Above the Gathering Storm, The National Academies Press, 2007. Also, id=11463. Or Foundations for Success: The National Mathematics Advisory Panel Final Report, U.S. Department of Education, 2008. Also, .

2Such a statement should not be misinterpreted to mean that this is the only reason for students' non-learning. There is enough blame to go around.

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is an integral part of life (you memorize your name, your home address, your cell phone, the password of your computer account, etc.), and you will have to memorize all the definitions we use. No apology will be offered. But the idea that a definition in mathematics is nothing but "one more thing to memorize" must be eradicated. In mathematics, precise definitions are the bedrock on which all logical reasoning rests because mathematics does not deal with vaguely conceived notions. These notes will respect this fundamental characteristic of mathematics by offering precise definitions for many concepts in the school curriculum usually used with no definitions: fraction, decimals, sum of fractions, product of fractions, ratio, percent, polygon, congruence, similarity, length, area, etc.

(II) Every statement should be supported by reasoning. There are no unexplained assertions in these notes. If something is true, a reason will be given. Although it takes some effort to learn the logical language used in mathematical reasoning, in the long run, the presence of reasoning in all we do eases the strain of learning and disarms disbelief. It also has the salutary effect of putting the learner and the teacher on the same footing, because the ultimate arbiter of truth will no longer be the teacher's authority but the compelling rigor of the reasoning.

(III) Mathematics is coherent. You will see that these notes unfold logically and

naturally rather than by fits and starts. On the one hand, each statement follows

logically from the preceding one, and on the other, the various statements form parts

of an unending story rather than a disjointed collection of disparate tricks and fac-

toids. A striking example of the failure of coherence is the common explanation of

the

theorem

on

equivalent

fractions,

which

states

that

m n

=

km kn

for

all

fractions

m n

and for all positive integer k. Most book would have you believe that this is true

because

m

m k m km

= 1? = ? =

n

n k n kn

Unfortunately, the last step depends on knowing how to multiply fractions. But

the definition of fraction multiplication is the most subtle among the four arithmetic

operations on fractions, whereas the theorem on equivalent fractions should be proved

as soon as fractions are defined. To invoke something not yet explained and technically

more complex to explain something logically simpler and more elementary is a blatant

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violation of the fundamental structure of mathematics. Such violations abound in current textbooks.

An example of the interconnections among seemingly different topics that hold the subject together is the fact that the concept of similarity in Chapter 6 relies on a knowledge of dividing fractions (Chapter 1) and congruence (Chapter 5). Another example is the key role played by congruence in the considerations of length, area, and volume (Chapter 7). And as a final example, you will notice that the division of whole numbers, of fractions (Chapter 1), and of rational numbers (Chapter 2) are conceptually identical.

I hope you find that these notes make more sense of the mathematics you know because they observe these basic principles. As far as this institute is concerned, however, what matters is that you can translate this new-found knowledge into better teaching in your classroom. I am counting on you to make this next step.

I am very grateful to Larry Francis for correcting a large number of linguistic infelicities and misprints.

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