## Algebra Survival Guide by Josh Rappaport

Algebra Survival Guide by Josh Rappaport

The Algebra Survival Guide comes with an Algebra Wilderness “Bored Game” that gives children a fun way to practice their lessons and also with a tear-out Emergency Fact Sheet poster with all of algebra’s secret rules and formulas at your fingertips. The book has been written by Josh Rappaport who has been a winner of the Outstanding Young Educator award, is a teacher and a tutor in Santa Fe, New Mexico. Josh runs an online community at www.mathkits.com to help students and parents to overcome the fears of algebra.

Algebra Survival Guide begins its conversation with What is algebra?, its Properties & Sets of Numbers and gradually steps further to Positive & Negative Numbers, Order of Operations & Like Terms, Absolute Value, Exponents, Radicals, Factoring, Cancelling, Equations, Coordinate Plane & Word Problems.

The book very well defines algebra as a branch of mathematics that performs the magic trick – it takes something that’s unknown and poof! – turns it into something known. Algebra does this by using letters (variables) to stand for mystery numbers, and giving you a process to let you discover the value of the variables. The book lays the following reasons to study & learn algebra:

• Studying algebra boosts your chances of going to college and succeeding in today’s world. Studies show that taking algebra, and following it with geometry, dramatically boosts a student’s chance of going to college. In fact, a 1990 College Board study found that students who take algebra and geometry stand a much greater chance of attending college than students who don’t take these courses. And, of course, to succeed in today’s high-tech/information-age economy, you must have a good education.
• Learning algebra can help you find a good-paying job. Anyone who wants to work in any field of science – computer science, astronomy, medicine, psychology, genetics, etc – needs to know algebra. That’s because all the science rely on algebra and on the higher maths.
• Knowing algebra will help you survive the “big, bad world.” Understanding algebraic ratios helps you become a smart comparison shopper; understanding percentages helps you make sense of the statistics thrown at you by the media. The ways to use algebra are numerous.
• Algebra teaches you to solve difficult problems. By developing your algebraic mental muscles, you strengthen your ability to tackle problems in life, for learning algebra improves your ability to think clearly.
• Learning algebra teaches you about yourself. Since algebra is a product of the human mind, studying algebra gives you insight into how your own mind works. Learning this subject shows the logical thought patterns that are part of you just because you’re human.

## How are expressions formed?

We know very well what a variable is. We use letters x, y, l, m, … etc. to denote variables. A variable can take various values. Its value is not fixed. On the other hand a constant has a fixed value. Examples of constant are: 4, 100, -17, etc.

We combine variables and constants to make algebraic expressions. For this we use the operations of addition, subtraction, multiplication and division. We have already come across expressions like 4x + 5, 10y -20. The expression 4x + 5 is obtained from the variable x, first by multiplying x by the constant 4 and then adding the constant 5 to the product. Similarly, 10y – 20 is obtained by multiplying constant 10 with variable y and then subtracting 20 from the product.

The above expressions were obtained by combining variables with constants. We can also obtain expressions by combining variables with themselves or with other variables.

Look at how the following expressions are obtained:

x2, 2y2, 3x2 – 5, xy, 4xy + 7

• The expression x2 is obtained by multiplying the variable x by itself;
x x x = x2

Just as 4 x 4 = 42, we write x x x = x2. It is commonly read as x squared. In the same manner, we can write
x x x x x = x3

Commonly, x3 is read as x cubed which can also be read as x raised to the power 3.

• The expression 2y2 is obtained from y: 2y2 = 2 x y x y
Here by multiplying y with y we obtained y2 and then we multiply y2 by the constant 2.
• In (3x2 – 5) we first obtain x2 and multiply it by 3 to get 3x2. From 3x2, we subtract 5 to finally arrive at 3x2 = 5.
• In xy, we multiply the variable x with another variable y. Thus, x x y = xy.
• In 4xy + 7, we first obtain xy, multiply it by 4 to get 4xy and add 7 to 4xy to get the expression.

## Intensely mathematical classifications

Math historians (if you thought regular math people were boring, you should get a load of these guys) generally agree that the earliest humans on the planet had a very simple number system that went like this: one, two, a lot. There was no need for more numbers. Lucky you – that’s not true any more. Here are the less familiar number classifications you’ll need to understand.

Natural numbers: The numbers 1, 2, 3, 4, 5, and so forth are called the natural (or counting) numbers. They’re the numbers you were first taught as a child when counting.

Whole numbers: Throw in the number 0 with the natural numbers and you get the whole numbers. That’s the only difference – 0 is a whole number but not a natural number. (That’s easy to remember, since a 0 looks like a drawing of a hole.)

Integers: Any number that has no explicit decimal or fractional component is an integer. Therefore, -4, 17, and 0 are integers, but 1.25 and 2/5 are not.

Rational numbers: If a number can be expressed as a decimal that either repeats infinitely or simply ends (called a terminating decimal), then the number is rational. Basically, either of those conditions guarantees one thing: That number is actually equivalent to a fraction, so all fractions are automatically rational.

(You can remember this using the mnemonic device “Rational means fractional.” The words sound roughly the same.) Using this definition, it’s easy to see that the numbers 1/3, 7.95, and 0838383838383… are all rational.

Irrational numbers: If a number cannot be expressed as a fraction, or its decimal representation goes on and on infinitely but not according to some obvious repeating pattern of digits, then the number is irrational. Although many radicals (square roots, cube roots and the like) are irrational, the most famous irrational number is pi = 3.141592653589793… No matter how many thousands (or millions) of decimal places you examine, there is no pattern to the numbers. In case you’re curious, there are far more irrational numbers that exist than rational numbers, even though the rationals include every conceivable fraction!

Real numbers: If you clump all of the rational and irrational numbers together, you get the set of real numbers. Basically, any number that can be expressed as a single decimal (whether it be repeating, terminating, attractive, or awkward looking but with a nice personality) is considered a real number.

Technically, 0 is divisible by 2, so it is considered even. However, 0 is not positive, nor it is negative – it’s just sort of hanging out there in mathematical purgoratory, and can be classified as both nonpositive and nonnegative.

Since every integer is divisible by 1, that means each can be written as a fraction (7 = 7/1). Therefore, every integer is also a rational number.

## Getting cozy with numbers

Getting cozy with the different types of numbers

Most people new to algebra view it as a disgusting, creeping disease whose sole purpose is to ruin everything they’ve ever known about math. They understand multiplication, and can even divide numbers containing decimals (as long as they can check their answers with a calculator or a nerdy friend), but algebra is an entirely different beast – it contains letters! Just when you feel like you’ve got a handle on math, suddenly all these x’s and y’s start sprouting up all over, like pimples on prom night.

Before we begin talking about those letters (they’re actually called variables), you’ve got to know a few things about those plain old numbers you’ve been dealing with all these years. Some of the things may sound familiar, but most likely some of it will also be new. In essence, you will gather here some necessary prealgebra skills; it’s one last chance to get to know your old number friends better, before we unceremoniously dump letters into the mix.

Classifying Number Sets

Most things can be classified in a bunch of different ways. For example, if you had a cousin named Scott, he might fall under the following categories: people in your family, your cousins, people with dark hair, and (arguably) people who could stand to brush their teeth a little more often. It would be unfair to consider only Scott’s hygiene (lucky for him); that’s only one classification. A broader picture is painted if you consider all of the groups he belongs to:

• People with dark hair
• Hygienically challenged people

The same goes for numbers. Numbers fall into all kinds of categories, and just because they belong to one group, it does not preclude them from belonging to others as well.

Familiar Classifications

## Algebra I for Dummies by Mary Jane Sterling

Algebra I for Dummies by Mary Jane Sterling

Algebra I for Dummies is a book by Mary Jane Sterling who has taught mathematics at Bradley University since 1979, and has also taught math at the high school and junior high school levels. Algebra I for Dummies provides a pain-free way to explore algebra and believe the reviewers, you would come out smiling. Does the word polynomial make your hair stand on end? Relax! Let Mary Jane Sterling show you the easy way to tackle algebra. This friendly guide explains the basic – and the toughest stuff – in easy-to understand, no-nonsense language. Whether you want to brush up on your math skills or help your children with their homework, this book gives you power – to the nth degree.

Algebra I for Dummies helps you discover how to:

• Find out about fractions and explore exponents
• Figure out factoring
• Solve linear equations
• Graph equations
• Solve story problems

## Video #1: Introduction to Algebra

Introduction to Algebra

## Postulates for the natural numbers

Natural numbers

Every natural number a has a successor a + 1. (For example, the successor of 5 if 6, the successor of 10 is 11, and so on.)

Every natural number a (except 1) has a predecessor a – 1. (The predecessor of 4 is 3, the predecessor of 9 is 8, and so on.)

The natural numbers can be put in order. In other words, if a and b are two natural numbers, then either a is greater than b (written a > b), a is less than b (a < b), or a is equal to b (a = b.)

The order doesn’t make a difference (commutative) property of addition: a + b = b + a (for any two given numbers a and b).

The order doesn’t make a difference (commutative) property of multiplication: a x b = b x a (for any two given numbers a and b).

The where you put the parentheses doesn’t make a difference property of addition: (a + b) + c = a + (b + c). For any three numbers a, b, and c. (This property is known as the associative property of addition, because it says that it doesn’t matter which numbers associate with each other.) For example,

(5 + 4) + 3 = 5 + (4 + 3)
(9) + 3 = 5 + (7)
12 = 12

The where you put the parentheses doesn’t make a difference (associative) property of multiplication: (a x b) x c = a x (b x c). For any three numbers, a b, and c. For example,

(2 x 6) x 4 = 2 x (6 x 4)
(12) x 4 = 2 + (24)
48 = 48

## Cartoons on Algebra

Find some cartoons below on algebra.

Algebra cartoon #1

Just a darn minute – yesterday you said that X equals two!

Algebra cartoon #2

Algebra is weightlifting for the brain!

Algebra cartoon #3

## Practical Algebra: A Self-Teaching Guide by Peter H. Selby

Practical Algebra by Peter Selby

If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you’re a student who needs an introduction to the subject, here’s the perfect book for you. Practical Algebra is an easy and fun-to-use workout program that quickly puts you in command of all the basic concepts and tools of algebra. With the aid of practical, real-life examples and applications, you’ll learn

• the basic approach and application of algebra to problem solving
• the number system (in a much broader way than you have known it from arithmetic)
• Monomials and polynomials; factoring algebraic expressions; how to handle algebraic fractions; exponents, roots, and radicals; linear and fractional equations
• Functions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and more.

Autors Peter Selby and Steve Slavin emphasize pratical algebra throughout by providing you with techniques for solving problems in a wide range of disciplines – from engineering, biology, chemistry, and the physical sciences, to psychology and even sociology and business administration. Step by step, Practical Algebra shows you how to solve algebraic problems in each of these areas, then allows you to tackle similar problems on your own, at your own pace. Self-tests are provided at the end of each chapter so you can measure your mastery.

Peter Selby (deceased) was Director of Educational Technology at Man Factors Associates, a human factors engineering consulting firm. He is the author of two other self-teaching guides: Quick Algebra Review: A Self-Teaching Guide and Geometry and Trigonometry for Calculus: A Self-Teaching Guide, both published by Wiley.

Steve Slavin, PhD, is Associate Professor of Economics at Union County College, Cranford, New Jersey. He has written over 300 newspaper and magazine articles, and is the author of four other books, including All the Math You’ll Ever Need: A Self-Teaching Guide and Economics: A Self-Teaching Guide, both published by Wiley.

## Definition of Algebra

Definition of Algebra

Whatever is capable of increase or diminution, is called magnitude, or quantity.

A sum of money therefore is a quantity, since we may increase it or diminish it. It is the same with a weight, and other things of this nature.

From this definition, it is evident, that the different kinds of magnitude must be so various, as to render it difficult to enumerate them: and this is the origin of the different branches of Mathematics, each being employed on a particular kind of magnitude. Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity.

Now, we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known, and pointing out their mutual relation. If it were proposed, for example, to determine the quantity of a sum of money, we should take some known pieces of money, as a louis, a crown, a ducat, or some other coin, and shew how many of these pieces are contained in the given sum. In the same manner, if it were proposed to determine the quantity of a weight, we should take a certain known weight; for example, a pound, an ounce, etc. and then shew how many times one of these weights is contained in that which we are endeavouring to ascertain. If we wished to measure any length or extension, we should make use of some known length, such as a foot.

So that the determination, or the measure of magnitude of all kinds, is reduced to this: fix at pleasure upon any one known magnitude of the same kind with that which is to be determined, and consider it as the measure or unit; then, determine the proportion of the proposed magnitude to this known measure. This proportion is always expressed by numbers; so that a number is nothing but the proportion of one magnitude to another arbitrarily assumed as the unit.