ALGEBRA. COPYRIGHT 1996 Mark Twain Media, Inc. ISBN Printing No EB

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1 ALGEBRA By Don Blattner and Myrl Shireman COPYRIGHT 1996 Mark Twain Media, Inc. ISBN Printing No EB Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing Company, Inc. The purchase of this book entitles the buyer to reproduce the student pages for classroom use only. Other permissions may be obtained by writing Mark Twain Media, Inc., Publishers. All rights reserved. Printed in the United States of America.

2 Table of Contents Table of Contents Introduction... iv...1 Real Numbers...2 Quotient Questions...3 Rational Numbers...4 Irrational Numbers...5 Quiz I:...7 Adding Positive and Negative Numbers...8 Subtracting Positive and Negative Numbers...9 Quiz II: Addition and Subtraction With Positive and Negative Numbers...10 Multiplication With Positive and Negative Numbers...11 More Multiplication With Positive and Negative Numbers...12 Division With Positive and Negative Numbers...13 Quiz III: Multiplication and Division With Positive and Negative Numbers...14 Mathematical Symbols...15 Variables...15 Equal, Less Than, Greater Than...15 More Mathematical Symbols...17 Math Problems With More Than One Variable...19 Word Problems Using Variables...20 Quiz IV: Mathematical Symbols...21 Understanding Polynomials...22 Definitions...22 Terms, Expressions, and Polynomials...23 Adding Polynomials...24 Subtracting Polynomials...26 Multiplying Polynomials...27 Dividing Polynomials...29 Quiz V: Understanding Polynomials...31 Learning About Equations...33 Written and Mathematical Expressions...33 Equations...35 Understanding Equations...36 Review of Equation Rules...38 Learning About Parentheses and Brackets...40 Order of Operations...42 Quiz VI: Learning About Equations...44 ii

3 Table of Contents Understanding and Using Exponents...45 Exponents, Bases, and Factors...46 Exponents in ic Expressions...47 Looking for Patterns When Using Exponents...48 Expressing Exponents in Words...49 Dealing With Negative Exponents and Negative Bases...50 Adding, Subtracting, and Multiplying Exponents...51 Quiz VII: Understanding and Using Exponents...53 Learning About Radicals and Roots...54 Learning How to Read and Write Radicals...55 Number of Factors...56 Adding Radicals...57 Subtracting Radicals...58 Multiplying Radicals...59 Dividing Radicals...60 Prime Factors and Simplifying Square Root Radicals...61 Rationalizing the Denominators in Radicals with Fractions...63 Learning About Radicals in Equations...64 Quiz VIII: Learning About Radicals and Roots...65 Learning About Linear Equations...66 Determining the Slope of a Graphed Linear Equation...67 Positive and Negative Slope...70 Formula for Determining the Slope of a Graphed Linear Equation...71 Graphing Linear Equations and Determing Their Slopes...72 Learning About the x-intercept and the y-intercept...74 Learning About the Slope Intercept Equation...76 Slope Intercept Exercises...78 Graphing Linear Equations...79 Reviewing What You Have Learned...80 Quiz IX: Learning About Linear Equations...81 Learning About Quadratic Equations...82 Graphing the Equation ax 2 + bx + c...83 Parabolas...85 Parabola Practice...87 Solving Quadratic Equations...88 Standard Form of a Quadratic Equation...89 Solving Quadratics by Factoring...90 Solving Quadratics by the Quadratic Formula...91 Quiz X: Learning About Quadratic Equations...93 Answer Keys...94 iii

4 Name Date In order to study algebra, one must understand the number system. Read the definition for each of the types of numbers and complete the exercise below each definition. Fill in the blanks with the numbers that should follow for the given definition. Remember each of the sets of numbers is infinite. Counting numbers: Sometimes called natural numbers, they are 1, 2, 3, 4, 5, 6 1) 1, 2, 3, 4,,,,,,,,,, 2),,, 54, 55, 56, 57,,,,,, 3) 1234,,,,, 1239,, Whole numbers: The whole numbers are 0, 1, 2, 3, 4, 5, 6 4) 0, 1, 2, 3, 4, 5, 6,,,,,,,,,,,, 5),,,,,, 30, 31,,,, 35, 6) 223, 224,,,,,,, Integer numbers: Integers are whole numbers together with their negatives -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 7),,,,,,,, -4, -3, -2, -1, 0, 1, 2, 3, 4,,,,,,,,,,,,... 8) - 10,009,,,,,,,,, - 10,000 In each of the above examples, three dots ( ) were used at the beginning or at the end of the set of numbers. These three dots are called ellipsis points and in mathematics they are the symbol for the term infinite. Infinite means without end. When used in conjunction with a number or a set of numbers, the ellipsis points indicate that the numbers could go on forever. For example, 6 means that the numbers are infinite. There is no end to the numbers that could follow the numeral six. If you started counting you could not count all of the numbers that would follow. None of the above number systems include fractions. When you are solving problems and you are limited to counting numbers, whole numbers, or integers, there are no fractions. 1

5 Name Date Real Numbers A common term used in mathematics is real number system. The real number system was developed by mathematicians to include fractions. Here is a quick review of fractions. Fractions have a numerator and a denominator. The numerator is the number above the line, and the denominator is the number below the line. The numerator may be any integer. The denominator may be any integer except zero (0). In the fraction, the three (3) is the numerator, and the four (4) is the denominator. The 3 and 4 are both integers, and the denominator is not zero. When any number is divided by zero there is no possible answer. For that reason the definition of a fraction states that the denominator of a fraction cannot be zero. Example: is not a fraction because the denominator is zero. Attempt to solve the problems below. Divide to the fourth decimal place. 1) = 3 0 = 2) = 5 0 = 3) = 0 2 = Look at the following numbers. Circle the ones that are fractions. 4) 5) 6) 7) 8) 9) 10) Real numbers are sometimes rational numbers, and sometimes they are irrational numbers. Real Numbers Rational Numbers Irrational Numbers Rational numbers are numbers that can be written as the quotient of fractions. Examples: a) = 3 4 = answer: 0.75 b) = 2 3 = answer: Notice the above fractions are quite different. In the first example, the quotient of 0.75 is said to be a non-repeating decimal, which means that it cannot be divided any further. In the second example, the answer is a repeating decimal. In other words, no matter how long you divided, you would continue to get The ellipsis points ( ) indicate that further division would produce the number 6 infinitely. However, both of these fractions are examples of rational numbers since the answers can be written as the quotient of a fraction. 2

6 Name Date Quotient Questions Complete the following. Express each answer as the quotient of the fraction. In the space provided, show all work for finding the answers to Questions 1, 2, and 3. Then use a calculator to find the solutions that are repeating decimals, and write their answers carried to eight places. 1) = 3 5 = 2) = 7 8 = 3) = 8 9 = You may use your calculator for answering the following. 4) = = 5) = = 6) = 7) = 8) = 9) = 10) = 11) = 12) = 13) = 14) = 15) = 16) = 17) List the above fractions that are non-repeating or terminating decimals. a) b) c) d) e) f) g) h) 18) List the above fractions that are repeating decimals. a) b) c) d) e) 19) Each of the above fractions is a rational number because all of the fractions can be written as a quotient that is either a or a decimal. 20) Any integer can be written as a rational number by writing the integer as the numerator of a fraction and writing the as 1. 3

Helping Students Understand Algebra

Helping Students Understand Algebra By Barbara Sandall, Ed.D., and Mary Swarthout, Ph.D. COPYRIGHT 2005 Mark Twain Media, Inc. ISBN 10-digit: 1-58037-293-7 13-digit: 978-1-58037-293-0 Printing No. CD-404020