PRE-ALGEBRA BY MYRL SHIREMAN

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1 PRE-ALGEBRA BY MYRL SHIREMAN COPYRIGHT 1994 Mark Twain Media, Inc. ISBN 10-digit: digit: Printing No. CD-1876 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 Introduction... ii... 1 Number Systems... 1 The Integer Number System... 2 Number Properties... 4 Whole Numbers... 5 Commutative Property for Multiplication... 6 Associative Property for Addition... 7 Associative Property for Multiplication... 8 Distributive Property of Multiplication Over Addition... 9 Properties of Subtraction and Division Properties of Zero/Identity Elements Order of Operations Integers Addition of Integers Addition of Integers: Exercises Subtraction of Integers Subtraction of Integers: Rule/Exercises Subtraction of Integers: Exercises/Rule Review Addition and Subtraction of Integers: Exercises Having Fun With Integers Multiplication of Integers Multiplication Chart Division of Integers Division of Integers: Rule Review Variables and Exponents Variables Variables and Multiplication Variables and Division/Variable Exercises Exponents Exponents: Maximum Power, Minimum Space Exponents: Rules to Remember Adding Exponents Reviewing Exponents Multiplying Exponents Zero and Negative Integer Exponents Scientific Notation Simplifying Large Numbers With Scientific Notation iii

3 TABLE OF CONTENTS (continued) Even and Odd Numbers Reviewing Even and Odd Numbers Factoring Learning About Factoring Learning About Prime Factors Finding Prime Factors Learning About the Greatest Common Factor Finding the Greatest Common Factor Learning About the Least Common Multiple Radicals; Square and Cube Roots Radicals and Roots Finding Square Roots Adding Radicals Subtracting Radicals Multiplying and Dividing Radicals Simplifying Radicals Multiplying and Simplifying Radicals Dividing and Simplifying Radicals Simple Equations Learning About Simple Equations Finding the Value of the Unknown I Finding the Value of the Unknown II Reviewing Simple Equations Solving Equations With Multiplication Solving Equations With Division Rectangular Coordinate Systems; Graphing Learning About the Rectangular Coordinate System Reviewing the Parts of the Rectangular Coordinate System Learning About Coordinates Plotting the Point The Rectangular Coordinate System and Equations Linear Equations With Two Variables Linear Equations With Two Variables Plotting Points for Linear Equations Finding the Slope and y Intercept of a Straight Line Answer Key iv

4 Number Systems You have been working with number systems for a long time. The number systems you are most familiar with are the counting numbers and the whole numbers. The counting numbers begin with the number 1, while the whole numbers being with 0. So the whole numbers, which you have used often, include the counting numbers plus 0. In your study of algebra you will need to get to know another number system to solve problems. These new numbers are called the integer numbers. The integer numbers include the whole numbers and a new set of numbers known as negative numbers. The integers are called signed numbers since they include both positive and negative numbers. Let s review the number systems we have talked about. 1, 2, 3, 4, 5, 6, 7, 8, 9... Counting Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9... Whole Numbers... -7, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, +7, +8, Integer Numbers You can see from the above that the integers include numbers to the right and left of 0. The numbers on the right are positive and are noted with the small plus (+) sign. The numbers on the left side of 0 are the negative numbers, and they are noted with the small minus (-) sign. Remember, the integers are known as signed numbers > You can see on the above number line that the integers to the right and left of 0 are exactly the same except for the sign. Any given number on the right has an opposite number on the left. The +5 integer has an opposite on the left, and it is -5. All other numbers except 0 also have an opposite. Complete the following: Counting numbers begin with the number (a). Whole numbers begin with (b). Whole numbers include the counting numbers plus (c). The integer number system includes positive and (d) the left of 0 have a (e) numbers. Those integers to sign, while those to the right have a (f) sign. Integer numbers have opposites. The opposite of +4 is (g). The opposite of -7 is (h). The opposite of +60 is (i). 1

5 The Integer Number System A number line can be very helpful in understanding the integer number system a number system that you use quite frequently in your daily life. Let s use the number line below to keep track of the yards lost and gained in a football game. The distance between each number represents 1 yard. 1st down gain > 2nd down loss Let s say your team has a first down and gains 7 yards. This is a gain, so it is represented on the number line by +7. Now on the second down your team loses 4 yards. A four-yard loss is a -4, so this is represented by showing a -4-yard move to the left from +7. So for the two downs your team has a net gain of +3 yards M T A thermometer, which is much like a number line, can be used to understand positive and negative numbers. Refer to the thermometer at the left, and read the following paragraph. Let s say you live where the winters are very cold. Monday you leave for school and the temperature is +20. Since the 20 has a positive sign it is 20 above 0, which has been marked with the letter M for Monday. Now on Tuesday you leave for school and note the mercury in the thermometer is at point T for Tuesday. Since the T is by the 10 below 0, it has a negative sign and you say it is 10 below zero. Let s practice with the number line below so that you become comfortable with using positive and negative numbers. Point A is at +5 on the number line. If you were told to move +3 places, you would move 3 places to the right of A and be at Point B or at +8. If you were told to move -3 places from Point A, you would be at Point C, which is located at +2. C -3 A +3 B Use the number line at the top of page 3 to locate the following: (a) +3 places from +10 is at (b) -4 places from +6 is at (c) +8 places from -9 is at (e) +10 places from -4 is at (f) -4 places from -2 is at (g) -6 places from -6 is at (d) -7 places from +3 is at 2

6 The Integer Number System (continued) Use the number line below to answer the questions that follow: To get from +4 to +9 you must move spaces in the positive direction. 2. To get from -3 to -9 you must move spaces in the direction. 3. To get from +3 to -4 you must move spaces in the direction. 4. To get from -2 to +6 you must move spaces in the direction. Review: Integers on a number line Those integers on a number line to the left of zero are called (a) numbers. The integers to the right of zero are called (b) numbers. Integers have opposites. The opposite of -7 is +7. What is the opposite of each of the following: (c) +2 is opposite (d) -40 is opposite (e) +800 is opposite (f) +6 is opposite (g) -249 is opposite (h) is opposite

7 Number Properties In algebra you will study number properties. These properties are known as commutative, associative, and distributive. The commutative property of addition says that the order in which you add whole numbers will not change the sum. For example, = 8 and = 8, so = In your study with algebra you will often see letters used to stand for numbers. These letters are called unknowns. You will see the commutative property for addition stated as a + b = b + a where a and b are unknown and can be any whole numbers. Complete the following: = so + 7 = 10 Solve so that a + b = b + a = 35 so 10 + = = so = = 104 so + 84 = = 90 so 65 + = 90 Complete the following using a + b = b + a when you fill in the blank for each problem. a + b = b + a = a=5 b= = + a=35 b= = + a=111 b= = + a=37 b= = + a=2101 b= = + a=10 b= = + a=12 b=23 4

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