Whole Numbers and Introduction to Algebra

Transcription

1 M01_TOBE7935_06_SE_C01.QXD 9/24/08 5:03 PM Page 1 CHAPTER Should I buy or lease a car? What are the benefits of each? Which choice fits my needs? Does leasing a car or buying a car save me the most money? Turn to Putting Your Skills to Work on page 62 to find out. Whole Numbers and Introduction to Algebra 1.1 UNDERSTANDING WHOLE NUMBERS ADDING WHOLE NUMBER EXPRESSIONS SUBTRACTING WHOLE NUMBER EXPRESSIONS MULTIPLYING WHOLE NUMBER EXPRESSIONS DIVIDING WHOLE NUMBER EXPRESSIONS EXPONENTS AND THE ORDER OF OPERATIONS 55 HOW AM I DOING? SECTIONS MORE ON ALGEBRAIC EXPRESSIONS INTRODUCTION TO SOLVING LINEAR EQUATIONS SOLVING APPLIED PROBLEMS USING SEVERAL OPERATIONS 82 CHAPTER 1 ORGANIZER 92 CHAPTER 1 REVIEW PROBLEMS 95 HOW AM I DOING? CHAPTER 1 TEST 101 1

2 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page UNDERSTANDING WHOLE NUMBERS Student Learning Objectives After studying this section, you will be able to: Understand place values of whole numbers. Write whole numbers in expanded notation. Write word names for whole numbers. Use inequality symbols with whole numbers. Round whole numbers. Often we learn a new concept in stages. First comes learning the new terms and basic assumptions. Then we have to master the reasoning, or logic, behind the new concept. This often goes hand in hand with learning a method for using the idea. Finally, we can move quickly with a shortcut. For example, in the study of stock investments, before tackling the question What is my profit from this stock transaction? you must learn the meaning of such terms as stock, profit, loss, and commission. Next, you must understand how stocks work (reasoning/logic) so that you can learn the method for calculating your profit. After you master this concept, you can quickly answer many similar questions using shortcuts. In this book, watch your understanding of mathematics grow through this same process. In the first chapter we review the whole numbers, emphasizing concepts, not shortcuts. Do not skip this review even if you feel you have mastered the material since understanding each stage of the concepts is crucial to learning algebra. With a little patience in looking at the terms, reasoning, and step-by-step methods, you ll find that your understanding of whole numbers has deepened, preparing you to learn algebra. Understanding Place Values of Whole Numbers We use a set of numbers called whole numbers to count a number of objects. The whole numbers are as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, Á There is no largest whole number. The three dots Á indicate that the set of whole numbers goes on forever. The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits. The position or placement of the digit in a number tells the value of the digit. For this reason, our number system is called a place-value system. For example, look at the following three numbers. 632 The 6 means 6 hundreds (600). 61 The 6 means 6 tens (60). 6 The 6 means 6 ones (6). To illustrate the values of the digits in a number, we can use the following place-value chart. Consider the number 847,632, which is entered on the chart. Place Value Chart Hundred millions Ten millions Millions Hundred thousands Ten thousands Thousands Hundreds Tens Ones, 8 4 7, Periods Millions Thousands Ones 2 The digit 8 is in the hundred thousands place. The digit 4 is in the ten thousands place. The digit 7 is in the thousands place. The digit 6 is in the hundreds place. The digit 3 is in the tens place. The digit 2 is in the ones place. When we write very large numbers, we place a comma after every group of three digits, moving from right to left. These three-digit groups are called periods. It is usually agreed that four-digit numbers do not have a comma, but numbers with five or more digits do.

3 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 3 Section 1.1 Understanding Whole Numbers 3 EXAMPLE 1 In the number 573,025: (a) In what place is the digit 7? (b) In what place is the digit 0? (a) 5 7 3,025 c ten thousands (b) 573, 0 25 c hundreds Teaching Example 1 In the number 63,752,481: (a) In what place is the digit 6? (b) In what place is the digit 2? Ans: (a) ten millions (b) thousands Practice Problem 1 In the number 3,502,781: (a) In what place is the digit 5? hundred thousands (b) In what place is the digit 0? ten thousands NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Writing Whole Numbers in Expanded Notation We sometimes write numbers in expanded notation to emphasize place value. The number 47,632 can be written in expanded notation as follows: 40, ten thousands thousands hundreds tens ones EXAMPLE 2 Write 1,340,765 in expanded notation. We write 1 followed by a zero for each of the remaining digits. T We write 1,340,765 as 1,000, , , We continue in this manner for each digit. Since there is a zero in the thousands place, we do not write it as part of the sum. Teaching Example 2 Write 509,637 in expanded notation. Ans: 500, Practice Problem 2 Write 2,507,235 in expanded notation. 2,000, , EXAMPLE 3 Jon withdraws $493 from his account. He requests the minimum number of bills in one-, ten-, and hundred-dollar bills. Describe the quantity of each denomination of bills the teller must give Jon. If we write $493 in expanded notation, we can easily describe the denominations needed hundreddollar bills tendollar bills onedollar bills Teaching Example 3 Tom withdraws $642 from his account. He requests the minimum number of bills in one-, ten-, and hundred-dollar bills. Describe the quantity of each denomination of bills the teller must give Tom. Ans: 6 hundred-dollar bills, 4 ten-dollar bills, 2 one-dollar bills Practice Problem 3 Christina withdraws $582 from her account. She requests the minimum number of bills in one-, ten-, and hundred-dollar bills. Describe the quantity of each denomination of bills the teller must give Christina. 5 hundred-dollar bills, 8 ten-dollar bills, 2 one-dollar bills

4 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 4 4 Chapter 1 Whole Numbers and Introduction to Algebra Understanding the Concept The Number Zero Not all number systems have a zero. The Roman numeral system does not. In our place-value system the zero is necessary so that we can write a number such as 308. By putting a zero in the tens place, we indicate that there are zero tens. Without a zero symbol we would not be able to indicate this. For example, 38 has a different value than 308. The number 38 means three tens and eight ones, while 308 means three hundreds and eight ones. In this case, we use zero as a placeholder. It holds a position and shows that there is no other digit in that place. Writing Word Names for Whole Numbers Sixteen, twenty-one, and four hundred five are word names for the numbers 16, 21, and 405. We use a hyphen between words when we write a two-digit number greater than twenty. To write a word name, start from the left. Name the number in each period, followed by the name of the period, and a comma. The last period name, ones, is not used. Teaching Example 4 Write a word name for each number. (a) 73,024 (b) 2,462,108 Ans: (a) Seventy-three thousand, twenty-four (b) Two million, four-hundred sixty-two thousand, one hundred eight NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 EXAMPLE 4 Write a word name for each number. (a) 2135 (b) 300,460 the period. (a) 2135 (b) 300,460 Look at the place-value chart on page 2 if you need help identifying The number begins with 2 in the thousands place. The word name is two thousand, one hundred thirty-five. c We use a hyphen here. The number begins with 3 in the hundred thousands place. Three hundred thousand, four hundred sixty c We place a comma here to match the comma in the number. Practice Problem 4 Write a word name for each number. (a) 4006 four thousand, six one million, two hundred twenty (b) 1,220,032 thousand, thirty-two CAUTION: We should not use the word and in the word names for whole numbers. Although we may hear the phrase three hundred and two for the number 302, it is not technically correct. As we will see later in the book, we use the word and for the decimal point when using decimal notation. Using Inequality Symbols with Whole Numbers It is often helpful to draw pictures and graphs to help us visualize a mathematical concept. A number line is often used for whole numbers. The following number line has a point matched with zero and with each whole number. Each number is equally spaced, and the : arrow at the right end indicates that the numbers go on forever. The numbers on the line increase from left to right If one number lies to the right of a second number on the number line, it is greater than that number. 4 lies to the right of 2 on the number line because 4 is greater than

5 M01_TOBE7935_06_SE_C01.QXD 10/14/08 2:48 PM Page 5 A number is less than a given number if it lies to the left of that number on the number line. Section 1.1 Understanding Whole Numbers 5 3 lies to the left of 5 on the number line because 3 is less than The symbol 7 means is greater than, and the symbol 6 means is less than. Thus we can write T T 4 is greater than 2. 3 is less than 5. The symbols 6 and 7 are called inequality symbols. The statements and are both correct. Note that the inequality symbol always points to the smaller number. EXAMPLE 5 Replace each question mark with the inequality symbol 6 or 7. (a) 1? 6 (b) 8? 7 (c) 4? 9 (d) 9? 4 (a) T 1 is less than 6. (b) T 8 is greater than 7. (c) T 4 is less than 9. (d) T 9 is greater than 4. Teaching Example 5 Replace each question mark with the inequality symbol 6 or 7. (a) 5? 9 (b) 3? 0 (c) 0? 3 (d) 11? 6 Ans: (a) (b) (c) (d) Replace each question mark with the inequality sym- Practice Problem 5 bol 6 or 7. (a) 3? 2 7 (b) 6? 8 6 (c) 1? 7 6 (d) 7? 1 7 EXAMPLE 6 (a) Five is less than eight. (a) Five is less than eight. T T T Rewrite using numbers and an inequality symbol. (b) Nine is greater than four. (b) Nine is greater than four. T T T Remember, the inequality symbol always points to the smaller number. Teaching Example 6 Rewrite using numbers and an inequality symbol. (a) Four is greater than two. (b) Six is less than ten. Ans: (a) (b) Practice Problem 6 Rewrite using numbers and an inequality symbol. (a) Seven is greater than two (b) Three is less than four Rounding Whole Numbers We often approximate the values of numbers when it is not necessary to know the exact values. These approximations are easier to use and remember. For example, if our hotel bill was $82.00, we might say that we spent about $80. If a car cost $14,792, we would probably say that it cost approximately $15,000. Why did we approximate the price of the car at $15,000 and not $14,000? To understand why, let s look at the number line. 14,792 13,000 14,000 15,000 16,000 The number 14,792 is closer to 15,000 than to 14,000, so we approximate the cost of the car at $15,000.

6 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 6 6 Chapter 1 Whole Numbers and Introduction to Algebra It would also be correct to approximate the cost at $14,800 or $14,790, since each of these values is close to 14,792 on the number line. How do we know which approximation to use? We specify how accurate we would like our approximation to be. Rounding is a process that approximates a number to a specific round-off place (ones, tens, hundreds, Á ). Thus the value obtained when rounding depends on how accurate we would like our approximation to be. To illustrate, we round the price of the car discussed above to the thousands and to the hundreds place. 14,792 rounded to the nearest thousand is 15,000. The round-off place is thousands. 14,792 rounded to the nearest hundred is 14,800. The round-off place is hundreds. We can use the following set of rules instead of a number line to round whole numbers. PROCEDURE TO ROUND A WHOLE NUMBER 1. Identify the round-off place digit. 2. If the digit to the right of the round-off place digit is: (a) Less than 5, do not change the round-off place digit. (b) 5 or more, increase the round-off place digit by Replace all digits to the right of the round-off place digit with zeros. Teaching Example 7 Round 237,069 to the nearest ten thousand. Ans: 240,000 NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Teaching Example 8 Round 14,075,829 to the nearest hundred thousand. Ans: 14,100,000 EXAMPLE 7 Do not change the round-off place digit. Round 57,441 to the nearest thousand. The round-off place digit is in the thousands place. 5 ~ 7, c c 57,000 ()* c 1. Identify the round-off place digit The digit to the right is less than Replace all digits to the right with zeros. We have rounded 57,441 to the nearest thousand: 57,000. This means that 57,441 is closer to 57,000 than to 58,000. Practice Problem 7 Round 34,627 to the nearest hundred. 34,600 EXAMPLE 8 Increase the round-off place digit by 1. Round 4,254,423 to the nearest hundred thousand. The round-off place digit is in the hundred thousands place. 4, ~ 2 5 4, Identify the round-off place digit 2. c 2. The digit to the right is 5 or more. c 4,300,000 ()* c 3. Replace all digits to the right with zeros. We have rounded 4,254,423 to the nearest hundred thousand: 4,300,000. Practice Problem 8 Round 1,335,627 to the nearest ten thousand. 1,340,000 CAUTION: The round-off place digit either stays the same or increases by 1. It never decreases.

7 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page EXERCISES Verbal and Writing Skills 1. Write the word name for (a) Eight thousand two (b) 802. Eight hundred two (c) 82. Eighty-two (d) What is the place value of the digit 0 in the number eight hundred twenty? One 2. Write in words. (a) Two is less than five. (b) Five is greater than two. (c) What can you say about parts (a) and (b)? We can use the inequality symbols to show the relationship between 5 and 2 in two different ways. 3. In the number 9865: (a) In what place is the digit 8? (b) In what place is the digit 5? Hundreds Ones 4. In the number 23,981: (a) In what place is the digit 2? (b) In what place is the digit 9? Ten thousands Hundreds 5. In the number 754,310: (a) In what place is the digit 4? (b) In what place is the digit 7? Thousands Hundred thousands 6. In the number 913,728: (a) In what place is the digit 9? (b) In what place is the digit 1? Hundred thousands Ten thousands 7. In the number 1,284,073: (a) In what place is the digit 1? (b) In what place is the digit 0? Millions Hundreds 8. In the number 3,098,269: (a) In what place is the digit 0? (b) In what place is the digit 8? Hundred thousands Thousands Write in expanded notation , , , , , , Damian withdraws $562 from his account. He requests the minimum number of bills in one-, ten-, and hundred-dollar bills. Describe the quantity of each denomination of bills the teller must give Damian. 5 hundred-dollar bills, 6 ten-dollar bills, and 2 one-dollar bills 16. Erin withdraws $274 from her account. She requests the minimum number of bills in one-, ten-, and hundreddollar bills. Describe the quantity of each denomination of bills the teller must give Erin. 2 hundred-dollar bills, 7 ten-dollar bills, and 4 one-dollar bills 17. Describe the denominations of bills for $46: (a) Using only ten- and one-dollar bills. 4 ten-dollar bills and 6 one-dollar bills (b) Using tens, fives, and only 1 one-dollar bill. 4 ten-dollar bills, 1 five-dollar bill, and 1 one-dollar bill; Answers may vary. Write a word name for each number Six thousand seventy-nine 18. Describe the denominations of bills for $96: (a) Using only ten- and one-dollar bills. 9 ten-dollar bills and 6 one-dollar bills (b) Using tens, fives, and only 1 one-dollar bill. 9 ten-dollar bills, 1 five-dollar bill, and 1 one-dollar bill; Answers may vary Four thousand thirty-two ,491 Eighty-six thousand, four hundred ninety-one ,224 Thirty-three thousand, two hundred twenty-four 7

8 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 8 8 Chapter 1 Whole Numbers and Introduction to Algebra 23. Fill in the check with the amount $ Fill in the check with the amount $379. James Hunt 4 Platt St. Mapleville, RI Hampton Apartments Six hundred seventy-two and 00 /100 PAY to the ORDER of Mason Bank California MEMO DATE DOLLARS Ellen Font 22 Rose Place Garden Grove, CA Atlas Insurance Three hundred seventy-nine and 00 /100 PAY to the ORDER of Mason Bank California MEMO DATE DOLLARS Replace each question mark with the inequality symbol 6 or ? ? ? ? ? ? ? ? ? ? ,647? 616, ,351? 101,251 7 Rewrite using numbers and an inequality symbol. 37. Five is greater than two Seven is less than ten Two is less than five Six is greater than four Automobile Prices The table lists the 2008 sticker prices on some popular vehicles. Use this table to answer exercises 41 and 42. Type of Automobile 2008 MSRP 2008 Ford Expedition XLT $30, Ford Supercab XLT $27, Dodge Charger SXT G $25, Dodge Caravan SXT K $26,805 Source: Replace the question mark with an inequality symbol to indicate the relationship between the prices of the vehicles. 41. Ford Expedition XLT? Ford Supercab XLT Dodge Caravan SXT K? Dodge Charger SXT G 7 Round to the nearest ten Round to the nearest hundred ,854 63, ,799 12, , , , ,500 Round to the nearest thousand ,431 38, ,312 56, , , , ,000 Round to the nearest hundred thousand ,254,423 5,300, ,395,999 1,400, ,007,601 9,000, ,116,201 3,100, The Sun The diameter of the sun is approximately 865,000 miles. Round this figure to the nearest ten thousand. 870,000 miles 60. Inches and Miles There are 3,484,800 inches in 55 miles. Round 3,484,800 to the nearest ten thousand. 3,480,000 inches One Step Further Round to the nearest hundred ,962 17, ,972 45,000

9 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 9 Section 1.1 Understanding Whole Numbers 9 Very large numbers are used in some disciplines to measure quantities, such as distance in astronomy and the national debt in macroeconomics. We can extend the place-value chart to include these large numbers. 63. Write 5,311,192,809,000 using the word name. Five trillion, three hundred eleven billion, one hundred ninety-two million, eight hundred nine thousand 64. Round 5,311,192,809,000 to the nearest million. 5,311,193,000,000 Hundred trillions Ten trillions Trillions Hundred billions Ten billions Billions 5, 3 1 1, Hundred millions Ten millions Millions Hundred thousands Ten thousands Thousands Hundreds Tens Ones, 8 0 9, To Think About Sometimes to get an approximation we must round to the nearest unit, such as a foot, yard, hour, or minute. 65. Train Travel Time A train takes 3 hours and 50 minutes to reach its destination. Approximately how many hours does the trip take? 4 hours 66. Automobile Travel Time An automobile trip takes 5 hours and 40 minutes. Approximately how many hours does the drive take? 6 hours 67. Fence Measurements The Nguyens backyard has a fence around it that measures 123 feet 5 inches. Approximately how many feet of fencing do the Nguyens have? 123 feet 68. Yardage Measurements Jessica has 15 yards 4 inches of material. Approximately how many yards of material does Jessica have? 15 yards Quick Quiz Write 6402 in expanded notation. 2. Replace each question mark with the appropriate symbol 6 or 7. (a) 0? 10 (b) 15? Round 154,627 to (a) the nearest ten thousand 150,000 (b) the nearest hundred 154, Concept Check Explain how to round 8937 to the nearest hundred. Answers may vary Classroom Quiz 1.1 You may use these problems to quiz your students mastery of Section Write 5301 in expanded notation. Ans: Replace each question mark with the appropriate symbol 6 or 7. (a) 8? 0 Ans: 7 (b) 2? 11 Ans: 6 3. Round 3571 to (a) the nearest thousand Ans: 4000 (b) the nearest ten Ans: 3570 Finding a Study Partner Attempt to make a friend in your class and become study partners. You may find that you enjoy sitting together and drawing support and encouragement from each other. You must not depend on a friend or fellow student to tutor you, do your work for you, or in any way be responsible for your learning. However, you will learn from each other as you seek to master the course. Studying with a friend and comparing notes, methods, and solutions can be very helpful. And it makes learning mathematics a lot more fun! Exercises 1. Exchange phone numbers with someone in class so you can call each other whenever you are having difficulty with your studying. 2. Set up convenient times to study together on a regular basis, to do homework, and to review for exams.

10 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page ADDING WHOLE NUMBER EXPRESSIONS Student Learning Objectives After studying this section, you will be able to: Use symbols and key words for expressing addition. Use properties of addition to rewrite algebraic expressions. Evaluate algebraic expressions involving addition. Add whole numbers when carrying is needed. Find the perimeters of geometric figures. Using Symbols and Key Words for Expressing Addition What is addition? We perform addition when we group items together. Consider the following illustration involving the sale of bikes. is equal to Bikes sold Saturday Bikes sold Sunday Total bikes sold bikes We see that the number 7 is the total of 4 and 3. That is, = 7 is an addition fact. The numbers being added are called addends. The result is called the sum = 7 addend addend sum In mathematics we use symbols such as + in place of the words sum or plus. The English phrase five plus two written using symbols is Writing English phrases using math symbols is like translating between languages such as Spanish and French. There are several English phrases that describe the operation of addition. The following table gives some of them and their translated equivalents written using mathematical symbols. English Phrase Six more than nine The sum of some number and seven Four increased by two Three added to a number One plus a number Translation into Symbols x n x When we do not know the value of a number, we use a letter, such as x, to represent that number. A letter that represents a number is called a variable. Notice that the variables used in the table above are different. We can choose any letter as a variable. Thus we can represent a number plus seven by x + 7, a + 7, n + 7, y + 7, and so on. Combinations of variables and numbers such as x + 7 and a + 7 are called algebraic expressions or variable expressions. Teaching Example 1 Translate each English phrase using numbers and symbols. (a) A number plus two (b) Twelve more than a number Ans: (a) n + 2 (b) x + 12 EXAMPLE 1 Translate each English phrase using numbers and symbols. (a) The sum of six and eight (b) A number increased by four (a) The sum of six and eight (b) A number increased by four 6 8 x 4 Although we used the variable x to represent the unknown quantity in part (b), any letter could have been used. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 10 Practice Problem 1 Translate each English phrase using numbers and symbols. (a) Five added to some number x + 5 (b) Four more than five 5 + 4

11 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 11 Section 1.2 Adding Whole Number Expressions 11 Using Properties of Addition to Rewrite Algebraic Expressions Most of us memorized some basic addition facts. Yet if we study these sums, we observe that there are only a few addition facts for each one-digit number that we must memorize. For example, we can easily see that when 0 items are added to any number of items, we end up with the same number of items: = 5, = 8, and so on. This illustrates the identity property of zero: a + 0 = a and 0 + a = a. EXAMPLE 2 Express 4 as the sum of two whole numbers. Write all possibilities. How many addition facts must we memorize? Why? Starting with 4 + 0, we write all the sums equal to 4 and observe any patterns. The numbers in this column decrease by 1. The last two rows of the pattern are combinations of the same numbers listed in the first three rows The numbers in this column increase by 1. We need to learn only two addition facts for the number four: and The remaining facts are either a repeat of these or use the fact that when 0 is added to any number, the sum is that number. Practice Problem 2 Express 8 as the sum of two whole numbers. Write all possibilities. How many addition facts must we memorize? Why? We need to learn only four addition facts for the number 8: 7 + 1, 6 + 2, 5 + 3, and The remaining facts are either a repeat of these or use the fact that when 0 is added to any number, the sum is that number. In Example 2 we saw that the order in which we add numbers doesn t affect the sum. That is, = 4 and = 4. This is true for all numbers and leads us to a property called the commutative property of addition. Teaching Example 2 Express 6 as the sum of two whole numbers. Write all possibilities. How many addition facts must we memorize? Why? Ans: We need to learn three addition facts for the number six: 5 + 1, 4 + 2, and The remaining facts are either a repeat of these, or use the identity property of zero. Teaching Tip Point out to students that Example 2 shows how thinking about numbers and trying to understand mathematical concepts is often easier than memorizing. Ask students to think about all those hours they spent using flash cards to memorize addition facts. COMMUTATIVE PROPERTY OF ADDITION a + b = b + a = = 13 Two numbers can be added in either order with the same result. EXAMPLE 3 Use the commutative property of addition to rewrite each sum. (a) (b) 7 + n (c) x + 3 (a) = (b) 7 + n = n + 7 (c) x + 3 = 3 + x Notice that we applied the commutative property of addition to the expressions with variables n and x. That is because variables represent numbers, even though they are unknown numbers. Teaching Example 3 Use the commutative property of addition to rewrite each sum. (a) (b) 11 + x (c) a + 15 Ans: (a) (b) x + 11 (c) 15 + a

12 M01_TOBE7935_06_SE_C01.QXD 9/22/08 5:11 PM Page Chapter 1 Whole Numbers and Introduction to Algebra Practice Problem 3 Use the commutative property of addition to rewrite each sum. (a) x x (b) 9 + w w + 9 (c) Teaching Example 4 If n = 532, then n =? Ans: 532 NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 EXAMPLE 4 If = 2725, then =? = 2725 Why? The commutative property states that the order in which we add numbers doesn t affect the sum. Practice Problem 4 If x + y = 6075, then y + x =? 6075 Teaching Example 5 Simplify x Ans: 12 + x or x + 12 To simplify an expression like x, we find the sum of 8 and x = 9 + x or x + 9 Simplifying x is similar to rewriting the English phrase 8 plus 1 plus some number as the simpler phrase 9 plus some number. Since addition is commutative, we can write this simplification as either 9 + x or x + 9. We choose to write this sum as x + 9, since it is standard to write the variable in the expression first. EXAMPLE 5 Simplify n To simplify, we find the sum of the known numbers n = 5 + n or = n + 5 We cannot add the variable n and the number 5 because n represents an unknown quantity; we have no way of knowing what quantity to add to the number 5. Teaching Tip Students often confuse the rules for simplifying 2 + 3x, 2x + 3x, and (2x)(3x). Introducing the concept through examples before a formal rule is less likely to confuse them. It makes sense to students that we cannot Practice Problem 5 Simplify x 9 + x or x + 9 add an unknown quantity to a number. Addition of more than two numbers may be performed in more than one manner. To add we can first add the 5 and 2, or we can add the 2 and 1 first. We indicate which sum we add first by using parentheses. We perform the operation inside the parentheses first = = = = = = 8 In both cases the order of the numbers 5, 2, and 1 remains unchanged and the sums are the same. This illustrates the associative property of addition. ASSOCIATIVE PROPERTY OF ADDITION 1a + b2 + c = a + 1b + c = = = 14 When we add three or more numbers, the addition may be grouped in any way.

13 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 13 Section 1.2 Adding Whole Number Expressions 13 EXAMPLE 6 Use the associative property of addition to rewrite the sum and then simplify. 1x x = x The associative property allows us to regroup. = x + 9 Simplify: = 9. Teaching Example 6 Use the associative property of addition to rewrite the sum and then simplify. 1a Ans: a + 6 Practice Problem 6 Use the associative property of addition to rewrite the sum and then simplify. 1w w + 5 Sometimes we must use both the associative and commutative properties of addition to rewrite a sum and simplify. In other words, we can change the order in which we add (commutative property) and regroup the addition (associative property) to simplify an expression. EXAMPLE 7 Use the associative and/or commutative property as necessary to simplify the expression n n + 72 = n2 = n = 12 + n 5 + 1n + 72 = n + 12 The commutative property allows us to change the order of addition. Regroup the sum using the associative property. Simplify. Write 12 + n as n Practice Problem 7 Use the associative and/or commutative property as necessary to simplify each expression. (a) 12 + x2 + 8 x + 10 (b) 14 + x x + 8 Understanding the Concept Addition Facts Made Simple There are many methods that can be used to add one-digit numbers. For example, if you can t remember that = 15 but can remember that = 14, just add 1 to 14 to get 15. Exercises 7+8=7+(7+1) =(7+7)+1 =14+1 =15 Another quick way to add is to use the sum = 10, since it is easy to remember. Let s use this to add =(2+5)+5 =2+(5+5) =2+10 =12 1. Use the fact that = 10 to add = = = = Use the fact that = 12 to add = = = = 14 Teaching Example 7 Use the associative and/or commutative property as necessary to simplify the expression a Ans: a + 15 Teaching Tip Many students need to review addition and multiplication facts. They often resist since they feel that they can use a calculator. This is a good time to let them know that in arithmetic and algebra, as well as in real-life situations, we need to be able to do calculations mentally without the use of a calculator. Remind them of the following situations that require quick mental math. 1. You often have just a few seconds to check the change you receive or to make change for others. 2. While shopping, you need to be sure that you have enough money to pay for the cost of several items. 3. In arithmetic, you need to know the multiplication facts when working with fractions. 4. In algebra, there are often many steps, thus using a calculator for every step of the process is slow and cumbersome. Emphasizing the importance of knowing the addition facts makes students more responsive to shortcuts that make the task easier. After discussing the Understanding the Concept, ask students to find other shortcuts.

14 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page Chapter 1 Whole Numbers and Introduction to Algebra Evaluating Algebraic Expressions Involving Addition We have already learned that when we do not know the value of a number, we designate the number by a letter. We call this letter a variable. We use a variable to represent an unknown number until such time as its value can be determined. For example, if 6 is added to a number but we do not know the number, we could write n + 6 where n is the unknown number. If we were told that n has the value 9, we could replace n with 9 and then simplify. n Replace n with 9. Simplify by adding. Thus n + 6 has the value 15 when n is replaced by 9. This is called evaluating the expression n + 6 if n is equal to 9. To evaluate an algebraic expression, we replace the variables in the expression with their corresponding values and simplify. An algebraic expression has different values depending on the values we use to replace the variable. Teaching Example 8 Evaluate m + n + 7 for the given values of m and n. (a) m is equal to 3 and n is equal to 1 (b) m is equal to 5 and n is equal to 9 Ans: (a) 11 (b) 21 EXAMPLE 8 Evaluate x + y + 3 for the given values of x and y. (a) x is equal to 6 and y is equal to 1 (b) x is equal to 4 and y is equal to 2 (a) x + y + 3 Replace x with and y with Simplify. When x is equal to 6 and y is equal to 1, x + y + 3 is equal to 10. (b) x + y Replace x with 4 and y with 2. Simplify. When x is equal to 4 and y is equal to 2, x + y + 3 is equal to 9. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Practice Problem 8 Evaluate x + y + 6 for the given values of x and y. (a) x is equal to 9 and y is equal to 3 18 (b) x is equal to 1 and y is equal to 7 14 Adding Whole Numbers When Carrying Is Needed Of course, we are often required to add numbers that have more than a single digit. In such cases we must: 1. Arrange the numbers vertically, lining up the digits according to place value. 2. Add first the digits in the ones column, then the digits in the tens column, then those in the hundreds column, and so on, moving from right to left. Sometimes the sum of a column is a multidigit number that is, a number larger than 9. When this happens we evaluate the place values of the digits to find the sum.

15 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 15 Section 1.2 Adding Whole Number Expressions 15 EXAMPLE Add We arrange numbers vertically and add the digits in the ones column first, then the digits in the tens column. 6 tens 8 ones 2 tens 5 ones 8 tens 13 ones We cannot have two digits in the ones column, so we must rename 13 as 1 ten and 3 ones. Teaching Tip Tell students that if they need more practice adding whole numbers, they can find more exercises in Appendix D. Teaching Example 9 Add Ans: tens+1 ten+3 ones 9 tens+3 ones=93 A shorter way to do this problem involves a process called carrying. Instead of rewriting 13 ones as 1 ten and 3 ones we would carry the 1 ten to the tens column by placing a 1 above the 6 and writing the 3 in the ones column of the sum ones+5 ones=13 ones Add 1 ten+6 tens+2 tens. Practice Problem 9 Add Often you must carry several times, by bringing the left digit into the next column to the left. EXAMPLE 10 A market research company surveyed 1870 people to determine the type of beverage they order most often at a restaurant. The results of the survey are shown in the table. Find the total number of people whose responses were iced tea, soda, or coffee. Teaching Example 10 Find the total number of people whose responses were milk, iced tea, or orange juice. Ans: 1118 We add whenever we must find the total amount Iced tea Soda Coffee We add 7+7+4=18. Since 18 equals 1 ten and 8 ones, we carry 1 ten placing a 1 at the top of the tens column. We add =21. Since 21 tens equals 2 hundreds and 1 ten, we carry 2 hundreds placing a 2 at the top of the hundreds column. We add 2+3+5=10. Since 10 hundreds equals 1 thousand and zero hundreds, we write 0 in the hundreds column and 1 in the thousands column = 1018 A total of 1018 people responded ice tea, soda, or coffee. Type of Beverage Soda Orange juice Coffee Iced tea Milk Other Number of Responses

16 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page Chapter 1 Whole Numbers and Introduction to Algebra NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Practice Problem 10 Use the survey results from Example 10 to answer the following: Find the total number of people whose responses were milk, orange juice, or other. 852 Finding the Perimeters of Geometric Figures Geometry has a visual aspect that many students find helpful to their learning. Numbers and abstract quantities may be hard to visualize, but we can take pen in hand and draw a picture of a rectangle that represents a room with certain dimensions. We can easily visualize problems such as what is the distance around the outside edges of the room (perimeter)? In this section we study rectangles, squares, triangles, and other complex shapes that are made up of these figures. A rectangle is a four-sided figure like the ones shown here. A rectangle has the following two properties: 1. Any two adjoining sides are perpendicular. 2. Opposite sides are equal. When we say that any two adjoining sides are perpendicular we mean that any two sides that join at a corner form an angle that measures 90 degrees (called a right angle) and thus forms one of the following shapes. When we say that opposite sides are equal we mean that the measure of a side is equal to the measure of the side across from it. When all sides of a rectangle are the same length, we call the rectangle a square. 7 Teaching Tip The formulas for perimeter are not used here because the goal is not to ask students to memorize formulas but to have them understand that perimeter is the distance around an object. Students will be introduced to perimeter formulas in a later section. If students are unfamiliar with the units of measure in either the U.S. or metric system, refer them to Appendix A Opposite sides are equal. A triangle is a three-sided figure with three angles All sides of a square are equal. The distance around an object (such as a rectangle or triangle) is called the perimeter. To find the perimeter of an object, add the lengths of all its sides.

17 M01_TOBE7935_06_SE_C01.QXD 9/22/08 5:11 PM Page 17 Section 1.2 Adding Whole Number Expressions 17 EXAMPLE 11 Find the perimeter of the triangle. (The abbreviation ft means feet). 5 ft 5 ft 7 ft We add the lengths of the sides to find the perimeter. Teaching Example 11 Find the perimeter of the rectangle. Ans: 20 in. 7 in. 3 in. 5 ft 5 ft 5 ft + 5 ft + 7 ft = 17 ft The perimeter is 17 ft. 7 ft Practice Problem 11 Find the perimeter of the square. 60 ft 15 ft If you are unfamilar with the value, meaning, and abbreviations for the metric and U.S. units of measure, refer to Appendix A, which contains a brief summary of this information. EXAMPLE 12 square. Find the perimeter of the shape consisting of a rectangle and a 150 ft 50 ft 65 ft 65 ft 215 ft Teaching Tip After completing Example 12, explain to students that we often do not have all the information we need to solve problems in real-life situations. We must use reasoning and logic skills to find this information, just as we did in Example 12. We want to find the distance around the figure. We look only at the outside edges since dashed lines indicate inside lengths. 65 ft 50 ft? ft 65 ft 150 ft? ft We cross off 65 ft since inside lengths are not included in the perimeter. Teaching Example 12 Find the perimeter of the shape consisting of a rectangle and a square. 215 ft Now we must find the lengths of the unlabeled sides. The shaded figure is a square since the length and width have the same measure. Thus each side of the shaded figure has a measure of 65 ft. 8 m 13 m 5 m 5 m 150 ft This side is 65 ft because the shaded figure is a square. 65 ft 65 ft 50 ft 65 ft 215 ft 115 ft This side equals or 115 ft because opposite sides of a rectangle have the same length. Ans: 62 m

18 M01_TOBE7935_06_SE_C01.QXD 9/22/08 5:11 PM Page Chapter 1 Whole Numbers and Introduction to Algebra Next, we add the length of the six sides to find the perimeter. 150 ft ft ft + 65 ft + 65 ft + 50 ft = 660 ft The perimeter is 660 ft. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Find the perimeter of the shape consisting of a rectan- Practice Problem 12 gle and a square. 450 ft 125 ft 40 ft 30 ft 30 ft 155 ft Teaching Tip After discussing the Developing Your Study Skills, explain to students that we sharpen our brain when we learn math because we develop the ability to reason and think logically. Tell students that we learn logic and basic number facts in different parts of the brain. Thus, we learn algebra and geometry in different parts of the brain algebra in the left part and geometry in the right part. Since some of us learn mainly on the left side of the brain and others on the right, we usually have a preference to either geometry or algebra. Then explain that this class includes both topics to enhance both sides of the brain. Exercising both sides of the brain sharpens us now and may even keep our brains functional in our old age. Understanding the Concept Using Inductive Reasoning to Reach a Conclusion When we reach a conclusion based on specific observations, we are using inductive reasoning. Much of our early learning is based on simple cases of inductive reasoning. If a child touches a hot stove or other appliance several times and each time he gets burned, he is likely to conclude, If I touch something that is hot, I will get burned. This is inductive reasoning. The child has thought about several actions and their outcomes and has made a conclusion or generalization. The following is an illustration of how we use inductive reasoning in mathematics. Find the next number in the sequence 10, 13, 16, 19, 22, 25, 28,... We observe a pattern that each number is 3 more than the preceding number: = 13; = 16, and so on. Therefore, if we add 3 to 28, we conclude that the next number in the sequence is 31. Exercise 1. For each of the following find the next number by identifying the pattern. (a) 8, 14, 20, 26, 32, 38, (b) 17, 28, 39, 50, 61, For more practice, complete exercises on page 22. Preparing to Learn Algebra Many people have learned arithmetic by memorizing facts and properties without understanding why the facts are true or what the properties mean. Learning strictly by memorization can cause problems. For example: Many of the shortcuts in arithmetic do not work in algebra. Memorizing does not help one develop reasoning and logic skills, which are essential to understanding algebra concepts. Memorization can eventually cause memory overload. Trying to remember a collection of unrelated facts can cause you to become anxious and discouraged. In this book you will see familiar arithmetic topics. Do not skip them, even if you feel that you have mastered them. The explanations will probably be different from those you have already seen because they emphasize the underlying concepts. If you don t understand a concept the first time, be patient and keep trying. Sometimes, by working through the material you will see why it works. Read all the Understanding the Concept boxes in the book since they will help you learn mathematics. Exercise 1. Write in words why the commutative property of addition reduces the amount of memorization necessary to learn addition facts. Since the order in which we add numbers does not affect the outcome, we need only learn one addition fact for every two sums.

19 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 19 Verbal and Writing Skills Write in words EXERCISES 10 + x 2. Ten plus a number. Answers may vary. n + 4 Some number plus four. Answers may vary. 3. Write in your own words the steps you must perform to find the answer to the following problem. Evaluate x + 6 if x is equal to 9. Replace x with 9, and then add 9 and Explain why the following statement is true. If x + y + z = 105, then z + y + x = 105. The commutative property allows us to change the order of addition without changing the value of the sum. State what property is represented in each mathematical statement = Associative property of addition 4 + 1x + 32 = x2 Commutative property of addition Translate using numbers and symbols. 7. A number plus two m The sum of five and y 5 + y or y Some number added to twelve 12 + m or m A number increased by seven m Two added to a number m The sum of eight and x 8 + x or x Twelve more than a number y A number plus four y + 4 Use the commutative property of addition to rewrite each sum a 16. y x 18. a y x x x If = 3758, then =? 20. If = 8947, then =? If 5 + n = 12, then n + 5 =? 22. If 8 + x = 31, then x + 8 =? Simplify. 23. x a x + 6 a y 27. x y + 8 x + 2 Use the associative property of addition to rewrite each sum, then simplify x x x + 3 x x n x n n n + 12 x x n n 1a a + 10 Use the associative and/or commutative property as necessary to simplify each expression x y x + 15 y n2 + 5 n x x2 40. x + 9 x a2 a

20 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page Chapter 1 Whole Numbers and Introduction to Algebra n n n + 9 n x x x + 14 x a a n n Evaluate y + 7 for the given values of y. 48. Evaluate n + 8 for the given values of n. (a) y is equal to 3 10 (a) n is equal to 4 12 (b) y is equal to 8 15 (b) n is equal to Evaluate x + y if x is 6 and y is Evaluate a + b if a is 6 and b is Evaluate a + b + c if a is 9, b is 15, and c is Evaluate x + y + z if x is 11, y is 18, and z is Evaluate n + m + 13 if n is 26 and m is Evaluate x + y + 21 if x is 32 and y is Yearly Bonus Pay For exercises 55 and 56, use the table and the formula Bonus x y 250 to calculate the yearly bonus for MJ Industry employees. Bonus = x + y R b x represents the number of productivity units earned. y represents the number of years of employment. Employee Name Employee Number Years of Employment Productivity Units Earned Julio Sanchez Mary McCab Jamal March Leo J. Cornell Calculate the yearly bonus for (a) Mary McCab. $442 (b) Leo J. Cornell. $ Calculate the yearly bonus for (a) Julio Sanchez. $415 (b) Jamal March. $393 Add. For more practice, refer to Appendix D , , , , , , , ,064

21 M01_TOBE7935_06_SE_C01.QXD 7/23/08 5:30 PM Page 21 Section 1.2 Adding Whole Number Expressions 21 Applications Exercises Answer each question. 75. Checking Account Angelica s check register indicates the deposits and debits (checks written or ATM withdrawals) for a 1-month period. 76. Checking Account The bookkeeper for the Spaulding Appliance Company examined the following record from the company account for the month of March. Date Deposits Debits 12/3/09 $159 12/9/09 $63 12/13/09 $241 12/15/09 $121 12/22/09 $44 Date Deposits Debits 3/6/08 $3477 3/9/08 $120 3/13/08 $3500 3/15/08 $4614 3/22/08 $1388 (a) What is the total of the deposits made to Angelica s checking account? $400 (b) What is the total of the debits made to Angelica s checking account? $228 (a) What is the total of the deposits in this time period? $8091 (b) What is the total of the debits in this time period? $ Apartment Expenses The rent on an apartment was $875 per month. To move in, Charles and Vincent were required to pay the first and last month s rent, a security deposit of $500, a connection fee with the utility company of $24, and a cable T.V. installation fee of $35. How much money did they need to move into the apartment? $ Car Expenses Shawnee found that for a 6-month period, in addition to gasoline, she had the following car expenses: insurance, $562; repair to brakes, $276; and new tires, $142. If gasoline for her car cost $495 for this time period, what was the total amount she spent on her car? $1475 Find the perimeter of each rectangle in. 36 in in. 16 in. 1 in. 5 in. Find the perimeter of each square ft 12 ft ft 32 ft Find the perimeter of each triangle in ft 4 in. 4 in. 3 ft 19 ft 6 in. 8 ft Find the perimeters of the shapes made of rectangles and squares ft 58 ft ft 112 ft 12 ft 4 ft 8 ft 24 ft 11 ft 7 ft 25 ft 17 ft