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Mathematical Excursions
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Chapter 1 / Exercise 10
Mathematical Excursions
Aufmann
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Unformatted text preview: Algebra and Integers The word algebra comes from the Arabic word al-jebr, which was part of the title of a book about equations and how to solve them. In this unit, you will lay the foundation for your study of algebra by learning about the language of algebra, its properties, and methods of solving equations. Chapter 1 The Tools of Algebra Chapter 2 Integers Chapter 3 Equations 2 Unit 1 Algebra and Integers Vacation Travelers Include More Families “Taking the kids with you is increasingly popular among Americans, according to a travel report that predicts an expanding era of kid-friendly attractions and services.” Source: USA TODAY, November 17, 1999 In this project, you will be exploring how graphs and formulas can help you plan a family vacation. Log on to . Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 1. Lesson Page 1-7 43 2-4 79 3-7 135 USA TODAY Snapshots® Spouses are top travel partners Spouses 58% 34% Children/grandchildren Friends 18% Other family members 14% Solo 13% Group tour 8% Source: Travel Industry Association of America By Cindy Hall and Sam Ward, USA TODAY Unit 1 Algebra and Integers 3 The Tools of Algebra • Lesson 1-1 Use a four-step plan to solve problems and choose the appropriate method of computation. • Lessons 1-2 and 1-3 Translate verbal phrases into numerical expressions and evaluate expressions. • Lesson 1-4 Identify and use properties of addition and multiplication. • Lesson 1-5 equations. Write and solve simple • Lesson 1-6 relations. Locate points and represent • Lesson 1-7 plots. Construct and interpret scatter Algebra is important because it can be used to show relationships among variables and numbers. You can use algebra to describe how fast something grows. For example, the growth rate of bamboo can be described using variables. You will find the growth rate of bamboo in Lesson 1-6. 4 Chapter 1 The Tools of Algebra Key Vocabulary • • • • • order of operations (p. 12) variable (p. 17) algebraic expression (p. 17) ordered pair (p. 33) relation (p. 35) Prerequisite Skills To To be be successful successful in in this this chapter, chapter, you’ll you'll need need to to master master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter X. 1. For Lesson 1-1 Add and Subtract Decimals Find each sum or difference. (For review, see page 713.) 1. 6.6  8.2 2. 4.7  8.5 3. 5.4  2.3 4. 8.6  4.9 5. 2.65  0.3 6. 1.08  1.2 7. 4.25  0.7 8. 4.3  2.89 9. 9.06  1.18 For Lessons 1-1 through 1-5 Estimate with Whole Numbers Estimate each sum, difference, product, or quotient. 10. 1800  285 11. 328  879 12. 22,431  13,183 13. 659  536 14. 68  12 15. 189  89 16. 3845  82 17. 21,789  97 18. $1951  49 For Lessons 1-1 through 1-5 Estimate with Decimals Estimate each sum, difference, product, or quotient. (For review, see pages 712 and 714.) 19. 8.8  5.3 20. 47.2  9.75 21. $7.34  $2.16 22. 83.6  75.32 23. 4.2  29.3 24. 18.8(5.3) 25. 7.8  2.3 26. 54  9.1 27. 21.3  1.7 Problem Solving Make this Foldable to help you organize your notes. Begin with a sheet of unlined paper. Fold Fold the short sides so they meet in the middle. Fold Again Fold the top to the bottom. Cut Unfold. Cut along the second fold to make four tabs. Label Label each of the tabs as shown. Explore Plan Examine Solve Reading and Writing As you read and study the chapter, you can write examples of each problem-solving step under the tabs. Chapter 1 The Tools of Algebra 5 Using a Problem-Solving Plan • Use a four-step plan to solve problems. • Choose an appropriate method of computation. Vocabulary • conjecture • inductive reasoning is it helpful to use a problem-solving plan to solve problems? The table shows the first-class mail rates in 2004. Weight (oz) Cost 1 $0.37 2 $0.60 3 $0.83 4 $1.06 5 $1.29 U.S. MAIL Source: a. Find a pattern in the costs. b. How can you determine the cost to mail a 6-ounce letter? c. Suppose you were asked to find the cost of mailing a letter that weighs 8 ounces. What steps would you take to solve the problem? FOUR-STEP PROBLEM-SOLVING PLAN It is often helpful to have an organized plan to solve math problems. The following four steps can be used to solve any math problem. 1. Explore • Read the problem quickly to gain a general understanding of it. • Ask, “What facts do I know?” “What do I need to find out?” • Ask, “Is there enough information to solve the problem? Is there extra information?” 2. Plan • • • • 3. Solve • Use your plan to solve the problem. • If your plan does not work, revise it or make a new plan. Study Tip Problem-Solving Strategies Here are a few strategies you will use to solve problems in this book. • Look for a pattern. • Solve a simpler problem. • Guess and check. • Draw a diagram. • Make a table or chart. • Work backward. • Make a list. 4. Examine • • • • Reread the problem to identify relevant facts. Determine how the facts relate to each other. Make a plan to solve the problem. Estimate the answer. Reread the problem. Ask, “Is my answer reasonable and close to my estimate?” Ask, “Does my answer make sense?” If not, solve the problem another way. Concept Check 6 Chapter 1 The Tools of Algebra Which step involves estimating the answer? Example 1 Use the Four-Step Problem-Solving Plan POSTAL SERVICE Refer to page 6. How much would it cost to mail a 9-ounce letter first class? Explore The table shows the weight of a letter and the respective cost to mail it first class. We need to find how much it will cost to mail a 9-ounce letter. Plan Use the information in the table to solve the problem. Look for a pattern in the costs. Extend the pattern to find the cost for a 9-ounce letter. Solve First, find the pattern. Weight (oz) Cost 1 2 3 4 5 $0.37 $0.60 $0.83 $1.06 $1.29  0.23  0.23  0.23  0.23 Each consecutive cost increases by $0.23. Next, extend the pattern. Weight (oz) Cost 5 6 7 8 9 $1.29 $1.52 $1.75 $1.98 $2.21  0.23 Study Tip Reasonableness Always check to be sure your answer is reasonable. If the answer seems unreasonable, solve the problem again.  0.23  0.23  0.23 It would cost $2.21 to mail a 9-ounce letter. Examine It costs $0.37 for the first ounce and $0.23 for each additional ounce. To mail a 9-ounce letter, it would cost $0.37 for the first ounce and 8  $0.23 or $1.84 for the eight additional ounces. Since $0.37  $1.84  $2.21, the answer is correct. A conjecture is an educated guess. When you make a conjecture based on a pattern of examples or past events, you are using inductive reasoning. In mathematics, you will use inductive reasoning to solve problems. Example 2 Use Inductive Reasoning a. Find the next term in 1, 3, 9, 27, 81, …. 1 3 9 27 3 3 3 81 3 ? 3 Assuming the pattern continues, the next term is 81  3 or 243. b. Draw the next figure in the pattern. In the pattern, the shaded square moves counterclockwise. Assuming the pattern continues, the shaded square will be positioned at the bottom left of the figure. Concept Check What type of reasoning is used when you make a conclusion based on a pattern? Lesson 1-1 Using a Problem-Solving Plan 7 CHOOSE THE METHOD OF COMPUTATION Choosing the method of computation is also an important step in solving problems. Use the diagram below to help you decide which method is most appropriate. Do you need an no exact answer? Estimate. yes Do you see a pattern or number fact? no yes Are there simple calculations to do? no Use a calculator. yes Use mental math. Use paper and pencil. Example 3 Choose the Method of Computation Log on for: • Updated data • More activities on Using a Problem-Solving Plan usa_today TRAVEL The graph shows the seating capacity of certain baseball stadiums in the United States. About how many more seats does Comerica Park have than Fenway Park? USA TODAY Snapshots® Fenway has baseball’s fewest seats Boston’s Fenway Park, opened in 1912, is Major League Baseball’s oldest and smallest stadium, with a capacity of 33,871. Baseball’s smallest stadiums in terms of capacity: 33,871 38,902 ton) s o B ( k r 0 a P y 40,00 icago) Fenwa ld (Ch ie F ,6 y ) 0 4 25 etroit Wrigle City) ark (D 0 P s a a s ic n r a 40,80 Come ium (K ) d o a c t S is an Franc Kauffm k (San ell Par B ic if c Pa Source: Major League Baseball By Ellen J. Horrow and Bob Laird, USA TODAY Explore You know the seating capacities of Comerica Park and Fenway Park. You need to find how many more seats Comerica Park has than Fenway Park. Plan The question uses the word about, so an exact answer is not needed. We can solve the problem using estimation. Estimate the amount of seats for each park. Then subtract. Solve Comerica Park: 40,000 → 40,000 Fenway Park: 33,871 → 34,000 Round to the nearest thousand. 40,000  34,000  6000 Subtract 34,000 from 40,000. So, Comerica Park has about 6000 more seats than Fenway Park. Examine 8 Chapter 1 The Tools of Algebra Since 34,000  6000  40,000, the answer makes sense. Concept Check 1. Tell when it is appropriate to solve a problem using estimation. 2. OPEN ENDED Write a list of numbers in which four is added to get each succeeding term. Guided Practice GUIDED PRACTICE KEY 3. TRAVEL The ferry schedule at the right shows that the ferry departs at regular intervals. Use the four-step plan to find the earliest time a passenger can catch the ferry if he/she cannot leave until 1:30 P.M. South Bass Island Ferry Schedule Find the next term in each list. 4. 10, 20, 30, 40, 50, … Departures Arrivals 8 :4 5 A.M. 9: 9 : 3 3 A.M. 1 10:21 A.M. 11:09 A.M. 5. 37, 33, 29, 25, 21, … 6. 12, 17, 22, 27, 32, … 7. 3, 12, 48, 192, 768, … Application 8. MONEY In 1999, the average U.S. household spent $12,057 on housing, $1891 on entertainment, $5031 on food, and $7011 on transportation. How much was spent on food each month? Round to the nearest cent. Source: Bureau of Labor Statistics Practice and Apply HEALTH For Exercises 9 and 10, use the table that gives the approximate heart rate a person should maintain while exercising at 85% intensity. For Exercises See Examples 9, 10 11–20 21–26 1 2 3 Age 20 25 30 35 40 45 Heart Rate (beats/min) 174 170 166 162 158 154 Extra Practice See page 724. 9. Assume the pattern continues. Use the four-step plan to find the heart rate a 15-year-old should maintain while exercising at this intensity. 10. What heart rate should a 55-year old maintain while exercising at this intensity? Find the next term in each list. 11. 2, 5, 8, 11, 14, … 12. 4, 8, 12, 16, 20, … 13. 0, 5, 10, 15, 20, … 14. 2, 6, 18, 54, 162, … 15. 54, 50, 46, 42, 38, … 16. 67, 61, 55, 49, 43, … 17. 2, 5, 9, 14, 20, … 18. 3, 5, 9, 15, 23, … GEOMETRY Draw the next figure in each pattern. 19. 20. 21. MONEY Ryan needs to save $125 for a ski trip. He has $68 in his bank. He receives $15 for an allowance and earns $20 delivering newspapers and $16 shoveling snow. Does he have enough money for the trip? Explain. Lesson 1-1 Using a Problem-Solving Plan 9 22. MONEY Using eight coins, how can you make change for 65 cents that will not make change for a quarter? 23. TRANSPORTATION A car traveled 280 miles at 55 mph. About how many hours did it take for the car to reach its destination? 24. CANDY A gourmet jelly bean company can produce 100,000 pounds of jelly beans a day. One ounce of these jelly beans contains 100 Calories. If there are 800 jelly beans in a pound, how many jelly beans can be produced in a day? 25. MEDICINE The number of different types of transplants that were performed in the United States in 1999 are shown in the table. About how many transplants were performed? Candy 1 In 1981, 3 tons of red, 2 blue, and white jelly beans were sent to the Presidential Inaugural Ceremonies for Ronald Reagan. Source: 26. COMMUNICATION A telephone tree is set up so that every person calls three other people. Anita needs to tell her co-workers about a time change for a meeting. Suppose it takes 2 minutes to call 3 people. In 10 minutes, how many people will know about the change of time? Transplant Number heart 2185 liver 4698 kidney 12,483 heart-lung 49 lung 885 pancreas 363 intestine 70 kidney-pancreas 946 Source: The World Almanac 27. CRITICAL THINKING Think of a 1 to 9 multiplication table. a. Are there more odd or more even products? How can you determine the answer without counting? b. Is this different from a 1 to 9 addition facts table? Answer the question that was posed at the beginning of the lesson. 28. WRITING IN MATH Why is it helpful to use a problem-solving plan to solve problems? Include the following in your answer: • an explanation of the importance of performing each step of the four-step problem-solving plan, and • an explanation of why it is beneficial to estimate the answer in the Plan step. Standardized Test Practice 29. Find the next figure in the pattern shown below. A B C D 30. A wagon manufacturing plant in Chicago, Illinois, can produce 8000 wagons a day at top production. Which of the following is a reasonable amount of wagons that can be produced in a year? A 24,000 B 240,000 C 2,400,000 D 240,000,000 Getting Ready for the Next Lesson BASIC SKILL Round each number to the nearest whole number. 31. 2.8 32. 5.2 33. 35.4 34. 49.6 35. 109.3 36. 999.9 10 Chapter 1 The Tools of Algebra Translating Expressions Into Words Translating numerical expressions into verbal phrases is an important skill in algebra. Key words and phrases play an essential role in this skill. The following table lists some words and phrases that suggest addition, subtraction, multiplication, and division. Addition Subtraction Multiplication Division plus sum more than increased by in all minus difference less than subtract decreased by less times product multiplied each of factors divided quotient per rate ratio separate A few examples of how to write an expression as a verbal phrase are shown. Expression Key Word Verbal Phrase 58 24 16  2 86 25 52 times sum quotient less than product less 5 times 8 the sum of 2 and 4 the quotient of 16 and 2 6 less than 8 the product of 2 and 5 5 less 2 Reading to Learn 1. Refer to the table above. Write a different verbal phrase for each expression. Choose the letter of the phrase that best matches each expression. 2. 9  3 a. the sum of 3 and 9 3. 3  9 b. the quotient of 9 and 3 4. 9  3 c. 3 less than 9 5. 3  9 d. 9 multiplied by 3 6. 9  3 e. 3 divided by 9 Write two verbal phrases for each expression. 7. 5  1 8. 8  6 9. 9  5 10. 2(4) 11. 12  3 20 12.  13. 8  7 14. 11  5 4 Reading Mathematics Translating Expressions Into Words 11 Numbers and Expressions • Use the order of operations to evaluate expressions. • Translate verbal phrases into numerical expressions. Vocabulary • numerical expression • evaluate • order of operations do we need to agree on an order of operations? Scientific calculators are programmed to find the value of an expression in a certain order. Expression 125 842 10  5  14  2 11 6 30 Value TEACHING TIP a. Study the expressions and their respective values. For each expression, tell the order in which the calculator performed the operations. b. For each expression, does the calculator perform the operations in order from left to right? c. Based on your answer to parts a and b, find the value of each expression below. Check your answer with a scientific calculator. 12  3  2 16  4  2 18  6  8  2  3 d. Make a conjecture as to the order in which a scientific calculator performs operations. ORDER OF OPERATIONS Expressions like 1  2  5 and 10  5  14  2 are numerical expressions . Numerical expressions contain a combination of numbers and operations such as addition, subtraction, multiplication, and division. When you evaluate an expression, you find its numerical value. To avoid confusion, mathematicians have agreed upon the following order of operations . Study Tip Grouping Symbols Grouping symbols include: • parentheses ( ), • brackets [ ], and • fraction bars, as 6 4 2 Order of Operations Step 1 Simplify the expressions inside grouping symbols. Step 2 Do all multiplications and/or divisions from left to right. Step 3 Do all additions and/or subtractions from left to right. in , which means (6  4)  2. Numerical expressions have only one value. Consider 6  4  3. 6  4  3  6  12  18 Multiply, then add. 6  4  3  10  3  30 Add, then multiply. Which is the correct value, 18 or 30? Using the order of operations, the correct value of 6  4  3 is 18. Concept Check 12 Chapter 1 The Tools of Algebra Which operation should you perform first to evaluate 10  2  3? Example 1 Evaluate Expressions Find the value of each expression. a. 3  4  5 3  4  5  3  20 Multiply 4 and 5.  23 Add 3 and 20. b. 18  3  2 Study Tip Multiplication and Division Notation A raised dot or parentheses represents multiplication. A fraction bar represents division. 18  3  2  6  2 Divide 18 by 3.  12 Multiply 6 and 2. c. 6(2  9)  3  8 6(2  9)  3  8  6(11)  3  8  66  3  8  66  24  42 Evaluate (2  9) first. 6(11) means 6  11. 3  8 means 3 times 8. Subtract 24 from 66. d. 4[(15  9)  8(2)] 4[(15  9)  8(2)]  4[6  8(2)]  4(6  16)  4(22)  88 Evaluate (15  9). Multiply 8 and 2. Add 6 and 16. Multiply 4 and 22. 53  15 e.  17  13 53  15   (53 + 15)  (17 – 13) 17  13  68  4  17 Rewrite as a division expression. Evaluate 53  15 and 17  13. Divide 68 by 4. TRANSLATE VERBAL PHRASES INTO NUMERICAL EXPRESSIONS You have learned to translate numerical expressions into verbal phrases. It is often necessary to translate verbal phrases into numerical expressions. Example 2 Translate Phrases into Expressions Study Tip Differences and Quotients In this book, the difference of 9 and 3 means to start with 9 and subtract 3, so the expression is 9  3. Similarly, the quotient of 9 and 3 means to start with 9 and divide by 3, so the expression is 9  3. Write a numerical expression for each verbal phrase. a. the product of eight and seven Phrase the product of eight and seven Key Word product Expression 8  7 b. the difference of nine and three Phrase the difference of nine and three Key Word difference Expression 9  3 Lesson 1-2 Numbers and Expressions 13 Use an Expression to Solve a Problem Example 3 TRANSPORTATION A taxicab company charges a fare of $4 for the first mile and $2 for each additional mile. Write and then evaluate an expression to find the fare for a 10-mile trip. Expression 4  $2 for each additional mile  and  $4 for the first mile  Words 29 4  2  9  4  18 Multiply.  22 Add. The fare for a 10-mile trip is $22. Concept Check 1. OPEN ENDED Give an example of an expression involving multiplication and subtraction, in which you would subtract first. 2. Tell whether 2  4  3 and 2  (4  3) have the same value. Explain. 3. FIND THE ERROR Emily and Marcus are evaluating 24  2  3. Emily Marcus 24 ÷ 2 x 3 = 12 x 3 = 36 24 ÷ 2 x 3 = 24 ÷ 6 = 4 Who is correct? Explain your reasoning. Guided Practice GUIDED PRACTICE KEY Name the operation that should be performed first. Then find the value of each expression. 4. 3  6  4 5. 32  24  2 6. 5(8)  7 7. 6(15  4) 10  4 8.  9. 11  56  (2  7) 12 Write a numerical expression for each verbal phrase. 10. the quotient of fifteen and five Application 11. the difference of twelve and nine 12. MUSIC Hector purchased 3 CDs for $13 each and 2 cassette tapes for $9 each. Write and then evaluate an expression for the total cost of the merchandise. Practice and Apply Find the value of each expression. For Exercises See Examples 13–28 31–38 39–42, 47, 48 1 2 3 Extra Practice See page 724. 13. 2  6  8 14. 12  3  3 15. 12  3  21 16. 9  18  3 17. 8  5(6) 18. 4(7)  11 15  9 19.  45  18 20.  21. 11(6  1) 22. (9  7)  13 23. 56  (7  2)  6 24. 75  (7  8)  3 32  20 93 25. 2[5(11  3)]  16 26. 5[4  (12  4)  2] 27. 9[(22  17)  5(1  2)] 28. 10[9(2  4)  6  2] 14 Chapter 1 The Tools of Algebra 29. Find the value of six ...
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