Revision Notes for Basic College Mathematics, 13th Edition

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INSTRUCTOR’SRESOURCEMANUALBASICCOLLEGEMATHEMATICSTHIRTEENTHEDITIONMarvin L. BittingerIndiana University Purdue University IndianapolisJudith A. BeecherBarbara L. JohnsonIvy Tech Community College of Indiana

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iGENERAL,FIRST-TIMEADVICEWe asked the contributing professors for words of advice to instructors who areteaching this course for the first time or for the first time in a long while. Theirresponses can be found on the following pages.Sally Jackman,Richland College1.This course can be a very challenging one toteach because of the variety of skill levelsthat the students bring to class. You will findthat many of your students know the materialand only need a review. This type of studentoften will want you to go faster. You willhave the student who knows some of thematerial, but not all of the material. Thisstudent may want you to move both fasterand the slower throughout the course. Youmay have the student who has never learnedthe material. This student usually wants youto go slower. Last but not least, there is thestudent who thinks they know the materialand something just happened when they tookthe placement test. This type of student canbe the most difficult because they want to doas little as possible: “they already know thematerial.”I have found that the best way to handle allthe different skill levels is to assume that theyall need to learn the material for the first timeand that it needs to be learned correctly. Iplan the entire course calendar before thesemester begins. I have the calendar set up asto which sections will be taught on whichday, and what days I will have exams. I makethis calendar part of my syllabus. I try tofollow this schedule as closely as possible tokeep my students on track. There have beentimes when I am not able to adhere to theschedule. When that happens, I try to getback on schedule or I do an addendum to theschedule and hand that out in class. I havetried teaching without a schedule and when Idid, I had a hard time finishing the material.You may say “What’s wrong with that?”My answer is that it hurts the students whenthey move to the next level and the materialnot covered is required for the next level.Thus, by having a calendar or schedule, ithelps to keep the class moving at a pace inwhich all levels should be able to learn.If this is your first time teaching the course,find out what sections you are required toteach. Ask an instructor who has taught thecourse before where more time is needed fora particular section or chapter. Plan theschedule and then have another instructorwho has taught the course before look at yourschedule. During the semester, keep notes onyour schedule as to where students neededmore time or less time than you planned. Thiswill help you the next time you teach thissame course.2.Students at this level need lots of practice onthese concepts. If they just watch youworking at the board, they will have a hardertime learning the concepts. You need to havethe students actively involved in the lecturebecause this helps them to remember better. Ido this several ways: groups, send them to theboard, practice in class, and tell me how towork a problem at the board.3.I tell students that “mathematics is its ownlanguage” and they must learn the languageof mathematics just like they would need tolearn any foreign language. So, I stressvocabulary. When explaining a concept, Ialways use correct terminology and I wantthem to use it. For example, when I talk aboutfractions I do not say “top and bottom.” I say“the numerator and the denominator.” Somany times, I have to help the studentsunlearnthe incorrect terminology that theyhave brought with them to class. We need touse correct terminology so that as studentsprogress in mathematics, they have thevocabulary that is needed to be successful.Remember that correct vocabulary starts withthe teacher.4.Many times, students at this level do notknow how to study mathematics. Thus, I feelthat I should help my students learn studyskills. I do this several ways.

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Resource ManualiiI begin discussing study skills by listinghelpful hints in my syllabus. Here are somethat I have in my syllabus:Take the chaptertest in the textbook for practice—if you havequestions, make sure you get help. Keep upwith the lessons and homework. Get togetherwith someone else in the class to talk throughproblems that seem to be taking a long timeto solve—Don’t spin your wheels.There is a video in our library on studentskills that I have students check out andwatch. I have them write a one-page report onstudy skills that I grade for extra credit.I have students keep a notebook for class. Inthe notebook, they keep class notes, handouts,assignment problems, and graded papers, toname a few items. Once in a while, I willcheck the notebooks by either collecting themor going around the room and looking at thenotebooks. If you are going to checknotebooks, you need to check them the withinthe first three weeks. If they are checkedearly, it helps the students to have a betternotebook later in the semester.I have students work in pairs a lot in class. Iexplain how to work a problem. Then inpairs, I have the students work someproblems. One student writes the problemwhile the other student explains how to workit. Then they reverse roles for the nextproblem. This approach is a good study skillbecause it helps students learn that if they canexplain how to work a problem, then they arelearning more themselves.5.Encourage students to write mathematics. Isometimes give a quiz or writing assignmentwhere they have to explain how to solve aproblem. For example: Explain in words howto find the answer to this problem:1124+.I do not want them to work it out and handthat in. I want them to write that they need tofind a common denominator and makeequivalent fractions. If they can explain theprocess, then they can solve almost anyproblem because they know the process.Too many times I have heard, “I never sawthat problem so how could I work it on thetest.” If they can explain the process, thenthey should be able to solve problems and nottry to memorize every problem they haveworked.Roger McCoach,County College of Morris1.The students who take this course needstructure. Specific homework must beassigned, giving the students some direction.Be careful that you don’t assign too manyproblems, which might overwhelm thestudents. If the course is run as a self-pacedcourse, then a timeline needs to be given aswell, in order to keep the students fromfalling behind.2.You, as an instructor, cannot assumeanything. Show the students how the authorexplains the material with examples withinthe text and then show them that similarexercises are provided for the student to do inthe margins.3.Making a big deal out of attendance isimportant.4.If students can do a problem “their way” it isfine—they don’t have to do the problem asthe text dictates. Mention that fractionsalwaysneed to be reduced, whether or not theproblem says it.5.The cumulative reviews are a very good toolfor a student when preparing for a final exam.6.Calculators are always a touchy issue. WecoverBasic Mathematics’ Chapters 1–6, partof 7, part of 9, and 10 in the course.Calculators are permitted beginning inChapter 5. The idea is that the students firstdemonstrate that they are able to add,subtract, multiply, and divide integers,fractions and decimals (Chapters 1–4) bythemselves—then in Chapter 5 calculatorsare allowed.7.The Synthesis part of the chapter tests at theend of the chapter might be skipped becausetime is such an issue in this course. Thissection may be appropriate for the well-aboveaverage student.8.When in doubt, an algebra approach to aproblem may be better than another method.It will be helpful when they go on to algebrain the next semester. It is highly recom-mended that the course include Chapter 10,which is actually the first chapter in thepublisher’s algebra text.

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General, First-Time Adviceiii9.To become a bit philosophical, I have alwaysbelieved that a good deal of this courseshould be about teaching the people in theclass how to become students and teachingthem skills that can be applied to all courses.Many people in the class have poor or non-existent study habits and need to be madeaware that they will not be passed through thecourse, as they have been passed throughmany courses in high school. Encouragementis important, but it must be clearly definedwhat is expected of the student and what willbe tolerated.10.I have found that a combination of mylecturing plus time where the students work inthe text by themselves (where I can assistthem) is the best approach. The students willoften “tune out” if there is too much lecturingand expecting the students to teach themselvesthe entire course is not realistic.11.Giving a test for each chapter is the bestoption. Giving a test on several chapterswould cover too much material at one time.Deloria Nanze-Davis,University of Texas, Brownsville,Texas Southmost CollegeThe teaching tips listed will help students learnterminology, concepts, and problem-solvingtechniques quicker.1.Always use correct terminology.2.Quiz students on definitions and terms.3.Have students go to the board to work andexplain problems to their classmates.4.Have students work in groups when it comesto solving algebraic word problems.5.Always assign homework.6.Assign the synthesis problems.7.Encourage the students to read the lessonbefore coming to class.Christy Peters,Edison Community CollegeAt this time in my career, I have enjoyed fifteenyears in the classroom. This has includedexperiences from K through undergraduate collegework. Each experience has provided me with somevaluable insights and wonderful memories that willlast my entire lifetime. The biggest tip I can givenew teachers is to handle each student with care.Take time to find out their strengths and encouragethem to do well! Each of us can be a shining star ifwe just have the right person to polish our skills.As William G. Spady stated, “All students canlearn and succeed, but not on the same day in thesame way.”Richard Semmler,Northern VirginiaCommunity CollegeThe most important tip is to plan out the course andprepare a syllabus for your students before youstart teaching. Plan to cover the course during theacademic term with the exception of the last day,which can be used as a review day or a make-upday in the event the college was closed or you wereill for a day.If the school has a mentoring program, takeadvantage of it so that you will have guidance froman experienced teacher. This can be importantsince many courses incorporate technology withthe instruction and you can benefit from the adviceof a master teacher. If the class meets for more thanone hour, plan a variety of activities to break up thelecture. This could include group work or havingstudents solve problems on the blackboard. This isquite useful if you plan to take a short breakbetween lectures.Sharon Testone,Onondaga Community College1.Students enrolled in a basic mathematicscourse at the community-college level areoften extremely math anxious. These studentsare different from those you may have taughtat four-year colleges/universities or in thesecondary schools. They are often returningadults and need to be treated with respect andunderstanding. A kind word of encouragementor a smile means the world to these students.Many of these people have accomplishedmany things in their lives and they just have

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Resource Manualivdifficulty with math. Instructors shouldencourage the students not to feel badly aboutbeing in this course and should help them tosee that their worth to society is not measuredby their mathematical ability.2.An important time-management tip is tocount the number of class periods, subtractthe number of testing days, and then subtractat least two class periods for review at the endthe semester. This result is the number ofinstructional periods. Divide the number ofinstructional periods into the number ofsections that need to be taught. This resultgives the approximate number of sections thatneed to be taught each class period. Thissimple calculation will help new faculty toavoid the pitfall of moving too quickly or tooslowly through the course. Additionally, thefaculty member should most likely completeall topics in the syllabus and not omit anyessential material.3.Always be on time to class and always endthe class on time. Our students have verybusy schedules and faculty need to respectthat fact.4.Always be prepared for class and alwayshand back graded homework, quizzes, andtests the very next class period.5.Before teaching the course for the first time,ask your department chairperson what theprerequisites are for this course. Thisinformation will assist you in gauging whatknowledge the students should have whenentering your classroom. They may not becompletely prepared, but you will know whatskills they are expected to possess and youwill not spend too much time reteachingmaterial from the previous course.Additionally, determine what course yourstudents will enroll in after completing yourcourse. Be sure that you prepare the studentsfor that course, but don’t teach the topicsfrom that course.6.Students at this level need to practice,practice, and practice. Develop additionalworksheets and allow time for group practicein class. Have individual students or arepresentative from the group show you atleast one completed problem before the endof the class period. (Another alternative is tohave a group representative put the solutionon the board.) This technique helps to ensurethat students will be able to do theirhomework.7.Often students at this level do not completetheir homework assignments and this leads tofailure. One option is to require that studentscomplete homework assignments in aseparate notebook. On test days, instructorscan review the notebook (without actuallygrading it) to determine if students arecompleting their assignments. Another optionis to collect homework daily or randomly andgrade it.8.Giving a five-minute quiz after reviewinghomework questions at the beginning of theclass period is often helpful to newinstructors. The quiz results will let both thestudents and the instructor know how they aredoing. If the whole class fails a quiz, thenmost likely the instructor needs to makeimprovements.9.Prepare handouts with matching overheads orPowerPoint slides. Students at this level oftenare not good note takers and have difficultylistening and writing at the same time.Handouts that include key concepts, one ortwo worked-out examples, and two or threeproblems for the students to completeimmediately are very useful.10.Have fun! Teaching basic math is not aboutthe material. It is about the students. Thechallenge is to present the material in a waythat the students can understand it and enjoylearning math. Since they may haveattempted to learn this material many, manytimes in the past, I try to present things in adifferent way.

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General, First-Time AdvicevCassondra Thompson,York Technical CollegeI have found that my students have difficulty withChapters 2 and 3 on fractions, so I teach Chapter 4on decimals at the beginning of the semester.Chapter 4 does not require as much teaching timeand allows the students a chance for success earlyin the semester. By teaching Chapter 4 at a fairlyquick pace, this allows for more teaching time onChapters 2 and 3 to help students to grasp theconcept of fractions.It is also wise to allow more teaching time forChapter 10 on real numbers and 11 on algebra. Ihave found that by teaching adult learners, some ofmy students are learning algebra for the first timeand that requires some extra time to explainconcepts with concrete examples.Roy West,Robeson Community CollegeWhen working a problem in class, think out loud.Tell students what you are thinking as it relates tothe problem, and how you came up with thedecisions of what to do next.Always be ready to answer “Why?” and ask“Why?” For example: When dividing 102.43 by1.2, “Why do you move the decimal in bothnumbers one place to the right?” Remember thatmany students in these developmental classes willbe helping their children with math. They need toknow why things are done the way they are tobetter their own understanding and to share withtheir children.Paige Wood,Kilgore College1.As a way of getting to know my students, Iask them to write a paper about themselves.The paper should include general informationabout the student: major, math background,work information, family, etc. The papershould be 1 page, double spaced, and nolarger than a 14-point font. If the studentschoose to handwrite their paper, I expect twosingle-spaced pages. I grade these accordingto length and effort. For the first few weeksor the semester, I look back at the papers as away to keep me acquainted with them.2.I also make it very clear to the students, thatalthough I assign “homework,” they shouldview the assignments as a way to get personalfeedback from the instructor. When gradingtheir assignments, I give them writtenresponses regarding their mistakes and showcorrect procedures in an attempt to help themlearn the new material. The conscientiousstudents really appreciate the extra pointersthat I write on their homework.3.I work in the math lab as part of mycontractual obligations to my institution. Thevalue of this assignment is that I amaccessible to my students. So, if you havetime, ask if you can volunteer in the learningcenter. Your students will be glad that youare here.4.I have found that I can keep the students’attention better if they complete problems ona worksheet page to turn in at the end of theclass period. They can take notes in theirspiral but this sheet is strictly for them topractice the new problems and proceduresbefore they leave to tackle the material ontheir own. I ask for volunteers to work theproblems on the board to make sure that thestudents got a correct answer or so that theycan correct their own work.

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Mini-Lecture 1.1BASIC COLLEGE MATHEMATICSML-1Standard NotationLearning Objectives:aGive the meaning of digits in standard notation.bConvert from standard notation to expanded notation.cConvert between standard notation and word names.Examples:1.What does the digit 5 mean in each number?a) 3654b) 265,812c) 56,203,4112.Write expanded notation for each number.a) 2756b) 6172c) 90,9433.Write a word name for each number.a) 325b) 60,448c) 9,542,0064.Write standard notation.a)two hundred fifty-threeb)seven thousand, ninety-eightc)three hundred forty million, one hundred thirty-two5.When do we use a hyphen when writing word names for whole numbers?Teaching Notes:•Students who do not speak English as their first language may need extra help learning the number periodvocabulary such as ones, thousands, millions, billions, etc. Refer them to thePlace-value Chartin thetextbook.•Many examples of tables are available on the Internet. Request that students obtain copies and ask them toexplain how to read the tables and write the numbers in words.Answers: 1a) 5 tens, b) 5 thousands, c) 5 ten millions; 2a) 2 thousands + 7 hundreds + 5 tens + 6 ones,b) 6 thousands + 1 hundred + 7 tens + 2 ones, c) 9 ten thousands + 0 thousands + 9 hundreds + 4 tens + 3 ones, or9 ten thousands + 9 hundreds + 4 tens + 3 ones; 3a) three hundred twenty-five, b) sixty thousand, four hundredforty-eight, c) nine million, five hundred forty-two thousand, six; 4a) 253, b) 7098, c) 340,000,132; 5) A hyphen isused when writing the numbers 21–99, except for numbers ending in zero.

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BASIC COLLEGE MATHEMATICSMini-Lecture 1.2ML-2AdditionLearning Objectives:aAdd whole numbers.bUse addition in finding perimeter.Examples:1.Add.a)5312b)1123345c)40, 00132, 44215,333d)9647e)56783574f)650517370441682. Find the perimeter of a figure with sides 52 ft, 80 ft, 25 ft, 41 ft, and 93 ft.3. Find the perimeter of a figure with sides 30 yd, 68 yd, 13 yd, 39 yd, 26 yd, and 7 yd.4. Find the perimeter of a soccer field.Teaching Notes:•Some students need to practice basic addition facts at home in order to master them.•Some students need to write the carry (or regrouping) digit in order to get the correct answer for additionwith carrying (or regrouping).•Most students find this section easy.•Point out that the commutative property deals with theorderof the addends, whereas the associativeproperty deals with thegrouping(order does not change) of the addends.•Remind students that units need to be included with perimeter answers.Answers: 1a) 65, b) 1468, c) 87,776, d) 143, e) 9252, f) 13,890; 2) 291 ft; 3) 183 yd; 4) 320 yd

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Mini-Lecture 1.3BASIC COLLEGE MATHEMATICSML-3SubtractionLearning Objective:aSubtract whole numbers.Examples:1.Subtract.a)9851-b)5422-c)66451-d)5112-e)1123345-f)40,00115,333-g)82734365-h)82,33318,888-i)30042805-2.In order to check the subtraction problem 523 – 276 = 247 by using addition, you wouldadd _____ and _____ to see if you get _____ .Teaching Notes:•Some students need to practice basic subtraction facts at home in order to master them.•Most students find subtraction without borrowing (or regrouping) easy.•Many students need to write the borrowing (or regrouping) step in order to get the correct answer forsubtraction with borrowing (or regrouping).•Some students have a lot of trouble using borrowing (or regrouping) when zeros are involved.•Remind students to always check subtraction problems by using addition.Answers: 1a) 47, b) 32, c) 613; d) 39, e) 778, f) 24,668 g) 3908, h) 63,445, i) 199; 2) Add 276 and 247 to see if youget 523.

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BASIC COLLEGE MATHEMATICSMini-Lecture 1.4ML-4MultiplicationLearning Objectives:aMultiply whole numbers.bUse multiplication in finding area.Examples:1.Multiply.a)4364b)1822c)53454d)4302107e)160200f)754938132.Find the area of a rectangle with length 63 ft and width 42 ft.3.Find the area of a rectangle with dimensions 48 mi by 39 mi.4.Find the area of a soccer field.Teaching Notes:•Some students need to practice basic multiplication facts at home in order to master them.•Some students do not know the different types of symbols that represent multiplication.•Some students are not sure how to multiply by factors which have zeros.•Some students are not sure how to align the products when a factor contains a digit of zero.•Remind students that square units, like sq ft or sq mi, need to be included with area answers.Answers: 1a) 1744, b) 396, c) 28,836, d) 460,314, e) 32,000, f) 28,784,337; 2) 2646 sq ft; 3) 1872 sq mi;4) 6000 sq yd

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Mini-Lecture 1.5BASIC COLLEGE MATHEMATICSML-5DivisionLearning Objective:aDivide whole numbers.Examples:1.Divide, if possible. If not possible, write “not defined.”a)61b)7 49c)90d)4 0e)568f)772.Divide.a)1899b)4 1290c)8 32, 222d)40 8258e)77737f)103 52673.How could you check the answer to this division problem?71 R26 428Teaching Notes:•Some students need to practice division facts at home in order to master them.•Many students do not know that division by zero is “not defined” and that zero divided by any nonzeronumber results in zero.•Some students do not know how to check a division problem if there is a remainder and must be shownseveral examples.Answers: 1a) 6, b) 7, c) not defined, d) 0, e) 7, f) 1; 2a) 21, b) 322 R2, c) 4027 R6, d) 206 R18, e) 21, f) 51 R14;3)71×6 = 426 and 426 + 2 = 428.

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BASIC COLLEGE MATHEMATICSMini-Lecture 1.6ML-6Rounding and Estimating; OrderLearning Objectives:aRound to the nearest ten, hundred, or thousand.bEstimate sums, differences, products, and quotients by rounding.cUse < or > forto write a true sentence in a situation like610.Examples:1.Round to the nearest ten.a) 212b) 3487c) 14d) 15,8612.Round to the nearest hundred.a) 312b) 1267c) 83d) 14,9613.Round to the nearest thousand.a) 3549b) 677c) 27,217d) 11714.Estimate the sum or difference by first rounding to the nearest hundred.a) 257 + 524 + 388 + 671b) 922 – 438c) 433 + 591 + 8275.Estimate the product or quotient by first rounding to the nearest ten.a) 67×72b) 19×93c) 52÷11d) 57÷216.Tickets for a lobster dinner in Maine cost $19 each. Ticket sales totaled $1425. Estimate the number ofpeople who attended the dinner by rounding the cost of a ticket to the nearest ten and the total sales to thenearest hundred.7.Use < or > forto write a true sentence.a)236227b)4535c)885975Teaching Notes:•Some students are unfamiliar with rounding and will need to be repeatedly reminded to look at the digit tothe right of the rounding position.•A common mistake students make is to leave the digits to the right of the rounding position intact instead ofchanging them to zeros after rounding.•Some students may need to use the number line to help them determine correct order.Answers: 1a) 210, b) 3490, c) 10, d) 15,860; 2a) 300, b) 1300, c) 100, d) 15,000; 3a) 4000, b) 1000, c) 27,000,d) 1000; 4a) 1900, b) 500, c) 1800; 5a) 4900, b) 1800, c) 5, d) 3; 6) 70 people; 7a) >, b) >, c) <

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Mini-Lecture 1.7BASIC COLLEGE MATHEMATICSML-7Solving EquationsLearning Objectives:aSolve simple equations by trial.bSolve equations like2854, 28168, and 98 2.txyExamples:1.Solve by trial.a)310xb)273w2.Solve. Be sure to check.a)47845421xb)81 12pc)3152xd)206128me)4585zf)6156yg)412747th)11905xTeaching Notes:•Emphasize that the Solutions by Trial technique results mostly in wrong solutions. However by usingalgebra, a correct answer is found more quickly and accurately.•Many students will want to skip steps to solve equations. Encourage them to carefully write each step tofind the solution.•Remind students to check their answers.Answers: 1a) 7, b) 9; 2a) 10,205, b) 972, c) 21, d) 78, e) 40, f) 26, g) 67, h) 238

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BASIC COLLEGE MATHEMATICSMini-Lecture 1.8ML-8Applications Problem SolvingLearning Objective:aSolve applied problems involving addition, subtraction, multiplication, or division of whole numbers.Examples:1.Which mathematical operation do each of the following indicator word(s) imply?a) more thanb) decreased byc) ofd) quotiente) is2.Solve.a)Students at a party ordered 8 large pizzas. Each pizza is cut into 6 slices. How many total slices arethere?b)If a new amusement park covers 54 acres and there are 44,010 square feet in 1 acre, how many squarefeet of land does the amusement park cover?c)The town of Chelmsford, Massachusetts, has a population of 45,322. The town ofBedford, Massachusetts, has a population of 31,004. What is the difference inpopulation between these two towns?3.Solve.a)Elizabeth bought six beach towels for $15 each, eight bottles of sunscreen for $5 each,and 2 pairs of sunglasses for $23 each. How much did she spend in all?b)Justin had a balance of $15 in his checking account. He made deposits of $784, $556,$413, and $50. He wrote checks for $551, $347, $12, and $81. When all thedeposits are recorded and all the checks clear, what balance will he have in hischecking account?c)Enrico’s car gets 32 miles to the gallon (mpg) in highway driving. How many gallonswill the car use for 5536 mi of highway driving?Teaching Notes:•Many students struggle with application problems.•Refer students to theFive Steps for Problem Solvingboxin the textbook.•Encourage students to use estimation to check whether answers to application problems are reasonable.Answers: 1a) addition, b) subtraction, c) multiplication, d) division, e) equals; 2a) 48 slices, b) 2,376,540 sq ft,c) 14,318 people; 3a) $176, b) $827, c) 173 gal

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Mini-Lecture 1.9BASIC COLLEGE MATHEMATICSML-9Exponential Notation and Order of OperationsLearning Objectives:aWrite exponential notation for products such as4 4 4.⋅⋅bEvaluate exponential notation.cSimplify expressions using the rules for order of operations.dRemove parentheses within parentheses.Examples:1.Write exponential notation.a)4 4 4⋅⋅b)2 2 2 2⋅⋅⋅c)1 1 1 1 1⋅ ⋅ ⋅ ⋅2.Evaluate.a) 32b) 23c) 44d) 103e) 71f) 903.Simplify.a)576´-b)28362´-¸c)()25435´-+d)()315389122¸´ ´¸-e)332231++f)()2499523´--¸+4.Find average of 42, 54, 68, and 24.5.Simplify.a)(){}[]16212421¸+-´-b)()(){}[]34165293´+-+-¸Teaching Notes:•Students who have never seen exponents before often write the exponent right next to the base, and same-sized, instead of right, higher up, and smaller.•Many students have trouble with order of operations.•Emphasize that the wordoras in multiplyordivide (or addorsubtract) implies whichever comes first.•Refer students to theOrder of Operationsbox in the textbook.Answers: 1a) 43, b) 24, c) 15; 2a) 9, b) 8, c) 256, d) 1000, e) 7, f) 1; 3a) 29, b) 69, c) 30, d) 90, e) 36, f) 43; 4) 47;5) 11, b) 18

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