Solution Manual for Intermediate Algebra with Integrated Review, 13th Edition

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RESOURCEMANUALINTERMEDIATEALGEBRATHIRTEENTHEDITIONMarvin L. BittingerIndiana University Purdue University IndianapolisJudith A. BeecherBarbara L. JohnsonIvy Tech Community College of Indiana

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1GENERALFIRST-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.David P. Bell,Florida CommunityCollege at Jacksonville1.So, you are considering teaching IntermediateAlgebra in college for the first time! Beforeyou even consider the syllabus, let’s try to geta picture of your students. The mostchallenging students will be those who areessentially totally unprepared for college.These students went through high schoolattending class at least some of the time andwaiting for the teacher to provide all therequired topics and knowledge in class. Thesestudents rarely if ever studied or didhomework. That certainly was a negativebeginning, wasn’t it. The fact is, if youprepare for these students, you will save timeand effort on your part and more importantlyfor the students who do come prepared forcollege. For many of these students, you aretheir first and most important link to collegelevel mathematics. These students need to betaught good study habits. They need to learnthe value of timely performance of homeworkassignments and they need to learn to read thealgebra textbook before each lesson. You stillwant to try mission impossible? That’s great.Let’s get started.2.You will want to speak with an experiencedfaculty member and discuss just how andwhere to get started with the course. Many ofthose I work with use the review chapter toset the stage for the concept of studentresponsibility and ownership of their success.The Bittinger Intermediate Algebra textprovides an excellent review of basic algebrain Chapter R. I skim the material in thischapter and stress the importance of thisbeing review material. I state, “If the materialand concepts in chapter R are new to you, oryou do not remember any of them, stop,think, and come see me or visit a counselor.You may be in the wrong class! If thematerial is familiar, but you have not used itfor some time, you have until the next classwhen we will test your knowledge of thismaterial. You and I need to know if you haveareas requiring extra review. Now is the timeto make certain that you have every chancefor success in the class!” Please don’t use thisstatement as a class opener. You must try togain some of the students’ trust before youcan be this honest with them.3.I’ve set the stage and given you somedisheartening details. Armed with thisknowledge, you now set up the first class.You will see every expression on the faces ofthese students the first day of class, from fearto outright boredom, as you enter theclassroom. This is your chance to get theirattention and make that first impression. DONOT pull out a stack of syllabi and startgoing over the details and requirements. Takethis opportunity to open a discussion betweenthe students and yourself. I have a groupsheet that I pass out to get things started thatincludes room for four students and somegeneral information. Notice that I have notmentioned algebra. The questions include:Why this college?; Where are you from?;What do you do?; What program are you in?;What do you expect from this class?; and e-mail and phone numbers. I then have thestudents break into groups and find studentswho reside in the same general area. Thisincreases the chance that they will gettogether later to study. I give them plenty oftime to gather the information knowing thatthey will each be introducing someone else inthe class. The fear of math is now gonebecause I just told them that they each have tostand and introduce someone they most likelynever met before. The icebreaker gives youthe opportunity to get the roll and a feel forthe class. Once the introductions are over, Iintroduce myself in the same format and weget down to business. Did I mentionattendance? You can not overstress the needto read and understand the syllabus (Thesyllabus must contain all data pertinent toclass management and student success forthis course.) and the need to get started

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2NOW! Did I mention attendance? You shouldmention attendance as a requirement as oftenas you can stand it. I use the rest of this firstsession to review topics from the reviewchapter. This is the last time I will cover thismaterial in class, but stress the need to visityou during office hours or visit school tutorsif available. I require each student to send mean e-mail from their school provided e-mailaccount. Did I mention attendance? Thesestudents are very mobile and the e-mails,addresses, and phone numbers they providethe first week of class change rapidly. Theirschool e-mail does not change. Thus, you willbe able to contact them should they not showup for class. The underlying theme you mustget out and repeat often is that to besuccessful, they will need to read, dohomework, and study. The more ways youfind to get this message out, the better.4.You should be able to procure an example ofan elementary algebra final exam. You canprovide this test to the students with a keyand allow them to test themselves to seewhether they are ready for this class. Youmay want to use the test as a quiz score orbonus just to emphasize to the student howimportant this material is. The importantthing to keep in mind is “Now is the best timeto remediate or relearn lost elementaryalgebra concepts.” I have found that studentswho are made aware of possible problemareas will rise to the occasion and seek helpearly to increase their chances of success.5.Once the first meeting is over, you alwayswant to begin the day with something tomake them think. I use quizzes, groupproblems, specific homework problems andanything else that I can to try to get theirattention early. If they know the first fewminutes are important, they will be less apt tobe tardy. I try not to present the veryexamples that are in the text. If they read thetextbook, presenting these often will createboredom that you do not want. I look forexamples similar to those presented in thetextbook. This provides the students withample resources to complete the homework.Speaking of homework, try to assign mostlyquestions that have the answers in thetextbook. Students can check the accuracy oftheir efforts. Some instructors collecthomework so that the student sees a need toactually do it. I find that the students oftenjust copy the solutions manual and turn it inas their own work. The contrast between thefirst test results and the homework is often afantastic learning tool for the student. Spend afew minutes each day covering homeworkquestions, but you control that time. Studentswould rather spend all day on homework thanthe new lesson.6.Tests are the primary measurement tool youhave to gauge student learning andcomprehension. Create the test so that itchecks understanding of the topics you choseto cover. I focus on open response andapplications questions. Build the test so thatthe better students finish early. Set a timelimit. These students need to learn timemanagement and there is no time like thepresent. Speaking of time management, youmight also want to talk about overall semesterload, work, family, study time, etc, and see ifthere really is time to get it all done. Thewebsites: http://www.askjeeves.com andhttp://www.purplemath.com have excellentinteractive study skills and time managementroutines and assistance.7.Timely return of test results is a must.Occasionally you will have students whoscore low for a number of silly errors. Letthem know that they could have done muchbetter if they had been a bit more careful.They like knowing that you care enough tonotice.8.The comprehensive final exam is your lastchance to see if you have prepared thesestudents well for college algebra or the nextcourse in the sequence at your college. I oftenoffer to replace each student’s lowest testscore with the final exam score. It provides abit of extra motivation and if the student doeswell on the final, I can rest assured that theyare as ready for college algebra as I can getthem.9.Throughout the course, try to use the classleaders to assist and motivate the studentswho are struggling. Everyone benefits fromthis exchange and the students get a sense ofteamwork for their efforts.10.Finally, on those occasions where you havegiven your best lecture with your bestdefinitions and you know the students shouldunderstand, but you see confusion in their

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3eyes, you may get frustrated. The frustrationcomes from the belief that these studentsshould have benefited from your presentation.Do your best to hide that frustration. Thestudents will pick up on it very quickly andinterpret that frustration as a personal issue.They feel you are frustrated with them andwe know the frustration is directed atyourself. If you slip, just explain to the classyour honesty with them will be a benefit later.Above all, keep an eye out for that light. Youwill see students suddenly perk up andprofess “I see it now.” It is so much funseeing these people of all ages, colors, creeds,races, religions, etc. turn into knowledgehungry students, and they pay you too.Best of luck.Chris Bendixen,Lake Michigan College1.Generally the first day of class is one of themost important classes in the semester, notfor the mathematical material, but for the“how to succeed” materials. The studentsneed the syllabus for the first class meeting.Go over the syllabus in great detail, stressingthat this is a contract between you (instructor)and them (students). Explain to the studentshow much material is generally covered ineach class period. Thus, if they miss a class,they know about how much was covered. Donot talk in a dry monotone voice for the entireclass. You need the students’ attention. Showenthusiasm in your teaching. Make it appearthat this is the best subject ever.2.Next item is the class roster. Go over theroster, trying to correctly pronounce allstudents names.3.I like to tell the students on day one thateverybody has a 100% in the class, now. It isup to you (the student) to maintain this grade.To maintain the grade, you should do thefollowing:Read the material before the material hasbeen covered. It may not make sense at thispoint, but when the instructor goes over it,you will recognize the terminology. Also,after the material has been covered in class,go back over the reading. Amazingly, a lotmore of the material will make sense.To be truly successful in math, you must dohomework, homework, homework—as soonas the material has been covered. Do not waituntil the day before the exam to work on theproblems, stay on top of it. Related to this isodd answers are generally listed in the backpart of the book. Tell the students to checkthe problems, and, if the answer is notcorrect, rework it until it is correct. DO notgive up after the first attempt. Next, tell thestudents to take advantage of all thesupplemental materials, such as: MathXL,InterAct Math, student solutions manuals, andadditional internet math resources.I try to get the students involved in studygroups with their peers. I volunteer to attendthe initial meeting of the study group. Afterthat, they are on their own.Always refresh yourself on prior materials.Math is a building block subject, you must beon top of all aspects of it throughout theentire course.Finally give the students a study guide for theexams. I personally give the students asample exam. I also post the sample exam onthe school’s BlackBoard site.4.I tell the class about special policies, like:•Attendance policy•Late exam policy•Cell phone policy (this is a must) makethe students turn off their cell phonesbefore class. If they go off during class,penalize the student. Cell phones are verydisruptive to the flow of the class.•Grading policy•When to expect examsSandy Berry,Hinds Community College“What do you teach?” Throughout my 35-yearcareer I have been asked that question many times.My response is that I try to teach students aboutmathematics. A teacher’s focus should be on theirstudents first. Treat each student with the utmostrespect. Take time to get to know your students aspeople. Learn their names as soon as possible. Callthem by name often. Remember what it was likethe first time you were a student in a course thatyou knew would be difficult for you. Try to help

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4your students understand that your purpose is tohelp them learn mathematics.Individual learning styles or learning modes-visual,auditory, and/or kinesthetic-play an important rolein student understanding and performance. Somestudents learn best through seeing problemsworked out, others through hearing a thoroughexplanation, and others by engaging in hands-onactivities. Most successful students utilize morethan one mode of learning.Itis important that you as an instructor becomeaware of your own learning and teaching styles andwork at developing techniques of presentingmaterial that will address all learning styles.Working through problems carefully andcompletely on the board while carefully tellingstudents what you are doing and thinking will serveto accommodate most learners. Incorporatinghands-on activities for kinesthetic learners willrequire some inventiveness on the part of theinstructor. Brain research shows that learners willnot develop appropriate neural networks forremembering how to do math problems withoutdoing them and doing them correctly. So take timein class to let your students practice what you aredemonstrating. Give them immediate feedback andhelp them correct their mistakes. Continuallysearch for better ways of communicating with yourstudents.Deanna L. Dick,Alvin CommunityCollege1.Try to remember to stick to a schedule,approximately 50 minutes per section in thetextbook.2.Always allow more time for rationalexpressions and word problems.3.Use fractions daily in class and the studentswill become much less afraid of them andwill be more successful with them. Just beprepared—they may never like them!4.Allow at most one-fourth of the class periodfor questions, and then move on to newmaterial. Students are great at wasting time ifthey can get away with it and teachers are badabout wanting to answer all the questions.Make them come to your office hours. One-on-one explanations are always betteranyway. If a class is persistent, answer thequestions and then give the lecture in the last10 minutes of class. They will never play thestall game again!5.Try to point out when a problem requirespreviously learned material to finish solvingit.Example: When working a problem likeExercise 34 in Section 5.5 (a rationalequation) remind the students that theylearned how to solve the resulting equation inSection 4.8. Sometimes it is hard for studentsto tie the sections together and to see how oneproblem relates to another.6.On word problems-make your studentsidentify the variable in words. Tell them to bespecific. If they don’t know what they arelooking for, it will be difficult for them tocome up with equations they can solve to findit!Example:x= # of bull-riders in Cheyenne,WyomingThis is more descriptive thanx= # men,which could mean bull-riders or just citizensof the city. This is especially important whenyou are dealing with systems of equationsthat may talk about both ideas.7.Have the students do the reviews and chaptertests when studying for their test.8.Remember, most developmental mathstudents don’t read the textbook, though theyshould always be encouraged to do so.Therefore, helpful hints come primarily fromthe instructor. If the text gives a good hint,make sure you also give it in your lecture.Otherwise they may never see it.9.Don’t work the examples in the textbook. Nocollege student wants to be read to. That iswhy they spent so much money on thetextbook! Give them new examples tosupplement what is in the text.10.Don’t panic or get upset if you make amistake on a problem or get stuck on one. Ithappens to everyone eventually and canactually be encouraging to your students.Students love to correct their teachers, andthe fact that they caught the mistake builds

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5their confidence. Also, they remember whenthey get stuck at home that we got stuck inclass, which can make it a little lessfrustrating. After all, if we can solve aproblem that gives us trouble with a littlemore time and thought, then that may be allthey need as well.11.Show your students how to use technology tocheck your results, but recognize that theywill never understand it if they don’t learn todo it by hand.12.My first semester teaching, I taught a classthat was almost all word problems, “finitemath.” It was an 8 A.M. class and I am not amorning person. I had a wonderful studentwho always asked questions and his favoritething to ask was why I set up the problem thisway instead of his way, and why his equationdidn’t work. Usually, I would erase thecorrect equation and write his in its place. Asstudents often find out, sometimes it is harderto troubleshoot an equation than to start againfrom scratch. I wasted a lot of time, andconfused students by focusing on the wrongequation. As a first time teacher I quicklylearned to do the problem correctly in class,and trouble-shoot their home work afterclass.Kathleen C. Ebert,Alfred State CollegeI value five basic things in all my classes and hereis my advice.1.When you teach a course for the first fewtimes, plan out the entire semester ahead.Additionally, until you get it down patinclude many extra review days at the end. Iam student centered and students always needmuch more time on a topic than I think. Ican’t always plan which topics they willstruggle on or be sick during.2.Summarize at the beginning of each class(what you’ve covered recently) and at the endof class (what we did today). It helps studentssee the big picture of how it all relates andcomes together.3.At the end of class I put 2-5problems on theboard (or on a handout) for them to work onbefore they leave. I walk around so I knowwho needs help and who doesn’t; they canwork together. Engaging the student is key.4.I do not collect homework regularly but I domake sure they review for tests. I collect (orlook through their notebooks) their reviewassignments and I count it as a quiz orhomework grade.5.Last but not least, I always make my studentsdo test corrections. I do not believe in addingpoints to their tests for it but I do count themas a quiz grade (percent corrected for allproblems they got wrong). I make them dothese on a separate sheet of paper or on a newtest (this makes it easier for me to grade).They can go to the lab, come to me, or find afriend that can help. They are able to continueto hand these in until all are correct. Often Imake them write a sentence about why theygot each one wrong. One other thing Ioccasionally do is offer extra credit. For extracredit, students complete a section in thebook that we did not cover. They have to readthe section, take good notes (that they couldteach from), do the odd problems (showingall work and either present the information ortake a quiz on it, or both. Somehow they findtime. They practice all the important stuff,reading a math text, taking their own notes,learning for understanding. I usually waitwith this offer until they start asking “is thereanything I can do...” Depending on howmuch they do, and how well determines howmuch credit I give them.Rosa Kavanaugh,Ozarks TechnicalCollege1.Since this is the second-level course in thetraditional algebra sequence, students shouldhave a basic familiarity with the introductoryconcepts of algebra. However, almost allstudents bring into this course a number ofdeficiencies in those basic concepts, andthose students who are marginally placed intothe course have some major holes in theirunderstanding of the foundations of algebra.Part of what the instructor should beattempting to do in this course is to helpstudents to identify and fill holes in theirfoundations as well as to learn the contentthat is new in this course.

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62.The instructor who is familiar with commonerrors and misunderstandings can be moreeffective in helping the students to besuccessful in repairing and building upon thisfoundation. The text is a good reference foraddressing many of these common errors. Aninstructor who chooses to include adiscussion of such an error in the classroomshould be very careful about what is writtenon the board. If the topic is avoiding“improper cancellation” in fractions, theinstructor should either be careful to write2434131xxxxxxWrong!as the author does in Chapter 5or should tellstudents to put down their pencils and simplywatch. If the lesson is being presented on adry-erase board, this is a very appropriatetime to use a red marker for emphasis.3.Mathematical language and notation areforeign and confusing for students who areseeing algebra for the first time. Much of themathematical language has meaning in themore familiar English language. Althoughdevelopmental mathematics students alsohave weak language skills, we can help themidentify words by associating them with theEnglish language. These associations canhelp our students make the connections thatwill give the mathematical language moremeaning to them. A good example is theword “distribute.” This is a good time toencourage some discussion of the meaning ofthe word in common everyday language andthen make the connections to the specificmathematical meaning.4.Perhaps the most troublesome counter-example to the notion that mathematicallanguage has meaning in the Englishlanguage is in the words “term” and “factor.”In the English language, these words do nothave the same specific meaning that they doin mathematics. When we say, “These werefactors in our decision,” that does not meanthat those factors were multiplied. When wesay, “These were the terms of ouragreement,” that does not mean that the termswere added. Yet these same words have thoseveryprecise meaning in mathematics. In theEnglish language we use the word term in avery generic way. The word term should beused very deliberately and precisely in themathematics classroom.5.Students often confuse problem types onexams. One reason is that they tend tooverlook the instructions as they dohomework since all of the problems in asection lend to be the same type. I believethat the instructor should not only emphasizethe meaning of the instructions but also writethe instructions as part of the example.Students in developmental mathematicsclasses tend to include in their notes much ofwhat is written on the board, but little of thewords that the instructor says, but does notwrite. Thus it is important that the instructornot only say, “Solve” but also write:“Solve3551xx.This kind of emphasis helps students avoidcommon confusion such as the differencebetween solve and simplify.6 . Mathematics instructors themselves mustmodel good mathematical notation. One ofmy notation “pet peeves” is improper use ofequal signs. Occasionally, adjunct facultyinsert errant equal signs as they performintermediate steps of a problem.7.Another more subtle but, I believe, equallyserious error is the omission of equals signsbetween simplification steps. Perhaps one ofthe most common occurrences of this is in thedemonstration of the factoring process. Forexample, I observe a number of faculty whowrite a factoring problem as:4444222231234322xyxyxyxyOmitting the equal signs between the steps isnot only mathematically unsound but alsoeliminates the rationale for checking theresult of a factoring problem by redistributingbecause they should be equal.8.Correct mathematical language can also be anissue. Some instructors say “LCM” whenthey mean “LCD.” Another common error isconfusion of “expression” and “equation.” Ifwe are to convince our student thatmathematics is a precise discipline withprecise language, our own language must alsomodel such precision.

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79.Students often have difficulty memorizingformulae. Whenever there are formulae thatstudents need to memorize, I give them astrategy for memorizing the formulae as theydo their homework. I tell them that each timethey use the formula, it is important that theywrite the formula first and then substitute thegiven info. Many students tend insteadmerely to write the result after substitutionand miss the learning benefits of the writingprocess. I explain that the repeated writing ofthe original formula as they use it actuallyhelps them to remember it.10.Instructors should be prepared not only todescribe to students what the process is butalso to explain why the process works. Thismay not be an issue for some students but isparticularly important to many adult learners.They may have heard the rules before but didnot buy into the validity of the process. Thisstumbling block is sometimes an impedimentto further learning in the course. Instructorsneed to be willing to provide suchexplanations during class if time permits. Forexample. some students never saw thevalidity of the rules for operations withsigned numbers. Others are still confusedabout why division by zero is undefined.Settling these kinds of questions allowsstudents to progress in topics where they wereconfused in the past.11.The text contains a good introduction to thebenefits of students’ recognizing and usingthe learning objectives of each section. Oneother use of these objectives that werecommend to students is in preparing forexams. So many students at this level havenever learned how to study. They tend tobelieve that the way to prepare for an exam isto work all of the several hundred problemsassigned for homework. We find Bittinger’sa, b, c, ... coding of learning objectives veryhelpful in helping students to identify thatthey do not have to rework all of theproblems in a given section but shouldinstead concentrate on one from eachobjective to identify areas of strength andweakness. Then they understand that if theyhave difficulty, they can return to thediscussion correlating to that objective in thesection/chapter.Susan Leland,Montana TechUsing silly, catchy phrases helps students toremember common pitfalls and procedures inalgebra. Some examples follow:1.Bells and whistles: as in “Bells and whistlesshould go off in your mind every time yousee a negative sign in front of parentheses!Change every sign inside.” This can be usedfor many other situations also. I have studentscome back to see me two and three years afterbeing in my class and they say they still hear“bells and whistles” when doing math.2.The rules never change: as in “This equationhas fractions in it—but the rules neverchange.” Keep properties, theorems,procedures, and rules very generic so thatthey fit every conceivable situation. Studentsat this level need to do problems the sameway every time. Try to avoid showing severaldifferent ways to do anyone type ofproblem—that just confuses manyintermediate algebra students. Those studentswho already know another way to approach agiven problem should be allowed to use adifferent, but correct, method, but don’tburden the rest of the class with more thanone method.3.Details. details, details: as in missing signs,incorrect arithmetic, and copying problemsincorrectly. If you can get students toconcentrate on all the details, grades improve!My students hear me say this at least amillion times in a semester—at leastaccording to them!4.Set the dumpster on fire, as in the followingjoke. A mathematician decided he wanted tochange jobs. He went to the fire chief andsaid. “Chief, I’d like to become a fireman!”The fire chief said, “Great! I need to give youa little test.” The chief took themathematician into the alley where there wasa dumpster, a hose on the ground, and a waterspigot. The chief said to the mathematician,“What would you do if you came out hereand found the dumpster on fire?” Themathematician immediately said, “I’d hookthe hose up to the spigot, turn the water on,and put out the fire.” The chief said, “Great!You’ll make a super fireman! Now, whatwould you do if you came out here and foundthe dumpster wasn’t on fire?” The

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8mathematician thought for a moment, thenbrightened and said, “I’d set it on fire!” Thechief said, “What? That’s terrible! Why in theworld would you set the dumpster on fire?”The mathematician replied, “Because then Iwould have reduced it to a problem I alreadyknow how to solve!” This joke is applicableto so many topics in algebra, especiallyequation solving. Every equation in the book-systems, quadratic, rational, radical-eventually “reduces” to a linear equation,something we already know how to solveafter Chapter One! So, “set the dumpster onfire” and get to a problem we already knowhow to do!Michael Montaño,Riverside CommunityCollege1.Eye contact with your audience is essentialwhen you are delivering a lecture.2.When a class does not reply to a question, donot be too anxious and proceed to answer thequestion yourself. Use the “dead air” time toyour advantage. A classroom does not need tobe dynamic at all times. Silence allowsstudents some time to organize and collecttheir thoughts.3.Definitions are very important. Make surethat students understand every aspect of thedefinition.4.Pattern recognition is a very powerfultechnique in the teaching of mathematics.Fitting the problem to the same format as therule can be very useful.Nancy Ressler,Oakton Community College1.The studentgrapevineis healthy andflourishing! If you habitually modify duedates for homework, exams, quizzes, projectsand content presentation, students will expectthat if they miss any of the above it isacceptable because youprobablywere goingto makeanotherchange anyway. The matterof fairness is prominent. Adjusting for onestudent means that it is necessary toadjustforothers. By doing so, you will havetwicethecourse duties that are necessary for aseasoned educator.2.Additionally, if you only address courseexpectations on the first class meeting, wordwill “get out” andfuturefirst class meetingswill not be well attended since students willexpect only a friendly discussion! Theyrationalize: theycan readthe syllabus bythemselves later! On the first class meetingbring a tablet and sharpened pencils (usuallyeach division or cluster office has supplies forfaculty) for those students that have broughtno supplies. Provide each with a pencil and afew sheets of paper as you begin your courselecture on day ONE! Ensuring good habits bya content focus on day ONE also allowsstudents to experience your teachingmethodology and style.3.Students will complain about the amount ofcourse content in most math courses. FacultyMUST prepare the students for success infuture courses. This can only be achieved bycovering all of the content described by thedepartment’s generic syllabus. You are hiredto teach the c1ass ... not to cater to those thatare better or less prepared. Individualizedhelp can be provided during your office hoursif a reasonable answer still leaves the studentconfused or if a better prepared student hasfuture/later content questions.4.Encourage students to visit with tutors. Inhigh school,needystudents or the labelednerdsfrequented tutoring. In college, thebright and astute attend tutoring. Thedifference between A’s and B’s and B’s andC’s become apparent by tutor visits. Invite acollege tutor to talk to your classes. Once thestudents meet the people they will be workingwith during the course it is easier for them tomake the tutoring appointment!Tomesa Smith,Wallace State CommunityCollegeFifteen years ago I began my teaching career at acommunity college. Considering the typicallydiverse nature of community college students, Idecided to have students fill out an index card ofinformation on the first day of class. On the card, Iasked for:Name (how it will be written on test papers,first and last); Student Number; Major; Phone(in case calculators, purses, books, etc. are left

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9in the room); Email; Family; Job (approximatenumber of hours worked and approximatetimes); Math history (high school and college);Goal for classThrough the years, I have changed the formatsome, but I continue to utilize the cards becausethey have proven to be very helpful. While studentscomplete their cards in class, I tell them someinformation about myself. After taking up thecards, I call roll by them immediately. This helpsme to begin connecting faces with names and toask for correct pronunciation if needed. I note thesethings on the card and then use the set as a deck ofcards throughout the term. I call roll by them,assign groups by them, and document absences orunusual circumstances on them. I think studentsappreciate that I strive to learn their names, that Iam concerned about them when they are absent andthat I want them to achieve their goals.I suppose through the years, I have incorporatedother strategies that have become equally routine tome. I’ve always tried to start class on time. Howcan I expect students to be on time if I’m not? Ialso try to utilize every minute of class time. Inever want students to leave my classroom feelingas though we could have accomplished more.Being organized is half the battle. I think studentsrespect a teacher who is prepared and enthusiasticabout each topic. I also try to foster respect in theclassroom. I give students my attention when theyare talking to me and expect them to give me theirattention when I am talking to them. Often studentswill say that they did not want to ask a questionbecause it was dumb. No questions are dumb ones,so I try to encourage students to ask me aboutanything they do not understand.Aside from the actual lecturing and learning in theclassroom, I have found that these tips have helpedme to become a better instructor through the years.To teach a subject seems the best way to learn it, sorepetition will help with the subject matter. To helpwith everything else, be prepared, punctual,respectful, cheerful, and willing to makeadjustments along the way. If you can do thesethings, you are destined to be a great teacher.Sharon Testone,Onondaga CommunityCollege1.Students enrolled in an intermediate algebracourse at the community college level areoften students who have just completed adevelopmental beginning algebra course orthey are students who have had intermediatealgebra in high school and just need to refreshtheir skills. These two groups of students arevery different from each other. The formerbeginning algebra students may be strugglingwith this new material. Meanwhile, thestudents who just need a refresher may bebored with the material. It is the instructor’srole to meet the needs of these diverse groups.2.Group work can help to alleviate the problemof diverse abilities in an intermediate algebracourse. The instructor should form the groupsby including a mixture of students who neverhave had intermediate algebra and studentswho are just reviewing the material. Thismethod will provide an avenue for students totruly help each other. Be sure to haveindividual students or a representative fromthe group show you at least one completedproblem before the end of the class period.(Another alternative is to have a grouprepresentative put the solution on the board.)This technique helps to assure that studentswill be able to do their homework.3.An important time management tip is to countthe number of class periods, subtract thenumber of “testing days” and subtract at leasttwo class periods for review at the end of thesemester. 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 facultyavoid the pitfall of moving too quickly or tooslowly through the course. Additionally, thefaculty member will most likely complete alltopics in the syllabus and not omit anyessential material.4.Always be on time to class and always endthe class on time. Our students have verybusy schedules and faculty needs to respectthat fact.

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105.Always be prepared for class and alwayshand back graded homework, quizzes, andtests the very next class period.6.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 much time reteaching materialfrom the previous course. Additionally,determine what course(s) your students willenroll in after completing your course. Besure that you prepare the students for thosecourses, but don’t teach the topics from them.7.Often intermediate algebra students do notcomplete their homework assignments andthis leads to failure. One option is to requirethat students complete homeworkassignments in a separate notebook. On testdays instructors can review the notebook(without actually grading it) to determine ifstudents are completing their assignments.Another option is to collect homework dailyor randomly and grade it.8.Giving a five-minute quiz after reviewinghomework questions at the beginning of theclass period is often helpful for 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 orPower Point slides. Students at this level areoften 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.Be sure to assign the synthesis problems ineach chapter as group work. These problemsare a little more challenging and they help thestudents increase their level of understanding.Roy West,Robeson Community CollegeRemember that the material being taught isdevelopmental. A lot of times college instructorsfeel that most students just need some extrapractice and they will catch on. This may be truefor some classes. You as an instructor need to feelthem out. I personally have found that nothingreplaces working problems for them in class andshowing the various situations that can occur. Theresponsible student will get the practice they needwhen they do the homework problems themselvesat home. Use your class time wisely.Rebecca Wyatt-Semple,Nash CommunityCollege1.Wish They Had Told Me ..Rule 1: For all instructors, whether full-timeor adjunct: There is never enough time!Prioritize your schedule so that you don’t getoverwhelmed. If you come into class too tiredto function, your value as an instructor will beminimal at best. Take care of yourself so thatyou will be able to take care of the studentsentrusted to you. We encourage students toset priorities in order to get their work done,so practice what you preach.Rule 2: You’ll work harder than any of yourstudents. You’ll have to be creative inpresentations, develop interestingassignments, challenge the most giftedstudents, and help the least gifted. You willbe many things to your students, but you arenot their buddy. You are a professional whoshould be clearly concerned with eachstudent’s welfare and progression ineducation.Rule 3: Use the text, and the supplementaryitems that are available, to your bestadvantage. Bittinger texts are extremely wellwritten with excellent Study Tips forstudents, generous margins and marginexercises (Yes! Encourage students to writein their books), Chapter Summaries, PracticeTests, warnings and cautions, and CumulativeReviews. Students can teach themselves fromthis text, especially if they take advantage ofthe Learning Resources such as theStudent’sSolutions Manual,videos, InterAct

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11Math®Tutorial Web site, and the on-line helpavailable through MyMathLab and MathXL®.Rule 4: The first week of class is critical. Setthe pace of the course and let students knowwhat is expected of them. Put your rules inwriting and give them out with the coursesyllabus. The first day of class is the best timeto have the students fill out an index cardwith information you may need: name,student ID number, e-mail, phone number,areas of interest, and intended major. Collectthe cards and review them before the nextscheduled meeting of the class. Try to learnthe students’ names as quickly as possible.Use the index cards in selecting problems touse in class and homework. For example, ifyou know that your class has a number ofstudents interested in the sciences ofmedicine, problems can be selected to reflecttheir interest.Rule 5: Know your school or department’spolicy on calculators in the classroom andtheir use on tests and exams. Find out whichcalculators are recommended for your courseand which, if any, are banned. Talk toinstructors who teach the courses followingyours to see what calculator skills they expectstudents to have upon entry into their classes,and be certain that your students developthese skills before the end of the course. It iseffective to split tests and exams in two parts:Part A covers basic concepts with minimalcalculations and no calculators allowed, andPart B emphasizes problems in which the useof calculators would be beneficial.Rule 6: If you assign homework, you’ll haveto grade homework. Students need a greatdeal of practice. You don’t need to becomebogged down in grading homework. Gradeapproximately one-fourth of the homeworkproblems by random selection, or by somemethod of selection of your own. Studentswill not know ahead of time which problemsare to be graded, and are responsible for allproblems assigned. Let students know thatyou are practicing sampling, an acceptablestatistical technique. If you fail to gradehomework, or fail to give some incentive fordoing it, most students will not do thepractice necessary to internalize concepts. Letstudents know from day one what your policyon homework will be, and stick to yourpolicy.Rule 7: Use proper mathematical notation,draw neat graphs, and label the axesappropriately. Do what you want yourstudents to do. Set high standards for yourwork and hold your students to thesestandards. It is far easier to develop goodhabits than to break bad habits. Being skilledin the reading and use of proper notation willallow your students to advance to higherlevels of mathematics, read, and understandthe texts.Rule 8: Most students do not know how toread math texts. Teach them. Structureassignments so that they must read the text,make use of examples, and write briefdiscussions. Assign discussion problems aswell as calculation problems. Have students“teach” a section to the class. Give groupassignments in which a student must explainconcepts to others in his group. Bittinger’smath texts always have an abundance ofdiscussion problems.Rule 9: Prepare yourself for the question:“When will I ever need to know this stuff?”Make a mental list of situations that requirealgebraic thought. At the beginning of eachchapter, think of ways the concepts of thatchapter may be encountered in the “realworld.” If you can’t think of many examples,do a little research or talk to other instructorsin the math area or in other departments.Rule 10:Multiple-choice tests should beused with caution. They are easy to grade, butyou should not fall into the trap of alwaystesting in this manner. Multiple-choice testsdo not always reveal what students actuallyknow, nor do they require critical thinkingskills. Students, by their own admission, do agreat deal of guessing. Mix multiple-choicequestions with questions requiring thatstudents show their work and thoughtprocesses.

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INTERMEDIATE ALGEBRAMini-Lecture R.1ML-1The Set of Real NumbersLearning Objectives:aUse roster notation and set-builder notation to name sets, and distinguish among various kinds of realnumbers.bDetermine which of two real numbers is greater and indicate which, using < and >; given an inequalitylikeab<, write another inequality with the same meaning; and determine whether an inequality like23-£or45>is true.cGraph inequalities on the number line.dFind the absolute value of a real number.Examples:1.Name all numbers from the set310,5,2, 0,5,104ìüïïïï---íýïïïïîþthat are:a) natural numbersb) whole numbersc) integersd) rational numberse) irrational numbersf) real numbers2.Use roster notation to name the set of positive integers less than 6.3.Use set-builder notation to name the set of all real numbers greater than 2.4.Use < or > forto write a true sentence.a)38b)4153c)10125.Write a different inequality with the same meaning.a)6xb)5yc)2.7z6.Graph each inequality.a)2xb)3x 7.Find the absolute value.a)3b)6c)08Teaching Notes:•Remind students that integers are rational numbers; any integer can be written as the ratio of itselfand 1.•Decimal numbers that terminate or repeat in a fixed block are both examples of rational numbers; askstudents to give examples of both.•The decimal form of an irrational number neither terminates nor repeats.•Help students see the relationships among subsets of the real numbers.Answers: 1a) 10, b) 0, 10, c)10,5,0,10--, d)310,2,5,0,104---, e)5, f)310,5 ,2,0,5,104---;2){1,2,3,4,5}; 3){}xx2>; 4a)38-<, b)4153-< -, c)1012> -; 5a)x6<, b)y5³ -, c)2.7z>;6a), b); 7a) 3, b) 6, c) 0

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Mini-Lecture R.2INTERMEDIATE ALGEBRAML-2Operations with Real NumbersLearning Objectives:aAdd real numbers.bFind the opposite, or additive inverse, of a number.cSubtract real numbers.dMultiply real numbers.eDivide real numbers.Examples:1.Find the opposite (additive inverse).a) 3b)34c) 0d)4y2.Add or subtract, as indicated.a)53 b)53 c)53d)2.17.3 e)231224f)1122g)35 h)35i)35 j)5.49.2k)2334l)13843.Multiply.a)  35b)35c) 35d) 35e)82.4f)2 153g)3289h)6824.Find the reciprocal of each number, if possible.a) 12b)56c)12d) 05.Divide, if possible.a)84 b)04 c)3.20d)142e)822yyf)3.61.2 g)315520h)2500.4 Teaching Notes:•Refer students to the boxes forRules for Addition of Real Numbers, Subtracting by Adding theOpposite, and to Multiply or Divide Two Real Numbers.•Many students find working with fractions difficult. It may be useful to review finding the leastcommon denominator.•Remind students that only nonzero numbers have reciprocals.•Refer students to theSign Changes in Fraction Notationbox in the text.Answers:1a)3-, b)34, c) 0, d)4 y; 2a)8-, b) 2, c)2-, d)9.4-, e)124-, f) 0, g) 8, h)8-, i) 2,j)3.8-, k)1712-, l)78 ; 3a) 15, b) 15, c)15-, d)15-, e)19.2, f)10-, g)112, h)96-; 4a)112, b) 65 ,c)2-, d) no reciprocal exists; 5a) 2, b) 0, c) not defined, d) 7, e) not defined, f)3-, g)45-, h) 625

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INTERMEDIATE ALGEBRAMini-Lecture R.3ML-3Exponential Notation and Order of OperationsLearning Objectives:aRewrite expressions with whole-number exponents, and evaluate exponential expressions.bRewrite expressions with or without negative integers as exponents.cSimplify expressions using the rules for order of operations.Examples:1.Write exponential notation.a)     44444b)2222c)xxxxx2.Evaluate.a)25b)32c)43d)25e)14f)32g)03h)43i)212j)31.23.Rewrite using a positive exponent. Evaluate, if possible.a)225b)5xc)41m4.Rewrite using a negative exponent.a)418b)21nc)5135.Simplify.a)5 246b)273 24c)396149d)29 629 e)2358 34 65 f)22691 36 Teaching Notes:•In Example 1b) some students answer42instead of42. Many students think these are equal.•Many students find Example 5 difficult and need a lot of practice.Answers:1a)54, b)()42-, c)5x; 2a) 25, b) 8, c) 81, d) 25, e) 4, f)8-, g) 1, h) 81, i) 14 , j)1.728-;3a)254, b)51x, c)4m; 4a)48-, b)2n-, c)()53--; 5a)4-, b) 1, c)7-, d)342-, e) 13, f)13-

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