Skip to content
Class Notes

Revision Notes for Basic College Mathematics, 13th Edition

Looking for organized chapter notes? Revision Notes for Basic College Mathematics, 13th Edition breaks down your lectures into concise summaries to enhance your understanding.

Preview 16 of 558 pages

See the first 16 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

     I NSTRUCTOR ’ S R ESOURCE M ANUAL B ASIC C OLLEGE M ATHEMATICS T HIRTEENTH E DITION Marvin L. Bittinger Indiana University Purdue University Indianapolis Judith A. Beecher Barbara L. Johnson Ivy Tech Community College of Indiana

  2. Page 2
  3. Page 3
    Read page 3 as text

    i G ENERAL, F IRST- T IME A DVICE We asked the contributing professors for words of advice to instructors who are teaching this course for the first time or for the first time in a long while. Their responses can be found on the following pages. Sally Jackman, Richland College 1. This course can be a very challenging one to teach because of the variety of skill levels that the students bring to class. You will find that many of your students know the material and only need a review. This type of student often will want you to go faster. You will have the student who knows some of the material, but not all of the material. This student may want you to move both faster and the slower throughout the course. You may have the student who has never learned the material. This student usually wants you to go slower. Last but not least, there is the student who thinks they know the material and something just happened when they took the placement test. This type of student can be the most difficult because they want to do as little as possible: “they already know the material.” I have found that the best way to handle all the different skill levels is to assume that they all need to learn the material for the first time and that it needs to be learned correctly. I plan the entire course calendar before the semester begins. I have the calendar set up as to which sections will be taught on which day, and what days I will have exams. I make this calendar part of my syllabus. I try to follow this schedule as closely as possible to keep my students on track. There have been times when I am not able to adhere to the schedule. When that happens, I try to get back on schedule or I do an addendum to the schedule and hand that out in class. I have tried teaching without a schedule and when I did, I had a hard time finishing the material. You may say “What’s wrong with that?” My answer is that it hurts the students when they move to the next level and the material not covered is required for the next level. Thus, by having a calendar or schedule, it helps to keep the class moving at a pace in which all levels should be able to learn. If this is your first time teaching the course, find out what sections you are required to teach. Ask an instructor who has taught the course before where more time is needed for a particular section or chapter. Plan the schedule and then have another instructor who has taught the course before look at your schedule. During the semester, keep notes on your schedule as to where students needed more time or less time than you planned. This will help you the next time you teach this same course. 2. Students at this level need lots of practice on these concepts. If they just watch you working at the board, they will have a harder time learning the concepts. You need to have the students actively involved in the lecture because this helps them to remember better. I do this several ways: groups, send them to the board, practice in class, and tell me how to work a problem at the board. 3. I tell students that “mathematics is its own language” and they must learn the language of mathematics just like they would need to learn any foreign language. So, I stress vocabulary. When explaining a concept, I always use correct terminology and I want them to use it. For example, when I talk about fractions I do not say “top and bottom.” I say “the numerator and the denominator.” So many times, I have to help the students unlearn the incorrect terminology that they have brought with them to class. We need to use correct terminology so that as students progress in mathematics, they have the vocabulary that is needed to be successful. Remember that correct vocabulary starts with the teacher. 4. Many times, students at this level do not know how to study mathematics. Thus, I feel that I should help my students learn study skills. I do this several ways.

  4. Page 4
    Read page 4 as text

    Resource Manual ii I begin discussing study skills by listing helpful hints in my syllabus. Here are some that I have in my syllabus: Take the chapter test in the textbook for practice—if you have questions, make sure you get help. Keep up with the lessons and homework. Get together with someone else in the class to talk through problems that seem to be taking a long time to solve—Don’t spin your wheels. There is a video in our library on student skills that I have students check out and watch. I have them write a one-page report on study skills that I grade for extra credit. I have students keep a notebook for class. In the notebook, they keep class notes, handouts, assignment problems, and graded papers, to name a few items. Once in a while, I will check the notebooks by either collecting them or going around the room and looking at the notebooks. If you are going to check notebooks, you need to check them the within the first three weeks. If they are checked early, it helps the students to have a better notebook later in the semester. I have students work in pairs a lot in class. I explain how to work a problem. Then in pairs, I have the students work some problems. One student writes the problem while the other student explains how to work it. Then they reverse roles for the next problem. This approach is a good study skill because it helps students learn that if they can explain how to work a problem, then they are learning more themselves. 5. Encourage students to write mathematics. I sometimes give a quiz or writing assignment where they have to explain how to solve a problem. For example: Explain in words how to find the answer to this problem: 1 1 2 4 + . I do not want them to work it out and hand that in. I want them to write that they need to find a common denominator and make equivalent fractions. If they can explain the process, then they can solve almost any problem because they know the process. Too many times I have heard, “I never saw that problem so how could I work it on the test.” If they can explain the process, then they should be able to solve problems and not try to memorize every problem they have worked. Roger McCoach , County College of Morris 1. The students who take this course need structure. Specific homework must be assigned, giving the students some direction. Be careful that you don’t assign too many problems, which might overwhelm the students. If the course is run as a self-paced course, then a timeline needs to be given as well, in order to keep the students from falling behind. 2. You, as an instructor, cannot assume anything. Show the students how the author explains the material with examples within the text and then show them that similar exercises are provided for the student to do in the margins. 3. Making a big deal out of attendance is important. 4. If students can do a problem “their way” it is fine—they don’t have to do the problem as the text dictates. Mention that fractions always need to be reduced, whether or not the problem says it. 5. The cumulative reviews are a very good tool for a student when preparing for a final exam. 6. Calculators are always a touchy issue. We cover Basic Mathematics ’ Chapters 1–6, part of 7, part of 9, and 10 in the course. Calculators are permitted beginning in Chapter 5. The idea is that the students first demonstrate that they are able to add, subtract, multiply, and divide integers, fractions and decimals (Chapters 1–4) by themselves—then in Chapter 5 calculators are allowed. 7. The Synthesis part of the chapter tests at the end of the chapter might be skipped because time is such an issue in this course. This section may be appropriate for the well-above average student. 8. When in doubt, an algebra approach to a problem may be better than another method. It will be helpful when they go on to algebra in the next semester. It is highly recom- mended that the course include Chapter 10, which is actually the first chapter in the publisher’s algebra text.

  5. Page 5
    Read page 5 as text

    General, First-Time Advice ii i 9. To become a bit philosophical, I have always believed that a good deal of this course should be about teaching the people in the class how to become students and teaching them skills that can be applied to all courses. Many people in the class have poor or non- existent study habits and need to be made aware that they will not be passed through the course, as they have been passed through many courses in high school. Encouragement is important, but it must be clearly defined what is expected of the student and what will be tolerated. 10. I have found that a combination of my lecturing plus time where the students work in the text by themselves (where I can assist them) is the best approach. The students will often “tune out” if there is too much lecturing and expecting the students to teach themselves the entire course is not realistic. 11. Giving a test for each chapter is the best option. Giving a test on several chapters would cover too much material at one time. Deloria Nanze-Davis , University of Texas, Brownsville, Texas Southmost College The teaching tips listed will help students learn terminology, concepts, and problem-solving techniques quicker. 1. Always use correct terminology. 2. Quiz students on definitions and terms. 3. Have students go to the board to work and explain problems to their classmates. 4. Have students work in groups when it comes to solving algebraic word problems. 5. Always assign homework. 6. Assign the synthesis problems. 7. Encourage the students to read the lesson before coming to class. Christy Peters , Edison Community College At this time in my career, I have enjoyed fifteen years in the classroom. This has included experiences from K through undergraduate college work. Each experience has provided me with some valuable insights and wonderful memories that will last my entire lifetime. The biggest tip I can give new teachers is to handle each student with care. Take time to find out their strengths and encourage them to do well! Each of us can be a shining star if we just have the right person to polish our skills. As William G. Spady stated, “All students can learn and succeed, but not on the same day in the same way.” Richard Semmler , Northern Virginia Community College The most important tip is to plan out the course and prepare a syllabus for your students before you start teaching. Plan to cover the course during the academic term with the exception of the last day, which can be used as a review day or a make-up day in the event the college was closed or you were ill for a day. If the school has a mentoring program, take advantage of it so that you will have guidance from an experienced teacher. This can be important since many courses incorporate technology with the instruction and you can benefit from the advice of a master teacher. If the class meets for more than one hour, plan a variety of activities to break up the lecture. This could include group work or having students solve problems on the blackboard. This is quite useful if you plan to take a short break between lectures. Sharon Testone , Onondaga Community College 1. Students enrolled in a basic mathematics course at the community-college level are often extremely math anxious. These students are different from those you may have taught at four-year colleges/universities or in the secondary schools. They are often returning adults and need to be treated with respect and understanding. A kind word of encouragement or a smile means the world to these students. Many of these people have accomplished many things in their lives and they just have

  6. Page 6
    Read page 6 as text

    Resource Manual iv difficulty with math. Instructors should encourage the students not to feel badly about being in this course and should help them to see that their worth to society is not measured by their mathematical ability. 2. An important time-management tip is to count the number of class periods, subtract the number of testing days, and then subtract at least two class periods for review at the end the semester. This result is the number of instructional periods. Divide the number of instructional periods into the number of sections that need to be taught. This result gives the approximate number of sections that need to be taught each class period. This simple calculation will help new faculty to avoid the pitfall of moving too quickly or too slowly through the course. Additionally, the faculty member should most likely complete all topics in the syllabus and not omit any essential material. 3. Always be on time to class and always end the class on time. Our students have very busy schedules and faculty need to respect that fact. 4. Always be prepared for class and always hand back graded homework, quizzes, and tests the very next class period. 5. Before teaching the course for the first time, ask your department chairperson what the prerequisites are for this course. This information will assist you in gauging what knowledge the students should have when entering your classroom. They may not be completely prepared, but you will know what skills they are expected to possess and you will not spend too much time reteaching material from the previous course. Additionally, determine what course your students will enroll in after completing your course. Be sure that you prepare the students for that course, but don’t teach the topics from that course. 6. Students at this level need to practice, practice, and practice. Develop additional worksheets and allow time for group practice in class. Have individual students or a representative from the group show you at least one completed problem before the end of the class period. (Another alternative is to have a group representative put the solution on the board.) This technique helps to ensure that students will be able to do their homework. 7. Often students at this level do not complete their homework assignments and this leads to failure. One option is to require that students complete homework assignments in a separate notebook. On test days, instructors can review the notebook (without actually grading it) to determine if students are completing their assignments. Another option is to collect homework daily or randomly and grade it. 8. Giving a five-minute quiz after reviewing homework questions at the beginning of the class period is often helpful to new instructors. The quiz results will let both the students and the instructor know how they are doing. If the whole class fails a quiz, then most likely the instructor needs to make improvements. 9. Prepare handouts with matching overheads or PowerPoint slides. Students at this level often are not good note takers and have difficulty listening and writing at the same time. Handouts that include key concepts, one or two worked-out examples, and two or three problems for the students to complete immediately are very useful. 10. Have fun! Teaching basic math is not about the material. It is about the students. The challenge is to present the material in a way that the students can understand it and enjoy learning math. Since they may have attempted to learn this material many, many times in the past, I try to present things in a different way.

  7. Page 7
    Read page 7 as text

    General, First-Time Advice v Cassondra Thompson , York Technical College I have found that my students have difficulty with Chapters 2 and 3 on fractions, so I teach Chapter 4 on decimals at the beginning of the semester. Chapter 4 does not require as much teaching time and allows the students a chance for success early in the semester. By teaching Chapter 4 at a fairly quick pace, this allows for more teaching time on Chapters 2 and 3 to help students to grasp the concept of fractions. It is also wise to allow more teaching time for Chapter 10 on real numbers and 11 on algebra. I have found that by teaching adult learners, some of my students are learning algebra for the first time and that requires some extra time to explain concepts with concrete examples. Roy West , Robeson Community College When working a problem in class, think out loud. Tell students what you are thinking as it relates to the problem, and how you came up with the decisions of what to do next. Always be ready to answer “Why?” and ask “Why?” For example: When dividing 102.43 by 1.2, “Why do you move the decimal in both numbers one place to the right?” Remember that many students in these developmental classes will be helping their children with math. They need to know why things are done the way they are to better their own understanding and to share with their children. Paige Wood , Kilgore College 1. As a way of getting to know my students, I ask them to write a paper about themselves. The paper should include general information about the student: major, math background, work information, family, etc. The paper should be 1 page, double spaced, and no larger than a 14-point font. If the students choose to handwrite their paper, I expect two single-spaced pages. I grade these according to length and effort. For the first few weeks or the semester, I look back at the papers as a way to keep me acquainted with them. 2. I also make it very clear to the students, that although I assign “homework,” they should view the assignments as a way to get personal feedback from the instructor. When grading their assignments, I give them written responses regarding their mistakes and show correct procedures in an attempt to help them learn the new material. The conscientious students really appreciate the extra pointers that I write on their homework. 3. I work in the math lab as part of my contractual obligations to my institution. The value of this assignment is that I am accessible to my students. So, if you have time, ask if you can volunteer in the learning center. Your students will be glad that you are here. 4. I have found that I can keep the students’ attention better if they complete problems on a worksheet page to turn in at the end of the class period. They can take notes in their spiral but this sheet is strictly for them to practice the new problems and procedures before they leave to tackle the material on their own. I ask for volunteers to work the problems on the board to make sure that the students got a correct answer or so that they can correct their own work.

  8. Page 8
    Read page 8 as text

    Mini-Lecture 1.1 BASIC COLLEGE MATHEMATICS ML-1 Standard Notation Learning Objectives: a Give the meaning of digits in standard notation. b Convert from standard notation to expanded notation. c Convert between standard notation and word names. Examples: 1. What does the digit 5 mean in each number? a) 3654 b) 265,812 c) 56,203,411 2. Write expanded notation for each number. a) 2756 b) 6172 c) 90,943 3. Write a word name for each number. a) 325 b) 60,448 c) 9,542,006 4. Write standard notation. a) two hundred fifty-three b) seven thousand, ninety-eight c) three hundred forty million, one hundred thirty-two 5. 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 period vocabulary such as ones, thousands, millions, billions, etc. Refer them to the Place-value Chart in the textbook. • Many examples of tables are available on the Internet. Request that students obtain copies and ask them to explain 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, or 9 ten thousands + 9 hundreds + 4 tens + 3 ones; 3a) three hundred twenty-five, b) sixty thousand, four hundred forty-eight, c) nine million, five hundred forty-two thousand, six; 4a) 253, b) 7098, c) 340,000,132; 5) A hyphen is used when writing the numbers 21–99, except for numbers ending in zero.

  9. Page 9
    Read page 9 as text

    BASIC COLLEGE MATHEMATICS Mini-Lecture 1.2 ML-2 Addition Learning Objectives: a Add whole numbers. b Use addition in finding perimeter. Examples: 1. Add. a) 53 12  b) 1123 345  c) 40, 001 32, 442 15,333  d) 96 47  e) 5678 3574  f) 6505 173 7044 168  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 addition with carrying (or regrouping). • Most students find this section easy. • Point out that the commutative property deals with the order of the addends, whereas the associative property deals with the grouping (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

  10. Page 10
    Read page 10 as text

    Mini-Lecture 1.3 BASIC COLLEGE MATHEMATICS ML-3 Subtraction Learning Objective: a Subtract whole numbers. Examples: 1. Subtract. a) 98 51 - b) 54 22 - c) 664 51 - d) 51 12 - e) 1123 345 - f) 40,001 15,333 - g) 8273 4365 - h) 82,333 18,888 - i) 3004 2805 - 2. In order to check the subtraction problem 523 – 276 = 247 by using addition, you would add _____ 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 for subtraction 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 you get 523.

  11. Page 11
    Read page 11 as text

    BASIC COLLEGE MATHEMATICS Mini-Lecture 1.4 ML-4 Multiplication Learning Objectives: a Multiply whole numbers. b Use multiplication in finding area. Examples: 1. Multiply. a) 436 4  b) 18 22  c) 534 54  d) 4302 107  e) 160 200  f) 7549 3813  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

  12. Page 12
    Read page 12 as text

    Mini-Lecture 1.5 BASIC COLLEGE MATHEMATICS ML-5 Division Learning Objective: a Divide whole numbers. Examples: 1. Divide, if possible. If not possible, write “not defined.” a) 6 1  b) 7 49 c) 9 0  d) 4 0 e) 56 8 f) 7 7  2. Divide. a) 189 9  b) 4 1290 c) 8 32, 222 d) 40 8258 e) 777 37  f) 103 5267 3. How could you check the answer to this division problem? 71 R2 6 428 Teaching 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 nonzero number results in zero. • Some students do not know how to check a division problem if there is a remainder and must be shown several 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.

  13. Page 13
    Read page 13 as text

    BASIC COLLEGE MATHEMATICS Mini-Lecture 1.6 ML-6 Rounding and Estimating; Order Learning Objectives: a Round to the nearest ten, hundred, or thousand. b Estimate sums, differences, products, and quotients by rounding. c Use < or > for  to write a true sentence in a situation like 6 10.  Examples: 1. Round to the nearest ten. a) 212 b) 3487 c) 14 d) 15,861 2. Round to the nearest hundred. a) 312 b) 1267 c) 83 d) 14,961 3. Round to the nearest thousand. a) 3549 b) 677 c) 27,217 d) 1171 4. Estimate the sum or difference by first rounding to the nearest hundred. a) 257 + 524 + 388 + 671 b) 922 – 438 c) 433 + 591 + 827 5. Estimate the product or quotient by first rounding to the nearest ten. a) 67 × 72 b) 19 × 93 c) 52 ÷ 11 d) 57 ÷ 21 6. Tickets for a lobster dinner in Maine cost $19 each. Ticket sales totaled $1425. Estimate the number of people who attended the dinner by rounding the cost of a ticket to the nearest ten and the total sales to the nearest hundred. 7. Use < or > for  to write a true sentence. a) 236 227  b) 45 35  c) 885 975  Teaching Notes: • Some students are unfamiliar with rounding and will need to be repeatedly reminded to look at the digit to the right of the rounding position. • A common mistake students make is to leave the digits to the right of the rounding position intact instead of changing 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) <

  14. Page 14
    Read page 14 as text

    Mini-Lecture 1.7 BASIC COLLEGE MATHEMATICS ML-7 Solving Equations Learning Objectives: a Solve simple equations by trial. b Solve equations like 28 54, 28 168, and 98 2 . t x y       Examples: 1. Solve by trial. a) 3 10 x   b) 27 3 w   2. Solve. Be sure to check. a) 4784 5421 x   b) 81 12 p   c) 31 52 x   d) 206 128 m   e) 45 85 z   f) 6 156 y   g) 41 2747 t   h) 1190 5 x   Teaching Notes: • Emphasize that the Solutions by Trial technique results mostly in wrong solutions. However by using algebra, 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 to find 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

  15. Page 15
    Read page 15 as text

    BASIC COLLEGE MATHEMATICS Mini-Lecture 1.8 ML-8 Applications Problem Solving Learning Objective: a Solve 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 than b) decreased by c) of d) quotient e) is 2. Solve. a) Students at a party ordered 8 large pizzas. Each pizza is cut into 6 slices. How many total slices are there? b) If a new amusement park covers 54 acres and there are 44,010 square feet in 1 acre, how many square feet of land does the amusement park cover? c) The town of Chelmsford, Massachusetts, has a population of 45,322. The town of Bedford, Massachusetts, has a population of 31,004. What is the difference in population 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 the deposits are recorded and all the checks clear, what balance will he have in his checking account? c) Enrico’s car gets 32 miles to the gallon (mpg) in highway driving. How many gallons will the car use for 5536 mi of highway driving? Teaching Notes: • Many students struggle with application problems. • Refer students to the Five Steps for Problem Solving box in 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

  16. Page 16
    Read page 16 as text

    Mini-Lecture 1.9 BASIC COLLEGE MATHEMATICS ML-9 Exponential Notation and Order of Operations Learning Objectives: a Write exponential notation for products such as 4 4 4. ⋅ ⋅ b Evaluate exponential notation. c Simplify expressions using the rules for order of operations. d Remove 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) 3 2 b) 2 3 c) 4 4 d) 10 3 e) 7 1 f) 9 0 3. Simplify. a) 5 7 6 ´ - b) 2 8 3 6 2 ´ - ¸ c) ( ) 2 5 4 3 5 ´ - + d) ( ) 3 15 3 8 9 12 2 ¸ ´ ´ ¸ - e) 3 3 2 2 3 1 + + f) ( ) 2 4 9 9 5 2 3 ´ - - ¸ + 4. Find average of 42, 54, 68, and 24. 5. Simplify. a) ( ) { } [ ] 16 2 12 4 2 1 ¸ + - ´ - b) ( ) ( ) { } [ ] 3 4 16 5 2 9 3 ´ + - + - ¸ 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 word or as in multiply or divide (or add or subtract) implies whichever comes first. • Refer students to the Order of Operations box in the textbook. Answers: 1a) 4 3 , b) 2 4 , c) 1 5 ; 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

542 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.