Solution manual of real number

RNUM: Real Numbers Workbook Pages …………………………………… 1-24 Building Blocks ……………………………………… ……… Factor Pairings ……………………………………… ……… Match Up on Fractions ……………………………… ……… Fractions Using a Calculator ………………………….……… Charting the Real Numbers ………………………… ……… Venn Diagram of the Real Numbers ……………… ……… Linking Rational Numbers with Decimals ………… ……… Using Addition Models ……………………………… ……… Match Up on Addition of Real Numbers …………… ……… 10 Scrambled Addition Tables ………………………… ……… 11 Signed Numbers Magic Puzzles …………………… ……… 12 Language of Subtraction …………………………… ……… 13 Match Up on Subtraction of Real Numbers ……… ……… 15 Signed Numbers Using a Calculator ……………… ……… 16 Pick Your Property …………………………………….……… 17 What’s Wrong with Division by Zero? ……………….……… 18 Match Up on Multiplying and Dividing Real Numbers …… 19 The Exponent Trio ……………………………… … ……… 20 Order Operation ……………………………………… ……… 21 Assess Your Understanding: Real Numbers ……… ……… 22 Metacognitive Skills: Real Numbers ………………….……… 23 Teaching Guides …………………………………… 25-44 Fractions ……………………………………………….……… The Real Numbers ………………………………… ……… Addition of Real Numbers ………………………… ……… Subtraction of Real Numbers ……………………….……… Multiplication and Division of Real Numbers ……….……… Exponents and Order of Operations ………….……….…… 25 30 33 35 37 40 Solutions to Workbook Pages …………………… 45-52   Student Activity RNUM-1 Building Blocks Fill In The Blanks: Fill in the missing numerators with whole numbers to build equivalent fractions to the fraction in the “Goal” box If there is not a whole number numerator that will work, then cross out the fraction Example: Goal: Goal: Goal: Goal: 3 Goal: 24 50 18 15 30 36 15 15 10 16 Goal: 16 32 80 1000 Goal: 48 32 60 99 Goal: 99 81 54 Goal: 25 35 14 Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Student Activity RNUM-2 Factor Pairings Directions: In each diagram, there is a number in the top box and exactly enough spaces beneath it to write all the possible factor-pairs involving whole numbers See if you can find all the missing factor-pairs The number 30 has been done for you 12 20 24 28 30 ⋅ 30 32 ⋅15 ⋅10 5⋅6 60 36 42 45 50 64 What is the largest number that is a factor of 20 and 30? _ Simplify: 20 30 What is the largest number that is a factor of 28 and 36? _ Simplify: 28 36 What is the largest number that is a factor of 36 and 60? _ Simplify: 36 60 What is the largest number that is a factor of 24 and 42? _ Simplify: 24 42 Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Student Activity RNUM-3 Match Up on Fractions Match-up: Match each of the expressions in the squares of the table below with its simplified value at the top If the solution is not found among the choices A through D, then choose E (none of these) A B C D E None of these ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ÷ ÷0 15 −1 ⎛1⎞ 6⎜ ⎟ ⎝6⎠ ÷ + 19 − 12 + 10 ( 0) + 2 1 ÷ 3 − 0÷ I thought we shared a common denominator, but he was only a fraction of the person I thought he was Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Student Activity RNUM-4 Fractions Using a Calculator When you input fractions into a calculator, you must be careful to tell the calculator which parts are fractions Each calculator has a set of algorithms that tell it what to first (later on, we will learn the mathematical order of operations, which is similar) In order to ensure that fractions are treated as fractions, for now, you need to tell your calculator which parts ARE fractions For example, first show that 15 ÷ is by hand: 15 on the calculator, we type 15 / or 15 ÷ (depending on the calculator) Practice by finding the decimal values for: To get the decimal value of 15 20 3 ÷ , but it without using any parentheses Do you get the decimal value equal to 15/8? Now try using your calculator to evaluate Find the button(s) on your calculator that allow you to input parentheses and write down how to use them on your calculator ⎛3⎞ ⎛2⎞ Try it on your calculator like this now: ⎜ ⎟ ÷ ⎜ ⎟ ⎝4⎠ ⎝5⎠ On my calculators, I type (3 / 4) /(2 / 5) or (3 ÷ 4) ÷ (2 ÷ 5) to enter this expression But each calculator is a little different When you have done it correctly, you should get 1.875 Write down how to it on your calculator: Now try these fraction problems using parentheses to tell your calculator which numbers represent fractions: ⋅ ÷ 15 + − 12 The operation in mixed numbers is addition, so when you input calculator, you must treat it like + (3 / 5) What is into your as a decimal? Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Guided Learning Activity RNUM-5 Charting the Real Numbers The set of natural numbers is { 1, 2, 3, 4, 5, } The set of whole numbers is { 0, 1, 2, 3, 4, 5, } The set of integers is { , − 4, − 3, − 2, − 1, 0, 1, 2, 3, 4, } The set of rational numbers consists of all numbers that can be expressed as a fraction (or ratio) of integers (except when zero is in the denominator) Note that all rational numbers can also be written as decimals that either terminate or repeat The set of irrational numbers consists of all real numbers that are not rational numbers The set of positive numbers consists of all the numbers greater than zero The set of negative numbers consists of all the numbers less than zero Part I: Using the definitions above, we will categorize each number below For each of the numbers in the first column, place an “X” in any set to which that number belongs a b c d −2 e f Natural Whole Integer Rational Irrational Positive Negative X X X X X g −1.4 h 0.75 π i Part II: Now we’ll it backwards Given the checked properties, find a number (try to use one that is different from one of the numbers in the previous table) that fits the properties If it is not possible to find a number with all these properties, write “impossible” instead Number a b c d e f g h Natural Whole Integer Rational Irrational Positive Negative X X X X X X X X X X X X X X X X X X X X X X Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Student Activity RNUM-6 Venn Diagram of the Real Numbers Directions: Place each number below in the smallest set in which it belongs For example, −1 is a real number, a rational number, and an integer, so we place it in the “Integers” box, but not inside the whole numbers or natural numbers −1.3 π 2.175 −7 1000 0.00005 −1 Natural numbers Integers Whole numbers Rational Numbers Irrational Numbers Real Numbers Given all possible real numbers, name at least one number that is a whole number, but not a natural number: _ Can a number be both rational and irrational? _ If yes, name one: Can a number be both rational and an integer? _ If yes, name one: Given all possible real numbers, name at least one number that is an integer, but not a whole number: _ Side note: Just for the record, this diagram in no way conveys the actual size of the sets In mathematics, the number of elements that belong to a set is called the cardinality of the set Technically (and with a lot more mathematics classes behind you) it can be proven that the cardinality of the irrational numbers (uncountable infinity) is actually larger than the cardinality of the rational numbers (countable infinity) Another interesting fact is that the cardinality (size) of the rational numbers, integers, whole numbers, and natural numbers are all equal This type of mathematics is studied in a course called Real Analysis (that comes after the Calculus sequence) Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Student Activity RNUM-7 Linking Rational Numbers with Decimals Let’s investigate why we say that decimals that terminate and repeat are really rational numbers You will need a calculator and some colored pencils for this activity Rational numbers consist of all numbers that can be expressed as a fraction (or ratio) of integers (except when zero is in the denominator) In the grid below are a bunch of fractions of integers Work out the decimal equivalents using your calculator If the decimals are repeating decimals, use an overbar to indicate the repeating sequence (like in the example that has been done for you) Shade the grid squares in which fractions were equivalent to repeating decimals in one color and indicate the color here: _ Shade the grid squares in which fractions were equivalent to terminating decimals in another color and indicate the color here: _ In the last row of the grid, write some of your own fractions built using integers and repeat the steps above = 0.6 = = = = = = 17 = 25 = 1000 = 12 = = 27 = 12 = = = Are there any fractions in the grid that were not shaded as either terminating or repeating? If you write one of these fractions as its decimal equivalent, what kind of decimal you get? Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Guided Learning Activity RNUM-8 Using Addition Models Part I: The first model for addition of real numbers that we look at is called the “colored counters” method Traditionally, this is done with black and red counters, but we make a slight modification here to print in black and white Solid counters (black) represent positive integers, +1 for each counter Dashed counters (red), represent negative integers, −1 for each counter When we look at a collection of counters (inside each rectangle) we can write an addition problem to represent what we see We this by counting the number of solid counters (in this case 3) and counting the number of dashed counters (in this case 5) So the addition problem becomes + ( −5 ) = To perform the addition, we use the Additive Inverse Property, specifically, that + ( −1) = By matching up pairs of positive and negative counters until we run out of matched pairs, we can see the value of the remaining result In this example, we are left with two dashed counters, representing the number −2 So the collection of counters represents the problem + ( −5 ) = −2 Now try to write the problems that represent the collections below a + = b + = c + = d + = e + = f + = Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-38 Examples: Identify the property of multiplication • ( −2 ) = −2 ( 3) (commutative) • • ⎛1⎞ + ⎜ ⎟ = + (inverse property) ⎝3⎠ 1⎞ ⎛ ⎛1 ⎞ ⎜ −4 ⋅ ⎟ ⋅ = −4 ⎜ ⋅ ⎟ (associative) 5⎠ ⎝ ⎝5 ⎠ • −6 + (1) = −6 + (identity) • ( ⋅ x ) 2⋅ = ( ⋅ x ) (commutative) ⋅ + ( −3) = + ( −3) (property of 0) • Student Activity: Pick Your Property In this activity, students practice identifying both the properties of addition and multiplication (associative, commutative, inverse, identity, and the zero property) (RNUM-17) Student Activity: What’s Wrong with Division by Zero? This activity will require a calculator, but should help students understand why division by zero has to be undefined without going into asymptotes and calculus (RNUM-18) Properties of Division: • • a a = a and = (where a ≠ ) a a Division Involving 0: = and is undefined (where a ≠ ) a Division Properties: Examples: Simplify • • • −3 −3 −5 −2 =1 • = −5 • (7) = −4 undefined =0 Multiplication or Division of Real Numbers: Multiply (or divide) the absolute values of the two numbers The result is a positive number Yes START HERE: Are the signs like? No Multiply (or divide) the absolute values of the two numbers Make the result negative Note: The above chart is for non-zero real numbers Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-39 Examples: Simplify • • • • • ( −3)( −7 ) = 21 = −60 −12 ( ) ( −2 )( −2 )( −3) = −12 • • −16 =2 −8 7⎛3⎞ − ⎜ ⎟ =− 9⎝2⎠ • • −56 = −7 ⎛4⎞ =− − ÷⎜ ⎟ ⎝9⎠ 18 = −6 −3 ( 3)( −9 ) = −27 Student Activity: Match up on Multiplying and Dividing Real Numbers This activity works well in groups or with students working in pairs at a whiteboard This activity also involves fractions, and a few problems with addition or subtraction, so students will get more practice with these topics (RNUM-19) Examples with expressions that look alike: Simplify • 8−2 ( −2 ) + ( −2 ) ÷ ( −2 ) −16 −4 • • −6 ( −3) −6−3 − + ( −3) − − ( −3 ) − ( −3) 18 −9 −9 −3 18 + ( −2 ) −2 − ( −2 ) ÷ ( −2 ) 0 ( −2 ) Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Teaching Guide RNUM-40 Exponents and Order of Operations Preparing for Your Class Common Vocabulary • Exponential notation, exponent, base, exponential expression • Order of operations ( ), [ ], • Grouping symbols: • • Innermost pair, outermost pair (as in parentheses or grouping symbols) Arithmetic mean, average , (fraction bar) Instruction Tips • A common student misconception is that ( −3) and −32 are the same Likely this is because at this point in the course, the students don’t think the parentheses in math really much of anything For example, we rewrite − as + ( −8 ) , but practically, the result does not change And so, students get the idea that parentheses don’t have much meaning, which is, of course, wrong If you continue to stress the expanded notation for these expressions: ( −3) = ( −3)( −3) and −32 = − ( 3)( 3) or − ( ⋅ 3) , your students will eventually understand the • • difference You should prepare yourself to re-explain this concept every time it comes up In fact, rather than explaining it yourself, ask the student who is confused to try to explain the difference to you instead Often instructors teach the order of operations with “Please Excuse My Dear Aunt Sally” to stand for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction Unfortunately, this leaves students with the mistaken impression that multiplication always comes before division, which is not true Multiplications and divisions are evaluated from left to right So perhaps it is better to say “Please Excuse My Dear Little Relative, Aunt Sally” to remind students about the left-to-right part Some expressions are more “tempting” to incorrectly evaluate than others These expressions give strong visual cues for the student to perform an incorrect operation first For example, in − ⋅ + , the − and the + are very tempting visually Consider that the students may be much more comfortable with addition and subtraction symbols than they are with the raised dot symbol for multiplication, and so their eyes are drawn into the wrong symbols first Also, the − and + symbols print much larger than the ⋅ symbol, and are naturally more noticeable Here’s a few more “tempting” examples and why they tempt students: o − − (doesn’t this just look like two side-by-side subtraction problems?) 2+6 (students really want to “cancel” the 2’s here) o ÷ ⋅16 (if students have a misconception of multiplication before division, they’ll this one wrong) o ⋅ 32 (now the ⋅ is more familiar than the 32 part, so they want to multiply first) o Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-41 Teaching Your Class Language of Exponents: • Use 52 and 34 to discuss the vocabulary base and exponent • The notation 52 means the factor is repeated times: ⋅ , the notation 34 means that the factor is repeated times: ⋅ ⋅ ⋅ • Discuss the base and exponent for ( −5 ) and −52 Examples: • 25 = ⋅ ⋅ ⋅ ⋅ = 32 • ( −3) = ( −3)( −3)( −3)( −3) = 81 • • • • ( −2 ) = ( −2 )( −2 )( −2 )( −2 )( −2 )( −2 ) = 64 ( −2 ) = ( −2 )( −2 )( −2 ) = −8 ( −3) = ( −3)( −3) = −32 = − ( ⋅ 3) = −9 Even and Odd Powers of a Negative Number: When a negative number is raised to an even power, the result is positive When a negative number is raised to an odd power, the result is negative Student Activity: The Exponent Trio This activity will give students practice with converting between expanded exponential expressions, exponential form, and evaluating exponential expressions This lays nice groundwork for eventually understanding why, for example, the solution to x = is both and −3 (RNUM-20) The Order of Operations: The diagram found on the last page of this teaching guide can be copied to an overhead, projected from a portable video projector, or projected from a computer and ceiling projector to a whiteboard Student Activity: Order Operation In this activity, there are many examples where students “jump” at an incorrect first step The idea is to carefully decide on a first step; then evaluate each expression Watch for those “tempting” mistakes that students make in this activity Each student will need a highlighter for this activity (RNUM-21) Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-42 Arithmetic mean (or average): • To find the mean of a set of values, divide the sum of the values by the number of values Example: Find the mean temperature for the following readings in degrees Celsius: −3°, 6°, 20°, and 5° (answer is 7°C ) Assess Your Understanding Real Numbers This activity is designed to help students see the big picture and to go back through all that they have learned in this chapter Students should try to write out, in words, how to start each problem, as if they were explaining it to a friend (RNUM-22) Metacognitive Skills Real Numbers This assessment provides a list of learning objectives for the chapter Students can use this to gain some insight into how well they actually know what they think they know (RNUM-23) Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning Order of Operations RNUM-43 Perform calculations inside parentheses (fraction bars, and absolute values) working from innermost to outermost pairs Evaluate exponential expressions Perform multiplications and divisions as they occur from left to right Perform additions and subtractions as they occur from left to right Please excuse my dear little relative, Aunt Sally Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning   RNUM-45 Solution Building Blocks (RNUM-1) Goal: Goal: Goal: 3 Goal: 24 36 75 50 12 9 15 15 12 18 10 15 16 Goal: 16 32 15 80 1500 1000 Goal: 30 48 20 32 60 99 99 Goal: 77 99 63 81 42 54 Goal: 35 25 49 35 14 Solution Factor Pairings (RNUM-2) 12 ⋅12 2⋅6 20 ⋅ 20 ⋅10 3⋅ 4⋅5 4⋅6 28 ⋅ 28 ⋅14 4⋅7 30 ⋅ 30 ⋅15 ⋅10 5⋅6 32 ⋅ 32 ⋅16 ⋅8 60 36 1⋅ 36 ⋅18 ⋅12 42 ⋅ 42 ⋅ 21 ⋅14 4⋅9 6⋅6 10, 24 1⋅ 24 ⋅12 3⋅8 6⋅7 4, 45 ⋅ 45 ⋅15 50 1⋅ 50 ⋅ 25 5⋅9 ⋅10 12, 6, ⋅ 60 ⋅ 30 ⋅ 20 ⋅15 64 ⋅ 64 ⋅ 32 ⋅16 ⋅12 ⋅10 8⋅8 Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-46 Solution Match up on Fractions (RNUM-3) B C C A E C A B D A E B C D Solution Fractions Using a Calculator (RNUM-4) 3 15 ÷ = ⋅ = 15 2 = 1.875; = 0.15; = 0.25; = 0.875; = 0.6666 = 0.6 20 3 No 4-5 Keystrokes will vary by calculator ⎛1⎞ ⎛4⎞ ⎛ ⎞ ⎛ ⎞ 45 = 5.625; ⎜ ⎟ ⋅ ⎜ ⎟ = = 0.2; ⎜ ⎟ ÷ ⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ 18 ⎝ ⎠ ⎝ 15 ⎠ ⎛ ⎞ ⎛ ⎞ 47 ⎛ ⎞ ⎛ ⎞ 23 = 1.175; ⎜ ⎟ − ⎜ ⎟ = = 0.383 ⎜ ⎟+⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ 40 ⎝ 12 ⎠ ⎝ ⎠ 60 2.6 Solution Charting the Real Numbers (RNUM-5) Part I: a b c d −2 e f Natural Whole Integer Rational Irrational Positive Negative X X X X X X X X X X X X X X X g −1.4 h 0.75 π i X X X X X X X X X X X Part II: Answers will vary; an example of each is given below Ask students to put all their answers on the board for parts a through h so that the class can discuss which are correct f impossible g h π Example answers: a −6 b 3.2 c d − e − Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-47 Solution Venn Diagram of the Real Numbers (RNUM-6) −1 1000 −7 π Natural numbers Whole numbers Integers −1.3 2.175 0.00005 Irrational Numbers Rational Numbers Real Numbers The only number that is a whole number and not a natural number is No Yes All integers are rational numbers, examples: = 15 , = 12 , − = −3 All of the negative integers are whole numbers, examples: , − 4, − 3, − 2, − Solution Linking Rational Numbers with Decimals (RNUM-7) 1-4 See grid answers below = 0.6 = 0.5 = 0.001 1000 = 0.2 Answers will vary = 0.25 4 = 0.4 = 0.416 12 12 = 2.4 Answers will vary = 0.875 8 = 1.6 = 0.75 4 = 1.3 Answers will vary = 0.5 17 = 0.68 25 = 0.259 27 = 0.16 Answers will vary No All the boxes in the grid should be shaded Fractions of integers will always be either a repeating or terminating decimal Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-48 Solution Using Addition Models (RNUM-8) Part I: a + ( −7 ) = −3 or −7 + = −3 b + ( −3) = or −3 + = c + ( −2 ) = or −2 + = d −5 + ( −3) = −8 or + ( −8 ) = −8 e + ( −4 ) = or −4 + = f + ( −6 ) = −5 or −6 + = −5 Part II: a − + = −7 −6 −5 −4 −3 d − + = −4 −2 −1 b − + ( −3) = −5 −7 −6 −5 −4 −3 −2 −1 −3 −3 −2 −1 e − + ( −2 ) + ( −1) = −5 c + ( −5 ) = −7 −6 −5 −4 −7 −6 −5 −4 −7 −6 −5 −4 −3 −2 −1 7 f + ( −8 ) + = −1 −2 −1 −7 −6 −5 −4 −3 −2 −1 Solution Match Up on Addition of Real Numbers (RNUM-10) B A C E B C E C B E A B D A A B C D E D Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-49 Solution Scrambled Addition Tables (RNUM-11) + −3 −2 −1 + 15 −5 20 −10 10 −3 −6 −5 −4 −3 −2 −1 −2 −5 −4 −3 −2 −1 0 10 25 20 30 25 −5 10 15 10 20 15 −1 −4 −3 −2 −1 −3 −2 −1 0 1 2 −5 −15 10 −10 −20 15 −15 −5 −25 −15 −10 −5 −2 −1 −1 0 1 2 3 4 20 35 15 15 −5 40 20 10 −10 20 25 30 10 3 15 30 10 35 15 20 25 + −4 −1 −10 15 −20 −15 −30 −10 −5 −20 −10 10 15 −5 15 25 −5 20 + −5 −3 −9 −6 −2 −5 −2 −6 −3 −2 6 −4 −1 5 10 −5 −20 −25 −10 −15 −5 10 5 Solution Signed Number Magic Puzzles (RNUM-12) Magic Puzzle #1 −2 10 −9 −3 Magic Puzzle #2 −1 -3 -1 -5 -2 -6 Magic Number = Magic Puzzle #4 Magic Puzzle #3 − 14 4 Magic Number = −1 12 12 − 12 14 −2 12 Magic Number = Magic Puzzle #5 −7 −2 −9 −4 10 −4 −5 −9 13 −6 −8 −3 −3 14 −11 −2 Magic Number = Magic Number = − 2 −5 −1 −8 −7 Instructor’s Resource Binder, M Andersen, Copyright 2011, Cengage Learning RNUM-50 Solution Language of Subtraction (RNUM-13) a 0