Introductory Algebra

Transcription

1 Introductory Algebra Student Media Workbook Manuscript An open source (CC-BY) media work book Adapted by Jesse Frausto and Elaine Pham Santiago Canyon College Continuing Education Division This is an adaptation of various OER textbooks listed on the next page. Sections have been rearranged, transformed, removed and added.

2 ACKNOWLEDGEMENT This workbook is made possible because of the exceptional work done by the following people listed below. We are deeply grateful. Introductory Algebra CK-12 flexbook Available for free download at: Introductory Algebra Student Workbook 6 th edition Scottsdale Community College Development Team Available for free download at: Arithmetic for College Readiness Student Workbook 1 st edition Scottsdale Community College Development Team Available for free download at: Understanding Algebra (with author s permission) James Brennan, Boise State University Available for free download at Beginning Algebra Darlene Diaz Santiago Canyon College Available for free download at html Beginning and Intermediate Algebra Tyler Wallace Available for free download at Numerous math video lessons from James Sousa at Tyler Wallace at Salman Khan at Larry Perez at Cover image Fibonacci spirals Photo by Aldo Cavini Benedetti Special thanks to Shannon Carter and Berenice Diaz for their help with typing up the answer key for this book. i

3 Copyright 2017, some rights reserved CC-BY Introductory Algebra is licensed under a Creative Commons Attribution 3.0 Unported License. You are free to share: copy, distribute and transmit the work remix: adapt the work Under the following conditions: Attribution: You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). With the understanding that Waiver: If you get permission from the copyright holder, any of the above conditions can be waived. Public Domain: Where the work or any of its elements is in the public domain under applicable law, that status is in no way affected by the license. Other Rights: In no way are any of the following rights affected by the license. Your fair dealing or fair use rights, or other applicable copyright exceptions and limitations; The author s moral rights; Rights other persons may have either in the work itself or in how the work is used such as publicity or privacy rights Notice: For any reuse or distribution, you must make clear to others the license term of this work. The best way to do this is with a link to the following web page: This is a human readable summary of the full legal code which can be read at the following URL: ii

4 TABLE OF CONTENTS CHAPTER 1: THE NUMBERS OF ARITHMETIC... 1 SECTION 1.1: THE REAL NUMBER SYSTEM... 4 SECTION 1.2: FACTORS AND DIVISIBILITY... 7 SECTION 1.3: FRACTIONS SECTION 1.4: DECIMALS SECTION 1.5: INTEGERS SECTION 1.6: ORDER OF OPERATIONS CHAPTER 2: INTRODUCTION TO VARIABLES AND PROPERTIES OF ALGEBRA SECTION 2.1 INTRODUCTION TO VARIABLES SECTION 2.2 PROPERTIES OF ALGEBRA CHAPTER 3: LINEAR EQUATIONS AND INEQUALITIES SECTION 3.1: LINEAR EQUATIONS SECTION 3.2: LINEAR INEQUALITIES SECTION 3.3: LITERAL EQUATIONS CHAPTER 4: LINEAR EQUATION APPLICATIONS SECTION 4.1: INTEGER PROBLEMS SECTION 4.2: MARK-UP AND DISCOUNT PROBLEMS SECTION 4.3: GEOMETRY PROBLEMS SECTION 4.4: VALUE AND INTEREST PROBLEMS SECTION 4.5: UNIFORM MOTION PROBLEMS SECTION 4.6 MIXTURE PROBLEMS CHAPTER 5: GRAPHING LINEAR EQUATIONS SECTION 5.1 GRAPHING AND SLOPE SECTION 5.2 EQUATIONS OF LINES SECTION 5.3 PARALLEL AND PERPENDICULAR LINES CHAPTER 6: SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES SECTION 6.1: SYSTEM OF EQUATIONS: GRAPHING SECTION 6.2: SYSTEMS OF EQUATIONS: THE SUBSTITUTION METHOD SECTION 6.3: SYSTEM OF EQUATIONS: THE ADDITION METHOD SECTION 6.4: APPLICATIONS WITH SYSTEMS OF EQUATIONS iii

5 CHAPTER 7: INTRODUCTION TO FUNCTIONS SECTION 7.1: RELATIONS AND FUNCTIONS SECTION 7.2: DOMAIN AND RANGE CHAPTER 8: EXPONENTS AND POLYNOMIALS SECTION 8.1: EXPONENTS RULES AND PROPERTIES SECTION 8.2 SCIENTIFIC NOTATION SECTION 8.3: POLYNOMIALS CHAPTER 9: FACTORING EXPRESSIONS AND SOLVING BY FACTORING SECTION 9.1: GREATEST COMMON FACTOR AND GROUPING SECTION 9.2: FACTORING TRINOMIALS OF THE FORM x 2 + bx + c SECTION 9.3: FACTORING TRINOMIALS OF THE FORM ax 2 + bx + c SECTION 9.4: SPECIAL PRODUCTS SECTION 9.5: FACTORING, A GENERAL STRATEGY SECTION 9.6: SOLVE BY FACTORING CHAPTER 10: RATIONAL EXPRESSIONS SECTION 10.1: REDUCE RATIONAL EXPRESSIONS SECTION 10.2: MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS SECTION 10.3 OBTAIN THE LOWEST COMMON DENOMINATOR SECTION 10.4: ADD AND SUBTRACT RATIONAL EXPRESSIONS CHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS SECTION 11.1: RATIONAL EQUATIONS SECTION 11.2: WORK-RATE PROBLEMS SECTION 11.3: UNIFORM MOTION PROBLEMS SECTION 11.4: REVENUE PROBLEMS CHAPTER 12: RADICALS SECTION 12.1 INTRODUCTION TO RADICALS SECTION 12.2: ADD AND SUBTRACT RADICALS SECTION 12.3: MULTIPLY AND DIVIDE RADICALS SECTION 12.4: RATIONALIZE DENOMINATORS SECTION 12.5: RADICAL EQUATIONS iv

6 CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS SECTION 13.1: THE SQUARE ROOT PROPERTY SECTION 13.2: COMPLETING THE SQUARE SECTION 13.3: QUADRATIC FORMULA SECTION 13.4: APPLICATIONS WITH QUADRATIC EQUATIONS v

7 CHAPTER 1: THE NUMBERS OF ARITHMETIC Chapter Objectives By the end of this chapter, students should be able to: The real number system Factors and divisibility Review operations with fractions Review operations with decimals Review operations with integers Review the order of operations Contents CHAPTER 1: THE NUMBERS OF ARITHMETIC... 1 SECTION 1.1: THE REAL NUMBER SYSTEM... 4 A. NATURAL NUMBERS... 4 B. WHOLE NUMBERS... 4 C. INTEGERS... 4 D. RATIONAL NUMBERS... 5 E. IRRATIONAL NUMBERS... 5 F. REAL NUMBERS... 6 SECTION 1.2: FACTORS AND DIVISIBILITY... 7 A. DIVISIBILITY... 7 B. FACTORS... 8 C. GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE... 9 D. PRIME AND COMPOSITE NUMBER E. PRIME FACTORIZATION, GCF, AND LCM EXERCISES SECTION 1.3: FRACTION A. WHAT IS A FRACTION? B. FRACTIONS IN CONTEXTS C. REPRESENTING UNIT FRACTIONS D. EQUIVALENT FRACTIONS E. WRITING FRACTIONS IN SIMPLEST FORM F. IMPROPER FRACTIONS AND MIXED NUMBERS G. OPERATIONS WITH FRACTIONS EXERCISES SECTION 1.4: DECIMALS A. INTRODUCTION TO DECIMALS B. OPERATIONS WITH DECIMALS

8 C. FRACTION AND DECIMAL CONNECTIONS EXERCISES SECTION 1.5: INTEGERS A. INTEGERS AND THEIR APPLICATIONS B. PLOTTING INTEGERS ON A NUMBER LINE C. ABSOLUTE VALUE AND NUMBER LINES D. OPPOSITES AND NUMBER LINES E. ORDERING INTEGERS USING NUMBER LINES F. REPRESENTING INTEGERS USING THE CHIP MODEL G. THE LANGUAGE AND NOTATION OF INTEGERS H. ADDING INTEGERS I. SUBTRACING INTEGERS J. CONNECTING ADDITION AND SUBTRACTION K. USING ALGORITHMS TO ADD AND SUBTRACT INTEGERS L. MULTIPLYING AND DIVIDE INTEGERS EXERCISES SECTION 1.6: ORDER OF OPERATIONS A. INTRODUCTION TO EXPONENTS B. THE ORDER OF OPERATIONS WITH ADDITION AND SUBTRACTION C. THE ORDER OF OPERATIONS WITH MULTIPLICATION AND DIVISION D. THE ORDER OF OPERATIONS FOR +,,, E. THE ORDER OF OPERATIONS WITH PARENTHESES F. PEMDAS AND THE ORDER OF OPERATIONS EXERCISES CHAPTER REVIEW

9 INTRODUCTION TO ALGEBRA Media Lesson Brief Origins of Algebra (Duration 7:17) View the video clip and answer the questions below. 1) Who wrote the 1 st book of Algebra? 2) What is the English translation of the title of the book? 3) Where and where was the book written? 4) What civilization were the stone tablets found exploring some of the fundamental ideas of algebra? When? 5) Who was Diophantus? 6) Who lived in India that also significantly contributed to Algebra? 7) CHAPTER 1: THE NUMBERS OF ARITHMETIC Media Lesson Different number systems (Duration 7:04) View the video and fill in the blanks: 1) As we journey through the rich and vibrant history of mathematics, we can see how ideas and creations grew out of 2) Through time, the mathematical explorations of men and women from around the globe have given us fascinating lenses that 3

10 SECTION 1.1: THE REAL NUMBER SYSTEM A. NATURAL NUMBERS The real number system evolved over time by expanding the notion of what we mean by the word number. At first, number meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers. 1, 2, 3, 4, 5,... The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever. B. WHOLE NUMBERS At some point, the idea of zero came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers. 0, 1, 2, 3, 4, 5,... Natural Numbers together with zero What is Zero? Getting Something from Nothing (Duration 3:52) 1) What are the two roles of 0? 2) Who defined 0 explicitly? 3) 0 and what number made up the binary numerical system formed the foundation for modern computer programing? (Optional) Discovery of Zero ( Duration 5:29) C. INTEGERS Even more abstract than zero is the idea of negative numbers. If, in addition to not having any sheep, the farmer owes someone 3 sheep, you could say that the number of sheep that the farmer owns is negative 3. It took longer for the idea of negative numbers to be accepted, but eventually they came to be seen as something we could call numbers. The expanded set of numbers that we get by including negative versions of the counting numbers is called the integers. Whole numbers plus negatives... 4, 3, 2, 1, 0, 1, 2, 3, 4,... The number zero is considered to be neither negative nor positive. About Negative Numbers How can you have less than zero? Well, do you have a checking account? Having less than zero means that you have to add some to it just to get it up to zero. And if you take more out of it, it will be even further less than zero, meaning that you will have to add even more just to get it up to zero. 4

11 D. RATIONAL NUMBERS Introduction to rational and irrational numbers (Duration 5:54) Rational numbers are numbers that can be written in the form of a b or numerator, where a and b denominator are integers (but b cannot be zero). Rational numbers include what we usually call fractions. Notice that the word rational contains the word ratio, which should remind you of fractions. Some decimals are also rational numbers because some decimals can be converted to fractions. RESTRICTION: The denominator cannot be zero! (But the numerator can) Examples: 3 = 0.75 Rational (terminates) = = 0. 6 Rational (terminates) 5 11 = = 0.45 Rational (terminates) There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational. NOTE: The denominator cannot be zero! (But the numerator can). A fraction has the denominator is undefined. For example, 5 0 = undefined. If the numerator is zero, then the whole fraction is just equal to zero. For example, 0 5 = 0 E. IRRATIONAL NUMBERS Irrational numbers are numbers: cannot be expressed as a ratio of integers. as decimals they never repeat or terminate (rational decimals always repeat or terminate) Examples: Irrational (Never repeats or terminates) 2 = π Irrational (Never repeats or terminates) Irrational (Never repeats or terminates) Below is an example of irrational numbers on a number line approximately: To get the exact location of on a number line, we can apply the Pythagorean Theorem to a right triangle with the length of each leg equal 1 to find the hypotenuse length of 2 like the diagram on the right. We can use the same method, apply the Pythagorean Theorem to other right triangles to find the exact location of 3, 4, 5 etc. on the number line. 5

12 Making sense of irrational numbers (Duration 4:41) (Skip 1:50 to 3:08) 1) Which of these numbers is/are rational: A) 3 B) 12.1 C) 5 A. A) only B. A) and B) only C. All of the above D. None of the above 2) What are irrational numbers? A. Numbers which do not make any sense to the common man B. Real numbers which can be expressed as a ratio of two integers in the form p/q and where the denominator is always non-zero C. Numbers which are exactly opposite to rational numbers D. Real numbers which cannot be expressed as a ratio of integers 3) Which of the following statements is true? A. Pi = 22/7 B. Pi = 355/13 C. Pi is the ratio of a circle s circumference to its diameter D. Pi is the ratio of a circle s diameter to its circumference E. Pi is a terminating recurring decimal F. REAL NUMBERS Rational + Irrational numbers All points on the number line When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers. 6

13 SECTION 1.2: FACTORS AND DIVISIBILITY A. DIVISIBILITY Divisible: When one number can be divided by another number and the result is an exact whole number that is there is no remainder left. Example: 12 is divisible by 3 because 12 3 = 4 exactly with no remainder. 13 is not divisible by 3 because 13 3 = 4 with remainder 1. Divisibility Rules Media Lesson Divisibility Rules (Duration 9:34) 1) A number is divisible by 2 if. Example: 512: Yes 431: No 2) A number is divisible by 4 if. Example: 3) A number is divisible by 8 if. Example: 4) A number is divisible by 3 if. Example: 5) A number is divisible by 6 if. Example: 6) A number is divisible by 9 if. Example: 7) A number is divisible by 10 if. Example: 8) A number is divisible by 5 if. Example: YOU TRY Determine if the given number is divisible by 2, 3, 4, 5, 6, 8, 9, 10. a) 8,064 b) 270 Yes/No Yes/No Yes/No Yes/No By 2 By 6 By 2 By 6 By 3 By 8 By 3 By 8 By 4 By 9 By 4 By 9 By 5 By 10 By 5 By 10 7

14 B. FACTORS Factors of a number are the numbers you multiply together to make that number. Example: 2 x 5 = 10, 2 and 5 are factors. Factor Factor Product 9 x 2 = 18 Media Lesson Factors (Duration 5:47) Here is another definition of factors. Factors are the numbers that divide evenly into a number. This means a factor divides into another number and there is no remainder. Example: The factor of 15 are 1, 3, 5, 15 since 1 15 = = 15 Determine the factors of each number. 1) 24 2) 54 3) 120 4) 23 FINDING FACTORS USING PERFECT SQUARES Media Lesson Finding all of the Factor of a Number using Perfect Squares (Duration 13:54) Method: To determine all of the factors of a whole number, we will find all the pairs of whole numbers whose product is the number. We will check all the numbers whose square is less than the number we are trying to factor. 8

15 Table of Perfect Squares 2 2 = = = = = = = = = = = = 169 Directions: Find all factors of the given numbers by finding factor pairs. Use the table of perfect squares to see what the largest number you have to check is. Write your final answer as a list of factors separated by commas. a) 18 Largest number you have to check: List of Factors: b) 90 Largest number you have to check: List of Factors: YOU TRY a) 84 Largest number you have to check: List of Factors: C. GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE Media Lesson Intro to Greatest Common Factor and Least Common Multiple (Duration: 9:54) a) You and your friends are sending care packages to military service members overseas. Each package will contain brownies and cookies. You have 20 brownies and 12 cookies. Every package made needs to be identical. What is the greatest number of packages you can send that meets this requirement? # of Packages # of Brownies # of Packages # of Cookies 9

16 b) Judy and Dan are running around a track. Judy can run one lap in 3 minutes while it takes Dan 4 minutes. If they both start at the same time, how many minutes will it take them to meet? Finding the Greatest Common Factor of Two Numbers (Duration 5:12) Common Factors of two numbers are factors that both numbers share. The Greatest Common Factor (GCF) of two numbers is the largest of these common factors. a) Find all factors of 36. Write your final answer as a list of factors separated by commas. List of Factors 36: b) Find all factors of 90. Write your final answer as a list of factors separated by commas. List of Factors of 90: c) List the common factors of 36 and 90: d) Identify the Greatest Common Factor (GCF) of 36 and 90: YOU TRY Finding the GCF of Two Numbers a) Find all factors of 24. Write your final answer as a list of factors separated by commas. List of Factors 24: b) Find all factors of 60. Write your final answer as a list of factors separated by commas. List of Factors of 60: c) List the common factors of 24 and 60: d) Identify the Greatest Common Factor (GCF) of 24 and 60: 10

17 Multiples, Common Multiples, and Least Common Multiple (Duration 3:26) A multiple of a number is a product of the number with any whole number. Common Multiples of two numbers are multiples that both numbers share. The Least Common Multiple (LCM) of two numbers is the least of these common multiples. a) The first six multiples of 8 are: b) The first six multiples of 12 are: c) Some common multiples of 8 and 12 are: d) The Least Common Multiple (LCM) of 8 and 12 is: YOU TRY The method above is called Finding LCM by using list of multiples. a) The first six multiples of 6 are: b) The first six multiples of 4 are: c) Some common multiples of 6 and 4 are: d) The Least Common Multiple (LCM) of 6 and 4 is: D. PRIME AND COMPOSITE NUMBER Prime and composite numbers (Duration 7:41) A prime number is a whole number greater than 1 whose factor pairs are only the number itself and 1. A composite number is a whole number greater than 1 which has at least one factor other than itself and 1. For example: 6 is not prime because it has factors 2 and 3 besides 1 and 6. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23,... To determine if a number is prime or composite, we only need to check to see if the number is divisible by the prime factors whose square is less than the number we are trying to factor. 2 2 = = = = = = = = 361 Verify that the following numbers are prime by checking to see if the number is divisible by any prime numbers whose square is less than the number given. a) 89 Largest prime you have to check: 11

18 b) 163 Largest prime you have to check: Notes: There are an infinite number of prime numbers (the list goes on forever). The numbers 0 and 1 are neither prime nor composite. All even numbers are divisible by 2 and so all even numbers greater than 2 are composite numbers. YOU TRY Verify that the following numbers are prime by checking to see if the number is divisible by any prime numbers whose square is less than the number given. a) 109 Largest prime you have to check: Determine a number is a prime or a composite (Duration 4:39) Determine whether the following numbers are prime, composite, or neither. 24: 2: 1: 17: The Fundamental Theorem of Arithmetic Any integer greater than 1 is either a prime number, or can be written as a unique product of prime numbers (ignoring the order). In other words, all composite numbers are unique products of smaller prime numbers (ignoring the order of the factors). This theorem is saying that prime numbers are building blocks of the integers, just like atoms are the building blocks of matter in chemistry and biology. Modern cryptography, which is absolute essential to computer security, relies on the fundamental theorem of arithmetic. By combining primes and forming very large composite numbers, code makers can create a key that makes a cipher virtually unbreakable. Knowing how to construct numbers has a very practical use in our world today. The Fundamental Theorem Of Arithmetic (Duration 3:51) Optional reading: What are prime numbers, and why are they so vital to modern life? 12

19 E. PRIME FACTORIZATION, GCF, AND LCM "Prime Factorization" is the process of finding which prime numbers multiply together to make the original number. In other words, prime factorization is the process of breaking down a number into its prime factors. Prime Factorization (Duration 5:34) Find the prime factorizations for the given numbers using factor trees. Write the final result in exponential form and factored form. a) 12 b) 75 c) 155 Factored Form: Factored Form: Factored Form: Exponential Form: Exponential Form: Exponential Form: YOU TRY a) 18 b) 84 c) 266 Factored Form: Factored Form: Factored Form: Exponential Form: Exponential Form: Exponential Form: DETERMINE GCF & LCM USING THE PRIME FACTORIZATION METHOD In this section, we are going to use prime factorization to find a more streamlined approach to finding the GCF and LCM of two numbers. First, let s review the method we used in Section C to find the GCF and LCM. A. To find the GCF of 8 and 12, we would follow the steps below. Step 1: Find all the factors of 8. Factors of 8: 1, 2, 4, 8 Step 2: Find all the factors of 12. Factors of 12: 1, 2, 3, 4, 6, 12 Step 3: The GCF of 8 and 12 is the largest factor they have in common. So the GCF is 4. 13

20 B. To find the LCM of 8 and 12, we would follow the steps below. Step 1. List some multiples of 8. Step 2. List some multiples of 12. Multiples of 8: 8, 16, 24, 32, 40, 48, Multiples of 12: 12, 24, 36, 48, 60, Step 3. The LCM of 8 and 12 is the smallest multiple they have in common. So the LCM is 24. Prime Factorization, GCF, and LCM (Duration 8:06) 1. Use the prime factorization method to determine the GCF and LCM of 8 and 12. a) Find the prime factorizations of 8 and 12 using factor trees and write the prime factorizations in factored form Factored Form: Factored Form: b) List of common prime factors: (include repeated factors) c) The product of the common prime factors of 8 and 12 is their GCF. Find the GCF. GCF of 8 and 12: d) The LCM of 8 and 12 is their product divided by their GCF. Find the LCM. Show all steps. LCM of 8 and 12: 2. Use the prime factorization method to determine the GCF and LCM of 54 and 90. a) Find the prime factorizations of 54 and 90 using factor trees and write the prime factorizations in factored form Factored Form: Factored Form: b) List of common prime factors: (include repeated factors) c) The product of the common prime factors of 54 and 90 is their GCF. Find the GCF. GCF of 54 and 90: d) The LCM of 54 and 90 is their product divided by their GCF. Find the LCM. Show all steps. LCM of 54 and 90: The method above is call Finding LCM by dividing GCF. 14

21 YOU TRY Use the prime factorization method to determine the GCF and LCM of 18 and 84. a) Find the prime factorizations of 18 and 84 using factor trees and write the prime factorizations in factored form Factored Form: Factored Form: b) List of common prime factors: (include repeated factors) c) The product of the common prime factors of 18 and 84 is their GCF. Find the GCF. GCF of 18 and 84: d) The LCM of 18 and 84 is their product divided by their GCF. Find the LCM. Show all steps. LCM of 18 and 84: Method 3: Finding LCM using the highest power of prime factors. Finding LCM (Duration 6:25) The method of writing out multiples is only practical for determining the LCM of small integers. As a result, we have a different way to build the LCM of two integers by analyzing prime factors. The LCM will be the product of the prime factors raised to the highest power of the prime factorization of the numbers. 36 = = = = LCM: Determine the LCM of 60 and = 72 = LCM = = 15

22 YOU TRY Find the LCM. a) 27, = 18 = b) 24, 32 LCM = = 24 = 32 = LCM = c) 35, = 25 = LCM = 16

23 EXERCISES Use the divisibility rules to answer the following questions. Explain your reasoning. 1) Is 18 divisible by 3? 2) Is 240 divisible by 5? 3) Is 22 divisible by 2? 4) Is 212 divisible by 4 and 6? 5) Is 44 divisible by 6? 6) Is 456 divisible by 6 and 3? 7) Is 112 divisible by 2 and 3? 8) Is 246 divisible by 2? 9) Is 27 divisible by 9 and 3? 10) Is 393 divisible by 3? 11) Is 219 divisible by 9? 12) Is 7450 divisible by 10? 13) Is 612 divisible by 2 and 3? 14) Is 884 divisible by 4? Find all factors of the given numbers by finding factor pairs. Use the table of perfect squares to see what the largest number you have to check is. Write your final answer as a list of factors separated by commas. Complete the Table of Perfect Squares 2 2 = 5 2 = 8 2 = 11 2 = 3 2 = 6 2 = 9 2 = 12 2 = 4 2 = 7 2 = 10 2 = 13 2 = 15) 12 Largest number you have to check: List of Factors 16) 48 Largest number you have to check: List of Factors 17) 185 Largest number you have to check: Fill in the blanks: List of Factors 18) A number is a whole number greater than 1 whose factor pairs are only the number itself and one. 19) A number is a whole number greater than 1 which has at least one factor other than itself and one. 20) The of a number is the number written as a product of only prime factors. 21) Common factors of two or more numbers are factors that both numbers. 22) The of two or more numbers is the largest of the two numbers common factors. 23) A of a number is a product of the number with any whole number. 17

24 24) The is the smallest multiple of two or more numbers. 25) Determine all of the prime numbers less than 50. Verify that the following numbers are prime by checking to see if the number is divisible by any prime numbers whose square is less than the number given. 26) ) ) 83 29) 39 Determine whether the numbers are prime or composite. If it is composite, show at least one factor pair of the number besides 1 and itself. If it is prime, show the numbers you tested and the results of your division. 30) ) 61 32) ) 139 Find the prime factorizations for the given numbers using factor trees. Write the final result in exponential form and factored form. 34) 32 35) ) 72 37) 280 Use two different factor trees to determine the prime factorizations of 90 and 150. Write the final result in exponential form and factored form. 38) Factored form: 90 = Exponential form: 90 = 39) Factored form: 150 = Exponential form: 150 = Find the GCF of the given numbers. 40) 8 and 20 41) 30 and ) 16 and 18 43) 22 and 25 44) 12, 8, 24 45) 120, 325, 525 Find the LCM for the given numbers using the listing of multiples method. 46) 4 and 6 47) 10 and 8 48) 15 and 9 49) 12 and 26 18

25 Find the prime factorizations using factor trees for the following pairs of numbers. Find the GCF Then find LCM by using the dividing the GCF method. 50) 4 and 6 51) 10 and 8 52) 15 and 9 53) 12 and 26 Find the LCM by using the highest power of prime factors method. 54) 125 and ) 156 and ) 126, 266 and 38 57) 36, 18 and 86 58) Penny and Sheldon are assembling hair clips. Penny can assemble a hair clip in 6 minutes and Sheldon can assemble a hair clip in 9 minutes. a) If they start making the hair clips at the same time, what is the least amount of minutes it will take for them finish a hair clip at the same time? b) After this amount of minutes, how many hair clips will Penny have made? c) After this amount of minutes, how many hair clips will Sheldon have made? 59) Kathryn is packing bags of food at the local food pantry. She has 24 jars of tomato sauce and 30 cans of soup. a) If she wants each bag to have the same numbers of tomato sauce and soup, what is the greatest number of bags she can pack? b) How many jars of tomato sauce will each bag have? c) How many cans of soup will each bag have? 60) Paige is buying hot dogs and buns for a family reunion. Each package of hot dogs contains 8 hot dogs. Each package of buns contains 10 buns. a) What is the least total amount of hot dogs and buns she needs to buy in order for the amounts to be equal? b) How many packages of hot dogs will she buy? c) How many packages of buns will she buy? 19

26 SECTION 1.3: FRACTION A. WHAT IS A FRACTION? There are many ways to think of a fraction. A fraction can be thought of as one quantity divided by another written by placing a horizontal bar between the two numbers such as 1 where 1 is called the numerator 2 and 2 is called the denominator. Or, we can think of fractions as a part compared to a whole such as 1 out of 2 cookies or 1 of the cookies. 2 In this lesson, we will look at a few other ways to think of fractions as well. numerator Officially, fractions are any numbers that can be written as but in this course, we will denominator consider fractions where the numerator and denominator are integers. These special fractions where the numerator and denominator are both integers are called rational numbers. Since rational numbers are indeed fractions, we will frequently refer to them as fractions instead of rational numbers. Language of Fractions (Duration 6:18) Each of the phrases below are one way we may indicate a fraction with words. Rewrite the phrases below in fraction form and write the fraction word name. Language Fraction Representation Fraction Word Name 20 divided by 6 8 out of 9 A ratio of 3 to 2 11 per 5 2 for every 7 20

27 B. FRACTIONS IN CONTEXTS Dividend Divisor Quotient 12 3 = 4 Fractions in Context: Four Models (Duration 8:12) In the next example, we will look at four different types of fractions in context. 1. Quotient Model (Division): Sharing equally into a number of groups 2. Part-Whole Model: A part in the numerator a whole in the denominator 3. Ratio Part to Part Model: A part in the numerator and a different part in the denominator 4. Rate Model: Different types of units in the numerator and denominator (miles and hours) Represent the following as fractions. Determine whether it is a quotient, part-whole, part to part, or rate model. a) Three cookies are shared among 6 friends. How many cookies does each friend get? b) Four out of 6 people in the coffee shop have brown hair. What fraction of people in the coffee shop have brown hair? c) Tia won 6 games of heads or tails and lost 3 games of heads or tails. What is the ratio of games won to games lost? d) A snail travels 3 miles in 6 hours. What fraction of miles to hours does he travel? What fraction of hours to miles does he travel? YOU TRY a) Jorge bikes 12 miles in 3 hours. What fraction of miles to hours does he travel? b) Callie has 5 pairs of blue socks and 12 pairs of grey socks. What fraction of blue socks to grey socks does she have? c) Callie has 5 pairs of blue socks and 12 pairs of grey socks. What fraction of all of her socks are blue socks? 21

28 NOTE: Let s revisit the note in the Rational number section. Dividing by zero is undefined. If we consider a fraction is a quotient that is sharing equally into a number of groups. For example, dividing 5 candies among 0 people, how much does each person get? The question does not make sense because we cannot share among 0 people, and we cannot divide by 0. When the numerator is zero, the whole fraction is equal to zero. For example, 0 = 0. When we do 5 not have any candies but want to share them with 5 people. Therefore, each of them will get 0 candy. C. REPRESENTING UNIT FRACTIONS A unit fraction is a fraction with a numerator of 1. In this section, we will develop the idea of unit fractions and use multiple representations of unit fractions. Representing unit fractions (Duration 4.37) View the video lesson, take notes and complete the problem below. a) Plot the following unit fractions on the number line: 1 2,1 4, 1. Label your points below the 5 number line. Representing unit fractions using area model (Duration 3.38) View the video lesson, take notes and complete the problem below. b) Represent the fractions using the area model. The unit is labeled in the second row of the table Representing unit fractions using the discrete objects (Duration 2.07 c) Represent the unit fractions using the discrete objects. The unit is all of the triangles in the rectangle. Represent 1 of the triangles. 4 22

29 D. EQUIVALENT FRACTIONS Media Lesson Equivalent fractions (Duration 7:37) Follow the video lesson, take notes and fill in the blanks below. Equivalent fractions are different fractions that are equal or represent the same amount To find equivalent fractions, or the numerator and denominator by the nonzero whole number. We would only the numerator and denominator by a whole number that is a of both the numerator and denominator. Once a fraction does not have any other than 1 between the numerator and denominator, a fraction is simplified or written in lowest terms

30 Define: Dark blue Rod = 1 1 Determine two equivalent fractions for each fractions for each fractions below Determine the numerator with given denominator Media Lesson Determine the numerator with given denominator (Duration 2:25) Example: Missing numerator Fill in the missing numerator in each fraction below to create equivalent fractions. 1 a) =? b) 14 =? c) 4 =? 7 49 d) =? 5 The denominator indicates how many part is the whole. The numerator indicates how many parts are being considered. 24

31 YOU TRY Rewrite the given fractions as equivalent fractions given the indicated denominator. 3 a) =? b) 12 =? 5 c) =? d) 6 =? e) 8 =? f) 3 =? 5 15 E. WRITING FRACTIONS IN SIMPLEST FORM What is a simplified Fraction (Duration 11:49 ) The simplest form of a fraction is the equivalent form of the fraction where the numerator and denominator are written as integers without any common factors besides 1. We may also request that a fraction be written in simplest form by using the equivalent language; simplify the fraction, reduce the fraction, or reduce the fraction completely. a) Write the fraction number for each diagram below the figure using one circle as the unit. b) What do the fractions have in common? c) Which fraction do you think is the simplest and why? d) Divide the numerators and denominators of the second and fourth fractions by 2. What do you notice? e) Rewrite the last three fractions below by writing their numerators and denominators in terms of their prime factorizations. Do you see any patterns? f) Simplify your fractions in part e by cancelling out all of the common factors (besides 1) that the numerators and denominators share. 25

32 Simplifying Fractions by Repeated Division and Prime Factorization Simplifying Fractions by Repeated Division and Prime Factorization (Duration 12:40) We can use two different methods to simplify a fraction: repeated division or prime factorization. 1. Repeated Division: Look for common factors between the numerator and denominator and divide both by the common factor. Continue this process until you are certain the numerator and denominator have no common factors. 2. Prime Factorization: Write the prime factorizations of the numerator and denominator and cancel out any common factors. Simplify the given fractions completely using both the repeated division and prime factorization methods. In each case, state which you think is easier and why. a) b) c) YOU TRY Simplify the given fractions completely using both the repeated division and prime factorization methods. In each case, state which you think is easier and why. a) 6 8 b) c)

33 F. IMPROPER FRACTIONS AND MIXED NUMBERS Media Lesson Improper fractions and mixed numbers (Duration 6:48) There are 3 types of fractions. 1. A proper fraction is a fraction with a smaller numerator than denominator. Proper fractions are less than 1. Example: 2. An improper fraction is a fraction, which the numerator is greater or equal to the denominator. Improper fractions are greater than or equal to 1. Example: 3. A mixed number is a number written as the sum of an integer and a proper fraction. Example: To convert from an improper fraction to a mixed number: a. b. c. d. Example: Convert 13 to a mixed number. 5 a. b. c. Shade the circles to represent The final answer is : 27

34 Example: Convert 32 6 to a mixed number. To convert a mixed number into an improper fraction Example: Convert 2 3 into an improper fraction. 8 Shade the circles to represent Example: Convert 7 5 into an improper fraction. 12 Why is converting a mixed number to an improper fraction important? It is important because you will have to convert mixed numbers to improper fractions in order to multiple or divide mixed numbers. YOU TRY Convert the following fractions to a mixed number if you can. a) 8 3 b) 15 8 c) d) e)

35 G. OPERATIONS WITH FRACTIONS Addition and Subtraction of Fractions Adding fractions (Duration 9:10) To add fractions with like denominators: 1) Add the numerators, keeping the same denominator. 2) Simplify, if possible = = To add fractions with unlike denominators: 1) Find the least common multiple of the denominators, that number is the least common denominator, LCD. The LCD is the least common multiple of the denominators. 2) Create equivalent fractions with the common denominator. 3) Add the numerators, keeping the common denominator the same. 4) Simplify if possible = ERROR! =

36 Add Adding mixed numbers (Duration 6:35) To add mixed numbers 1. Obtain a (the least) common denominator. 2. Add. 3. If the sum has an improper fraction, convert it to a mixed number and add it to the whole number. 4. Simplify if possible

37 YOU TRY Add the following fractions and reduce your answer to the lowest term if possible. a) b) Subtract fractions with like and unlike denominators (Duration 8:29) To subtract fractions with like denominators: 1) Keep the denominator the same and subtract the numerators. 2) Simplify, if possible = = To subtract fractions with unlike denominators: 1) Find the least common multiple of the denominators, that number is the least common denominator, LCD. 2) Create equivalent fractions with the common denominator. 3) Keep the common denominator the same and subtract the numerators 4) Simplify, if possible. Ex: = = Subtract. 5 a)

38 b) c) d) Subtract mixed numbers (Duration 7:15) To subtract mixed numbers 1. Obtain a (the least) common denominator. 2. Check to see is you need to borrow. 3. Subtract the fractions and then the whole numbers. 4. Simplify, if needed. Subtract

39 YOU TRY Subtract and reduce your answer to the lowest term if possible. a) b) c) d) Multiplication and Division of Fractions Multiply fractions (Duration 6:41) To multiply fractions 1. Multiply the numerators and multiply the denominators. 2. Write the product in simplest form or in lowest terms. A fraction is simplified or in lowest terms when the only common factor between the numerator and denominator is 1. We can simplify before multiplying or after multiplying. Example: Multiply. If applicable, write your answer as both an improper fraction and a mixed number. 1 a. 3 5 b c d

40 YOU TRY Multiply. If applicable, write your answer as both an improper fraction and a mixed number. 2 a) 3 b) c) d) To divide fractions: 1. Change the second fraction (the divisor) to its reciprocal. 2. Simplify or cancel common factors if possible. 3. Multiply. Divide fractions (Duration 3:06) Example: To divide mixed numbers: 1. Rewrite mixed numbers and whole numbers as improper fractions. 2. Change the second fraction (the divisor) to its reciprocal. 3. Simplify or cancel common factors if possible. 4. Multiply. Note: A pair of numbers are reciprocals if their product is one. Each number is considered the reciprocal of the other. For example, the reciprocal of 2 3 is 3 because their product is

41 Divide mixed numbers (Duration 3:47) Example: YOU TRY Divide. Write your answers in simplest form. a) b) c) d) e) f) g) h)

42 EXERCISES Each of the phrases below are one way we may indicate a fraction with words. Rewrite the phrases below in fraction form and write the fraction word name. Language Fraction Representation Fraction Word Name 1) A ratio of 5 to 10 2) 9 for every 10 3) 4 out of 7 4) 15 per 2 5) 16 divided by 5 For each problem below, write the fraction that best describes the situation. Be sure to reduce your final result. 6) John had 12 marbles in his collection. Three of the marbles were Comet marbles. What fraction of the marbles were Comet marbles? What fraction were NOT Comet marbles? 7) Jorge s family has visited 38 of the 50 states in America. What fraction of the states have they visited? 8) In a given bag of M & M s, 14 were yellow, 12 were green, and 20 were brown. What fraction were yellow? Green? Brown? 9) Donna is going to swim 28 laps. She has completed 8 laps. What fraction of laps has she completed? What fraction of her swim remains? 10) Last night you ordered a pizza to eat while watching the football game. The pizza had 12 pieces of which you ate 6. Today, two of your friends come over to help you finish the pizza and watch another game. What is the fraction of the LEFTOVER pizza that each of you gets to eat (assuming equally divided)? What is the fraction of the ORIGINAL pizza that each of you gets to eat (also assuming equally divided)? 11) Represent the unit fraction 1 8 a) Number line using each of the representations below. 36

43 b) Area models. Use the unit labeled in the second row of the table. c) Discrete objects 12) Create two fractions equivalent to the given fraction by cutting the given representations into a different number of equal pieces. a) Given fraction 3 4 b) Given fraction is equivalent to the fraction: 3 5 is equivalent to the fraction: Rewrite the given fractions as equivalent fractions given the indicated numerator or denominator 13) Rewrite 2 3 with a denominator of ) Rewrite 6 7 with a numerator of ) Rewrite 5 6 with a numerator of ) Rewrite 4 8 with a denominator of 2. 17) Rewrite with a numerator of 9. 18) Rewrite 5 1 with a denominator of 5. 37

44 Simplify each of the following fractions if possible. 5 19) 20) ) ) ) ) 1 0 Simplify the given fractions completely using both the repeated division and prime factorization methods. In each case, state which method you think is easier and why. Fraction Repeated Division Method Prime Factorization Method 25) 26) ) Complete the table below. Convert. Improper Fraction Mixed Number 28) ) ) ) ) ) Use the diagrams given to represent the values in the addition problem and find the sum. Then perform the operation algebraically. 34) Tom had 1 6 of a carrot cake last night and 2 6 of a carrot cake today. How much of one whole carrot cake did Tom have? 38

45 35) Ava walked 3 8 of a mile to the store and then ran another 7 8 of a mile to school. How far did she travel in total? 36) Add or subtract each of the following. Be sure to leave your answer in simplest (reduced) form. If applicable, write your answer as both an improper fraction and a mixed number. 37) ) ) ) ) ) ) 44) ) ) Divide or multiply. Simplify your result if necessary. Write any answers greater than 1 as both an improper fraction and a mixed number. 47) ) ) ) ) ) ) ) ) 57) ) )

46 Solve the following problems. When necessary, write your final answers as both mixed numbers and improper fractions in the simplest term. 59) If Josh ate 1 of a pizza, what fraction of the pizza is left? 4 60) If I drove miles one day and miles the second day and 8 1 miles the third day, how far did I 5 drive? 61) Melody bought a 2-liter bottle of soda at the store. If she drank 1 of the bottle and her brother drank 8 2 of the bottle, how much of the bottle is left? 7 62) James brought a small bag of carrots for lunch. There are 6 carrots in the bag. Is it possible for him to eat 2 6 of the bag for a morning snack and 5 of the bag at lunch? Why or why not? 6 63) Maureen went on a 3 day, 50 mile biking trip. The first day she biked biked miles. How many miles did she bike on the 3 rd day? miles. The second day she 64) Scott bought a 5 lb bag of cookies at the bakery. He ate 2 5 of a bag and his sister ate 2 9 of a bag. What fraction of the bag did they eat? What fraction of the bag remains? 65) Suppose your school costs for this term were $2500 and financial aid covered 3 4 much did financial aid cover? of that amount. How 4 66) If, on average, about of the human body is water weight how much water weight is present in a 7 person weighing 182 pounds? 67) If, while training for a marathon, you ran 920 miles in 3 1 months, how many miles did you run each 2 month? (Assume you ran the same amount each month) 68) On your first math test, you earned 75 points. On your second math test, you earned 6 as many points 5 as your first test. How many points did you earn on your second math test? 69) You are serving cake at a party at your home. There are 12 people in total and 2 3 cakes. (You ate 4 some before they got there!). If the cakes are shared equally among the 12 guests, what fraction of a cake will each guest receive? 40

47 SECTION 1.4: DECIMALS A. INTRODUCTION TO DECIMALS Introduction to Decimals (Duration 8:03) Decimal notation is used to write numbers according to place value in base -10. A decimal point is used to separate whole numbers from numbers from numbers less than 1. It costs $2.89 per gallon of gas. The runner finished in 10.3 seconds. The Dow was up points today. Ten Thousands Thousands Hundreds Tens Ones And Tenths Hundredths Thousandths Ten Thousandths 100,000 1, Read the following number using place values. Write the decimal in using fraction notation , Comparing using >, <, or = Order each list from least to greatest , 3.1, -3,04, 3.11, , , ,

48 Rounding Decimal Numbers Procedure 1. Identify the round-off place digit. 2. If the digit to the right of the round-off place digit is: Less than 5, do not change the round of place digit. 5 or more, increase the round-off place digit by In either case, drop all digits to the right of the round-off place digit. Round each number to the indicated place value to the hundredths to the tenths to the ten thousandths YOU TRY a) Order the numbers from least to greatest: 2.8, 2.08, 2.88, 2.088, 2.008, 2.808, b) Order from smallest to largest: 4.25, 0.425, 4.05, 4.2, c) Round 3.24 to the nearest tenth. d) Round to the nearest hundredth. e) Round to the nearest tenth. f) Round to the nearest thousandth. 42

49 B. OPERATIONS WITH DECIMALS Adding and subtracting decimals Add and subtract decimals (Duration 5:35) Procedure 1. Write the numbers vertically and line up the decimal points. Add zeros to the right as needed so each number has the same number of digits to the right of the decimal points. 2. Bring the decimal point down into the sum or difference. 3. Add or subtract as you normally would. Example: YOU TRY Perform the operations indicated below. Be sure to show your work. a) b) c) d) e) f)

50 Multiplying decimals Multiply decimals (Duration 7:54) Procedure 1. Multiply the numbers just as you would multiply whole numbers. 2. Find the sum of the decimal places in the factors. 3. Place the decimal point in the product so that the product has the same number of decimal places as the sum of the decimal places. You may need to write zeros to the left of the number. Example: Multiply. 5.2 x x x Multiplying a decimal by a power of ten To multiply a decimal by a power of 10, move the decimal point to the right the same number of places as the number of zeros in the power of 10. It may be necessary to add zeros at the end of the number. Example: x 100. Why? Multiply ,000 10,000 x YOU TRY Multiply the numbers by the given powers of 10 by moving the decimal point the appropriate number of places. a) = b) = c) = d) = 44

51 Multiply the decimals. e) = f) = g) = h) = Dividing decimals Dividing decimals (Duration 9:11) Dividing a decimal by a whole number: 1. Place the decimal point in the answer directly above the decimal point in the dividend. 2. Divide as if there were no decimal point involve. Example: Dividing a decimal by a decimal 1. If the divisor is a decimal, change it to a whole number by moving the decimal point to the right as many places as necessary. 2. Then move the decimal point in the dividend to the right the same number of places. 3. Place the decimal point in the answer directly above the decimal point in the dividend. 4. Divide until the remainder becomes zero or the remainder repeats itself, or the desired number of decimal places it achieved. 45

52 Dividing a decimal by a powers of ten Dividing decimal by Powers of Ten (Duration 5:09) Divide the numbers by the given powers of 10 on your calculator then look for patterns to make a general strategy. a) = b) = c) = d) = e) = f) = g) Look for patterns in the examples above and complete the statement below. To divide a decimal number by a power of 10, you move the decimal place. YOU TRY Divide the decimals. a) = b) = c) d) Divide the numbers by the given powers of 10 by moving the decimal point the appropriate places. e) = f) = g) = 46

53 C. FRACTION AND DECIMAL CONNECTIONS Converting Fractions to Decimals Method 1: To convert the denominator to a power of 10. Then write the number using the correct place value. Converting a Fraction to a Decimal (Power of 10) (Duration 5:50) Ten Thousands Thousands Hundreds Tens Ones And Tenths Hundredths Thousandths Ten Thousandths 100,000 1, ,000 Example: Convert the fraction to the decimal by converting the denominator to a Power of

54 Method 2: Perform long division. Remember a fraction bar is a division symbol. a b = a b b Converting a fraction to a decimal (long division) (Duration 6:05) Convert the fractions to decimals. a) 7 25 b) 2 5 c) 7 8 d) 3 11 Converting decimals to fractions Converting decimals to fractions (Duration 5:10) When we rewrite a decimal as a simplified fraction, we will start by writing it as a fraction based on its place value, a power of ten. Observe that 10 s prime factorization is 2 5. So any power of 10 is just a product of 2 s and 5 s. This will make the process of simplification easier because we will only have to check the numerator for factors of 2 s and 5 s. Complete the table below. Show all of your work for simplifying the fraction. Decimal Fraction Simplified Fraction a) 0.8 b) 0.65 c) 0.44 d)

55 YOU TRY Convert the following fractions to decimals using the indicated method below. Show all of your work. Fraction Powers of 10 method Long division method 7 a) 25 b) 3 5 c) 3 8 d) Convert the following decimals to fractions and simplify your answer. Decimal Fraction Simplified Fraction e) 0.6 f) 0.85 g) h)

56 EXERCISES Perform the following operations with decimals. Show your work. 1) ) ) ) ) ) ) ) ) ) ) ) ) ) Multiply or divide the numbers by the given powers of 10 by moving the decimal point the appropriate number of places. 15) = 16) = 17) = 18) = 19) = 20) = 21) Sylvia just received her monthly water usage data from her local water department. For the past 6 months, her water used (in thousands of gallons) was 19.9, 25.6, 28.8, 22.5, 20.3, and What was her average usage during this time? (Round to the nearest tenth) 22) Glenn normally earns $8.50 per hour in a given 40-hour work-week. If he works overtime, he earns time and a half pay per hour. During the month of October, he worked 40 hours, 50 hours, 45 hours, and 42 hours for the four weeks. How much did he earn total for October? 23) Dave is making a gazebo for his yard. He has a piece of wood that is 13 feet long and he needs to cut it into pieces of length 1.5 inches. How many pieces of this size can he cut from the 13 foot piece of wood? 24) Callie ordered 4 items online. She is charged $2.37 per pound per shipping. The items weighed 3.2 lbs., 4.6 lbs., 9.2 lbs. and 1.5 lbs. How much will be charged for shipping? (Round to the nearest cent). 25) Mary s son wants to go on the Gadget s Go Coaster ride at Disneyland. The height requirement for the ride is 35 inches. He is 29.3 inches tall now. How many inches more does he need in order to get into the ride? 50

57 Complete the table below. Show all of your work for simplifying the fraction. Decimal Fraction Simplified Fraction 26) ) ) ) Complete the table below. Show all of your work for simplifying the fraction. Fraction Powers of 10 method if possible Long division method 30) ) ) )

58 SECTION 1.5: INTEGERS INTRODUCTION Now that we have discussed the Base-10 number system including whole numbers and place value, we can extend our knowledge of numbers to include integers. The first known reference to the idea of integers occurred in Chinese texts in approximately 200 BC. There is also evidence that the same Indian mathematicians who developed the Hindu-Arabic Numeral System also began to investigate the concept of integers in the 7th century. However, integers did not appear in European writings until the 15th century. After conflicting debate and opinions on the concept of integers, they were accepted as part of our number system and fully integrated into the field of mathematics by the 19th century. A. INTEGERS AND THEIR APPLICATIONS Integers and their Applications (Duration 3:24) Definition: The integers are all positive whole numbers and their opposites and zero.... 4, 3, 2, 1, 0, 1, 2, 3, 4 The numbers to the left of 0 are negative numbers and the numbers to the right of 0 are positive numbers. We denote a negative number by placing a symbol in front of it. For positive numbers, we either leave out a sign altogether or place a + symbol in front of it. Determine the signed number that best describes the statements below. Circle the word that indicates the sign of the number. Statement a) Tom gambled in Vegas and lost $52. Signed Number b) Larry added 25 songs to his playlist. c) The airplane descended 500 feet to avoid turbulence. YOU TRY Determine the signed number that best describes the statements below. Circle the word that indicates the sign of the number. Statement Signed Number a) A balloon dropped 59 feet. 52

59 b) The altitude of a plane is 7500 feet. c) A submarine is 10,000 feet below sea level. B. PLOTTING INTEGERS ON A NUMBER LINE Number lines are very useful tools for visualizing and comparing integers. We separate or partition a number line with tick marks into segments of equal length so the distance between any two consecutive major tick marks on a number line are equal. Plotting Integers on a Number Line (Duration 6:05) Plot the negative numbers that correspond to the given situations. Use a to mark the correct quantity. Also label all the surrounding tick marks and scale the tick marks appropriately. a) The temperature in Greenland yesterday was 5 What does 0 represent in this context? b) The altitude of the plane decreased by 60 feet. What does 0 represent in this context? YOU TRY Plot the negative numbers that correspond to the given situations. Use a to mark the correct quantity. Also label all the surrounding tick marks and scale the tick marks appropriately. Akara snorkeled 30 feet below the surface of the water. What does 0 represent in this context? What does 0 represent in this context? 53

60 C. ABSOLUTE VALUE AND NUMBER LINES Absolute Value and Number Lines (Duration 4:50) Definition: The absolute value of a number is the positive distance of the number from zero. Notation: Absolute value is written by placing a straight vertical bar on both sides of the number. 50 = 50 Read the absolute value of 50 equals = 50 Read the absolute value of 50 equals 50 Answer the questions below based on the given example. The submarine dove 15 meters below the surface of the water. a) What integer best represents the submarine s location relative to the surface of the water? b) What word indicates the sign of this number? c) What does 0 represent in this context? d) Plot your number from part a and 0 on the number line below. e) Draw a line segment that represents this value s distance from zero on the number line below. f) Write the symbolic form of the absolute value representation. When we want to talk about how large a number is without regard as to whether it is positive or negative, we use the absolute value function to represent the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number. In short: If a number is positive (or zero), the absolute value function does nothing to it: 4 = 4. If a number is negative, the absolute value function makes it positive: - 4 = 4. 54

61 YOU TRY Answer the questions below based on the given example. The temperature dropped 8 Celsius overnight. a) What integer best represents the change in temperature? b) What word indicates the sign of this number? c) What does 0 represent in this context? d) Plot your number from part a and 0 on the number line below. e) Draw a line segment that represents this value s distance from zero on the number line below. f) Write the symbolic form of the absolute value representation. Find the absolute value of: g) 5 = h) 5 = i) 120 = D. OPPOSITES AND NUMBER LINES Opposites and number lines (Duration 4:48) Definition: The opposite of a nonzero number is the number that has the same absolute value of the number, but does not equal the number. Another useful way of thinking of opposites is to place a negative sign in front of the number. The opposite of 4 is (4) = 4 The opposite of 4 is ( 4) = 4 Answer the questions below to use number lines to find the opposite of a number. a) Plot the number 5 on the number line below. b) Draw an arrow that shows the reflection of 5 about the reflection line to find 5 s opposite c) The opposite of 5, or (5) is. d) Draw an arrow that shows the reflection of 5 about the reflection line to find 5 s opposite e) The opposite of 5, or ( 5) is. f) Based on the pattern above, what do you think ( ( 5)) equals? 55

62 YOU TRY Answer the questions below to use number lines to find the opposite of a number. a) Plot the number 4 on the number line below. b) Draw an arrow that shows the reflection of 4 about the reflection line to find 4 s opposite. c) The opposite of 4 is. E. ORDERING INTEGERS USING NUMBER LINES Ordering Integers Using Number Lines (Duration 5:12) Fact: If two numbers are not equal, one must be less than the other. One number is less than another if it falls to the left of the other on the number line. Equivalently, if two numbers are not equal, one must be greater than the other. One number is greater than another if it falls to the right of the other on the number line. Notation: We use inequality notation to express this relationship. 2 < 5, read 2 is less than 5 6 > 3, read 6 is greater than 3 Although we typically read the < sign as less than and the > sign as greater than because of the equivalency noted above, we can also read them as follows: 2 < 5, is equivalent to 5 is greater than 2 6 > 3, is equivalent to 3 is less than 6 Plot the given numbers on the number line. Determine which number is greater and insert the correct inequality symbol in the space provided. a) Plot 5 and 3 on the number line below. Write the number that is further to the right: Insert the correct inequality symbol in the space provided: b) Plot 4 and 7 on the number line below. Write the number that is further to the right: Insert the correct inequality symbol in the space provided:

63 YOU TRY Plot the given numbers on the number line. Determine which number is greater and insert the correct inequality symbol in the space provided. Plot 8 and 2 on the number line below. Write the number that is further to the right: Insert the correct inequality symbol in the space provided: F. REPRESENTING INTEGERS USING THE CHIP MODEL Representing integers using manipulatives (Duration 3:11) Observe the two images below. Although they both have a total of 5 chips, the chips on the left are marked with " + " signs and the chips on the right are marked with " " signs. This is how we indicate the sign each chip represents Determine the value indicated by the sets of integer chips below. a) b) Number: c) Number: d) Number: Number: YOU TRY Determine the value indicated by the sets of integer chips below. a) b) Number: Number: 57

64 G. THE LANGUAGE AND NOTATION OF INTEGERS The Language and Notation of Integers (Duration 6:28) The + symbol: 1. In the past, you have probably used the symbol + to represent addition. Now it can also represent a positive number such as + 4 read positive Let s agree to say the word plus when we mean addition and positive when we refer to a number s sign. The symbol: 1. In the past, you have probably used the symbol to represent subtraction. Now it can also mean a negative number such as 4 read negative 4 or the opposite of Let s agree to say the word minus when we mean subtraction and negative when we refer a number s sign. Write the given numbers or mathematical expressions using correct language using the words opposite of, negative, positive, plus or minus. Number or Expression Written in Words a) 6 b) ( 6) c) d) 3 ( 4) YOU TRY Write the given numbers or mathematical expressions using correct language using the words opposite of, negative, positive, plus or minus. Number or Expression Written in Words a) 3 b) ( 7) c) 4 + ( 2) d) 1 ( 5) 58

65 H. ADDING INTEGERS Adding integers using manipulatives (Duration 8:24) We call the numbers we are adding in an addition problem the addends. We call the simplified result the sum. a) Using integer chips, represent positive 5 and positive 3. Find their sum by combining them into one group. Addend Addend Sum = b) Using integer chips, represent negative 5 and negative 3. Find their sum by combining them into one group. Addend Addend Sum ( 5) + ( 3) = c) Using integer chips, represent positive 5 and negative 3. Find their sum by combining them into one group. Addend Addend Sum 5 + ( 3) = d) Using integer chips, represent negative 5 and positive 3. Find their sum by combining them into one group. Addend Addend Sum ( 5) + 3 = 59

66 Results: Addends have the same sign = 8 ( 5) + ( 3) = 8 Addends have different signs 5 + ( 3) = 2 ( 5) + 3 = 2 Summary of the Addition of Integers When adding two numbers with the same sign, 1. Add the absolute values of the numbers. 2. Keep the common sign of the numbers. When adding two numbers with different signs, 1. Find the absolute value of the numbers. 2. Subtract the smaller absolute value from the larger absolute value. 3. Keep the original sign of the number with the larger absolute value. YOU TRY a) Using integer chips, represent negative 6 and negative 4. Find their sum by combining them into one group. Addend Addend Sum ( 6) + ( 4) = b) Using integer chips, represent negative 6 and positive 4. Find their sum by combining them into one group. Addend Addend Sum ( 6) + 4 = Adding Integers Using a Number Line (Duration 3:05) Use a number line to represent and find the following sums. a) = 60

67 b) ( 5) + ( 3) = c) 5 + ( 3) = d)( 5) + 3 = YOU TRY Use a number line to represent and find the following sums. a) 7 + ( 3) = b) ( 7) + 3 = I. SUBTRACING INTEGERS The Language of Subtraction (Duration 4:38) Minuend Subtrahend Difference 9 6 = 3 Symbolic Minus Language Subtracted from Language 5 3 Less than Language Decreased by Language 5 ( 3) 61

68 YOU TRY Symbolic Minus Language Subtracted from Language Less than Language Decreased by Language 6 ( 5) Subtracting integers with manipulatives Take away method (Duration 4:02) Using integer chips and the take away method, represent the following numbers and their difference. a) 5 3 Minuend Subtrahend Take Away Simplified Difference b) ( 5) ( 3) 5 3 = Minuend Subtrahend Take Away Simplified Difference ( 5) ( 3) = Subtracting integers with manipulatives Comparison method (Duration 4:19) Using integer chips and the comparison method, represent the following numbers and their difference. a) 3 5 Minuend Subtrahend Comparison Simplified Difference 3 5 = 62

69 b) 5 ( 3) Minuend Subtrahend Comparison Simplified Difference 5 ( 3) = c) ( 5) 3 Minuend Subtrahend Comparison Simplified Difference ( 5) 3 = YOU TRY Using integer chips and the method indicated, represent the following numbers and their difference. a) ( 6) ( 2) Minuend Subtrahend Take Away Simplified Difference ( 6) ( 2) = b) 3 4 Minuend Subtrahend Comparison Simplified Difference 3 4 = J. CONNECTING ADDITION AND SUBTRACTION You may have noticed that we did not write a set of rules for integer subtraction like we did with integer addition. The reason is that the set of rules for subtraction is more complicated than the set of rules for addition and, in general, wouldn t simplify our understanding. However, there is a nice connection between integer addition and subtraction that you may have noticed. We will use this connection to rewrite integer subtraction as integer addition. Fact: Subtracting an integer from a number is the same as adding the integer s opposite to the number. Rewriting Subtraction as Addition (Duration 4:38) Rewrite the subtraction problems as equivalent addition problems and use a number line to compute the result. a) 4 7 Rewrite as addition: 63

70 b) 6 ( 2) Rewrite as addition: YOU TRY Rewrite the subtraction problems as equivalent addition problems and use a number line to compute the result. a) 4 6 Rewrite as addition: b) 3 ( 4) Rewrite as addition: K. USING ALGORITHMS TO ADD AND SUBTRACT INTEGERS Using Algorithms to Add and Subtract Integers (Duration 6:30) Thus far, we have only added and subtracted single digit integers. Now we will use base blocks and the ideas developed in this lesson to add and subtract larger numbers. We will follow the protocol below. 1. If given a subtraction problem, rewrite it as an addition problem. 2. Use the rules for addition to add the signed numbers as summarized below. 3. Use regrouping or decomposing from Lesson 1 to carry in addition when necessary or borrow in subtraction when necessary. 4. Write the associated standard algorithm that represents this process. Summary of the Addition of Integers When adding two numbers with the same sign, 1. Add the absolute values of the numbers. 2. Keep the common sign of the numbers. When adding two numbers with different signs, 1. Find the absolute value of the numbers. 2. Subtract the smaller absolute value from the larger absolute value. 3. Keep the original sign of the number with the larger absolute value. 64

71 Use the Standard Algorithms to solve the addition and subtraction problems below. a) b) c) Use your results from above and your knowledge of integer addition and subtraction to find the following. ( 275) + ( 308) = 275 ( 308) = ( 275) = ( 275) ( 308) = YOU TRY ( 308) = ( 275) 308 = Use the Standard Algorithms to solve the addition and subtraction problems below. a) b) c) Use your results from above and your knowledge of integer addition and subtraction to find the following. ( 137) + ( 324) = 137 ( 324) = ( 324) = ( 137) 324 = L. MULTIPLYING AND DIVIDE INTEGERS To multiply or divide two signed numbers 1. Multiply or divide the absolute values. 2. If both signs are the same, the sign of the result is always positive. 3. If the signs are different, the sign of the result is negative. Remember the following rules: (+)(+)=(+) (+)(-)=(-) (-)(+)=(-) (-)(-)=(+) Using algorithms to multiply integers (Duration 2:05) a) Use the Standard Algorithm to find b) Use your results from above and your knowledge of integer multiplication to find the following = = 14( 23) = ( 14)( 23) = 65

72 YOU TRY a) Use the Standard Algorithm to find b) Use your results from above and your knowledge of integer multiplication to find the following = 25( 32) = = ( 25)( 32) = Using algorithms to divide integers (Duration 3:02) a) Find using the Standard Algorithm. b) Use your results from above and your knowledge of integer division to find the following. c) 564 ( 4) = 564 = ( 564) ( 4) = 4 YOU TRY a) Find using the Standard Algorithm. b) Use your results from above and your knowledge of integer division to find the following ( 3) = = 3 = 66

73 EXERCISES 1) Determine the signed number that best describes the statements below. Statement Signed Number a) The boiling point of water is 212 o F. b) Carlos snorkeled 40 feet below the surface of the water. c) Jack lost 32 pounds. d) Jill gained 5 pounds. e) The company suffered a net loss of twelve million dollars. f) The elevation of Death Valley is about 280 feet below sea level. g) The elevation of Longs Peak is about 14,000 feet above sea level. 2) Plot the numbers 4 and 1 on the number line below. 3) Plot the numbers 4 and 1 on the number line below. 4) Plot the numbers 20, 5, and 30 on the number line below. 5) Label the following number line so that it includes 0 and the integers from 3 to 7. 6) Label the following number line so that it includes 0 and the integers from 8,000 to 12,000. 7) Plot the numbers that correspond to the given situations. Use a to mark the correct quantity. Also label all the surrounding tick marks. Make sure to include 0 on your number line and scale the tick marks appropriately. a) Shelby lost 8 pounds. 67

74 b) Juan snorkeled 25 feet below the surface of the water. 8) Consider the number line shown below. Elevation (in meters) relative to sea level a) What does 3 represent in this situation? b) What does 2 represent in this situation? c) What does 0 represent in this situation? 9) Use the number line to plot the given number and use the reflection line to find the opposite. a) Plot the number 2. Make sure to scale the tick marks on your number line appropriately. The opposite of 2 is b) Plot the number 30. Make sure to scale the tick marks on your number line appropriately. The opposite of 30 is 10) Plot the number 8. Make sure to scale the tick marks on your number line appropriately. The opposite of 8 is 11) Insert the correct <, >, = symbol in the space provided. a) 3 9 g) = l) 5 5 b) 5 1 c) 0 8 d) h) i) j) 8 5 m) 0 21 n) o) 1,213 1,123 68

75 e) 8 2 k) 4 0 p) 4,651 4,650 f) 400 > ) Write TRUE or FALSE in the space provided. a) If two numbers are positive, the one that is closest to zero is greater. b) If two numbers are negative, the one that is closest to zero is greater. c) If one number is positive and one number is negative, the positive number is greater. 13) Camden, SC had a record low temperature of -19 F on Jan 21, 1985, and Monahans, TX had a record low temperature of -23 F on Feb 8, (Data Source Wikipedia: a) Plot these numbers on the number line below, and label all the surrounding tick marks. Make sure to include 0 on your number line and scale the tick marks appropriately. b) Write an inequality statement that compares the two numbers. c) Which of the two temperatures was colder? 14) Liquid hydrogen evaporates at about 400. Liquid nitrogen evaporates at about 300. a) Plot these numbers on the number line below, and label all the surrounding tick marks. Make sure to include 0 on your number line and scale the tick marks appropriately. b) Write an inequality statement that compares the two numbers. c) Which liquid has the lower evaporating temperature? 15) Determine the value indicated by the sets of integer chips below. Chip Representation Number a) b) c) 69

76 d) 16) Write the following numbers from least to greatest. Ordering from least to greatest: 17) Write + or in the blank next to each of the following words. negative opposite plus positive minus 18) Write the given numbers or mathematical expressions using correct language using the words opposite of, negative, positive, plus, or minus. Number or Expression a) 5 b) ( 5) c) +5 d) 5 3 e) (+2) f) g) h) 4 + ( 9) i) (5 1) Written in Words 19) Complete the table. Write the symbolic operations in words using the indicated language below. Symbolic Minus Language Subtracted from Language Less than Language Decreased by Language ( 4) 70

77 20) Using integer chips, represent the expressions and their combined amount. Use the table to show how you did this using + for positive chips and for negative chips and find the sum. Addend Addend Sum a) b) 4 + ( 2) c) -3 + (-3) d) e) 6 + ( 4) f) g) ) Using integer chips, represent the following numbers and their difference. Use the table to show how you did this using + for positive chips and for negative chips. Circle Subtrahend Taken Simplified Minuend Subtrahend Away from Minuend Difference a) 5 3 b) 5 ( 3) c) 2 6 d) 6 2 e) 5 ( 4) 22) Perform the indicated operations. a) b) 3 + ( 1) c) d) 5 ( 5) e) 5 ( 5) f) 5 5 g) 5 5 h) 2 ( 5) i) 2 ( 5) j) 5 + ( 5) k) 3 + ( 3) l) m) 2 5 n) 2 5 o) 5 2 p) 8 + ( 9) q) r) 41 ( 41) s) t) 5 ( 2) u) 5 ( 2) 71

78 v) 35 ( 22) w) x) ( 200) 23) Perform the indicated operations. a) 6 2 b) 4( 2) c) 3 1 d) e) 2( 5) f) ( 24) ( 4) g) 20 4 h) ( 9) ( 9) i) 32 ( 4) j) 5( 6) k) 8 ( 4) l) ( 12) ( 2) m) 5 x 3 n) 6 x 3 o) 8 ( 4) p) 10 ( 5) q) 18 3 r) 7 4 Represent the application problem in symbolic form and evaluate. Then write your answer as a complete sentence. (Note: Make sure to use an addition statement even though a subtraction statement may apply as well). 24) Sara hiked down a mountain for 3 hours. Each hour, her elevation decreased by 30 meters. Compute her change in elevation in meters relative to her starting point. 25) Tom gained 10 pounds and then lost 12 pounds. What is his total change in weight relative to his original weight? 26) Joanne lost 3 pounds per month for 6 months. Find Joanne s total change in weight relative to her original weight. 27) Carlos lowers the temperature of his freezer by 7 degrees. It was originally set to 4 degrees Celsius. What is the new temperature of the freezer in degrees Celsius? 28) Leslie bought coffee 8 days this month and charged it to her checking account. She spent 6 dollars each time she visited the store. Determine the change in dollars in her checking account. 29) Malala's pool was filled 9 inches below the top of the pool. She drained the pool 5 inches. What is the water level relative to the top of the pool? 30) A total of 10 friends have a debt of 50 dollars. If they share the debt equally, what number represents the change in dollars for each friend? 31) Allie had 5 dollars in her debit account. She returned an internet purchase and they removed a charge of 10 dollars from her debit account. 32) The temperature in Minneapolis changed by 32 degrees in 8 days. If the temperature changed by the same amount each day, what was the change in temperature per day? 33) Kayla camped at 9 miles relative to sea level. She then hiked 4 miles upwards. What is her current altitude relative to sea level? 34) Tally bought 50 packages of printer paper for her business. Each package contained 300 sheets of paper. How many sheets of paper is this in total? 72

79 35) Sheldon has 140 dollars in his checking account and Penny has 150 dollars in her checking account. How much did they have all together? 36) Ken had 15 dollars in his checking account and wrote a check for 21 dollars. What is the balance in his checking account in dollars? 73

80 SECTION 1.6: ORDER OF OPERATIONS A. INTRODUCTION TO EXPONENTS Introduction to Exponents (Duration 5:54) Solve the problem below. Use the rectangle below to represent the problem visually. a) Don makes a rectangular 20 square foot cake for the state fair. After he wins his award, he wants to share it with the crowd. First he cuts the cake into 2 pieces. Then he cuts the 2 pieces into 2 pieces each. Then he cuts all of these pieces into two pieces. He continues to do this a total of 5 times. How many pieces of cake does he have to share? b) Write a mathematical expression that represents the total number of pieces in which Don cut the cake. Terminology We will use exponential expressions to represent problems such as the last one. Exponents represent repeated multiplication just like multiplication represents repeated addition as shown below. Multiplication: 5 2 = = 10 Exponents: 2 5 = = 32 Language and Notation of Exponents (Duration 7:32) View the video lesson, take notes and complete the problems below In the exponential expression, is called the base 5 is called the exponent We will say 2 5, as 2 raised to the fifth power or 2 to the fifth Since exponents represent repeated multiplication, and we call the numbers we multiply factors, we will also use this more meaningful language when discussing exponents. 2 5 means 5 factors of 2 or We also have special names for bases raised to the second or third power. a) For 3 2, we say 3 squared or 3 to the second power b) For 4 3, we say 4 cubed or 4 to the third power 74

81 Represent the given exponential expressions in the four ways indicated. a) 6 2 b) 6 2 Expanded Form Expanded Form Word Name Word Name Factor Language Factor Language Math Equation Math Equation c) ( 6) 2 d) ( 5) 3 Expanded Form Expanded Form Word Name Word Name Factor Language Factor Language Math Equation Math Equation YOU TRY Represent the given exponential expressions in the four ways indicated. a) 7 2 b) ( 7) 2 Expanded Form Expanded Form Word Name Word Name Factor Language Factor Language Math Equation Math Equation 75

82 B. THE ORDER OF OPERATIONS WITH ADDITION AND SUBTRACTION The Order of Operations with Addition and Subtraction (Duration 4:29) Solve the problem below. Be sure to indicate every step in the process of your solution. a) Suppose on the first day of the month you start with $150 in your bank account. You make a debit transaction on the second day for $60 and then make a deposit on the third day for $20. What is the balance in your account on the third day? b) What string of operations (written horizontally) can be used to determine the amount in your account? Rule 1: When we need to add or subtract 2 or more times in one problem, we will perform the operations from left to right. The Order of Operations with Addition and Subtraction (cont.) (Duration 5:35) Use a highlighter to highlight the operations in the problem. Determine the number of operations to be performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) # of operations b) 12 ( 5) ( 1) # of operations YOU TRY Use a highlighter to highlight the operations in the problem. Determine the number of operations to be performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) ( 2) # of operations b) 8 + ( 5) 6 ( 2) + 9 # of operations 76

83 C. THE ORDER OF OPERATIONS WITH MULTIPLICATION AND DIVISION The Order of Operations with Multiplication and Division (Duration 4:22) Solve the problem below. Be sure to indicate every step in the process of your solution. a) Suppose you and your three siblings inherit $40,000. You divide it amongst yourselves equally. You then invest your portion and make 5 times the amount of your portion. How much money do you have? Be sure to indicate every step in your process. b) What string of operations (written horizontally) can be used to determine the result? Rule 2: When we need to multiply or divide 2 or more times in one problem, we will perform the operations from left to right. The Order of Operations with Multiplication and Division (Cont.) (Duration 3:27) Use a highlighter to highlight the operations in the problem. Determine the number of operations to be performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) 6 4 ( 2) 2 # of operations b) ( 3) # of operations YOU TRY Use a highlighter to highlight the operations in the problem. Determine the number of operations to be performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) 8( 2) ( 4) ( 2) # of operations b) ( 1)(2) # of operations 77

84 D. THE ORDER OF OPERATIONS FOR +,,, Order of Operations for +,,, (Duration 5:21) Solve the two problems below. Be sure to indicate every step in your process a) Bill went to the store and bought 3 six-packs of soda and an additional 2 cans. How many cans did he buy in total? What string of operations (written horizontally) can be used to represent this problem? b) Amber went to the store and bought 3 six-packs of cola and an additional 2 six-packs of diet cola. How many cans did she buy in total? What string of operations (written horizontally) can be used to represent this problem? Rule 3: Unless otherwise indicated by parentheses, we perform multiplication and division before addition and subtraction. We continue to perform the operations from left to right. Order of Operations for +,,, (cont. ) (Duration 4:04) Use a highlighter to highlight the operations in the problem. Determine the number of operations to be performed in the problem. Perform the operations in the appropriate order.. a) ( 3) # of operations b) ( 3) # operations YOU TRY Use a highlighter to highlight the operations in the problem. Determine the number of operations to be performed in the problem. Perform the operations in the appropriate order. Show all intermediary steps. a) ( 3) # of operations b) ( 3 )( 4) # operations 78

85 E. THE ORDER OF OPERATIONS WITH PARENTHESES There are cases when we want to perform addition and subtraction before multiplication and division in the order of operations. So we need a method of indicating we want to make such a modification. In the next media problem, we will discuss how to show this change. Grouping symbols ( ), { }, [ ] If there are several parentheses in a problem, we will start with the inner most parenthesis and work our way out as we apply order of operations to the expression. To avoid confusion with multiple parentheses, we use different types of grouping symbols such as { } and [ ] and ( ). These grouping symbols all mean the same thing and imply the expression inside must be evaluated first. Rule 4: If we want to change the order in which we perform operations in an arithmetic expression, we can use parentheses to indicate that we will perform the operation(s) inside most parentheses first. Order of operations with grouping symbols ( ), { }, [ ] (4:42) View the video lesson, take notes and complete the problems below Example: Different types of parentheses: Always do first! (4 + 2) [ 5 2 (2 + 3) ] 7 { [20 (4 + 6) ] } YOU TRY Simplify the expression completely: 2{8 2 7[32 4( ) ] ( 1) } 79