4/27/2010 ALGEBRA 1 CHAPTER 8 FACTORING PAR Activities Cara Boening
Table of Contents Preparation Strategies... 3 Scavenger Hunt... 4 Graphic Organizer Chapter 8.... 6 Word Inventory... 8 TEASE... 12 Anticipation Guide Chapter 8... 13 Assistance Strategies... 15 Study Guide... 16 Vocabulary Guide Section One... 18 QAR for Section 2... 20 Jot Chart-Sections 3 and 4... 22 Algebra Lab Difference of Squares... 24 Guide-O-Rama Section 6... 26 Preparation Strategies... 27 Brainstorming... 28 Mid-Chapter Quiz... 29 BINGO!... 31 Quizlet... 33 Authentic Assessment... 34 Chapter 8 Test Factoring... 37 Readability... 41 Book Search... 42
Preparation Strategies Scavenger Hunt... Error! Bookmark not defined. Graphic Organizer Chapter 8... Error! Bookmark not defined. Word Inventory... Error! Bookmark not defined. K-W-L... Error! Bookmark not defined. TEASE... Error! Bookmark not defined. Anticipation Guide Chapter 8...Error! Bookmark not defined.
Scavenger Hunt DIRECTIONS: Answer each question; then identify the part of the book that helped you to find the answer. 1. Where can symbols and formulas be found? 2. At the end of every chapter, what is included to help you prepare for a test? 3. How many chapters are in this textbook? 4. What valuable information can be found at the beginning of each chapter? 5. If you are having trouble with an odd-numbered problem, where can the answer be located? 6. What is the definition of hypotenuse? 7. In what chapter can you find how to factor trinomials? 8. Who are the authors of this book? 9. Where was the picture on the front cover taken? 10. In almost every chapter, there are key concepts included in a box. Find one key concept and list the information and what it is used for (make sure to include the page number.) 11. What are the key vocabulary words for chapter 6?
ANSWERS PARTS OF THE BOOK SEARCH: Reading to Learn in the Content Areas 1. Inside the back cover. 2. Study Guide and Review 3. 12 4. The knowledge and skills that will be covered, key vocabulary, a FOLDABLES Study Organizer, and a Quick Quiz and Quick Review. 5. In the Answer Appendix 6. The side opposite the right angle in a right triangle. 7. Chapter 8 8. Holliday, Luchin, Cuevas, Carter, Marks, Day, Casey, Hayek 9. At the International El Paso Balloon Festival. 10. Can be multiple answers. Check page number given to make sure it is done correctly. 11. Compound Inequality, Intersection, Set-Builder Notation, Union
Graphic Organizer Chapter 8 Define the following terms or answer the question. Factoring Monomials & Factoring Distributive Property Factoring Trinomials Factoring Differences of Squares Perfect Squares & Factoring Perfect Sqaures & Factoring -Prime Factorization -Factored Form -Greatest Common Factor -Grouping x^2+bx+c What happens when: -b and c are Positive -b is Negative and c is Postive ax^2+bx+c -Prime Polynomial a^2-b^2 a^2+2ab+b^2 -c is Negative
Chapter 8 Define the following terms or answer the question. Factoring Monomials & Factoring Distributive Property Factoring Trinomials Factoring Differences of Squares Perfect Squares & Factoring Perfect Sqaures & Factoring -Prime Factorization Number expressed in prime factors -Factored Form Monomial -Greatest Common Factor The product of the common prime factors -Grouping A way of factoring using the distributive property when a polynomial has 4 or more terms x^2+bx+c What happens when: -b and c are Positive All coefficients are positive -b is Negative and c is Postive Both factors are negative -c is Negative One factor is positive and one is negative ax^2+bx+c -Prime Polynomial A polynomial that cannot be written as a product of two polynomials a^2-b^2 (a+b)(a-b) a^2+2ab+b^2 (a+b)^2 (a-b)^2
Word Inventory Directions: Determine which of the following words you are familiar with and you know the definition. Beside the words you do know, indicate this by placing a in the blank. If you are not sure if you know the definition or the meaning of the word, place a in the blank. For those words you have no idea what they mean place a in the blank. 1. Prime Number 2. Composite Number 3. Prime Factorization 4. Factored Form 5. Greatest Common Factor 6. Factoring 7. Factoring By Grouping 8. Zero Products Property 9. Roots 10. Prime Polynomial 11. Difference of Squares 12. Perfect Square Trinomials
Word Inventory Determine which of the following words you are familiar with and you know the definition. Beside the words you do know, indicate this by placing a in the blank. If you are not sure if you know the definition or the meaning of the word, place a in the blank. For those words you have no idea what they mean place a in the blank. 1. Prime Number 2. Composite Number 3. Prime Factorization 4. Factored Form 5. Greatest Common Factor 6. Factoring 7. Factoring By Grouping 8. Zero Products Property 9. Roots 10. Prime Polynomial 11. Difference of Squares 12. Perfect Square Trinomials
K-W-L Directions: Fill in the columns with what you know about factoring and what you would like to learn about factoring. After the chapter is completed, the forms will be returned and you will fill in what you learned from the unit. What We Know What We Want to Know What We Learned
K-W-L Directions: Fill in the columns with what you know about factoring and what you would like to learn about factoring. After the chapter is completed, the forms will be returned and you will fill in what you learned from the unit. What We Know What We Want to Know What We Learned We know how to multiply a polynomial by a monomial. We know how to multiply polynomials. We know FOIL We know how to find squares of polynomials. We know how to use the Distributive Property with polynomials.
TEASE Directions: I have completed a video for students to watch before the first day of instruction. It shows a relationship between puzzles and factoring. It is much easier to complete a puzzle if you separate the different pieces first by edge and corner, then by color. It is also much easier if you are able to have someone help you and if you are able to look at the picture you are creating on the box. This short video shows the relationship. Have a short discussion that involves these relationships before beginning the first day of instruction. Puzzles & Factoring
Anticipation Guide Chapter 8 Directions: Read each statement before beginning chapter 8. Determine if you agree or disagree by placing an A or a D by each statement. After covering chapter 8, return and read the statements. Repeat the process and see how your opinions have changed or remained the same. If you still disagree with the statements after reading the chapter, correct them at the bottom of this page. Pre- Reading A or D Statement Post- Reading A or D 1. The number 2 is a prime number because the only factors of 2 are 1 and itself. 2. The prime factorization of a number is expressed as a product of factors that are all prime. 3. If the greatest common factor of two numbers is 1, they are said to be relatively prime. 4. If the product of two factors is 0, they must be negative inverses of each other. 5. To find the solution of, you can take the square root of each side. 6. The polynomial is not factorable. 7. The numbers 36,49, 64, and 81 are perfect squares.
Anticipation Guide Chapter 8 Directions: Read each statement before beginning chapter 8. Determine if you agree or disagree by placing an A or a D by each statement. After covering chapter 8, return and read the statements. Repeat the process and see how your opinions have changed or remained the same. If you still disagree with the statements after reading the chapter, correct them at the bottom of this page. Pre- Reading A or D Statement Post- Reading A or D 1. The number 2 is a prime number because the only factors of 2 are 1 and itself. A 2. The prime factorization of a number is expressed as a product of factors that are all prime. A 3. If the greatest common factor of two numbers is 1, they are said to be relatively equal. D 4. If the product of two factors is 0, they must be negative inverses of each other. D 5. To find the solution of, you can take the square root of each side. D 6. The polynomial is not factorable. A 7. The numbers 36,49, 64, and 81 are perfect squares. A 3. If the greatest common factor of two numbers is 1, they are said to be relatively prime. 4. If the product of two factors is 0, at least one of them must be 0. 5. To find the solution of, you must first subtract 2x from each side. Then you must factor out an x and solve from there.
Assistance Strategies Study Guide... Error! Bookmark not defined. Vocabulary Guide Section One... Error! Bookmark not defined. QAR for Section 2... Error! Bookmark not defined. Jot Chart-Sections 3 and 4... Error! Bookmark not defined. Algebra Lab Difference of Squares... Error! Bookmark not defined. Guide-O-Rama Section 6... Error! Bookmark not defined.
Study Guide Directions: Keep this study guide throughout the unit. We will fill in the information as we cover the information in each chapter. DO NOT LOSE THIS PAPER! Number of Terms Factoring Technique Example 2 or more greatest common factor 2 Difference of Squares Perfect Square Trinomial 3 wh en and when and. Then use factoring by grouping 4 or more Factoring by grouping
Study Guide Directions: Keep this study guide throughout the unit. We will fill in the information as we cover the information in each chapter. DO NOT LOSE THIS PAPER! Number of Terms Factoring Technique Example 2 or more greatest common factor 3 6 15 3 2 5 2 Difference of Squares 4 25 2 5 2 5 Perfect Square Trinomial 6 9 3 4 4 1 2 1 3 wh en and 9 20 4 5 when and. Then use factoring by grouping 6 2 6 3 4 2 3 2 1 2 2 1 2 1 3 2 4 or more Factoring by grouping 3 6 5 10 3 6 5 10 3 2 5 2 2 3 5
Vocabulary Guide Section One Directions: Before you read section one, fill in what you believe is a possible definition of each vocabulary word. As you read, locate the vocabulary words, fill in the page number, and write the given definition in the section. Word Page Possible Definition Verified Definition Prime Number Composite Number Prime Factorization Factored Form Greatest Common Factor (GCF)
Vocabulary Guide Section One Directions: Before you read section one, fill in what you believe is a possible definition of each vocabulary word. As you read, locate the vocabulary words, fill in the page number, and write the given definition in the section. Word Page Possible Definition Verified Definition Prime Number 414 A number that cannot be divided by any number other than 1 A whole number, greater than 1, for which the only factors are 1 and itself Composite Number 414 Can be divided by numbers other than 1 A whole number, greater than 1, that has more than two factors Prime Factorization 414 The prime numbers of something A whole number expressed as the product of prime factors Factored Form 415 Something in its lowest form A monomial when expressed as the product of prime numbers and variables, and no variable has an exponent greater than 1 Greatest Common Factor (GCF) 415 The biggest number two numbers have in common The greatest number that is a factor of both original numbers
QAR for Section 2 Directions: Complete the questions as you begin to think in depth about polynomials and factoring using the Distributive Property and Grouping. Question Answer 1. List the two methods of factoring discussed in Section 2. 2. Why do you think it is important to know the distributive property when x and y are in an equation? 3. Compare and contrast the Distributive Property and Grouping method for factoring. 4. Create an example of an equation in which you would use grouping and explain how you would solve the equation.
QAR for Section 2 Directions: Complete the questions as you begin to think in depth about polynomials and factoring using the Distributive Property and Grouping. Question Answer 1. List the two methods of factoring discussed in Section 2. Distributive Property Grouping 2. Why do you think it is important to know the distributive property when x and y are in an equation? When two variables are involved, the values may need to be known in order to solve the equation. If you can isolate some of the variables, you can find the values of them. 3. Compare and contrast the Distributive Property and Grouping method for factoring. Distributive property is used to remove common factors from an equation. The grouping method takes like terms and removes the GCF, then the equations can be solved. 4. Create an example of an equation in which you would use grouping and explain how you would solve the equation. 15 3 4 20 15 3 4 20 3 5 4 5 3 4 5
Jot Chart-Sections 3 and 4 Directions: As we complete sections 3 and 4, fill in the possible factors when standard form is used in factoring. Standard Form Possible Factors, is prime, is composite
Jot Chart-Sections 3 and 4 Directions: As we complete sections 3 and 4, fill in the possible factors when standard form is used in factoring. Standard Form Possible Factors and add to and multiply to and add to and multiply to and add to and multiply to and add to and multiply to, is prime and add to, and multiply to, is composite (,,
Algebra Lab Difference of Squares Directions: Follow the instructions given to you step-by-step. Begin by cutting out the large square then wait for directions. When you are done with the lab, answer the following questions on a separate piece of paper. 1. Write an exp ression representing the area of the rectangle. 2. Explain why.
Algebra Lab Difference of Squares Directions: 1. Give students the handout without these instructions so students will discover the mystery. 2. Cut out the largest square. This is of length a. The area of the large squar e is and the area of the small square is. 3. Cut the small square from the large square. The area of the new figure is. 4. Cut the remaining figure into two congruent figures (cut on the dotted line.) 5. Arrange the pieces to form a rectangle with lengths and. Have students answer the questions on a separate sheet of paper. 1. Write an expres sion representing the area of the rectangle a. 2. Explain why. a. Students may say something about the fact that when a smaller square is removed from a larger square, the remaining shape forms a rectangle. The area of a rectangle is length times width. If is multiplied out, 0.
Guide-O-Rama Section 6 1. Can you list the square roots of 4, 9, 16, 25, 36, etc.? 2. Notice the definition of perfect square trinomials. 3. Look over and make sure you understand the information in the table at the bottom of page 448. The table at the top of the next page is the same information in general form. 4. Make sure you know the three questions you should ask about each trinomial in order to determine if it is a perfect square trinomial. 5. HINT!!! The concept summary is information that will be on the test! 6. See Example 2. Remember on all trinomials to look for a GCF before beginning factoring. 7. The square root property will help save time if there is only an x^2 and an integer in an equation! 8. Do the assigned homework problems to turn in tomorrow!
Preparation Strategies Brainstorming... Error! Bookmark not defined. Mid-Chapter Quiz... Error! Bookmark not defined. BINGO!... Error! Bookmark not defined. Quizlet... Error! Bookmark not defined. Authentic Assessment... Error! Bookmark not defined. Chapter 8 Test Factoring... Error! Bookmark not defined.
Brainstorming Directions: Students have learned to factor using the Distributive Property and using Grouping. For this brainstorming activity, students will be asked several questions to make them consider how you would solve an equation that involved an equal sign. Students will be split into groups of three or four and will be given the following equations and see what they would do to solve them. Reminder: The students have only covered section one and part of section two. Don t make the questions too advanced for where you are in the chapter. Questions: 1. 2. 3. Students will be given 5-10 minutes to discuss what steps need to be taken to find the answers. At this point, we will discuss in class the correct steps to solving the equations. Each group will be given an opportunity to share their answers and their logic behind them. Answers: 1. 2. 3.
Mid-Chapter Quiz Directions: Factor or solve each monomial or polynomial completely. SHOW YOUR WORK! 1. 35 6. 8 5 4 2. 78 1. 10 3 3. What is the GCF of 15 & 5? 2. 4 3 4. 4 3. 13 36 5. 6 16 15 40 4. 2 6 74
Mid-Chapter Quiz Directions: Factor or solve each monomial or polynomial completely. SHOW YOUR WORK! 1. 35 3 5 6. 8 5 4 0 8 5 0 4 0 8 5 4 5 8 4 2. 78 2 3 13 7. 10 3 10 3 0 10 3 0 0 3 10 3. What is the GCF of 15 & 5? 5 8. 4 3 3 3 3 3 4. 4 4 1 9. 13 36 4 9 36 4 9 4 9 4 5. 6 16 15 40 2 3 8 5 3 8 2 5 3 8 10. 2 6 7 4 2 80 0 10 8 80 0 10 8 10 0 8 10 0 8 10
BINGO! Directions: Fill in the spaces using the words from the word list. You may want to write equations in different forms or make notes before we start! B I N G O FREE! Word List: 2 2 Square of a Negative Binomial: Roots Composite Number Prime Factorization 5 7 7 7 Factored Form 12, 2 4 terms GCF 3 8 2 6 100 Factoring 36 Prime Polynomial Square of a Positive Binomial: BINGO! Prime Number Zero Products Property
Directions: Give students 5 minutes to fill in the Bingo board and to look over terms. The first 3 times you play, allow students to raise their hand and answer questions on at a time. The next two times, play where students may not talk during play. Make sure to allow time for students to think through the problems. Here are the answers and the clues to be given. Clue A nswer 1. What is the standard form of the difference of squares? 2. What is the expanded form of? 2 3. What is the expanded form of? 2 4. What is the standard form of a trinomial? 5. The solutions are 8 3, 3. 3 8 6 6. The solutions are 10. 100 7. The factored standard form of where is prime. 8. The factored standard form of where is composite. 9. The prime factorization of 35. 5 7 10. The GCF of these two terms is 2. 12, 2 11. This term describes when a polynomial is expressed as the product of monomials an d polynomials. Factoring 12. Expressed in standard form as 2. Square of a Positive Binomial: 13. Expressed in standard form as 2. Square of a Negative Binomial: 14. The solutions of a quadratic equation. Roots 15. A specific example of factored difference of squares. 7 7 16. This is the perfect time to use Factoring by Grouping. 4 terms 17. An example of a perfect square trinomial. 36 18. A whole number, greater than 1, with only factors that are 1 and itself. Prime Number 19. A whole number, greater than 1, that has more than two factors. Composite Number 20. A whole number expressed as a product of factors that are all prime numbers. Prime Factorization 21. A monomial expressed as a product of prime numbers and variables and no variable has an exponent Factored Form greater than 1. 22. The product of the prime factors common to two or more integers. GCF 23. A polynomial that cannot be written as a product of two polynomials with integral coefficients. Prime Polynomial 24. This states that if the product of two factors is 0, then at least one of the factors must be 0. Zero Products Property
Directions: Quizlet Step 1: Log into Quizlet.com. Username: algebrafactoring Password: factoring Step 2: Click the option "My Dashboard" Step 3: Click on the title "Factoring Trinomials" Step 4: Use the flashcards to practice what we have learned in class. Some of these may appear on the exam. Step 5: Choose the option "Test" and work the problems. Print the graded test sheet. This will be turned in for a quiz grade.
Authentic Assessment Directions: Students will be split into groups of three or four. Each group will be given a career to research. The students will find how each profession uses factoring in their workplace. The careers will include, but won't be limited to, engineering, physics, biology, and landscape designer. Students will need to write a paper that explains what the career is, how factoring is used, and how factoring can be used in the students' everyday lives. A graphic representation should be created to show how factoring is used.
Authentic Assessment Rubric CATEGORY 4 3 2 1 Explanation Explanation Explanation shows complete shows substantial shows some understanding of understanding of understanding of the mathematical the mathematical the mathematical concepts used to concepts used to concepts needed solve the solve the to solve the problem(s). problem(s). problem(s). Mathematical Concepts Explanation Working with Others Mathematical Terminology and Notation Neatness and Organization Diagrams and Sketches Explanation is detailed and clear. Student was an engaged partner, listening to suggestions of others and working cooperatively throughout lesson. Correct terminology and notation are always used, making it easy to understand what was done. The work is presented in a neat, clear, organized fashion that is easy to read. Diagrams and/or sketches are clear and greatly add to the reader's understanding of the procedure(s). Explanation is clear. Student was an engaged partner but had trouble listening to others and/or working cooperatively. Correct terminology and notation are usually used, making it fairly easy to understand what was done. The work is presented in a neat and organized fashion that is usually easy to read. Diagrams and/or sketches are clear and easy to understand. Explanation is a little difficult to understand, but includes critical components. Student cooperated with others, but needed prompting to stay on task. Correct terminology and notation are used, but it is sometimes not easy to understand what was done. The work is presented in an organized fashion but may be hard to read at times. Diagrams and/or sketches are somewhat difficult to understand. Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written. Explanation is difficult to understand and is missing several components OR was not included. Student did not work effectively with others. There is little use, or a lot of inappropriate use, of terminology and notation. The work appears sloppy and unorganized. It is hard to know what information goes together. Diagrams and/or sketches are difficult to understand or are not used.
Completion All problems are completed. All but one of the problems are completed. All but two of the problems are completed. Several of the problems are not completed. Mathematical Errors 90 100% of the steps and solutions have no mathematical errors. Almost all (85 89%) of the steps and solutions have no mathematical errors. Most (75 84%) of the steps and solutions have no mathematical errors. More than 75% of the steps and solutions have mathematical errors.
Chapter 8 Test Factoring Directions: Complete the following problems. Be sure to show your work. (4.5 pts each) Factor Each Monomial Completely. Factor eac h polynomial. 1. 33 7. 28 60 2. 150 8. 3 10 8 Find the GCF of each set of monomials. 9. 21 20 3. 12, 18, 40 10. 13 30 4. 16, 30 11. 5 41 8 5. 5 30 12. 4 27 18 6. 8 12 13. 16
Distribute and Simplify 14. 3 9 15. 6 7 2 3 Solve the equation. 18. 4 1 3 5 0 16. 7 70 175 19. 9 20 0 17. 4 20 96 20. 0 21. A gymnast jumps on a trampoline traveling at 12 feet per second. Her height in feet above the trampoline after seconds is given by the formula 12 16. How long is the gymnast in the air before returning to the trampoline? 22. A boulder breaks loose from the face of a mountain and falls toward the water 576 feet below The distance that the boulder falls in seconds is given by the equation 16. How long does it take the boulder to hit the water?
Chapter 8 Test Factoring Directions: Complete the following problems. Be sure to show your work. (4.5 pts each) Factor Each Monomial Completely. Factor eac h polynomial. 1. 33 1 3 11 7. 28 60 30 2 2. 150 2 3 5 5 8. 3 10 8 3 2 4 Find the GCF of each set of monomials. 9. 21 20 20 1 3. 12, 18, 40 2 10. 13 30 15 2 4. 16, 30 2 11. 5 41 8 5 1 8 Factor each Polynomial 5. 5 30 5 6 12. 4 27 18 4 3 6 6. 8 12 4 2 3 13. 16 16
Distribute and Simplify 14. 3 9 9 3 27 6 27 15. 6 7 2 3 12 18 14 21 12 32 21 Solve the equation. 18. 4 1 3 5 0 1 4, 5 3 16. 7 70 175 7 70 175 0 7 10 25 0 7 5 0 5 17. 4 20 96 4 20 96 0 4 5 24 0 4 8 3 0 8, 3 21. A gymnast jumps on a trampoline traveling at 12 feet per second. Her height in feet above the trampoline after seconds is given by the formula 12 16. How long is the gymnast in the air before returning to the trampoline? 0 12 16 0 4 3 4 3 4 19. 9 20 0 5 4 0 5, 4 20. 0 1 3 1 3 22. A boulder breaks loose from the face of a mountain and falls toward the water 576 feet below The distance that the boulder falls in seconds is given by the equation 16. How long does it take th e boulder to hit the water? 576 16 36 6 0
Readability Algebra 1 Use the information about sirens at the left. A care and an emergency vehicle are heading toward each other. The car is traveling at a speed of 30 miles per hour or about 44 feet per second. The emergency vehicle is traveling at a speed of 50 miles per hour or about 74 feet per second. If the vehicles are 1000 feet apart and the conditions are ideal, in how many seconds will the driver of the car first hear the siren? Under ideal conditions, a siren can be heard from up to 440 feet. However, under normal conditions, a page 122 Sentences: 6.38 Syllables: 158 The Caverns of Sonora, in Sonora, Texas, is one of the most active caves in the world, with more than 90% of the formations in the cave still "growing." There are two different tours of the caves, as shown in the table. The total length of both tours combines is 3.25 miles. The Crystal Palace tour is twice the distance of the Horseshoe Lake tour minus 0.5 mile. How could you use a system of equations to determine the lengths of the tours? You have learned five methods for solving systems of linear equations. The table summarizes the methods and page 278 Sentences: 6.43 Syllables: 142 The President of the United States is elected every four years, and senators are elected every six years. A certain senator is elected in 2004, the same year as a presidential election, and is reelected in subsequent elections. In what year is the senator's reelection the same year as a presidential election? The least number of years that will pass until the next election for both a specific senator and the President is the least common multiple of 4 and 6. The least common multiple (LCM) is the least number that is a multiple of two or more numbers. You page 606 Sentences: 5.09 Syllables: 164 The book is at a 10th grade reading level so most students should be able to read the text without many modifications.
Book Search DIRECTIONS: Answer each question; then identify the part of the book that helped you to find the answer. 12. Where can symbols and formulas be found? 13. At the end of every chapter, what is included to help you prepare for a test? 14. How many chapters are in this textbook? 15. What valuable information can be found at the beginning of each chapter? 16. If you are having trouble with an odd-numbered problem, where can the answer be located? 17. What is the definition of hypotenuse? 18. In what chapter can you find how to factor trinomials? 19. Who are the authors of this book? 20. Where was the picture on the front cover taken? 21. In almost every chapter, there are key concepts included in a box. Find one key concept and list the information and what it is used for (make sure to include the page number.) 22. What are the key vocabulary words for chapter 6?
ANSWERS PARTS OF THE BOOK SEARCH: Reading to Learn in the Content Areas 12. Inside the back cover. 13. Study Guide and Review 14. 12 15. The knowledge and skills that will be covered, key vocabulary, a FOLDABLES Study Organizer, and a Quick Quiz and Quick Review. 16. In the Answer Appendix 17. The side opposite the right angle in a right triangle. 18. Chapter 8 19. Holliday, Luchin, Cuevas, Carter, Marks, Day, Casey, Hayek 20. At the International El Paso Balloon Festival. 21. Can be multiple answers. Check page number given to make sure it is done correctly. 22. Compound Inequality, Intersection, Set-Builder Notation, Union