Geometric Reasoning 2A Inductive and Deductive S E C T I O N 2A Reasoning Inductive and Deductive Reasoning 2B Mathematical Proof S E C T I O N 2B

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1 Geometric Reasoning ECTION 2A nductive and Deductive easoning On page 02, students use inductive and deductive reasoning to decipher he wordplay in Lewis Carroll s lice s Adventures in Wonderland. xercises designed to prepare tudents for success on he Multi-Step Test Prep an be found on pages 8, 85, 92, and 00. ECTION 2B athematical Proof On page 26, students use mathematical proof to investigate the eometric properties of intersecting ighways. xercises designed to prepare tudents for success on he Multi-Step Test Prep an be found on pages 09, 5, and 24. 2A Inductive and Deductive Reasoning 2- Using Inductive Reasoning to Make Conjectures 2-2 Conditional Statements 2-3 Using Deductive Reasoning to Verify Conjectures Lab Solve Logic Puzzles 2-4 Biconditional Statements and Definitions 2B Mathematical Proof 2-5 Algebraic Proof 2-6 Geometric Proof Lab Design Plans for Proofs 2-7 Flowchart and Paragraph Proofs Ext Introduction to Symbolic Logic Winning Strategies Mathematical reasoning is not just for geometry. It also gives you an edge when you play chess and other strategy games. KEYWORD: MG7 ChProj 70 Chapter 2 Winning Strategies About the Project In the Chapter Project, students use logical reasoning to develop strategies for a paperand-pencil game and to solve a mathematical puzzle. Project Resources All project resources for teachers and students are provided online. KEYWORD: MG7 ProjectTS

2 Angle Relationships Select the best description for each labeled angle pair. 6. Vocabulary Match each term on the left with a definition on the right.. angle B A. a straight path that has no thickness and extends forever 2. line A B. a figure formed by two rays with a common endpoint 3. midpoint F C. a flat surface that has no thickness and extends forever 4. plane C D. a part of a line between two points 5. segment D E. names a location and has no size F. a point that divides a segment into two congruent segments 7. linear pair or adjacent angles or supplementary angles or vertical angles vertical angles complementary angles lin. pair vert. comp. Classify Real Numbers Tell if each number is a natural number, a whole number, an integer, or a rational number. Give all the names that apply natural, whole, rational. 3 integer, rational integer, rational rational Points, Lines, and Planes 5 9. Possible answers: B BD Name each of the following. 5. a point 6. a line 7. a ray CA 8. a segment CD 9. a plane plane F 3_ 8 rational whole, integer, rational Solve One-Step Equations Solve x = y = = 6s p - 7 = z_ 25 = = -.2r -7 5 Organizer Objective: Assess students understanding of prerequisite skills. Prerequisite Skills Angle Relationships Classify Real Numbers Points, Lines, and Planes Solve One-Step Equations Assessing Prior Knowledge INTERVENTION Diagnose and Prescribe Use this page to determine whether intervention is necessary or whether enrichment is appropriate. Resources Are You Ready? Intervention and Enrichment Worksheets Are You Ready? CD-ROM Are You Ready? Online Geometric Reasoning 7 NO INTERVENE Diagnose and Prescribe YES ENRICH ARE YOU READY? Intervention, Chapter 2 Prerequisite Skill Worksheets CD-ROM Online Angle Relationships Skill 25 Activity 25 Classify Real Numbers Skill 7 Activity 7 Points, Lines, and Planes Skill 22 Activity 22 Solve One-Step Equations Skill 68 Activity 68 Diagnose and Prescribe Online ARE YOU READY? Enrichment, Chapter 2 Worksheets CD-ROM Online

3 2 Study Guide: Preview Organizer Objective: Help students organize the new concepts they will learn in Chapter 2. Online Edition Multilingual Glossary esources Previously, you studied relationships among points, lines, and planes. identified congruent segments and angles. examined angle relationships. used geometric formulas for perimeter and area. Key Vocabulary/Vocabulario conjecture counterexample deductive reasoning inductive reasoning polygon proof quadrilateral conjetura contraejemplo razonamiento deductivo razonamiento inductivo polígono demostración cuadrilátero theorem teorema Multilingual Glossary Online KEYWORD: MG7 Glossary nswers to ocabulary Connections. Possible answer: a number that is not positive, such as Possible answer: a general conclusion 3. Possible answer: You start with general principles to get to a specific conclusion. 4. Possible answer: Polygon might mean a figure with many. You will study inductive and deductive reasoning. using conditional statements and biconditional statements. justifying solutions to algebraic equations. writing two-column, flowchart, and paragraph proofs. You can use the skills learned in this chapter when you write proofs in geometry, algebra, and advanced math courses. when you use logical reasoning to draw conclusions in science and social studies courses. when you assess the validity of arguments in politics and advertising. triangle triángulo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like.. The word counterexample is made up of two words: counter and example. In this case, counter is related to the Spanish word contra, meaning against. What is a counterexample to the statement All numbers are positive? 2. The root of the word inductive is ducere, which means to lead. When you are inducted into a club, you are led into membership. When you use inductive reasoning in math, you start with specific examples. What do you think inductive reasoning leads you to? 3. The word deductive comes from de, which means down from, and ducere, the same root as inductive. What do you think the phrase lead down from would mean when applied to reasoning in math? 4. In Greek, the word poly means many, and the word gon means angle. How can you use these meanings to understand the term polygon? 72 Chapter 2

4 2 Reading Strategy: Read and Interpret a Diagram A diagram is an informational tool. To correctly read a diagram, you must know what you can and cannot assume based on what you see in it. Collinear points Betweenness of points Coplanar points Straight angles and lines Adjacent angles Linear pairs of angles Vertical angles Points A, B, and C are collinear. Points A, B, C, and D are coplanar. B is between A and C. AC is a line. ABD and CBD are adjacent angles. ABD and CBD form a linear pair. Measures of segments Measures of angles Congruent segments Congruent angles Right angles If a diagram includes labeled information, such as an angle measure or a right angle mark, treat this information as given. Try This CBD is acute. ABD is obtuse. AB BC List what you can and cannot assume from each diagram.. 2. Geometric Reasoning 73 Organizer Objective: Help students apply strategies to understand and retain key concepts. Online Edition Resources Chapter 2 Resource Book Reading Strategies Reading Strategy: Read and Interpret a Diagram ENGLISH LANGUAGE LEARNERS Discuss Students will write geometric proofs in Lessons 2-6 and 2-7. Emphasize that knowing what they can and cannot assume from a diagram will be essential to their success in writing proofs and in solving a variety of geometric problems throughout the course. Extend As students work through the problems in Chapter 2, have them discuss what information can be assumed from the diagrams. Ask them to list the information in their journals and to refer to the list when writing proofs. They might find it helpful to create a separate list of what cannot be assumed. Answers to Try This. Possible answer: Can assume: W, A, and Y are collinear. X, A, and Z are collinear. All the pts. are coplanar. A is between W and Y. A is between X and Z. XZ is a line. WY is a line. XAW and WAZ are adj.. WAZ and ZAY are adj.. ZAY and YAX are adj.. YAX and XAW are adj.. XAW and WAZ form a lin. pair. WAZ and ZAY form a lin. pair. ZAY and YAX form a lin. pair. YAX and XAW form a lin. pair. XAW and ZAY are vert.. WAZ and YAX are vert.. Cannot assume: anything about the measures of the ; anything about the measures of the segs.; XAW WAZ; XAW WAZ; YA AZ ; YA AZ. 2. See p. A.

5 SECTION 2A Inductive and Deductive Reasoning One-Minute Section Planner Lesson Lab Resources Materials Lesson 2- Using Inductive Reasoning to Make Conjectures Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures. SAT-0 NAEP ACT SAT SAT Subject Tests Lesson 2-2 Conditional Statements Identify, write, and analyze the truth value of conditional statements. Write the inverse, converse, and contrapositive of a conditional statement. SAT-0 NAEP ACT SAT SAT Subject Tests Lesson 2-3 Using Deductive Reasoning to Verify Conjectures Apply the Law of Detachment and the Law of Syllogism in logical reasoning. SAT-0 NAEP ACT SAT SAT Subject Tests 2-3 Geometry Lab Solve Logic Puzzles Use tables to solve logic puzzles. Use networks to solve logic puzzles. SAT-0 NAEP ACT SAT SAT Subject Tests Lesson 2-4 Biconditional Statements and Definitions Write and analyze biconditional statements. SAT-0 NAEP ACT SAT SAT Subject Tests Geometry Lab Activities 2-2 Geometry Lab Geometry Lab Activities 2-3 Lab Recording Sheet Geometry Lab Activities 2-4 Geometry Lab Optional toothpicks, science textbook, magazine Optional magazine or newspaper advertisements Optional globe Optional dictionary, reversible vest or jacket MK = Manipulatives Kit

6 Section Overview Inductive Reasoning Lesson 2- Scientists use inductive reasoning when they form hypotheses to test by experiment. Inductive reasoning is used to make conjectures and continue patterns. By observing the triangles, you can make a conjecture about the pattern. Specific observation Generalized conclusion A generalized conclusion is a conjecture. To disprove a conjecture, you need only one counterexample. Conjecture: The color alternates between red and blue, and the triangle rotates 90 clockwise each time. Based on the conjecture, the next triangle in the pattern is the following: Conditionals and Deductive Reasoning Lessons 2-2, 2-3 Deductive reasoning is the basis for proof in mathematics. Lawyers use deductive reasoning when presenting cases in court. Deductive reasoning is the process of using logic to draw conclusions. A conditional statement is an if-then statement. It has a hypothesis and a conclusion. If p, then q. p q Conditional: p q Converse: q p Inverse: ~p ~q Contrapositive: ~q ~p Logically equivalent Law of Detachment If p q is a true statement and p is true, then q is true. Law of Syllogism If p q and q r are true statements, then p r is a true statement. Biconditionals and Definitions Lesson 2-4 Definitions must be precise in order for people to communicate effectively. A biconditional statement is an if-and-only-if statement. p if and only if q. p q This means both p q and q p. Biconditionals are used to write precise definitions.

7 2- Organizer Pacing: Traditional 2 day Block 4 day Objectives: Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures. Online Edition Tutorial Videos Countdown to Testing Week 3 Objectives Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures. Vocabulary inductive reasoning conjecture counterexample 2- Using Inductive Reasoning to Make Conjectures Who uses this? Biologists use inductive reasoning to develop theories about migration patterns. Biologists studying the migration patterns of California gray whales developed two theories about the whales route across Monterey Bay. The whales either swam directly across the bay or followed the shoreline. Warm Up Complete each sentence..? points are points that lie on the same line. Collinear 2.? points are points that lie in the same plane. Coplanar 3. The sum of the measures of two? angles is 90. complementary Also available on transparency ome patterns have more than one orrect rule. For example, the pattern, 2, 4, can be extended with 8 by multiplying each term by 2) or 7 by adding consecutive numbers to ach term). EXAMPLE Identifying a Pattern Find the next item in each pattern. A Monday, Wednesday, Friday, Alternating days of the week make up the pattern. The next day is Sunday. B 3, 6, 9, 2, 5, Multiples of 3 make up the pattern. The next multiple is 8. C,,, In this pattern, the figure rotates 45 clockwise each time. The next figure is.. Find the next item in the pattern 0.4, 0.04, 0.004, When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture. EXAMPLE 2 Making a Conjecture Complete each conjecture. A The product of an even number and an odd number is?. List some examples and look for a pattern. (2)(3) = 6 (2)(5) = 0 (4)(3) = 2 (4)(5) = 20 The product of an even number and an odd number is even. 74 Chapter 2 Geometric Reasoning Introduce Motivate Ask students to describe a science experiment in which they collected data and formed a hypothesis based on their data. Explain that this kind of reasoning, in which generalizations are based on examples, is called inductive reasoning. KEYWORD: MG7 Resources Explorations and answers are provided in the Chapter 2 Resource Book.

8 Complete each conjecture. B The number of segments formed by n collinear points is?. Draw a segment. Mark points on the segment, and count the number of individual segments formed. Be sure to include overlapping segments. Points Segments = = = 0 The number of segments formed by n collinear points is the sum of the whole numbers less than n. 2. Complete the conjecture: The product of two odd numbers is?. odd EXAMPLE 3 Biology Application To learn about the migration behavior of California gray whales, biologists observed whales along two routes. For seven days they counted the numbers of whales seen along each route. Make a conjecture based on the data. Numbers of Whales Each Day Direct Route Shore Route More whales were seen along the shore route each day. The data supports the conjecture that most California gray whales migrate along the shoreline. Female whales are longer than male whales. 3. Make a conjecture about the lengths of male and female whales based on the data. Average Whale Lengths Length of Female (ft) Length of Male (ft) To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. A counterexample can be a drawing, a statement, or a number. Inductive Reasoning. Look for a pattern 2. Make a conjecture. 3. Prove the conjecture or find a counterexample. 2- Using Inductive Reasoning to Make Conjectures 75 When testing conjectures about numbers, students may fail to find a counterexample because they try only the same type of number. Remind students to try various types of numbers, such as whole numbers and fractions, positive numbers, negative numbers, and zero. Example Find the next item in each pattern. A. January, March, May,... July B. 7, 4, 2, 28, C. Additional Examples Example 2 Complete each conjecture. A. The sum of two positive numbers is?. positive B. The number of lines formed by 4 points, no three of which are collinear, is?. 6 Example 3 The cloud of water leaving a whale s blowhole when it exhales is called its blow. A biologist observed blue-whale blows of 25 ft, 29 ft, 27 ft, and 24 ft. Another biologist recorded humpback-whale blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a conjecture based on the data. The height of a blue whale s blow is greater than a humpback whale s. Also available on transparency 2 Teach Guided Instruction Many of the examples and exercises in this lesson use the vocabulary learned in Chapter. Review terms such as collinear and coplanar, the different types of angles, linear pairs of angles, and complementary and supplementary angles. Science You may want to use a science textbook so you can review the steps of the scientific method. Relate the lesson to students experiences doing experiments in their science classes. Through Cooperative Learning Have students work in small groups. The first student writes a number or draws a shape. The next student writes or draws a second item, beginning a pattern. Have them continue until each student has contributed to the pattern. Then ask the first student to describe a rule for the pattern. Have the groups repeat this activity until each student has gone first. INTERVENTION Questioning Strategies EXAMPLE Do you have to find a general rule to find the next item in a pattern? EXAMPLE 2 How many examples do you need to look at to complete a conjecture? Explain. EXAMPLE 3 How do you read the data to find what conjecture is supported?

9 Additional Examples Example 4 Show that each conjecture is false by finding a counterexample. Possible answers: A. For every integer n, n 3 is positive. n =-3 B. Two complementary angles are not congruent. 45 and 45 C. Based on the data in Example 4C, the monthly high temperature in Abilene is never below 90 F for two months in a row. Jan Feb Also available on transparency NTERVENTION uestioning Strategies EXAMPLE 4 How do you know which numbers to test when trying to find a counterexample for an algebraic conjecture? EXAMPLE 4 Finding a Counterexample 4b. Possible answer: Show that each conjecture is false by finding a counterexample. _ A For all positive numbers n, n n. Pick positive values for n and substitute them into the equation to see if the conjecture holds. Let n =. Since _ n = and, the conjecture holds. Let n = 2. Since _ n = _ 2 and _ 2, the conjecture holds. 2 Let n = _ 2. Since _ n = _ = 2 and 2 _, the conjecture is false. _ 2 2 n = _ is a counterexample. 2 B For any three points in a plane, there are three different lines that contain two of the points. Draw three collinear points. If the three points are collinear, the conjecture is false. C The temperature in Abilene, Texas, never exceeds 00 F during the spring months (March, April, and May). Monthly High Temperatures ( F) in Abilene, Texas Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec The temperature in May was 07 F, so the conjecture is false. Show that each conjecture is false by finding a counterexample. _ 4a. For any real number x, x 2 x. Possible answer: x = 2 4b. Supplementary angles are adjacent. 4c. The radius of every planet in the solar system is less than 50,000 km. Jupiter or Saturn Planets Diameters (km) Mercury Venus Earth Mars Jupiter Saturn Uranus Nepture Pluto ,00 2, ,000 2,000 5,00 49, THINK AND DISCUSS. Can you prove a conjecture by giving one example in which the conjecture is true? Explain your reasoning. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the steps of the inductive reasoning process. 76 Chapter 2 Geometric Reasoning 3 Answers to Think and Discuss Close. No; possible answer: a conjecture ummarize eview with students the three steps of the nductive reasoning process: Look for a pattern. Make a conjecture. Prove the conjecture or find a counterexample. xplain to students that they will learn to rove a conjecture later in the chapter. and INTERVENTION Diagnose Before the Lesson 2- Warm Up, TE p. 74 Monitor During the Lesson Check It Out! Exercises, SE pp Questioning Strategies, TE pp Assess After the Lesson 2- Lesson Quiz, TE p. 79 Alternative Assessment, TE p. 79 cannot be proven true just by giving examples, no matter how many. 2. See p. A2.

10 SEE EXAMPLE p. 74 SEE EXAMPLE 2 p. 74 SEE EXAMPLE 3 p. 75 SEE EXAMPLE 4 p The 3 pts. are collinear. Independent Practice For See Exercises Example Extra Practice Skills Practice p. S6 Application Practice p. S Each term is the square of the previous term; 256, 65, Possible answer: each term is the previous term _ multiplied by 2 ; 6, The terms are multiples of 3 with alternating signs; -5, 8. Exercises KEYWORD: MG7 2-. Possible answer: A conjecture is based on observation and is not true GUIDED PRACTICE until proven true in every case.. Vocabulary Explain why a conjecture may be true or false. KEYWORD: MG7 Parent Find the next item in each pattern. 2. March, May, July, 3. _ 3, 2_, 3_, September 6 Complete each conjecture. 5. The product of two even numbers is?. even 6. A rule in terms of n for the sum of the first n odd positive integers is?. 2 n 7. Biology A laboratory culture contains 50 bacteria. After twenty minutes, the culture contains 300 bacteria. After one hour, the culture contains 200 bacteria. Make a conjecture about the rate at which the bacteria increases. The number of bacteria doubles every 20 minutes. Show that each conjecture is false by finding a counterexample. 8. Kennedy is the youngest U.S. president to be inaugurated. Roosevelt was inaugurated at age Three points on a plane always form a triangle. 0. For any real number x, if x 2, then x. Possible answer: x =-3 PRACTICE AND PROBLEM SOLVING President Washington T. Roosevelt Truman Kennedy Clinton Age at Inauguration Find the next item in each pattern.. 8 A.M., A.M., 2 P.M., 2. 75, 64, 53, 3.,,, 5 P.M. 42 Complete each conjecture. 4. A rule in terms of n for the sum of the first n even positive integers is?. n(n + ) 5. The number of nonoverlapping segments formed by n collinear points is?. n - 6. Industrial Arts About 5% of the students at Lincoln High School usually participate in the robotics competition. There are 526 students in the school this year. Make a conjecture about the number of students who will participate in the robotics competition this year. About 26 students will participate. Show that each conjecture is false by finding a counterexample. 7. If - y > 0, then 0 < y <. Possible answer: y =- 8. For any real number x, x 3 x 2. Possible answer: x =- 9. Every pair of supplementary angles includes one obtuse angle. m = m 2 = 90 Make a conjecture about each pattern. Write the next two items , 4, 6, 2. _ 2, _, _, 22. 3, 6, 9, 2, Draw a square of dots. Make a conjecture about the number of dots needed to increase the size of the square from n n to (n + ) (n + ). 2n + 4_ Exercises Assignment Guide Assign Guided Practice exercises as necessary. If you finished Examples 2 Basic 5, 20 22, 3 33 Average 5, 20 22, 28 33, 4 Advanced 5, 20 23, 28 33, 4 43 If you finished Examples 4 Basic 27, 3 33, 36 39, Average 22, 24 29, 3, 32, 34 40, Advanced 2, 4, 6, 8, Homework Quick Check Quickly check key concepts. Exercises: 2, 4, 6, 8, 24, 26, 32 Communicating Math For Exercises 3, have students describe each pattern in words. 2- Using Inductive Reasoning to Make Conjectures 77 KEYWORD: MG7 Resources

11 34, some students may have trouble understandng the information given in the text nd the table. Explain the concept of turnaround date and ow this relates to the umbers in the table. ENGLISH LANGUAGE LEARNERS Exercise 36 involves interpreting text from Alice s Adventures in onderland and translating these ords into a mathematical pattern. his exercise prepares students for he Multi-Step Test Prep on page 02. nswers 26. Possible answer: = 0. 09, = 0. 2 = 0. 8, 27, ; the fraction pattern is multiples of, and the decimal pattern is repeating multiples of = 3 + 3; 8 = 5 + 3; 0 = 5 + 5; 2 = 7 + 5; 4 = , 55, 89; each term is the sum of the 2 previous terms. 32. The middle number is the mean of the other 2 numbers. 35. Possible answer: Even numbers are divisible by 2, but odd numbers are not. So the conjecture, while true for even numbers, does not necessarily hold true for all numbers. 2- PRACTICE A 2- PRACTICE C 2- PRACTICE B, Math History Goldbach first stated his conjecture in a letter to Leonhard Euler in 742. Euler, a Swiss mathematician who published over 800 papers, replied, I consider [the conjecture] a theorem which is quite true, although I cannot demonstrate it. 34. Feb. 9; possible answer: the weather or the whales health 78 Chapter 2 Geometric Reasoning Determine if each conjecture is true. If not, write or draw a counterexample. 24. Points X, Y, and Z are coplanar. T 25. If n is an integer, then n is positive. F; possible answer: n = In a triangle with one right angle, two of the sides are congruent. F 27. If BD bisects ABC, then m ABD = m CBD. T 28. Estimation The Westside High School Day Money Raised ($) band is selling coupon books to raise money for a trip. The table shows the amount of money raised for the first four days of the sale. If the pattern continues, estimate the amount of money raised during the sixth day. about $ Write each fraction in the pattern _, 2_, 3_, as a repeating decimal. Then write a description of the fraction pattern and the resulting decimal pattern. 30. Math History Remember that a prime number is a whole number greater than that has exactly two factors, itself and. Goldbach s conjecture states that every even number greater than 2 can be written as the sum of two primes. For example, 4 = Write the next five even numbers as the sum of two primes. 3. The pattern,, 2, 3, 5, 8, 3, 2, is known as the Fibonacci sequence. Find the next three terms in the sequence and write a conjecture for the pattern. 32. Look at a monthly calendar and pick any three squares in a row across, down, or diagonal. Make a conjecture about the number in the middle. 33. Make a conjecture about the value of 2n - when n is an integer. odd 34. Critical Thinking The turnaround date for migrating gray whales occurs when the number of northbound whales exceeds the number of southbound whales. Make a conjecture about the turnaround date, based on the table below. What factors might affect the validity of your conjecture in the future? Migration Direction of Gray Whales Feb. 6 Feb. 7 Feb. 8 Feb. 9 Feb. 20 Feb. 2 Feb. 22 Southbound Northbound Write About It Explain why a true conjecture about even numbers does not necessarily hold for all numbers. Give an example to support your answer. 36. This problem will prepare you for the Multi-Step Test Prep on page 02. a. For how many hours did the Mock Turtle do lessons on the third day? 8 b. On what day did the Mock Turtle do hour of lessons? tenth READING STRATEGIES (4, 4) (3, 3) (2, 2) (, ) And how many hours a day did you do lessons? said Alice, in a hurry to change the subject. Ten hours the first day, said the Mock Turtle: nine the next, and so on RETEACH The Granger Collection, New York

12 52. (-, ), (0, 3), and (4, 2) 53. (3, -2), (4, 0), and (8, -) 37. Which of the following conjectures is false? If x is odd, then x + is even. The sum of two odd numbers is even. The difference of two even numbers is positive. If x is positive, then x is negative. 38. A student conjectures that if x is a prime number, then x + is not prime. Which of the following is a counterexample? x = x = 6 x = 3 x = The class of 2004 holds a reunion each year. In 2005, 87.5% of the 20 graduates attended. In 2006, 90 students went, and in 2007, 75 students went. About how many students do you predict will go to the reunion in 200? CHALLENGE AND EXTEND 40. Multi-Step Make a table of values for the rule x 2 + x + when x is an integer from to 8. Make a conjecture about the type of number generated by the rule. Continue your table. What value of x generates a counterexample? 4. Political Science Presidential elections are held every four years. U.S. senators are elected to 6-year terms, but only of the Senate is up for election every two 3 years. If of the Senate is elected during a presidential election year, how many 3 years must pass before these same senate seats are up for election during another presidential election year? 2 years 42. Physical Fitness Rob is training for the President s Challenge physical fitness program. During his first week of training, Rob does 5 sit-ups each day. He will add 20 sit-ups to his daily routine each week. His goal is to reach 50 sit-ups per day. a. Make a table of the number of sit-ups Rob does each week from week through week 0. b. During which week will Rob reach his goal? c. Write a conjecture for the number of sit-ups Rob does during week n. 43. Construction Draw AB. Then construct point C so that it is not on AB and is the same distance from A and B. Construct AC and BC. Compare m CAB and m CBA and compare AC and CB. Make a conjecture. SPIRAL REVIEW Determine if the given point is a solution to y = 3x - 5. (Previous course) 44. (, 8) no 45. (-2, -) yes 46. (3, 4) yes 47. (-3.5, 0.5) no Find the perimeter or circumference of each of the following. Leave answers in terms of x. (Lesson -5) 48. a square whose area is x 2 4x 49. a rectangle with dimensions x and 4x a triangle with side lengths of x a circle whose area is 9π x 2 0x - 6 3x + 6 6πx A triangle has vertices (-, -), (0, ), and (4, 0). Find the coordinates for the vertices of the image of the triangle after each transformation. (Lesson -7) 52. (x, y) (x, y + 2) 53. (x, y) (x + 4, y - ) 2- Using Inductive Reasoning to Make Conjectures 79 If students do not recognize the pattern in Exercise 3, give them the hint that for each term they should look at the two previous terms. In Exercise 38, point out to students that the hypothesis if x is a prime number eliminates choice G. They can use the given values of x in choices F, H, and J to determine whether x + is prime. Answers 40, See p. A. Journal Have students write a conjecture about numbers and then use examples to determine whether it is true. Have students find a pattern in a magazine and describe it in words. Have students make up one conjecture about numbers that is true and one that is false, giving a counterexample to disprove it. 2- Find the next item in each pattern.. 0.7, 0.07, 0.007, PROBLEM SOLVING 2- CHALLENGE Figure Figure 2 Figure 3 Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. T 4. Every prime number is odd. F; 2 5. Two supplementary angles are not congruent. F; 90 and The square of an odd integer is odd. T Also available on transparency

13 Number Theory See Skills Bank pages S53 and S8 Pacing: Traditional 2 day Block 4 day Objective: Apply reasoning skills to drawing Venn diagrams of number sets. Organizer Online Edition each emember tudents review sets of numbers. NTERVENTION For addiional review and practice on Venn iagrams, see Skills Bank page S8. or practice with classifying numers, see Skills Bank page S53. Communicating Math Show students diagrams of two concentric circles, two verlapping circles, and two circles hat do not intersect. Have students escribe sets of everyday things, uch as animals, for each Venn iagram. lose ssess ave students draw a Venn diagram hat shows the relationship between ntegers and rational numbers. Number Theory See Skills Bank pages S53 and S8 Venn Diagrams Recall that in a Venn diagram, ovals are used to represent each set. The ovals can overlap if the sets share common elements. The real number system contains an infinite number of subsets. The following chart shows some of them. Other examples of subsets are even numbers, multiples of 3, and numbers less than 6. Set Description Examples Natural numbers The counting numbers, 2, 3, 4, 5, Whole numbers The set of natural numbers and 0 0,, 2, 3, 4, Integers The set of whole numbers and their opposites, -2, -, 0,, 2, Rational numbers Irrational numbers Example The set of numbers that can be written as a ratio of integers The set of numbers that cannot be written as a ratio of integers Draw a Venn diagram to show the relationship between the set of even numbers and the set of natural numbers. The set of even numbers includes all numbers that are divisible by 2. This includes natural numbers such as 2, 4, and 6. But even numbers such as 4 and 0 are not natural numbers. So the set of even numbers includes some, but not all, elements in the set of natural numbers. Similarly, the set of natural numbers includes some, but not all, even numbers. - 3_, 5, -2, 0.5, 0 4 π, 0, Draw a rectangle to represent all real numbers. Draw overlapping ovals to represent the sets of even and natural numbers. You may write individual elements in each region. Try This Draw a Venn diagram to show the relationship between the given sets.. natural numbers, 2. odd numbers, 3. irrational numbers, whole numbers whole numbers integers 80 Chapter 2 Geometric Reasoning Answers to Try This KEYWORD: MG7 Resources

14 2-2 Conditional Statements Objectives Identify, write, and analyze the truth value of conditional statements. Write the inverse, converse, and contrapositive of a conditional statement. Vocabulary conditional statement hypothesis conclusion truth value negation converse inverse contrapositive logically equivalent statements Why learn this? To identify a species of butterfly, you must know what characteristics one butterfly species has that another does not. It is thought that the viceroy butterfly mimics the bad-tasting monarch butterfly to avoid being eaten by birds. By comparing the appearance of the two butterfly species, you can make the following conjecture: If a butterfly has a curved black line on its hind wing, then it is a viceroy. Conditional Statements DEFINITION SYMBOLS VENN DIAGRAM A conditional statement is a statement that can be written in the form if p, then q. The hypothesis is the part p of a conditional statement following the word if. The conclusion is the part q of a conditional statement following the word then. p q By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion. EXAMPLE Identifying the Parts of a Conditional Statement Identify the hypothesis and conclusion of each conditional. If p, then q can also be written as if p, q, q, if p, p implies q, and p only if q. A If a butterfly has a curved black line on its hind wing, then it is a viceroy. Hypothesis: A butterfly has a curved black line on its hind wing. Conclusion: The butterfly is a Viceroy. B A number is an integer if it is a natural number. Hypothesis: A number is a natural number. Conclusion: The number is an integer.. Identify the hypothesis and conclusion of the statement A number is divisible by 3 if it is divisible by 6. Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence s hypothesis and conclusion by figuring out which part of the statement depends on the other. 2-2 Organizer Pacing: Traditional day Block 2 day Objectives: Identify, write, and analyze the truth value of conditional statements. Write the inverse, converse, and contrapositive of a conditional statement. Geometry Lab In Geometry Lab Activities Online Edition Tutorial Videos, Interactivity Countdown to Testing Week 3 Warm Up Determine if each statement is true or false.. The measure of an obtuse angle is less than 90. F 2. All perfect-square numbers are positive. T 3. Every prime number is odd. F 4. Any three points are coplanar. T Also available on transparency Teacher: Which month has 28 days? Student: All of them! 2-2 Conditional Statements 8 Introduce Motivate Have students bring in advertisements that promise certain results if you buy a particular product. Ask students to restate the advertising claims in the form If, then. Explain to students that statements of this form are called conditional statements. Explorations and answers are provided in the Chapter 2 Resource Book. KEYWORD: MG7 Resources