70 Chapter 2 2A Inductive and Deductive Reasoning 2-1 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 Geometric Reasoning KEYWORD: MG7 ChProj Winning Strategies Mathematical reasoning is not just for geometry. It also gives you an edge when you play chess and other strategy games. 70 Ch 2 S ECTION 2A Inductive and Deductive Reasoning On page 102, stu- dents use inductive and deductive rea- soning to decipher the wordplay in Lewis Carroll’s Alice’s Adventures in Wonderland. Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 78, 85, 92, and 100. S ECTION 2B Mathematical Proof On page 126, students use math- ematical proof to investigate the geometric properties of intersecting highways. Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 109, 115, and 124. About the Project In the Chapter Project, students use logical reasoning to develop strategies for a paper- and-pencil game and to solve a mathemati- cal puzzle. Winning Strategies Project Resources All project resources for teachers and stu- dents are provided online. KEYWORD: MG7 ProjectTS
Transcript
70 Chapter 2
2A Inductive and Deductive Reasoning
2-1 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
Geometric Reasoning
KEYWORD: MG7 ChProj
Winning StrategiesMathematical reasoning is not just for geometry. It also gives you an edge when you play chess and other strategy games.
70 Ch 2
S E C T I O N 2AInductive and Deductive Reasoning
On page 102, stu-dents use inductive and deductive rea-soning to decipher
the wordplay in Lewis Carroll’s Alice’s Adventures in Wonderland.
Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 78, 85, 92, and 100.
S E C T I O N 2BMathematical Proof
On page 126, students use math-ematical proof to investigate the
geometric properties of intersecting highways.
Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 109, 115, and 124.
About the ProjectIn the Chapter Project, students use logical reasoning to develop strategies for a paper-and-pencil game and to solve a mathemati-cal puzzle.
Winning Strategies
Project ResourcesAll project resources for teachers and stu-dents are provided online.
KEYWORD: MG7 ProjectTS
Geometric Reasoning 71
Angle RelationshipsSelect the best description for each labeled angle pair.
6. 7. 8.
linear pair or adjacent angles or supplementary angles orvertical angles vertical angles complementary angles
Classify Real NumbersTell if each number is a natural number, a whole number, an integer, or a rational number. Give all the names that apply.
9. 6 10. –0.8 11. –3
12. 5.2 13. 3_8
14. 0
Points, Lines, and PlanesName each of the following.
15. a point
16. a line
17. a ray
18. a segment
19. a plane
Solve One-Step EquationsSolve.
20. 8 + x = 5 21. 6y = -12 22. 9 = 6s
23. p - 7 = 9 24. z_5
= 5 25. 8.4 = -1.2r
VocabularyMatch each term on the left with a definition on the right.
1. angle
2. line
3. midpoint
4. plane
5. segment
A. a straight path that has no thickness and extends forever
B. a figure formed by two rays with a common endpoint
C. a flat surface that has no thickness and extends forever
D. a part of a line between two points
E. names a location and has no size
F. a point that divides a segment into two congruent segments
B
A
F
C
D
lin. pair vert. � comp. �
natural, whole, integer, rational
rational integer, rational
rational whole, integer, rationalrational
B� ⎯ � BD
⎯⎯ � CA−−
CDplane F
-7
-3 -2 1.5
16 25
15–19. Possible answers:
d ? 71
NOINTERVENE Diagnose and Prescribe
ARE YOU READY? Intervention, Chapter 2
Prerequisite Skill Worksheets CD-ROM Online
Angle Relationships Skill 25 Activity 25
Diagnose andPrescribe Online
Classify Real Numbers Skill 17 Activity 17
Points, Lines, and Planes Skill 22 Activity 22
Solve One-Step Equations Skill 68 Activity 68
YESENRICH
ARE YOU READY?Enrichment, Chapter 2
WorksheetsCD-ROMOnline
OrganizerObjective: Assess students’ understanding of prerequisite skills.
Prerequisite Skills
Angle Relationships
Classify Real Numbers
Points, Lines, and Planes
Solve One-Step Equations
Assessing Prior Knowledge
INTERVENTIONDiagnose and Prescribe
Use this page to determine whether intervention is necessary or whether enrichment is appropriate.
ResourcesAre You Ready? Intervention and Enrichment Worksheets
Are You Ready? CD-ROM
Are You Ready? Online
72 Chapter 2
Previously, you • studied relationships among
points, lines, and planes.
• identified congruent segments and angles.
• examined angle relationships.
• used geometric formulas for perimeter and area.
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.
KeyVocabulary/Vocabularioconjecture conjetura
counterexample contraejemplo
deductive reasoning razonamiento deductivo
inductive reasoning razonamiento inductivo
polygon polígono
proof demostración
quadrilateral cuadrilátero
theorem teorema
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.
1. 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 Ch 2
Study Guide: Preview2
OrganizerObjective: Help students organize the new concepts they will learn in Chapter 2.
Online EditionMultilingual Glossary
Resources
Multilingual Glossary Online
KEYWORD: MG7 Glossary
Answers toVocabulary Connections
1. Possible answer: a number that is not positive, such as -3
2. Possible answer: a general con-clusion
3. Possible answer: You start with general principles to get to a specific conclusion.
4. Possible answer: Polygon might mean “a figure with many �.”
✔ Collinear points
✔ Betweenness of points
✔ Coplanar points
✔ Straight angles and lines
✔ Adjacent angles
✔ Linear pairs of angles
✔ Vertical angles
✘ Measures of segments
✘ Measures of angles
✘ Congruent segments
✘ Congruent angles
✘ Right 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.
✘ ∠CBD is acute.
✘ ∠ABD is obtuse.
✘ −− AB � −− BC
Geometric Reasoning 73
Reading Strategy: Read and Interpret a DiagramA 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.
If a diagram includes labeled information, such as an angle measure or a right angle mark, treat this information as given.
Try This
List what you can and cannot assume from each diagram.
1. 2.
2
di d i i h 73
OrganizerObjective: Help students apply strategies to understand and retain key concepts.
Online Edition
ResourcesChapter 2 Resource Book
Reading Strategies
Reading Strategy:
ENGLISHLANGUAGELEARNERS
Read and Interpret a DiagramDiscuss Students will write geo-metric 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 solv-ing 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 1. Possible answer:
Can assume: W, A, and Y are col-linear. 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. �. ∠YAXand ∠XAW are adj. �. ∠XAW and ∠WAZ form a lin. pair. ∠WAZand ∠ZAY form a lin. pair. ∠ZAYand ∠YAX form a lin. pair. ∠YAXand ∠XAW form a lin. pair. ∠XAWand ∠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. A11.
74A Ch 2
SECTION
2A
Lesson Lab Resources Materials
Lesson 2-1 Using Inductive Reasoning to Make Conjectures • Use inductive reasoning to identify patterns and make conjectures.
• Find counterexamples to disprove conjectures.
□✔ SAT-10 □✔ NAEP □ ACT □✔ SAT □ SAT Subject Tests
Optionaltoothpicks, science textbook, magazine
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-10 □✔ NAEP □ ACT □ SAT □ SAT Subject Tests
Geometry Lab Activities2-2 Geometry Lab
Optionalmagazine or newspaperadvertisements
Lesson 2-3 Using Deductive Reasoning to Verify Conjectures • Apply the Law of Detachment and the Law of Syllogism in logical
reasoning.
□✔ SAT-10 □✔ NAEP □ ACT □✔ SAT □ SAT Subject Tests
Optionalglobe
2-3 Geometry Lab Solve Logic Puzzles • Use tables to solve logic puzzles.
• Use networks to solve logic puzzles.
□ SAT-10 □ NAEP □ ACT □ SAT □ SAT Subject Tests
Geometry Lab Activities2-3 Lab Recording Sheet
Lesson 2-4 Biconditional Statements and Definitions • Write and analyze biconditional statements.
□ SAT-10 □✔ NAEP □ ACT □ SAT □ SAT Subject Tests
Geometry Lab Activities2-4 Geometry Lab
Optionaldictionary, reversible vest or jacket
MK = Manipulatives Kit
One-Minute Section Planner
Inductive and Deductive Reasoning
74B
Inductive Reasoning Lesson 2-1
Scientists use inductive reasoning when they form hypotheses to test by experiment.
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.
Biconditionals and Definitions Lesson 2-4
Definitions must be precise in order for people to communicate effectively.
Law of DetachmentIf p → q is a true statement and p is true, then q is true.
Inductive reasoning is used to make conjecturesand continue patterns.
A generalized conclusion is a conjecture.To disprove a conjecture, you need only one counterexample.
A conditional statement is an if-then statement. It has a hypothesis and a conclusion.
If p, then q.p → q
Law of SyllogismIf p → q and q → r are true statements, then p → r is a true statement.
A biconditional statement is an if-and-only-if statement.p if and only if q.
p ↔ qThis means both p → q and q → p.
Biconditionals are used to write precise definitions.
Specificobservation
Generalizedconclusion
Logically equivalent
By observing the triangles, you can make a conjecture about the pattern.
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:
74 Chapter 2 Geometric Reasoning
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.
1E X A M P L E 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, 12, 15, …Multiples of 3 make up the pattern. The next multiple is 18.
C ←, ↖, ↑, …In this pattern, the figure rotates 45° clockwise each time. The next figure is ↗.
1. 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 .
2E X A M P L E 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) = 10 (4)(3) = 12 (4)(5) = 20
The product of an even number and an odd number is even.
ObjectivesUse inductive reasoning to identify patterns and make conjectures.
Objectives: Use inductive reasoning to identify patterns and make conjectures.
Find counterexamples to disprove conjectures.
Online EditionTutorial Videos
Countdown to Testing Week 3
Warm UpComplete each sentence.
1. ? 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
Some patterns have more than one correct rule. For example, the pattern 1, 2, 4, … can be extended with 8 (by multiplying each term by 2) or 7 (by adding consecutive numbers to each term).
MotivateAsk students to describe a science experiment in which they collected data and formed a hypoth-esis based on their data. Explain that this kind of reasoning, in which generalizations are based on examples, is called inductive reasoning.
Introduce1
Explorations and answers are provided in the Chapter 2 Resource Book.KEYWORD: MG7 Resources
2- 1 Using Inductive Reasoning to Make Conjectures 75
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
2 1
3 2 + 1 = 3
4 3 + 2 + 1 = 6
5 4 + 3 + 2 + 1 = 10
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 −−− ? .
3E X A M P L E 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 1 3 0 2 1 1 0
Shore Route 7 9 5 8 8 6 7
More whales were seen along the shore route each day. The data supports the conjecture that most California gray whales migrate along the shoreline.
3. Make a conjecture about the lengths of male and female whales based on the data.
Average Whale Lengths
Length of Female (ft) 49 51 50 48 51 47
Length of Male (ft) 47 45 44 46 48 48
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
1. Look for a pattern
2. Make a conjecture.
3. Prove the conjecture or find a counterexample.
odd
Female whales are longer than male whales.
2 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 1
Find the next item in each pattern.
A. January, March, May, ... July
B. 7, 14, 21, 28, ... 35
C.
Example 2
Complete each conjecture.
A. The sum of two positive num-bers 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 con-jecture based on the data. The height of a blue whale’s blow is greater than a humpback whale’s.
Also available on transparency
Additional Examples
INTERVENTION Questioning Strategies
EXAMPLE 1
• 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 conjec-ture? Explain.
EXAMPLE 3
• How do you read the data to find what conjecture is supported?
Guided InstructionMany of the examples and exercises in this lesson use the vocabulary learned in Chapter 1. Review terms such as collinearand coplanar, the different types of angles, linear pairs of angles, and complementaryand 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 theirscience classes.
Teach2
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 con-tributed 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.
76 Chapter 2 Geometric Reasoning
4E X A M P L E Finding a Counterexample
Show that each conjecture is false by finding a counterexample.
A For all positive numbers n, 1_n ≤ n.
Pick positive values for n and substitute them into the equation to see if the conjecture holds.
Let n = 1. Since 1_n = 1 and 1 ≤ 1, the conjecture holds.
Let n = 2. Since 1_n = 1_2
and 1_2
≤ 2, the conjecture holds.
Let n = 1_2
. Since 1_n = 1_
1_2
= 2 and 2 1_
2 , the conjecture is false.
n = 1_2
is a counterexample.
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 100°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
88 89 97 99 107 109 110 107 106 103 92 89
The temperature in May was 107°F, so the conjecture is false.
Show that each conjecture is false by finding a counterexample.
4a. For any real number x, x2 ≥ x.
4b. Supplementary angles are adjacent.
4c. The radius of every planet in the solar system is less than 50,000 km.
Planets’ Diameters (km)
Mercury Venus Earth Mars Jupiter Saturn Uranus Nepture Pluto
THINK AND DISCUSS 1. 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.
Possible answer: x = 1_2
Jupiter or Saturn
4b. Possible answer:
76 Ch 2
Example 4
Show that each conjecture is false by finding a counter-example. Possible answers:
A. For every integer n, n3 is posi-tive. n = -3
B. Two complementary angles are not congruent. 45° and 45°
C. Based on the data in Example4C, the monthly high tempera-ture in Abilene is never below 90°F for two months in a row. Jan–Feb
Also available on transparency
Additional Examples
INTERVENTION Questioning Strategies
EXAMPLE 4
• How do you know which numbers to test when trying to find a counterexample for an algebraic conjecture?
Assess After the Lesson2-1 Lesson Quiz, TE p. 79
Alternative Assessment, TE p. 79
Monitor During the LessonCheck It Out! Exercises, SE pp. 74–76Questioning Strategies, TE pp. 75–76
Diagnose Before the Lesson2-1 Warm Up, TE p. 74
and INTERVENTIONSummarizeReview with students the three steps of the inductive reasoning process:
• Look for a pattern.
• Make a conjecture.
• Prove the conjecture or find a counterexample.
Explain to students that they will learn to prove a conjecture later in the chapter.
Close3 Answers to Think and Discuss 1. No; possible answer: a conjecture
cannot be proven true just by giving examples, no matter how many.
2. See p. A2.
2- 1 Using Inductive Reasoning to Make Conjectures 77
ExercisesExercisesKEYWORD: MG7 2-1
KEYWORD: MG7 Parent
GUIDED PRACTICE 1. Vocabulary Explain why a conjecture may be true or false.
SEE EXAMPLE 1 p. 74
Find the next item in each pattern.
2. March, May, July, … 3. 1_3
, 2_4
, 3_5
, … 4.
SEE EXAMPLE 2 p. 74
Complete each conjecture.
5. The product of two even numbers is −−− ? .
6. A rule in terms of n for the sum of the first n odd positive integers is −−− ? .
SEE EXAMPLE 3 p. 75
7. Biology A laboratory culture contains 150 bacteria. After twenty minutes, the culture contains 300 bacteria. After one hour, the culture contains 1200 bacteria. Make a conjecture about the rate at which the bacteria increases.
SEE EXAMPLE 4 p. 76
Show that each conjecture is false by finding a counterexample.
8. Kennedy is the youngest U.S. president to be President
Age at Inauguration
WashingtonT. RooseveltTrumanKennedyClinton
5742604346
inaugurated.
9. Three points on a plane always form a triangle.
10. For any real number x, if x2 ≥ 1, then x ≥ 1.
PRACTICE AND PROBLEM SOLVING
For See Exercises Example
11–13 1 14–15 2 16 3 17–19 4
Independent Practice Find the next item in each pattern.
14. A rule in terms of n for the sum of the first n even positive integers is −−− ? .
15. The number of nonoverlapping segments formed by n collinear points is −−− ? .
Show that each conjecture is false by finding a counterexample.
17. If 1 - y > 0, then 0 < y < 1.
18. For any real number x, x3 ≥ x2 .
19. Every pair of supplementary angles includes one obtuse angle.
Make a conjecture about each pattern. Write the next two items.
20. 2, 4, 16, … 21. 1_2
, 1_4
, 1_8
, … 22. –3, 6, –9, 12, …
23. 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 + 1) × (n + 1) .
Skills Practice p. S6Application Practice p. S29
Extra Practice 16. 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.
n 2
Possible answer: x = -3
September
4_6
Roosevelt was inaugurated at age 42.
even
5 P.M. 42
n(n + 1)
n - 1
About 26 students will participate.
Possible answer: y = -1
Possible answer: x = -1
m∠1 = m∠2 = 90°
2n + 1
The number of bacteria doubles every 20 minutes.
9. The 3 pts. are collinear.
20. Each term is the square of the previous term; 256, 65,536.21. Possible answer: each term is the previous term multiplied by 1_
2 ; 1_
16 , 1_
32.
22. The terms are multiples of 3 with alternating signs; -15, 18.
1. Possible answer: A conjecture is based on observation and is not true until proven true in every case.
2 77
ExercisesExercises
Assignment Guide
Assign Guided Practice exercises as necessary.
If you finished Examples 1–2 Basic 11–15, 20–22, 31–33 Average 11–15, 20–22, 28–33,
41 Advanced 11–15, 20–23, 28–33,
41–43
If you finished Examples 1–4 Basic 11–27, 31–33, 36–39,
Communicating Math For Exercises 11–13, have students describe each
pattern in words.
KEYWORD: MG7 Resources
“And how many hours aday did you do lessons?” said Alice, in a hurry tochange the subject.“Ten hours the first day,” said the Mock Turtle: “nine the next, and so on.”
78 Chapter 2 Geometric Reasoning
Goldbach first stated his conjecture in a letter to Leonhard Euler in 1742. 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.”
Math History
Determine if each conjecture is true. If not, write or draw a counterexample.
24. Points X, Y, and Z are coplanar.
25. If n is an integer, then –n is positive.
26. In a triangle with one right angle, two of the sides are congruent.
27. If ��� BD bisects ∠ABC, then m∠ABD = m∠CBD.
28. Estimation The Westside High School Day Money Raised ($)
1 146.25
2 195.75
3 246.25
4 295.50
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.
29. Write each fraction in the pattern 1_11
, 2_11
, 3_11
, … 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 1 that has exactly two factors, itself and 1. Goldbach’s conjecture states that every even number greater than 2 can be written as the sum of two primes. For example, 4 = 2 + 2. Write the next five even numbers as the sum of two primes.
31. The pattern 1, 1, 2, 3, 5, 8, 13, 21, … 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 - 1 when n is an integer.
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?
31. 34, 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 num-bers are not. So the conjecture, while true for even numbers, does not necessarily hold true for all numbers.
45°22.5°
11.25°
3 6 10
1 1
1
1
1 3
11
1
1
1
1 4
11
1 5
1
11
1 6
2-1 RETEACH
0
654321
1 2 3 4 5 6
(1, 1)(2, 2)
(3, 3)(4, 4)
2-1 READING STRATEGIES
,
2-1 PRACTICE C
2-1 PRACTICE B
2-1 PRACTICE A
2- 1 Using Inductive Reasoning to Make Conjectures 79
37. Which of the following conjectures is false?
If x is odd, then x + 1 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 + 1 is not prime. Which of the following is a counterexample?
x = 11 x = 6 x = 3 x = 2
39. The class of 2004 holds a reunion each year. In 2005, 87.5% of the 120 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 2010?
12 15 24 30
CHALLENGE AND EXTEND 40. Multi-Step Make a table of values for the rule x2 + x + 11 when x is an integer
from 1 to 8. Make a conjecture about the type of number generated by the rule. Continue your table. What value of x generates a counterexample?
41. Political Science Presidential elections are held every four years. U.S. senators are elected to 6-year terms, but only 1__
3 of the Senate is up for election every two
years. If 1__3 of the Senate is elected during a presidential election year, how many
years must pass before these same senate seats are up for election during another presidential election year?
42. Physical Fitness Rob is training for the President’s Challenge physical fitness program. During his first week of training, Rob does 15 sit-ups each day. He will add 20 sit-ups to his daily routine each week. His goal is to reach 150 sit-ups per day.
a. Make a table of the number of sit-ups Rob does each week from week 1 through week 10.
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 REVIEWDetermine if the given point is a solution to y = 3x - 5. (Previous course)
Find the perimeter or circumference of each of the following. Leave answers in terms of x. (Lesson 1-5)
48. a square whose area is x2 49. a rectangle with dimensions x and 4x - 3
50. a triangle with side lengths of x + 2 51. a circle whose area is 9π x2
A triangle has vertices (-1, -1) , (0, 1) , and (4, 0) . Find the coordinates for the vertices of the image of the triangle after each transformation. (Lesson 1-7)
52. (x, y) → (x, y + 2) 53. (x, y) → (x + 4, y - 1)
no yes yes
3x + 6 6πx
no
4x10x - 6
12 years
52. (-1, 1), (0, 3), and (4, 2)
53. (3, -2), (4, 0), and (8, -1)
2 79
2-1 PROBLEM SOLVING
Figure 1 Figure 2 Figure 3
2-1 CHALLENGE
If students do not recognize the pat-tern in Exercise 31, 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 + 1 is prime.
Answers40, 42–43. See p. A11.
JournalHave students write a conjecture about numbers and then use exam-ples to determine whether it is true.
Have students find a pattern in a magazine and describe it in words. Have students make up one conjec-ture about numbers that is true and one that is false, giving a counter-example to disprove it.
2-1
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 0.0007
2.
Determine if each conjecture is true. If false, give a counter-example.
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 90°
6. The square of an odd integer is odd. T
Also available on transparency
80 Chapter 2 Geometric Reasoning
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.
Whole numbers The set of natural numbers and 0 0, 1, 2, 3, 4, …
Integers The set of whole numbers and their opposites …, -2, -1, 0, 1, 2, …
Rational numbers The set of numbers that can be written as a ratio of integers
- 3_4
, 5, -2, 0.5, 0
Irrational numbers The set of numbers that cannot be written as a ratio of integers
π, √ � 10 , 8 + √ � 2
Example
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 –10 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.
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.
Objective: Apply reasoning skills to drawing Venn diagrams of number sets.
Online Edition
TeachRememberStudents review sets of numbers.
INTERVENTION For addi-tional review and practice on Venn diagrams, see Skills Bank page S81. For practice with classifying num-bers, see Skills Bank page S53.
Communicating Math Show students diagrams of two concentric circles, two
overlapping circles, and two circles that do not intersect. Have students describe sets of everyday things, such as animals, for each Venn diagram.
CloseAssessHave students draw a Venn diagram that shows the relationship between integers and rational numbers.
KEYWORD: MG7 Resources
Organizer
Answers to Try This 1.
2.
3.
2- 2 Conditional Statements 81
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.
DEFINITION SYMBOLS VENN DIAGRAM
A conditional statement is a statement that can be written in the form “if p, then q.”
p → qThe 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.
Conditional Statements
By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.
1E X A M P L E Identifying the Parts of a Conditional Statement
Identify the hypothesis and conclusion of each conditional.
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.
1. 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.
ObjectivesIdentify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.
“If p, then q” can also be written as “if p, q,” “q, if p,”“p implies q,” and “p only if q.”
2 2 81
2-2 OrganizerPacing: Traditional 1 day
Block 1__2 day
Objectives: Identify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.
Geometry LabIn Geometry Lab Activities
Online EditionTutorial Videos, Interactivity
Countdown to Testing Week 3
Warm UpDetermine if each statement is true or false.
1. 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!
MotivateHave students bring in advertisements that prom-ise 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 conditionalstatements.
Introduce1
Explorations and answers are provided in the Chapter 2 Resource Book. KEYWORD: MG7 Resources
