Lesson 2-1

Inductive Reasoning and Conjecture

5-Minute Check on Chapter 15-Minute Check on Chapter 15-Minute Check on Chapter 15-Minute Check on Chapter 1 Transparency 2-1

1. Find the value of x if R is between Q and T, QR = 3x + 5,RT = 4x – 9, and QT = 17.

2. Find the distance between A(–3, 7) and B(1, 4).

3. Find mC if C and D are supplementary, mC = 3y – 5, and mD = 8y + 20.

__4. Find SR if R is the midpoint of SU.

5. Find n if WX bisects VWY.

___6. Find the coordinates of the midpoint of MN if

M(3, 6) and N(9, -4).Standardized Test Practice:

A CB D(12, 2) (6, 5) (1, 6) (6, 1)

5-Minute Check on Chapter 15-Minute Check on Chapter 15-Minute Check on Chapter 15-Minute Check on Chapter 1 Transparency 2-1

1. Find the value of x if R is between Q and T, QR = 3x + 5,RT = 4x – 9, and QT = 17. 3

2. Find the distance between A(–3, 7) and B(1, 4). 5

3. Find mC if C and D are supplementary, mC = 3y – 5, and mD = 8y + 20. 40

__4. Find SR if R is the midpoint of SU. 22

5. Find n if WX bisects VWY. 10

___6. Find the coordinates of the midpoint of MN if

M(3, 6) and N(9, -4).Standardized Test Practice:

A CB(12, 2) (6, 5) (1, 6) (6, 1)D

Objectives

• Make conjectures based on inductive reasoning

• Find counterexamples

Vocabulary

• Conjecture – an educated guess based on known information

• Inductive reasoning – reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction

• Counterexample – a false example

See the pattern in the following series:

a. 1, 4, 7, 10, 13, ______ b. 1, -2, 4, -8, 16, ____

Find the pattern in the following series:

a. 2, 4, 8, 16, 32, ____ b. 1, 4, 9, 16, 25, _____

c. 3, 1, 5, 3, 7, 5, ____ d. 1, 2, 3, 5, 7, 11, _____

e. 10, 12, 6, 8, 2, 4, ____ f. 5, 3, 10, 6, 15, 9, ____

+3 +3+3+3 •(-2) •(-2)

-3216

Series

64 36

9 13

-2 20

Make a conjecture about the next number based on the pattern.

2, 4, 12, 48, 240

Answer: 1440

Find a pattern:

2 4 12 48 240

×2

The numbers are multiplied by 2, 3, 4, and 5.

Conjecture: The next number will be multiplied by 6. So, it will be or 1440.

×3 ×4 ×5

Example 1

Make a conjecture about the next number based on the pattern.

Answer: The next number will be

Example 2

UNEMPLOYMENT Based on the table showing unemployment rates for various cities in Kansas, find a counterexample for the following statement: “The unemployment rate is highest in the cities with the most people.”

County Civilian Labor Force Rate

Shawnee 90,254 3.1%

Jefferson 9,937 3.0%

Jackson 8,915 2.8%

Douglas 55,730 3.2%

Osage 10,182 4.0%

Wabaunsee 3,575 3.0%

Pottawatomie 11,025 2.1%

So

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Mar

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Ser

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Example 3

Examine the data in the table. Find two cities such that the population of the first is greater than the population of the second while the unemployment rate of the first is less than the unemployment rate of the second. Shawnee has a greater population than Osage while Shawnee has a lower unemployment rate than Osage.

Answer: Osage has only 10,182 people on its civilian labor force, and it has a higher rate of unemployment than Shawnee, which has 90,254 people on its civilian labor force.

Example 3 cont

DRIVING The table below shows selected states, the 2000 population of each state, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, find a counterexample for the following statement:

“The greater the population of a state, the lower the number of drivers per 1000 residents.”

State Population Licensed Drivers per 1000

Alabama 4,447,100 792

California 33,871,648 627

Texas 20,851,820 646

Vermont 608,827 831

West Virginia 1,808,344 745

Wisconsin 5,363,675 703 So

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Alm

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Boo

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003

Answer: Alabama has a greater population than West Virginia, and it has more drivers per 1000 than West Virginia.

Example 4

Summary & Homework

• Summary:– Conjectures are based on observations and

patterns – Counterexamples can be used to show that a

conjecture is false

• Homework: – pgs. 64-5: 4,5,11,13,15,17,21,23,29