Transcript Lesson 1 Contents - Headlee's Math Mansion

Lesson 2-1
Inductive Reasoning and
Conjecture
Transparency 2-1
5-Minute Check on Chapter 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 mC if C and D are supplementary, mC = 3y – 5,
and mD = 8y + 20.
__
4. Find SR if R is the midpoint of SU.
5. Find n if WX bisects VWY.
6.
Standardized Test Practice:
___
Find the coordinates of the midpoint of MN if
M(3, 6) and N(9, -4).
A
B (6, 5)
(12, 2)
C
(1, 6)
D
(6, 1)
Transparency 2-1
5-Minute Check on Chapter 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 mC if C and D are supplementary, mC = 3y – 5,
and mD = 8y + 20. 40
__
4. Find SR if R is the midpoint of SU. 22
5. Find n if WX bisects VWY. 10
6.
Standardized Test Practice:
___
Find the coordinates of the midpoint of MN if
M(3, 6) and N(9, -4).
A
B (6, 5)
(12, 2)
C
(1, 6)
D
(6, 1)
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
Series
See the pattern in the following series:
16
a. 1, 4, 7, 10, 13, ______
-32
b. 1, -2, 4, -8, 16, ____
•(-2)
+3 +3+3+3
•(-2)
Find the pattern in the following series:
a.
64
2, 4, 8, 16, 32, ____
36
b. 1, 4, 9, 16, 25, _____
c.
9
3, 1, 5, 3, 7, 5, ____
13
d. 1, 2, 3, 5, 7, 11, _____
e.
-2
10, 12, 6, 8, 2, 4, ____
20
f. 5, 3, 10, 6, 15, 9, ____
Example 1
Make a conjecture about the next number based on
the pattern.
2, 4, 12, 48, 240
Find a pattern:
2
4
×2
12
×3
48
×4
240
×5
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.
Answer: 1440
Example 2
Make a conjecture about the next number based on
the pattern.
Answer: The next number will be
Example 3
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%
3,575
3.0%
11,025
2.1%
Wabaunsee
Pottawatomie
Source: Labor Market Information Services–
Kansas Department of Human Resources
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.”
Example 3 cont
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 4
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:
State
Population
Licensed Drivers
per 1000
Alabama
4,447,100
792
California
33,871,648
627
Texas
20,851,820
646
608,827
831
West Virginia
1,808,344
745
Wisconsin
5,363,675
703
Vermont
Source: The World Almanac and Book of Facts 2003
“The greater the population of a state, the lower the number of
drivers per 1000 residents.”
Answer: Alabama has a greater population than West Virginia, and it
has more drivers per 1000 than West Virginia.
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