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HW-2.1 Practice B & pgs. 84-85 (3-11, 13-21, 24-36)
2.1-2.5 Quiz TUESDAY 10-15-13
www.westex.org HS, Teacher Websites
10-8-13
Warm up—Geometry H
Find the next item in the pattern.
1. 1, 5, 9, 13, …
Determine if each statement is true or false.
2. The measure of an obtuse angle is less than 90°.
3. All perfect-square numbers are positive.
4. Every prime number is odd.
5. Any three noncollinear points have exactly one
plane in common.
HW-2.1 Practice B & pgs. 84-85 (3-11, 13-21, 24-36)
2.1-2.5 Quiz TUESDAY 10-15-13
www.westex.org
HS, Teacher Websites
Name _________________________
Geometry H
2-1 Use Inductive Reasoning to make Conjectures
GOAL:
I will be able to:
1. use inductive reasoning to find patterns and make conjectures.
2. find counterexamples to disprove conjectures.
Date ________
Example 1: Find a Pattern
Find the next item in the pattern.
January, March, May, ...
You Try:
Find the next item in the pattern.
1. 7, 14, 21, 28, …
2.
3. 0.4, 0.04, 0.004, …
When several examples form a pattern and you assume the pattern will continue, you are
applying _______________ ____________. 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 __________________.
Example 2: Making a Conjecture
The sum of two positive numbers is ? .
You Try:
Complete the conjecture.
1. The number of lines formed by 4 points, no three of which are collinear, is ? .
2. The product of two odd numbers is ? .
Example 3: Biology Application
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 ____________________.
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.
Example 4: Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
For every integer n, n3 is positive.
You Try:
Show that the conjecture is false by finding a counterexample.
1. Two complementary angles are not congruent.
2. The monthly high temperature in Abilene is never below 90°F for two months in a row.
Monthly High Temperatures (ºF) 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
3. For any real number x, x2 ≥ x.
4. Supplementary angles are adjacent.
5. 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
Neptune
Pluto
4880
12,100
12,800
6790
143,000
121,000
51,100
49,500
2300
2-2 Conditional Statements
GOAL:
I will be able to:
1. identify, write and analyze the truth value of conditional statements.
By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and
conclusion.
Example 1: Identifying the Parts of a Conditional Statement
Identify the hypothesis and conclusion of each conditional statement.
A.
If today is Thanksgiving Day, then today is Thursday.
B. A number is a rational number if it is an integer.
You Try:
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.
Example 2: Writing a Conditional Statement
Write a conditional statement from the following:
An obtuse triangle has exactly one obtuse angle.
You Try:
1. Write a conditional statement from the following:
2. Write a conditional statement from the sentence “Two angles that are complementary are
acute.”
A conditional statement has a ____________________ of either true (T) or false (F). It is
false only when the hypothesis is true and the conclusion is false.
To show that a conditional statement is false, you need to find only ___________________
where the hypothesis is true and the conclusion is false.
Example 3: Analyzing the Truth Value of a Conditional Statement
Determine if the conditional is true. If false, give a counterexample.
If this month is August, then next month is September.
You Try:
Determine if the conditional is true. If false, give a counterexample.
1. If two angles are acute, then they are congruent.
2. If an even number greater than 2 is prime, then 5 + 4 = 8.
3. Determine if the conditional “If a number is odd, then it is divisible by 3” is true. If false,
give a counterexample.
Remember!
If the hypothesis is false, the conditional statement is
true, regardless of the truth value of the conclusion.
EXIT TICKET
Name _______________________ 10-8-13
How many counterexamples are necessary to show a conjecture is false? Explain why.
What is the result of the entire conditional statement if the hypothesis is false? What is the
one way for a conditional statement to be false?
EXIT TICKET
Name _______________________ 10-8-13
How many counterexamples are necessary to show a conjecture is false? Explain why.
What is the result of the entire conditional statement if the hypothesis is false? What is the
one way for a conditional statement to be false?
EXIT TICKET
Name _______________________ 10-8-13
How many counterexamples are necessary to show a conjecture is false? Explain why.
What is the result of the entire conditional statement if the hypothesis is false? What is the
one way for a conditional statement to be false?