Transcript (a) Statement

Inductive Reasoning
• Inductive Reasoning: The process of using
examples, patterns or specific
observations to form a conclusion,
conjecture, or generalization.
Inductive Reasoning
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1. The sun rose this morning.
2. The sun rose yesterday morning.
3. The sun rose two days ago.
4. The sun rose three days ago.
What conclusion can we make?
More Examples
You go to the cafeteria every Friday for the
first 12 weeks of school. Every Friday they
have burgers on the grill. What conjecture can
you make about what you will see when you
go in there on Friday of the 13th week?
Is this necessarily true?
More Examples
The Bad News Bears have lost every game
this season going into the last week. What
would you expect to happen their last game of
the season?
More Examples
Can you think of your own example?
Deductive Reasoning
• The process of using accepted definitions
or theories to form more specific
conclusions.
Deductive Reasoning
Examples:
1. All students eat pizza.
Rachel is a student at Hiram.
Therefore, Rachel eats pizza.
2. All athletes work out in the gym.
Matt Ryan is an athlete.
Therefore, Matt Ryan works out
in the gym.
EOCT Practice
All of the following are examples of inductive reasoning except:
a) Every time John goes to a Hillgrove basketball game they win. He
concludes that they are going to win the next game he goes to.
b) If the last digit of a number is divisible by 5, then the number itself is
divisible by 5.
c) McDonald’s has 49 cent burgers every Tuesday for the first 9
weeks of the year. Next time I go there on a Tuesday, I expect to
be able to buy some for 49 cents.
d) It snows every Christmas in Boise, Idaho where Tina’s
grandparents live. She expects to see snow again this year when
she goes to see them for Christmas.
b
EOCT Practice
Goldbach’s conjecture states: every even number greater than 2 can be
written as the sum of two primes. Which sum for 30 supports his conjecture?
a) 2 + 28
b) 12 + 18
c) 15 + 15
d) 17 + 13
d
EOCT Practice
For the past 6 weeks, your aunt has asked you to watch your cousin on
Wednesday night. You conclude that you are will be asked to watch your
cousin next Wednesday. Which of the following did you use to reach this
conclusion?
a) Inductive reasoning
b) Deductive reasoning
c) Indirect proof
d) Conjecture
a
Conditional Statements
• Conditional statement: A statement usually
written in “If-Then” form that has two parts:
• Hypothesis: The condition
• Conclusion: What follows when the
condition is met
• Example 1: If it is noon in Florida, then it is 9:00 AM
in California.
• Hypothesis: “it is noon in Florida”
• Conclusion: “it is 9:00 AM in CA”
Practice 1: Identify Hypothesis /
Conclusion
• If the weather is warm, then we should go
swimming.
• Hypothesis:
______________________________
• Conclusion:
______________________________
• If the sun is shining, then we should go to
the beach.
• Hypothesis:
___________________________
• Conclusion:
__________________________
Rewrite Into If-Then Form
• Example : “Today is Monday if yesterday was
Sunday.”
• If-Then form: If yesterday was Sunday, then
today is Monday
• Practice : “An angle is acute if it measures less
than 90°.”
• If-Then form: ______________________
___________________________________
Counterexamples
• Conditional statements can be TRUE or
FALSE
• To show a statement is TRUE, you must
prove it for ALL cases
• To show a statement is FALSE, you must
show only ONE counterexample
Provide a Counterexample
Decide if the statement is true or false
• If false, provide a counterexample
• (a) Statement: If x² = 16, then x = 4.
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True / False? ____________
• Counterexample:
_____________________________
Provide a Counterexample
• (b) Statement: If it is February 14,
then it is Valentine’s Day.
True / False? ____________
• Counterexample:
_____________________________
Provide a Counterexample
• (a) Statement: If you visited New York,
then you visited the Statue of Liberty.
True / False? ________________
• Counterexample:
___________________________________
Provide a Counterexample
• (b) Statement: If a number is odd, then it
is divisible by 3.
True / False? _____________
• Counterexample:
_________________________________
Other Forms of If-Then
Statements
• Statement: If you like volleyball, then you like
to be at the beach.
• Converse: (Think of “FLIP FLOPS”) SWITCH
hypothesis & conclusion
• If you like to be at the beach, then
you like volleyball.
Other Forms of If-Then
Statements
• Statement: If you like volleyball, then you like to be
at the beach.
• Inverse: NEGATE hypothesis & conclusion
• If you do not like volleyball, then you do
not like to be at the beach.
Other Forms of If-Then
Statements
• Statement: If you like volleyball, then you like
to be at the beach.
• Contrapositive: SWITCH and NEGATE!!!
• If you do not like to be at the beach, then
you do not like volleyball.
• Write the converse, inverse, and contrapositive
• If x is odd, then 2x is even.
• Converse:
_________________________________
• Inverse:
_________________________________
• Contrapositive:
_________________________________
• If there is snow, then flowers are not in
bloom.
• Converse:
_________________________________
• Inverse:
_________________________________
• Contrapositive:
_________________________________
• If an angle measures 90°, then it is a right
angle.
• Converse:
_________________________________
• Inverse:
_________________________________
• Contrapositive:
_________________________________
Equivalent Statements
• Two conditional statements are equivalent
if they are BOTH true or BOTH false
• A conditional statement and its
contrapositive are equivalent statements
• The inverse and converse are equivalent
• The statements themselves are not the
same, but their logic is the same
Statement: If you like volleyball, then you
like to be at the beach.
Converse: If you like to be at the beach,
then you like volleyball.
Inverse: If you do not like volleyball, then
you do not like to be at the beach.
Contrapositive: If you do not like to be at
the beach, then you do not like volleyball.
Ticket out the Door!
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Closure
If you do homework, then you will succeed.
Hypothesis: _____________________________
Conclusion: _____________________________
Converse: ______________________________
Inverse: _________________________________
Contrapositive:
________________________________________
Homework
Geometry Worksheet