1. Do Now: Notice any patterns for the multiplication table for 9s?
Aim: What is Deductive and Inductive Reasoning and how can we use them to solve problems?
Do Now: Notice any patterns for the multiplication table for 9s?
1 + 8 = 9
The sum of the digits to the
right of the equality add to 9
2 + 7 = 9
3 + 6 = 9
This pattern of adding after multiplying by 9 generates a sequence called the 9 pattern.
Will it always be the case?

2. Will it always be the case?
the ‘9’ pattern
Will it always be the case?

3. Inductive Reasoning
Inductive reasoning is a type of reasoning that allows you to reach conclusions, (conjectures) based on a pattern of specific examples or past events. The more occurances observed, the better generalization can be made.
It is sometimes called the scientific method.
Ex. Find the next two terms is this sequence: 2, 4, 6, 8, And describe the pattern.
10, Add 2 to each term.
WEAKNESS - one counterexample can show a conclusion to be false.

4. Predict the next two numbers in this pattern: 4, 12, 36, 108, . .
324, 972
Candice examined five different examples and came up with this conjecture: “If any two positive numbers are multiplied, their product is always greater than either of the two numbers.” Is her conjecture correct? Explain why or why not.
No - counterexample: 1/2 x 1/2 = 1/4

5. Use inductive reasoning to find the sum of the first 20 odd numbers.
(Hint: Find the first few sums and see if there are any patterns.)
Understand the problem
Devise a plan
Carry out the plan
Look back
1 = = = = 16
Conjecture: the sum of the first 20 odd number would be 202, or 400.

6. Maria’s parents tells her she can go to the mall with her friends if she finishes her homework. Maria shows her parents her completed homework Is this a case for inductive reasoning? What conclusion can you make?
Maria’s going to the mall.
Solve for x x + 4 = 5x - 10
-3x x
4 = 2x - 10
14 = 2x
x = 7

7. Properties of Equality/Congruence
Deductive Reasoning
Deductive reasoning involves reaching a conclusion by using a formal structure based on a set of undefined terms and a set of unproved axioms or premises (facts). Conclusions are said to be proved using these facts and called theorems.
Properties of Equality/Congruence
Reflexive Property a = a, A  A Symmetric Property if a = b, then b = a Transitive Property if a = b and b = c, then a = c.

8. Deductive Reasoning Terminology
If you read the Times, then you are well informed.
You read the Times.
Therefore, you are well informed.
1, 2 & 3 are statements called an argument
If you accept 1 & 2, called the hypotheses or premises of the argument, as true, then statement 3, called the conclusion, must be true and the reasoning of the argument is said to be valid. The conclusion is inescapable. 8

9. Sentences are either open or closed.
Logic
The study of reasoning
Sentences (complete thoughts) are the building blocks for the study of logic.
Sentences are either open or closed.
Closed sentences or statements state facts that are either true or false.
Open sentences contain a variable (pronoun) that has an indeterminate truth value.
Note: Phrases, commands, & questions are not sentences  not part of Logic

10. Closed Sentences or Statements
Julia Roberts is a movie star. T
Patrick Ewing plays basketball for the NY Knicks. F
Open Sentences
She is a movie star. ?
He plays basketball for the NY Knicks. ?
Non-Logical
Be a star!!
Play ball!!

11. Sentence (Complete Thought) Open Closed Truth value Truth value ?
T or F
Open
Truth value ?
Contains a variable
Statement

12. Identify each sentence as open or closed.
Barack Obama is President of the U.S.
Tu Pac is dead.
They eat meat.
3 + 6 = 9
She loves music.
We hate homework. C C O C O O

13. This school is in Staten Island. It is the capitol of the U.S.
Replace the variable (pronoun) in each sentence to make the sentence a true statement.
He is a singer.
This school is in Staten Island.
It is the capitol of the U.S.
Replacement (Domain) Set
Elements that can be used in place of the variable.
Truth Set
Elements that replace the variable and make the sentence a true statement.

14. How do we change this false statement to a true one?
Negation
Negation changes the truth value of a closed sentence (statement) to its opposite truth value ~
Bon Jovi is an opera singer. F
How do we change this false statement to a true one?
By inserting the word “not”.
Bon Jovi is not an opera singer. T
Derek Jeter is not a basketball player. T
Derek Jeter is a basketball player. F

15. Negation – Symbolic Notation Otto is telling the truth.~
Let t represent the statement: Otto is telling the truth Translate ~ t
Otto is not telling the truth.
It is not the case that
Otto is telling the truth.
Negate: All students have pencils.
No students have pencils.
Not all students have pencils At least one student doesn’t have a pencil It is not the case that all students have pencils. Some students do not have pencils.

16. Negation – Symbolic Notation and Truth Tables
Definition of Negation p ~p T F F T

17. PR and PT are equal in length
Use deductive reasoning to reach a conclusion.
Rachel is older than Michelle and Hector is younger than Michelle.
Rachel is older than Michelle
x = and = 5
x = 5
A circle is a set of points that are all the same distance from a single point called the center. PR has one end point at the center and the other on the circle and is called a radius. PT is also a radius. P R T
PR and PT are equal in length

18. Inductive or Deductive? Explain.
It has snowed every New Year’s Day for the past 4 years. Akiko says it will snow on New Year’s Day this year.
Band members are admitted free to all football games. Rachel plays flute in the band. She gets into the football game free.
Every customer who came into Joe’s clothing was wearing a raincoat. Joe decided it was raining.