Course: Math Literacy Aim: Inductive/Deductive Reasoning Aim: What is Deductive and Inductive Reasoning and how can we use them to solve problems? Do

Course: Math LiteracyAim: Inductive/Deductive Reasoning

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 9 9

2 9 18

3 9 27

4 9 36

5 9 45

6 9 54

7 9 63

8 9 72

9 9 81

1 + 8 = 9

2 + 7 = 9

3 + 6 = 9

The sum of the digits to theright of the equality add to 9

4 5 9

This pattern of adding after multiplying by 9 generates a sequence called the 9 pattern.

Will it always be the case?


11 9 99

12 9 108

13 9 117

14 9 126

15 9 135

16 9 144

17 9 153

18 9 162

19 9 171

20 9 180

the ‘9’ patternWill it always be the case?

21 9 189

22 9 198

23 9 207

24 9 216

25 9 225

26 9 234

27 9 243

28 9 252

29 9 261

30 9 270

31 9 279

32 9 288

33 9 297

34 9 306

35 9 315

36 9 324

37 9 333

38 9 342

39 9 351

40 9 360


Inductive ReasoningInductive 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.

Ex. Find the next two terms is this sequence: 2, 4, 6, 8, . . . . . And describe the pattern.

10, 12 - Add 2 to each term.

WEAKNESS - one counterexample can show a conclusion to be false.

It is sometimes called the scientific method.


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


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.)

1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16

Conjecture: the sum of the first 20 odd number would be 202, or 400.

12

22

32 42

Understand the problemDevise a plan

Carry out the planLook back


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 3x + 4 = 5x - 10

-3x -3x 4 = 2x - 10

+10 +10

14 = 2x x = 7


Deductive ReasoningDeductive 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

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.

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.


Deductive Reasoning TerminologyDeductive Reasoning Terminology

1. If you read the Times, then you are well informed.

2. You read the Times.3. 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.


LogicThe study of reasoning

Sentences (complete thoughts) are the building blocks for the study of logic.

Sentences are either open or closed.

•Open sentences contain a variable (pronoun) that has an indeterminate

truth value.

•Closed sentences or statements state facts that are either true or false.

Note: Phrases, commands, & questions are not sentences not part of Logic


Closed Sentences or Statements

Julia Roberts is a movie star.

Patrick Ewing plays basketball for the NY Knicks.

T

F

Open Sentences

She is a movie star.

He plays basketball for the NY Knicks.

?

?

Non-Logical

Be a star!!

Play ball!!


Sentence

(Complete Thought)

Sentence

(Complete Thought)

Closed

Truth value

T or F

Closed

Truth value

T or F

Statement

Open

Truth value

?

Open

Truth value

?

Contains a variable


Identify each sentence as open or closed.

1. Barack Obama is President of the U.S.

2. Tu Pac is dead.

3. They eat meat.

4. 3 + 6 = 9

5. She loves music.

6. We hate homework.

O

C

O

O

C

C


Replace the variable (pronoun) in each sentence to make the sentence a true statement.

1. He is a singer.

2. This school is in Staten Island.

3. It is the capitol of the U.S.

Replacement (Domain) Set

Elements that can be used in place of the variable.

Truth SetElements that replace the variable and

make the sentence a true statement.


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

~


Negation – Symbolic Notation

Negation ~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.


Negation – Symbolic Notation and Truth Tables

p ~p

Definition of Negation

T

T

F

F


Use deductive reasoning to reach a conclusion.

Rachel is older than Michelle and Hector is younger than Michelle.

Rachel is older than Michelle

x = 3 + 2 and 3 + 2 = 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


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.

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Course: Math Literacy Aim: Inductive/Deductive Reasoning Aim: What is Deductive and Inductive Reasoning and how can we use them to solve problems? Do