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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?
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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.
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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.
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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.
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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!!
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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.
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
~
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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.
Course: Math LiteracyAim: Inductive/Deductive Reasoning
Negation – Symbolic Notation and Truth Tables
p ~p
Definition of Negation
T
T
F
F
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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
Course: Math LiteracyAim: Inductive/Deductive Reasoning
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.