The document outlines objectives related to inductive reasoning, including identifying patterns, making conjectures, and analyzing conditional statements. It emphasizes the importance of counterexamples in disproving conjectures and provides examples of sequences, conditional statements, and truth values. Lastly, it discusses the concepts of converses, inverses, and contrapositives of statements, along with their truth values.

1. Obj. 9 Inductive Reasoning
Objectives:
The student is able to (I can):
• Use inductive reasoning to identify patterns and make
conjectures
• Find counterexamples to disprove conjectures
• Identify, write, and analyze the truth value of conditional
statements.
• Write the inverse, converse, and contrapositive of a
conditional statement.

2. Find the next item in the sequence:
1. December, November, October, ...
SeptemberSeptemberSeptemberSeptember
2. 3, 6, 9, 12, ...
15151515
3. , , , ...
4. 1, 1, 2, 3, 5, 8, ...
13131313 ———— This is called the FibonacciThis is called the FibonacciThis is called the FibonacciThis is called the Fibonacci
sequence.sequence.sequence.sequence.

3. inductive
reasoning
conjecture
Reasoning that a rule or statement is true
because specific cases are true.
A statement believed true based on
inductive reasoning.
Complete the conjecture:
The product of an odd and an even number
is ______ .
To do this, we consider some examples:
(2)(3) = 6 (4)(7) = 28 (2)(5) = 10
eveneveneveneven

4. counterexample
If a conjecture is true, it must be true for
every case. Just one exampleJust one exampleJust one exampleJust one example for which the
conjecture is false will disprove it.
A case that proves a conjecture false.
Example: Find a counterexample to the
conjecture that all students who take
Geometry are 10th graders.

5. Examples
To Use Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample to disprove it.
Show that each conjecture is false by giving
a counterexample.
1. The product of any two numbers is
greater than the numbers themselves.
((((----1)(5) =1)(5) =1)(5) =1)(5) = ----5555
2. Two complementary angles are not
congruent.
45º and 45º45º and 45º45º and 45º45º and 45º

6. Sometimes we can use inductive reasoning
to solve a problem that does not appear to
have a pattern.
Example: Find the sum of the first 20 odd
numbers.
Sum of first 20 odd numbers?
1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1
4
9
16
12
22
32
42
202 = 400

7. These patterns can be expanded to find the “nth” term using
algebra. When you complete these sequences by applying a
rule, it is called a functionfunctionfunctionfunction.
Examples: Find the missing terms and the rule.
To find the pattern, the coefficient of n is the difference
between each term, and the value at 0 is what is added or
subtracted.
1 2 3 4 5 … 8 … 20 … n
-3 -2 -1 0 1 4 16 n — 4
1 2 3 4 5 … 8 … 20 … n
32 39 46 53 60 81 165 7n+25

8. conditional
statement
hypothesis
conclusion
A statement that can be written as an
“if-then” statement.
Example: IfIfIfIf today is Saturday, thenthenthenthen we
don’t have to go to school.
The part of the conditional following the
word “if”.
“today is Saturday” is the hypothesis.
The part of the conditional following the
word “then”.
“we don’t have to go to school” is the
conclusion.

9. Notation
Examples
Conditional statement: p → q, where
p is the hypothesis and
q is the conclusion.
Identify the hypothesis and conclusion:
1. If I want to buy a book, then I need
some money.
2. If today is Thursday, then tomorrow is
Friday.
3. Call your parents if you are running late.

10. Examples
To write a statement as a conditional,
identify the sentence’s hypothesis and
conclusion by figuring out which part of the
statement depends on the other.
Write a conditional statement:
• Two angles that are complementary are
acute.
If two angles are complementary, then theyIf two angles are complementary, then theyIf two angles are complementary, then theyIf two angles are complementary, then they
are acute.are acute.are acute.are acute.
• Even numbers are divisible by 2.
If a number is even, then it is divisible by 2.If a number is even, then it is divisible by 2.If a number is even, then it is divisible by 2.If a number is even, then it is divisible by 2.

11. truth value T if a conditional is true,
F if a conditional is false.
The statement is false only when theThe statement is false only when theThe statement is false only when theThe statement is false only when the
hypothesis is true and the conclusion ishypothesis is true and the conclusion ishypothesis is true and the conclusion ishypothesis is true and the conclusion is
false.false.false.false. To show that a conditional is false,
you need only find one counterexample
where the hypothesis is true and the
conclusion is false.
Hypothesis Conclusion Statement
T T T
TTTT FFFF FFFF
F T T
F F T

12. Examples Determine if each conditional is true. If
false, give a counterexample.
1. If your zip code is 76012, then you live in
Texas.
TrueTrueTrueTrue
2. If a month has 28 days, then it is
February.
September also has 28 days, whichSeptember also has 28 days, whichSeptember also has 28 days, whichSeptember also has 28 days, which
proves the conditional false.proves the conditional false.proves the conditional false.proves the conditional false.
3. If 14 is evenly divisible by 3, then
tomorrow is Tuesday.
The hypothesis is false, so theThe hypothesis is false, so theThe hypothesis is false, so theThe hypothesis is false, so the
conditional isconditional isconditional isconditional is truetruetruetrue....
Texas
76012

13. negation of p “Not p”
Notation: ~p
Example: The negation of the statement
“Blue is my favorite color,” is “Blue is notnotnotnot
my favorite color.”
Related Conditionals Symbols
Conditional p → q
Converse q → p
Inverse ~p → ~q
Contrapositive ~q →~p

14. Example Write the conditional, converse, inverse, and
contrapositive of the statement:
“A cat is an animal with four paws.”
Type Statement
Truth
Value
Conditional
(p → q)
If an animal is a cat, then it
has four paws.
T
Converse
(q → p)
If an animal has four paws,
then it is a cat.
F
Inverse
(~p → ~q)
If an animal is not a cat,
then it does not have four
paws.
F
Contrapos-
itive
(~q → ~p)
If an animal does not have
four paws, then it is not a
cat.
T

15. Example Write the conditional, converse, inverse, and
contrapositive of the statement:
“When n2 = 144, n = 12.”
Type Statement
Truth
Value
Conditional
(p → q)
If n2 = 144, then n = 12.
F
(n = —12)
Converse
(q → p)
If n = 12, then n2 = 144. T
Inverse
(~p → ~q)
If n2 ≠ 144, then n ≠ 12 T
Contrapos-
itive
(~q → ~p)
If n ≠ 12, then n2 ≠ 144
F
(n = —12)