1. Inductive and Deductive Reasoning Objectives: The student is
able to (I can): Use inductive reasoning to identify patterns and
make conjecturesconjectures Understand the differences between
inductive and deductive reasoning Use properties of algebra and
deductive reasoning to create algebraic proofs
2. Find the next item in the sequence: 1. December, November,
October, ... 2. 3, 6, 9, 12, ... 3. , , , ... 4. 1, 1, 2, 3, 5, 8,
...
3. 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 Fibonacci sequence.This is called the Fibonacci
sequence.This is called the Fibonacci sequence.This is called the
Fibonacci sequence.
4. inductiveinductiveinductiveinductive
reasoningreasoningreasoningreasoning reasoning that a rule or
statement is true because specific cases are true.
conjectureconjectureconjectureconjecture a statement believed true
based on inductive reasoning. Complete the conjecture: The product
of an odd and an even number is ______ .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
5. inductiveinductiveinductiveinductive
reasoningreasoningreasoningreasoning reasoning that a rule or
statement is true because specific cases are true.
conjectureconjectureconjectureconjecture a statement believed true
based on inductive reasoning. Complete the conjecture: The product
of an odd and an even number is ______ .evenevenevenevenThe 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
6. 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.a counterexample. 1. The product of any
two numbers is greater than the numbers themselves. 2. Two
complementary angles are not congruent.
7. 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.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. 45 and 4545 and 4545 and 4545 and 45
8. 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. 1 1 + 3 1 + 3 + 5 1 4 9 Sum of first 20
odd numbers? 1 + 3 + 5 1 + 3 + 5 + 7 9 16
9. 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. 1 1 + 3 1 + 3 + 5 1 4 9 12 22 32 Sum of
first 20 odd numbers? 1 + 3 + 5 1 + 3 + 5 + 7 9 16 32 42 202 =
400
10. 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. 1 2 3 4 5 8 20 n -3 -2 -1 0 1 To find
the pattern when the difference between each term is the same, 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 32 39 46 53
60
11. 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. 1 2 3 4 5 8 20 n -3 -2 -1 0 1 4 16 n 4
To find the pattern when the difference between each term is the
same, 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 32
39 46 53 60 81 165 7n+25
12. deductivedeductivedeductivedeductive
reasoningreasoningreasoningreasoning the process of using logic to
draw conclusions from given facts, definitions, and properties.
Inductive reasoning uses specific cases and observations to form
conclusions about general ones (circumstantial evidence). Deductive
reasoning uses facts about general cases to form conclusions about
specific cases (direct evidence).
13. Example Decide whether each conclusion uses inductive or
deductive reasoning. 1. Police arrest a person for robbery when
they find him in possession of stolen merchandise. 2. Gunpowder
residue tests show that a2. Gunpowder residue tests show that a
suspect had fired a gun recently.
14. Example Decide whether each conclusion uses inductive or
deductive reasoning. 1. Police arrest a person for robbery when
they find him in possession of stolen merchandise. Inductive
reasoningInductive reasoningInductive reasoningInductive reasoning
2. Gunpowder residue tests show that a2. Gunpowder residue tests
show that a suspect had fired a gun recently. Deductive
reasoningDeductive reasoningDeductive reasoningDeductive
reasoning
15. Transitive Property of Equality If a = b, and b = c, then a
= c. Ex: If 1 dime = 2 nickels, and 2 nickels = 10 pennies, then 1
dime = 10 pennies. Substitution Property of Equality If a = b, then
b can be substituted for a in any expression. Ex: If x = 5, then we
can substitute 5 for x in the equation Ex: If x = 5, then we can
substitute 5 for x in the equation y = x + 2, which means y = 7. We
will also use the Distributive Property: a(b + c) = ab + ac and ab
+ ac = a(b + c)
16. proofproofproofproof an argument which uses logic,
definitions, properties, and previously proven statements to show
that a conclusion is true. algebraic proofalgebraic proofalgebraic
proofalgebraic proof a proof which uses algebraic properties When
you write a proof, you must give a justification (reason) for each
step to show that it is valid. For each(reason) for each step to
show that it is valid. For each justification, you can use a
definition, postulate, property, or a piece of given
information.
17. Ex.: Solve the equation x 1 = 5, and write a justification
for each step. x 1 = 5 Given x 1 + 1 = 5 + 1 Addition prop. = x = 6
Simplify Ex: Solve the equation 4m + 8 = 12, and write the
justification for each step.justification for each step. 4m + 8 =
12 Given 4m + 8 8 = 12 8 Subtraction prop. = 4m = 20 Simplify
Division prop. = m = 5 Simplify = 4 20 4 4 m
18. As a general rule, we do not write out the actions that we
use to solve the equation, we just write the result of each action.
Example: Solve the equation 4x = 2x + 8. Write a justification for
each step. 4x = 2x + 8 Given equation 2x = 8 Subtraction prop. =2x
= 8 Subtraction prop. = x = 4 Division prop. =
19. We can also apply the definitions and properties we learned
in Chapter 1: Solve for x and write a justification for each step.
MY = MA + AY Segment Addition Postulate M A Y 9x 4 5x + 7 2x + 3 MY
= MA + AY Segment Addition Postulate 9x 4 = 5x + 7 + 2x + 3
Substition prop. = 9x 4 = 7x + 10 Simplify 9x = 7x + 14 Addition
prop. = 2x = 14 Subtraction prop. = x = 7 Division prop. =
