Warm-up
August 22, 2011
• Evaluate the following expressions
1) x 2 when x  .5, 1, - 1
2)(x  1)(x - 1), when x  1, 2, 3
x 1
3)
, when x  1, 2, 3
2
4)
1
x , when x  , 4, 9
4
Outcomes
I will be able to:
• 1) Explain Inductive Reasoning by identifying
the three steps.
• 2) Identify patterns in number sequences and
pictures.
• 3) Make Conjectures based on observed data.
• 4) Define and Create Counterexamples.
White Boards
• Find the next three number in the sequence:
• 1) 1, 2, 4, 7, 11…
• 2) 1, 1, 2, 3, 5, 8, 13…
• 3) 2, 4, 8, 16…
• 4) 23, 19, 15, 11…
Inductive Reasoning
• Conjecture – an unproven statement based on
observations. Conjectures can be modified
until they are concrete
• ***The process of describing what is being
observed
Inductive Reasoning
• Inductive Reasoning – Observing data,
recognizing patterns, and making
generalizations about that data
• We use inductive reasoning everyday
• Can you think of a few examples where you
may use inductive reasoning?
Inductive Reasoning
• 3 Stages of Inductive Reasoning
• 1) Look for a pattern – Look at examples and
use diagrams, tables, and pictures to help
discover a pattern.
• 2) Make a conjecture - Use your observations
to make “guess” about the pattern.
• 3) Verify the conjecture - Use logical reasoning
skills to decide if your conjecture is valid.
Using Inductive Reasoning
Using Inductive Reasoning
Using Inductive Reasoning
• Example 3. Predict the next three numbers.
• (a) 17, 15, 12, 8,….
• (b) 64, 16, 4, 1, ¼ ,….
• (c) 48, 16, 16/3, 16/9, ….
• (d) 4, -6, 8, -10, …
Finding and Describing Patterns
on White Boards
Ex 1:
What will the 5th and 6th shapes in the
pattern look like?
Describe the pattern that is happening
Finding and Describing Patterns
on White Boards
Ex 2: What will the 4th term in the
sequence look like?
Describe the pattern
Finding and Describing Patterns
on White Boards
• Describe the pattern in the sequence below.
Predict the next number
• a) 1, 4, 16, 64…
• b) -5, -2, 4, 13…
Making A Conjecture
• Example 1. Pick a secret number. Add the next
highest number to it. Add 9. Divide by 2.
Subtract your secret number. What is your
conjecture?
Making a Conjecture
• Example 2. Pick a secret number. Add 5.
Multiply by 2. Subtract 4. Divide by 2. Subtract
your secret number. What is your conjecture?
Finding a Counterexample
On White Boards
• Example:
• For all real numbers x, the expression x³ is
greater than or equal to x.
• When does this not happen?
Counterexample
• Counterexample – an example that shows
that a conjecture is false
• Unproven or undecided conjectures –
Conjectures that have not been proven true or
false
Finding Counterexamples
• Example 1. Show the conjecture is false by
finding a counterexample: “The difference of
two positive integers is always positive.”
Finding Counterexamples
• Example 2. Show the conjecture is false by
finding a counterexample: “For all real
numbers x, the expression x2 is greater than or
equal to x.”
Finding a Conjecture based on Pattern
• Example 3. Complete the conjecture: The sum of the
first n odd positive integers is ____________________.
•
•
•
•
•
•
•
•
first odd positive integer:
sum of first two odd positive integers:
sum of first three odd positive integers:
sum of first four odd positive integers:
sum of first five odd positive integers:
Exit Quiz
• Find the next two terms in each pattern
• 1)
2)
• 3) Find the missing term
• 4) Make a conjecture about the sums of any
two odd numbers
• 1+1 = 2
3 + 7 = 10
• 1 + 3 = 4 5 + 9 = 14
• 3 + 5 = 8 7 + 9 = 16