Download 1-1 Patterns and Inductive Reasoning

1-1 Patterns and Inductive Reasoning
Objectives:
• Define:
– Conjectures
– Inductive reasoning
– Counterexamples
• Make conjectures based on inductive
reasoning
• Find counterexamples to prove that a
conjecture is false
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Definition of Conjecture, Inductive Reasoning
• Conjecture – An unproven guess based
on observations
• Inductive Reasoning – Reasoning that is
based on patterns you observe
Example 0: Sketch the next figure in the
pattern
?
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Solution to Example 0
Problem solving methodology:
• Look for a pattern.
• Make a conjecture based on the pattern.
• Verify that the conjecture is true in all cases.
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Quick Check 1a
• Write the next two terms in the sequence:
Step 1. Look for a pattern.
1
1+1=2
1x2=2
2 2+2=4 2x2=4
4 4 + 3 = 7 4 x 2 = 8 (NO)
Step 2. Make a conjecture.
7 7 + 4 = 11
Step 3. Verify that the conjecture is true for all
11 11 + 5 = 16
cases
16 16 + 6 = 22
22
22 + 7 = 29
?
29 + 8 = 37
??
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Example 2. Using Inductive Reasoning
• Make a conjecture about the sum of the
first 30 odd numbers.
Step 1. Look for a pattern.
1 = 12
1 + 3 = 4 = 22
1 + 3 + 5 = 9 = 32
1 + 3 + 5 + 7 = 16 = 42
Step 2. Make a conjecture.
1 + 3 + 5 + 7 +……+ 59 = 302 = 30 x 30 = 900
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Example 3a. Finding a Counterexample to Disprove a
Conjecture
•
•
Not all conjectures are true.
You can prove that a conjecture is false
by finding one counterexample, which is
an example that contradicts the
conjecture.
Conjecture a.: The square of any number is
greater than the original number.
22 = 2 x 2 = 4, 32 = 3 x 3 = 9,
12 = 1 x 1 = 1 (Counterexample; proves the
conjecture is false)
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Example 3b. Finding a Counterexample
• Conjecture b. You can connect any three
points to form a triangle.
Counterexample
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Unproven Conjectures
• Not every conjecture in mathematics has
been proven.
http://en.wikipedia.org/wiki/Goldbach’s_conjecture
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Learning Check and Summary
• What type of reasoning is based on
patterns you observe?
• An unproven guess you reach using
inductive reasoning is called a ?
• To prove that a conjecture is false, what do
you try to find?
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Assignment
• Class Work
– Workbook:
• Daily Notetaking Guide 1-1 p. 2; Omit Ex. 2, 3
• Practice 1-1 p. 249, Omit 7-12, 19, 20
• Homework
– Workbook:
• Daily Notetaking Guide 1-3 p. 10
• Daily Notetaking Guide 1-4 p. 13
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