1-1 Patterns and Inductive Reasoning

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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
• 1
• 2
• 4
• 7
• 11
• 16
• 22
• ?
• ??

Step 1. Look for a pattern.
1 1 2 1 x 2 2
2 2 4 2 x 2 4
4 3 7 4 x 2 8 (NO)
Step 2. Make a conjecture.
7 4 11
Step 3. Verify that the conjecture is true for
all cases
11 5 16
16 6 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.
• 1 12
• 1 3 4 22
• 1 3 5 9 32
• 1 3 5 7 16 42

Step 1. Look for a pattern.
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.

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Unproven Conjectures

• Not every conjecture in mathematics has been
  proven.
• http//en.wikipedia.org/wiki/Goldbachs_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