Download Inductive Reasoning

2.1 Inductive Reasoning
Objectives:
• I CAN use patterns to make
conjectures.
• I CAN disprove geometric conjectures
using counterexamples.
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Example #1
Describe how to sketch the
4th figure. Then sketch it.
Each circle is divided into twice as
many equal regions as the figure
number. The fourth figure should be
divided into eighths and the section
just above the horizontal segment
on the left should be shaded.
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Example #2
Describe the pattern. Write
the next three numbers.
7,
 21,
3
 63,
3
 189,...
3
Multiply by 3 to get the next
number in the sequence.
189  3  567
567  3  1701
1701 3  5103
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What comes next? (look @ notes)
How do we know what comes next?
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What is a conjecture?
What is inductive reasoning?
Write in your foldable
Conjecture: conclusion made based
on observation
Inductive Reasoning: conjecture
based on patterns
Proving conjectures TRUE
is very hard.
Proving conjectures FALSE
is much easier.
What is a counterexample?
Counterexample: example that
How do you disprove a conjecture?
shows a conjecture is false
What are the steps for inductive
reasoning?
How do you use inductive
reasoning?
Steps for Inductive Reasoning
1. Find pattern.
2.Make a conjecture.
3.Test your conjecture
or find a counterexample.
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Inductive Reasoning
Look @ notes
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Example #3
Make and test a conjecture
about the sum of any 3
consecutive numbers.
(Consecutive numbers are
numbers that follow one
after another like
3, 4, and 5.)
3  4  5  12  4  3
6  7  8  21  7  3
8  9 10  27  9  3
11 12 13  36 12  3
Conjecture:
The sum of any 3 consecutive numbers
is 3 times the middle number.
1  0  1  0  0  3
20  21  22  63  21 3
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Counterexample
Look @ notes
Example #4
Conjecture:
The sum of two numbers is
always greater than the
larger number.
True or false?
2  0  2
sum > larger number
2  0
A counterexample was found,
so the conjecture is false.
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
9
Examples
• 1. Describe how to sketch the fourth figure in the
pattern. Then sketch the fourth figure.
Example