Download 2.1 Using Inductive Reasoning to Make Conjectures

2.1
Using Inductive
Reasoning to Make
Conjectures
Inductive Reasoning
• Inductive Reasoning- the process of reasoning that a
rule or statement is true because specific cases are
true.
Inductive Reasoning
1) Look for a pattern.
2) Make a conjecture.
3) Prove the conjecture true or find a
counterexample.
Patterns
• Pattern- an arrangement or sequence in objects or
events
• Examples: Find the next item in each pattern.
1. January, March, May, …
2. 7, 14, 21, 28, …
3.
,
,
…
4. 0.4, 0.04, 0.004, …
Conjecture
• Conjecture: A statement you believe to be true based
on inductive reasoning
• Examples: Complete each conjecture.
1. The sum of two positive number is ___________.
2. The number of lines formed by 4 points, no three of
which are collinear, is ______________.
3. The product of two odd numbers is ____________.
Example:
• The cloud of water leaving a whale’s blowhole when
it exhales is called its blow. A biologist observed
blue-whale blows of 25 ft, 29 ft, 27 ft, and 24 ft.
Another biologist recorded humpback-whale blows
of 8 ft, 7 ft, 8 ft, and 9 ft. Make a conjecture based
on the data.
True or False Conjectures
• To show a conjecture is true, you must prove it.
• To show a conjecture is false, you must find only
one example that is not true
• Counterexample: An example that proves a
conjecture or example is false.
• Counterexamples can be a drawing, a statement, or a
number
Examples:
• Show that each conjecture is false by finding a
counterexample
1.
2.
3.
4.
For every integer n, n3 is positive.
Two complementary angles are not congruent.
For any real number x, x2 > x.
Supplementary angles are adjacent.