1. 2.1 Using Inductive Reasoning to Make Conjectures

2. 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.
Make a conjecture.
3) Prove the conjecture true or find a
counterexample.

3. Patterns Pattern- an arrangement or sequence in objects or events
Examples: Find the next item in each pattern.
January, March, May, …
7, 14, 21, 28, …
, , …
0.4, 0.04, 0.004, …

4. Conjecture
Conjecture: A statement you believe to be true based on inductive reasoning
Examples: Complete each conjecture.
The sum of two positive number is ___________.
The number of lines formed by 4 points, no three of which are collinear, is ______________.
The product of two odd numbers is ____________.

5. 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.

6. 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

7. Examples:
Show that each conjecture is false by finding a counterexample
For every integer n, n3 is positive.
Two complementary angles are not congruent.
For any real number x, x2 > x.
Supplementary angles are adjacent.