1. Inductive Reasoning & Conjecture What is a Conjecture? What is inductive reasoning?

2. Definitions Inductive reasoning – is reasoning that uses a number of examples to arrive at a plausible generalization or prediction. Looking at several situations to arrive at a conjecture is called inductive reasoning. To prove a conjecture is false, you must provide only one false example. The false example is called the counterexample. A conjecture is an educated guess based on known information. It is made based on past observations.

3. Patterns and Conjecture The numbers represented below are called triangular numbers. Make a conjecture about the next triangular number based on the pattern. Observe – each triangle is formed by adding another row of dots. 1361015

4. Patterns and Conjecture The numbers represented below are called triangular numbers. Make a conjecture about the next triangular number based on the pattern. Find a pattern 1 361015 Conjecture – the next number will increase by 6. So the next number will be 15 + 6 = 21. 1361015 +2 +3 +4+5

5. Geometric Conjecture For points P, Q, and R, PQ = 9, QR = 15, and PR = 12. Make a conjecture and draw a figure to illustrate your conjecture. Given: points P, Q, and R, PQ = 9, QR = 15, and PR = 12 Examine the measures of the segments. Since PQ + QR ≠ PR, the points cannot be collinear. Conjecture: P, Q, and R are not collinear. R 15 Q P 12 9

6. Find Counterexamples A conjecture based on several observations may be true in most circumstances, but false in others. It takes only one false example to show that a conjecture is not true. The false example is called a counterexample.

7. Find Counterexamples Find a counterexample of the following statements Given: <A and <B are supplementary Conjecture: <A and <B are not congruent Counterexample: <A = 90° and <B = 90° <A + <B = 180°. They are supplementary and congruent - by definition thus proving the conjecture false!

8. Example- Determine if the conjecture is true or false. If false, give a counterexample. Given: m<A is greater than m<B; and m<B is greater than m<C Conjecture: m<A is greater than m<C. Solution: Draw a picture Think of examples that prove the conjecture true….can you think of a counterexample to prove the conjecture false? TRUE