Inductive Reasoning & Conjecture

What is a Conjecture?What is inductive reasoning?

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.

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.

1 3 6 10 15

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 pattern1 3 6 10 15

• Conjecture – the next number will increase by 6. So the next number will be 15 + 6 = 21.

1 3 6 10 15

+2 +3 +4 +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

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.

Find Counterexamples Find a counterexample of the following statements

Given: <A and <B are supplementaryConjecture: <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!

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<CConjecture: 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