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Inductive Reasoning & Conjecture. What is a Conjecture ? What is inductive reasoning?. Definitions. A conjecture is an educated guess based on known information. It is made based on past observations. - PowerPoint PPT Presentation
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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
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