5-1 Inductive Reasoning, Conjectures, & Counter Examples (6 Per Page)

Board Work - Solution:

Board Work ~ Logic Problem: Solve the following Logic Problem: Three friends, Mike, Hari, and Carl, each have a sister. The girls are Amy, Sarah, and Tia. The six friends play mixed doubles tennis, but no siblings are ever partnered together. Partners have been as follows: •Mike & Amy •Carl & Tia •Carl & Sarah •Hari & Tia Æ What are the three sets of siblings? ___________________________________________________________________________ Pure Math 20

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Unit #5: Formal Reasoning

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Inductive Reasoning,   Conjectures, &  Counterexamples

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Inductive reasoning involves the determination of a pattern from a set of data or examples, and then the creation of an educated guess about what should follow. This educated guess is called a conjecture. Inductive reasoning does not prove something to be true. It can only provide examples that support a conjecture.

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Inductive reasoning, then, forms the basis of science with its “trial & error” process To prove a conjecture we use deductive reasoning (next class). Once a conjecture is proven, it can be called a theorem. The theorem can then be applied to other scientific situations.

The Mathematical Process: 1. Perform an investigation. 2. Use inductive reasoning to write a conjecture. 3. Use deductive reasoning to prove the conjecture. 4. Use theorem to solve problems. ___________________________________________________________________________ Pure Math 20

Inductive Reasoning Statement: All sheep I have seen are white Conjecture: All sheep everywhere are white. We may forever see white sheep, but we cannot say with 100% certainty that the conjecture is true because there may exist a non white sheep. ___________________________________________________________________________ Pure Math 20

Inductive Reasoning Example #1: {

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Inductive Reasoning Example #2:

Given the following, make a conjecture about the data.

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What conjecture can we make about multiplying an odd number by an even number?

Choose a natural number Double it Add five Add your original number Add seven Divide by three Subtract the original number

Remember, though, we may not be certain that it

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Inductive Reasoning Example #3: x x x x

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Inductive Reasoning Example #3…:

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The diagrams above show the number of fence posts used to enclose a square region. The fence posts are placed one metre apart.

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Conjecture:

Make a conjecture about the number of fence posts needed to enclose a square region with side length s metres. Test your conjecture for a square with side length 6m.

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Inductive Reasoning Example #3…: { Check if the conjecture is true for a side length of 6m.

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Remember that although we may be able to provide many examples that will support our conjecture, we are not able to prove that the conjecture is true using inductive reasoning. We are, however, able to prove a conjecture is false. This is done by providing a counterexample. A counterexample is an example that shows a conjecture is false. If the conjecture was that all sheep everywhere are white, a black sheep will show that this is a false conjecture. We only need ONE counterexample to disprove a

conjecture. ___________________________________________________________________________ Pure Math 20

Inductive Reasoning Example #4: { Disprove the following conjecture: Every natural number can be written as the sum of consecutive numbers that are natural. Counterexample: Examples: 9=4+5 State a natural number that cannot 6=1+2+3 be written as the sum of consecutive numbers. 12 = 3 + 4 + 5 10 = 1 + 2 + 3 + 4

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Inductive Reasoning Example #5: { Disprove the following conjecture: Multiples of 4 are divisible by 8. Examples: 8/8 = 1 16/8 = 2 24/8 = 3 32/8 = 4

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Inductive Reasoning Example #6: { Disprove the following conjecture: A number is more than its square root.

Inductive Reasoning Your Task: { {

Examples: 36 > 6 49 > 7 81 > 9 25 > 5

Counterexample:

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Counterexample:

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pg 340-341, #1-15 pg 345, #1-17 reasoning booklet, pg 2-7

Next Class: Deductive Reasoning (Section 6-3 in textbook)

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