Questions and Answers
The statement $3 + 3 = 6$ serves as a/an _______ to the conjecture that the sum of two odd numbers is an odd number.
Counterexample
Which president has been younger than 65 at the time of his inauguration?
A, B, C, & D
Find a counterexample to show that the following statement is false: If a number is multiplied by itself, the result is even.
Arriving at a general conclusion based on observations of specific examples is called _______.
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Which type of reasoning is illustrated when concluding that all colleges and universities have increased tuition based on recent articles?
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Is the nurse in the emergency room using inductive or deductive reasoning in the following scenario? The nurse notifies the doctor about a specific pattern in heart rate that is indicative of heart attacks.
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What number comes next in the pattern 3, 9, 36, 180, 1080?
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What is the next number in the pattern 2, 10, 18, 26, 34?
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What is the next number in the pattern 75, 64, 53, 42, 31?
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What is the next number in the pattern 2, 4, 8, 16, 32?
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What is the next number in the pattern 4, 8, 13, 19, 26, 34?
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Identify the reasoning process in the example: The course policy states that work turned in late will be marked down a grade. I turned in my report a day late, so it was marked down from a B to a C. This is an example of __________.
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Study Notes
Counterexamples and Inductive Reasoning
- A counterexample disproves a conjecture by showing an exception, such as "3 + 3 = 6" refuting the idea that two odd numbers sum to an odd number.
- In U.S. presidential history, President ages at inauguration disprove the claim that no president has been younger than 65, exemplified by Theodore Roosevelt, John F. Kennedy, Barack Obama, and Harry S. Truman.
- A counterexample for the false statement regarding multiplication gives "5 X 5 = 25," demonstrating the product is odd despite being multiplied by itself.
Types of Reasoning
- Inductive reasoning involves forming general conclusions based on specific observations. An example is drawing a conclusion about all colleges increasing tuition from selected articles.
- Deductive reasoning applies general principles to specific cases, showcased when nurses notify doctors about a patient's heart rate pattern indicative of potential heart attacks.
Number Patterns
- The next number in the sequence "3, 9, 36, 180, 1080" is "7560," indicating growth by multiplication.
- In the pattern "2, 10, 18, 26, 34," the next number is "42," revealing a consistent increment.
- The sequence "75, 64, 53, 42, 31" leads to "20," displaying a decrement pattern.
- "2, 4, 8, 16, 32" progresses to "64," by doubling each previous number.
- The progression in "4, 8, 13, 19, 26, 34" results in "43," indicating a mixed increment approach.
Reasoning Processes in Studies
- A flu shot study involving 1328 patients demonstrates inductive reasoning, concluding that all patients should get vaccinated based on specific positive outcomes observed within the study group.
- Deductive reasoning is illustrated by the course policy on late submission penalties affecting report grades, concluding that a report turned in late is marked down following a general rule.
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Description
Explore the concepts of counterexamples and different types of reasoning in mathematics, including inductive and deductive reasoning. This quiz features scenarios from U.S. presidential history and numerical patterns that illustrate these principles. Test your understanding of how counterexamples can challenge conjectures and the role of reasoning in problem-solving.
