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Exploring Conjectures and Reasoning in Geometry

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This guide delves into the key concepts of conjectures and reasoning in geometry. It explains the difference between inductive and deductive reasoning, providing clear definitions and examples. Learn how to make and test conjectures about odd/even integers, and discover counterexamples that refute common misconceptions, such as the belief that all odd numbers are prime. The text also covers conditional statements, converses, and equivalent statements, emphasizing their importance in logical reasoning and mathematical proofs.

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Exploring Conjectures and Reasoning in Geometry

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  1. Geometry

  2. Vocabulary Conjecture- an unproven statement that is based on observation. Inductive Reasoning- when you find a pattern in specific cases and then write a conjecture for the general case. Deductive Reasoning- Uses facts, definitions, and /or accepted properties Counter example- a specific case for which a conjecture is false.

  3. Make and test a conjecture about the product of any two odd integers. Step 1: find a pattern using a few groups of small numbers

  4. 3 x 13 = 39 7 x 21 = 147 5 x 9 = 45 11 x 9 = 99 Conjecture:

  5. Step 2: Test your conjecture using other numbers. For example, test that it works with pairs 17, 19, and 23, 31.

  6. Make and test a conjecture about the product of any two even numbers.

  7. Make and test a conjecture about the sum of an even integer and an odd integer

  8. Find a counterexample to show that the conjecture is false. Conjecture: All odd numbers are prime. Solution: To find a counterexample, you need to find an odd number that is a composite number.

  9. Find a counterexample to show that the conjecture is false. Conjecture: The difference of two positive numbers is always positive.

  10. Page: red book 217-219 1-21

  11. Conditional Statement- a logical statement that has two parts, a hypothesis ( the if part) and a conclusion ( the then part). p → q Example: If today is Tuesday, then tomorrow is Wednesday. If it is 12:01 pm, then I am hungry.

  12. Converse- formed by switching the hypothesis and the conclusion. q → p If tomorrow is Wednesday, then today is Tuesday.

  13. Negation- the opposite of the original statement. ~ • Inverse- Negates both the hypothesis and the conclusion. ~p → ~q • If today is not Tuesday, then tomorrow is not Wednesday.

  14. Contrapositive- Switch the hypothesis and conclusion and negate both of them. If tomorrow is not Wednesday, then today is not Tuesday. ~q →~p

  15. Equivalent statements- Two statements that are both true or both false. ****A conditional and its contrapositive are equivalent statement.******** Perpendicular lines- Two lines that intersect to form a right angle.

  16. Biconditional statement- A statement that contains the phrase “if and only if”. • Today is Tuesday if and only if tomorrow is Wednesday. • Tomorrow is Wednesday if and only if today is Tuesday. • Red book page 224-2

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