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Use inductive reasoning to identify patterns and make conjectures.

Objectives. Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures. Example 1A: Identifying a Pattern. Find the next item in the pattern. January, March, May,. Example 1B: Identifying a Pattern. Find the next item in the pattern.

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Use inductive reasoning to identify patterns and make conjectures.

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  1. Objectives Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures.

  2. Example 1A: Identifying a Pattern Find the next item in the pattern. January, March, May, ...

  3. Example 1B: Identifying a Pattern Find the next item in the pattern. 7, 14, 21, 28, …

  4. Example 1C: Identifying a Pattern Find the next item in the pattern.

  5. _________________- the process of reasoning that a rule or statement is true because specific cases are true. _________________- a statement you believe to be true based on inductive reasoning. When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. You may use inductive reasoning to draw a conclusion from a pattern.

  6. Example 2A: Making a Conjecture Complete the conjecture. The sum of two positive numbers is ? . List some examples and look for a pattern. 1 + 1 = 2 3.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917

  7. Example 2B: Making a Conjecture Complete the conjecture. The number of lines formed by 4 points, no three of which are collinear, is ? . Draw four points. Make sure no three points are collinear. Count the number of lines formed:

  8. Example 2C Complete the conjecture. The product of two odd numbers is ? .

  9. Example 3 Make a conjecture about the lengths of male and female whales based on the data.

  10. To show that a conjecture is always true, you must prove it. _________________ - one example in which to show that a conjecture is false. A counterexample can be a drawing, a statement, or a number.

  11. Example 4A: Finding a Counterexample Show that the conjecture is false by finding a counterexample. For every integer n, n3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds.

  12. Example 4B: Finding a Counterexample Show that the conjecture is false by finding a counterexample. Two complementary angles are not congruent.

  13. Example 4C: Finding a Counterexample Show that the conjecture is false by finding a counterexample. The monthly high temperature in Abilene is never below 90°F for two months in a row.

  14. Check It Out! Example 4C Show that the conjecture is false by finding a counterexample. For any real number x, x2 ≥ x.

  15. Check It Out! Example 4D Show that the conjecture is false by finding a counterexample. Supplementary angles are adjacent.

  16. Check It Out! Example 4c Show that the conjecture is false by finding a counterexample. The radius of every planet in the solar system is less than 50,000 km.

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