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Conjecture and Inductive Reasoning: Patterns in Geometry Sequences

Explore inductive reasoning in geometry sequences to make conjectures and find counterexamples. Learn to identify patterns and make predictions using examples and real-world data. Homework included.

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Conjecture and Inductive Reasoning: Patterns in Geometry Sequences

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  1. Geometry Lesson 2 – 1 Inductive Reasoning and Conjecture Objective: Make conjectures based on inductive reasoning. Find counterexamples.

  2. Inductive Reasoning • Reasoning that uses a number of specific examples to arrive at a conclusion. • Reasoning that assumes a pattern will continue. • Conjecture – a conclusion you reach using inductive reasoning.

  3. Write a conjecture that describes the pattern. Then use your conjecture to find the next item in the sequence. • Movie show times 8:30 a.m., 9:45 a.m., 11:00 a.m., 12:15 p.m. … • 2, 4, 12, 48, 240 Movies show every 1 hour 15 minutes Next showing 1:30 p.m. x 5 x 2 x 3 x 4 Each term is being multiplied by 1 more than the previous term Next term 240(6) = 1440

  4. Write a conjecture that describes the pattern. Then use your conjecture to find the next item in the sequence. +6 +8 +10 +12 The patter is starting at 6 and increasing by 2. +14 will be next. Next figure will have 54 segments.

  5. Write a conjecture that describes the pattern. Then use your conjecture to find the next item in the sequence. • Follow up visits: Dec., May, Oct., March… • 10, 4, -2, -8 … Every 5 months Next month Aug. Minus 6 each time. Next term -14

  6. Make a conjecture about each value or geometric relationship. List or draw examples that support your answer. • The sum of two odd numbers • segments joining opposite vertices of a rectangle. Examples: 1+3 3+5, 15 + 17 The sum of two odd numbers is an even number Draw several rectangles… notice anything? The segments joining opposite vertices of a rectangle are congruent

  7. Real World • The table shows the price of postage for the years 1982 through 2007 • Construct a statistical graph that best displays the data Continued…

  8. Real world continued… • Predict the postage rate in 2013 based on the graph. From ’97 to ’07 cost Increased by about 10 cents. From ’07 to ’13 is 6 years Estimated 46 cents in 2013 *The cost in 2010 was 44 cents.

  9. Continued… • Does it make sense that the pattern of the data will continue over time? If not, how will it change? Explain your reasoning. Yes it makes sense that the pattern would continue

  10. Counterexample • Counterexample • A specific example that shows a conjecture is false.

  11. Find a counterexample to show that the conjecture is false. Counterexample: n = 1 12 > 1 • If n is a real number, then n2 > n. • If JK = KL, then K is the midpoint of JL. J K L

  12. Find a counterexample to show that the conjecture is false. • In n is a real number, then –n is negative. Sample: n = -2 -(-2) = 2

  13. Homework • Pg. 92 1 – 13 all, 14 – 42 EOE, 60 – 68 EOE

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