1. Warm-up August 22, 2011 Evaluate the following expressions

2. Outcomes I will be able to: 1) Explain Inductive Reasoning by identifying the three steps. 2) Identify patterns in number sequences and pictures. 3) Make Conjectures based on observed data. 4) Define and Create Counterexamples.

3. White Boards Find the next three number in the sequence: 1) 1, 2, 4, 7, 11… 2) 1, 1, 2, 3, 5, 8, 13… 3) 2, 4, 8, 16… 4) 23, 19, 15, 11…

4. Inductive Reasoning Conjecture – an unproven statement based on observations. Conjectures can be modified until they are concrete ***The process of describing what is being observed

5. Inductive Reasoning Inductive Reasoning – Observing data, recognizing patterns, and making generalizations about that data We use inductive reasoning everyday Can you think of a few examples where you may use inductive reasoning?

6. Inductive Reasoning 3 Stages of Inductive Reasoning 1) Look for a pattern – Look at examples and use diagrams, tables, and pictures to help discover a pattern. 2) Make a conjecture - Use your observations to make “guess” about the pattern. 3) Verify the conjecture - Use logical reasoning skills to decide if your conjecture is valid.

7. Using Inductive Reasoning

9. Example 3. Predict the next three numbers. (a) 17, 15, 12, 8,…. (b) 64, 16, 4, 1, ¼,…. (c) 48, 16, 16/3, 16/9, …. (d) 4, -6, 8, -10, …

10. Finding and Describing Patterns on White Boards Ex 1: What will the 5 th and 6 th shapes in the pattern look like? Describe the pattern that is happening

11. Finding and Describing Patterns on White Boards Ex 2: What will the 4 th term in the sequence look like? Describe the pattern

12. Finding and Describing Patterns on White Boards Describe the pattern in the sequence below. Predict the next number a) 1, 4, 16, 64… b) -5, -2, 4, 13…

13. Making A Conjecture Example 1. Pick a secret number. Add the next highest number to it. Add 9. Divide by 2. Subtract your secret number. What is your conjecture?

14. Making a Conjecture Example 2. Pick a secret number. Add 5. Multiply by 2. Subtract 4. Divide by 2. Subtract your secret number. What is your conjecture?

15. Finding a Counterexample On White Boards Example: For all real numbers x, the expression x³ is greater than or equal to x. When does this not happen?

16. Counterexample Counterexample – an example that shows that a conjecture is false Unproven or undecided conjectures – Conjectures that have not been proven true or false

17. Finding Counterexamples Example 1. Show the conjecture is false by finding a counterexample: “The difference of two positive integers is always positive.”

18. Finding Counterexamples Example 2. Show the conjecture is false by finding a counterexample: “For all real numbers x, the expression x 2 is greater than or equal to x.”

19. Finding a Conjecture based on Pattern Example 3. Complete the conjecture: The sum of the first n odd positive integers is ____________________. first odd positive integer: sum of first two odd positive integers: sum of first three odd positive integers: sum of first four odd positive integers: sum of first five odd positive integers:

20. Exit Quiz Find the next two terms in each pattern 1) 2) 3) Find the missing term 4) Make a conjecture about the sums of any two odd numbers 1+1 = 2 3 + 7 = 10 1 + 3 = 4 5 + 9 = 14 3 + 5 = 8 7 + 9 = 16