1. 2.1 Inductive Reasoning Review Geometry R/H

3. Inductive Reasoning When you make a prediction based on several examples, you are applying inductive reasoning. Formal Definition: Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases (examples that you observe) are true.

4. Making a Conjecture A statement you believe to be true based on inductive reasoning is called a conjecture. Example 2: 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 The sum of two positive numbers is positive.

5. Counterexamples To show that a conjecture is true you must prove it, but to show a conjecture is false, you need only to find one example for which the conjecture is false. This is called a counterexample. In the following conjecture, show that it is not true by finding a counterexample. For any real number x, x 2 ≥ x A counterexample could be x = ½ x 2 = (½) 2 = ¼ ¼ ≥ ½ therefore, the conjecture is not true.

7. Steps in Inductive Reasoning Step 1: Look for a Pattern Step 2: Make a Conjecture Step 3: Is it True? Verify the Conjecture Find a Counter- example yesno

8. Find the next item in each pattern. 1. 0.7, 0.07, 0.007, … 2. Review of 2.1 0.0007 Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. 4. Every prime number is odd. 5. Two supplementary angles are not congruent. 6. The square of an odd integer is odd. false; 2 true false; 90° and 90° true

9. Go over Sequences Packet Arithmetic, Quadratic, Triangular, Geometric Sequences ** (NOT in Book)

10. 2.2 Logic and Venn Diagrams Geometry R/H

12. Conjunction and Disjunction Let p: I am tired. Let q: I will fall asleep

13. Now you try! Let p: Let q:

14. A Venn Diagram is a diagram made of inter- locking circles that we can use to solve a logic problem. Let’s try solving the following problem using a Venn Diagram: In a class of 50 students, 18 take Chorus, 26 take Band, and 2 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band? What is a Venn Diagram?

15. Here’s the Venn Diagram: First we start with a rectangle to represent the whole class of 50 Add a circle for the students in Chorus (18 people will be in this circle.) Add a circle for the students taking Band (26 people will be in this circle.) Chorus Band Total = 50

16. How do we solve the problem? Start with the number of students taking both classes – that will have to be inside both circles. Since we now have 2 people inside the circle of chorus, how many more do we need to add to How many do we need to add to have a total of 26 in band? make a total of 18? Chorus Band 224 16 Total = 50

17. How do we solve the problem? We now have a total of 16 + 2 + 24 or 42 students that are either taking band or chorus. There are 50 students in the class, so that means that 8 people are not enrolled in either band or chorus. We place those 8 people in this area. 8 Chorus Band 224 16 Total = 50

18. 19 1855 ChocolateVanilla 12 We add all the numbers together to get the total number of people in the survey. That would be? A survey is taken at an ice cream parlor. People are asked to list their two favorite flavors. 74 list vanilla as one of their favorite flavors while 37 list chocolate. If 19 list both flavors and 12 list neither of these two flavors, how many people participated in the survey? 104

19. That is 25. But how can we divide 25 into the two circles? Thirty students went to lunch. Pizza and ham- burgers were offered. 15 students chose pizza, and 13 chose hamburgers. 5 students chose neither. How many students chose only hamburgers? If we subtract 5 from 30 we will get the number of people that should be in one of the circles. 5 Pizza Burgers Total = 30

20. If we add the 15 people who wanted pizza to the 13 who wanted hamburgers, we get 28. This is 3 more than the total of 25 that we should have in one of the circles This means that those 3 people are being counted twice and so is the number that belongs in the center. 5 Pizza Burgers Total = 30 Finish the diagram, and answer the question. The students that chose only hamburgers is 10. 3 1012