1. 2.1 Inductive Reasoning and Conjecture

2. Objectives Make conjectures based on inductive reasoning Find counterexamples Describing Patterns: Visual patterns are made by shapes Number patterns are made by numbers

3. Making Conjectures Conjecture – an educated guess based on known information It can be true or false; it is a unproven statement Inductive Reasoning – reasoning based on patterns you observe We use inductive reasoning to create a conjecture.

4. Making Conjectures Inductive Reasoning: Allows you to reach conclusions based on a pattern of specific examples. Ex: Each morning you see Miss Kaur drinking iced coffee. You see this everyday for 3 weeks. The next day what do you expect Miss Kaur to be drinking? *Does that mean it will definitely happen?

5. Make a conjecture about the next number based on the pattern. 2, 4, 12, 48, 240 Find a pattern: 2 4 12 48 240 ×2×2 The numbers are multiplied by 2, 3, 4, and 5. Conjecture: The next number will be multiplied by 6. So, it will be 6*240 or 1440. ×3×3×4×4×5×5 Example 1: Answer: 1440

6. Make a conjecture about the next number based on the pattern. **Hint** What is the relationship between 2 and 4? 3 and 9? Answer: The next number will be Your Turn:

7. Example 2 Make a conjecture about the next term based on the pattern. Answer:

8. Given: points L, M, and N; Examine the measures of the segments. Since the points can be collinear with point N between points L and M. Answer: Conjecture: L, M, and N are collinear. For points L, M, and N, and, make a conjecture and draw a figure to illustrate your conjecture. Example 2:

9. ACE is a right triangle with Make a conjecture and draw a figure to illustrate your conjecture. Answer: Conjecture: In  ACE,  C is a right angle and is the hypotenuse. Your Turn:

10. Using Inductive Reasoning What conjecture can you make about the number of regions after 20 diameters are drawn? ANSWER: Each circle has twice as many regions as diameter. Twenty diameters form: 2*20 = 40.

11. Make a Conjecture! R, W, B, R, W, B, R, W, B, … What will the 21 st term be? Every 3 rd term is B, so the 21 st term will be B. How can you tell if a term will be an R?

12. Collecting Info to Make a Conjecture What conjecture can you make about the sum of the first 30 even numbers? # of TermsSum 12= 2 = 1*2 22 + 4= 6 = 2*3 32 + 4 + 6= 12 =3*4 4 2 + 4 + 6 + 8 = 20 = 4*5 **Each sum is the product of the number of terms and the number of terms plus one. What about the first 30? 30*31 = 930

13. Collecting Info to Make a Conjecture What conjecture can you make about the product of two odd numbers?

14. Homework Page 85: 6-15 all 20-24 all

15. Use the Data Sales of backpacks at a nationwide company decreased over a period of six consecutive months. What conjecture can you make about the number of backpacks the company will sell in May? Answer: You can predict they will sell 8000 backpacks

16. Counterexamples Recall, conjectures are based on multiple observations. Whenever we are able to find an instance in which the conjecture is false, the entire conjecture is untrue. This false example is referred to as a counterexample. You only need one counterexample to prove a conjecture false!!!!

17. True or False All teachers are taller than students. Can anyone prove this? Counterexample: is something that disproves a conjecture. A counterexample can be a drawing, a statement, or a number. -Only one counterexample is needed to disprove a statement.

18. Prove me Wrong! If the name of a month starts with the letter J, it is a summer month. You can connect any three points to form a triangle.

19. Guided Practice Statement : If Jerry lives in North America, then he lives in Pennsylvania. Counterexample: Jerry can live in New Jersey Statement: When you multiply a number by 3, the product is divisible by 6? Counterexample When you multiply by any odd number, it is not divisible by 6

20. Examples: What are the next two terms in each sequence? 1) 7, 13, 19, 25,… 2) 3) What is a counterexample for the following conjecture? All four-sided figures are squares.

21. Homework Page 86: 31-45 all