1. Chapter 1 Problem Solving Section 1-1 The Nature of Mathematical Reasoning
Objectives:
Identify and explain differences between types of reasoning
Use inductive reasoning to form conjectures
Find counterexamples
Use deductive reasoning to prove conjectures

2. Vocabulary
Inductive reasoning – the process of reasoning that arrives at a general conclusion based on the observation of specific examples
Deductive reasoning – the process of reasoning that arrives at a conclusion based on previously accepted general statements (facts). It does not rely on specific examples.

3. Example 1: Using Inductive Reasoning to Find a Pattern
Use inductive reasoning to find a pattern, and then find the next three numbers by using that pattern. 1, 2, 4, 5, 7, 8, 10, 11, 13, … 1, 4, 2, 5, 3, 6, 4, 7, 5, …

4. Example 2: Using Inductive Reasoning to Find a Pattern
Make a reasonable conjecture for the next figure in the sequence.

5. Example 3: Using Inductive Reasoning to Make a Conjecture
When two odd numbers are added, will the result always be an even number? Use inductive reasoning to determine your answer. If two odd numbers are multiplied, is the result always odd, always even, or sometimes odd and sometimes even? Use inductive reasoning to answer.

6. Example 4: Using Inductive Reasoning to Test a Conjecture
Use inductive reasoning to decide if the following conjecture is true: If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.

7. Example 5: Finding a Counterexample
Find a counterexample that proves the conjecture below is false. Conjecture: A number is divisible by 3 if the last two digits are divisible by 3. Conjecture: The name of every month in English contains either the letter y or the letter r.

8. Example 6: Making and Testing a Conjecture
Use inductive reasoning to make a conjecture about the number of sections a circle is divided into when a given number of points on the circle are connected by chords. Then test the conjecture with one further example.

9. Example 7: Using Deductive Reasoning to Prove a Conjecture
Consider the following problem: Think of any number. Multiply that number by 2, then add 6, and divide the result by 2. Next subtract the original number. What is the result?
Use inductive reasoning to make a conjecture for the answer.
Use deductive reasoning to prove your conjecture.

10. Example 7: Using Deductive Reasoning to Prove a Conjecture
Consider the following problem: Think of any number. Multiply that number by 3, then add 30, and divide the result by 3. Next subtract the original number. What is the result?
Use inductive reasoning to make a conjecture for the answer.
Use deductive reasoning to prove your conjecture.

11. Example 8: Using Deductive Reasoning to Prove a Conjecture
Use inductive reasoning to arrive at a general conclusion, and then prove your conclusion is true by using deductive reasoning. Select a number: Add 50: Multiply by 2: Subtract the Original Number: Result:

12. Example 9: Comparing Inductive and Deductive Reasoning
Determine whether the type of reasoning used is inductive or deductive. The last six times we played our archrival in football, we, won, so I know we’re going to win on Saturday. There is no mail delivery on holidays. Tomorrow is Thanksgiving, so I know my student loan check won’t be delivered.

13. Example 10: Comparing Inductive and Deductive Reasoning
Determine whether the type of reasoning used is inductive or deductive. The syllabus states that any final average between 80% and 90% will result in a B. If I get 78% on my final, my overall average will be 80.1%, so I’ll get a B. Everyone I know in my sorority got at least a 2.5 GPA last semester, so I’m sure I’ll get at least a 2.5 this semester.

14. On the SAT

15. On the SAT

16. On the SAT

17. Homework
P. 12 #7, 8, 9, 13, 14, 15, 17, 19, 32, 41, 43, 48, 69ab