glesson2.1.ppt
- 1. Chapter 2 Reasoning and Proof Lesson 2.1 Use Inductive Reasoning
- 2. What will you learn? • To describe patters and use inductive reasoning, so you can make predictions • CA Standard 1 • Students demonstrate understanding by identifying and giving examples of inductive reasoning • CA Standard 3 • Students give counterexamples to disprove a statement
- 3. Describing a visual pattern • Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure • Each circle is divided into twice as many regions as the figure number
- 4. Describing a number pattern • Describe a pattern in the numbers -7, -21, -63, -189, … and write the next three numbers in the pattern • Each number is 3 times the previous number • The next three numbers are -567, -1701, and -5103
- 5. Conjecture • An unproven statement that is based on observations
- 6. Inductive Reasoning • When you find a pattern in specific cases and then write a conjecture for the general case
- 7. Example • Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points. • Conjecture: You can connect five collinear points 6 + 4 or 10 different ways
- 8. Example • Numbers such as 3, 4 and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers • Find a pattern using small groups of numbers • 3 + 4 + 5 = 12 = 4 ● 3 • 7 + 8 + 9 = 24 = 8 ● 3 • 10 + 11 + 12 = 33 = 11 ● 3 • Conjecture: The sum of any three consecutive integers is three times the second number • Test your conjecture • -1 + 0 + 1 = 0 = 0 ● 3 √ • 100 + 101 + 102 = 303 = 101 ● 3 √
- 9. Counterexample • A specific case for which the conjecture is false
- 10. Disproving conjectures • To show that a conjecture is true, you must show that it is true for all cases • You can show that a conjecture is false by simply finding one counterexample
- 11. Example • Conjecture: The sum of two numbers is always greater than the larger number • Find a counterexample to disprove the student’s conjecture • -2 + -3 = -5 • -5 > -2
- 12. Example • Which conjecture could a high school athletic director make based on the graph at the right? 1. More boys play soccer than girls 2. More girls are playing soccer today than in 1995 3. More people are playing soccer today than in the past because the 1994 World Cup games were held in the US 4. The number of girls playing soccer was more in 1995 than in 2001
- 13. You now know, • What is a conjecture • Unproven statement based on observations • What is inductive reasoning • Reasoning in which you find patterns and make conjectures • What is a counterexample • A specific case that proves a conjecture false