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Geometry Section 2-1 1112

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Inductive Reasoning and Conjecture

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Geometry Section 2-1 1112

1. 1. CHAPTER 2 Reasoning and Proof Wednesday, October 15, 14
2. 2. SECTION 2-1 Inductive Reasoning and Conjecture Wednesday, October 15, 14
3. 3. ESSENTIAL QUESTIONS How do you make conjectures based on inductive reasoning? How do you find counterexamples? Wednesday, October 15, 14
4. 4. VOCABULARY 1. Inductive Reasoning: 2. Conjecture: 3. Counterexample: Wednesday, October 15, 14
5. 5. VOCABULARY 1. In d u c t i v e R e a s o n i n g : Coming to a conclusion based off of specific examples and observations 2. Conjecture: 3. Counterexample: Wednesday, October 15, 14
6. 6. VOCABULARY 1. In d u c t i v e R e a s o n i n g : Coming to a conclusion based off of specific examples and observations 2. C o n j e c t u r e : The conclusion reached from using inductive reasoning 3. Counterexample: Wednesday, October 15, 14
7. 7. VOCABULARY 1. In d u c t i v e R e a s o n i n g : Coming to a conclusion based off of specific examples and observations 2. C o n j e c t u r e : The conclusion reached from using inductive reasoning 3. C o u n t e r e x a m p l e : A false example that shows the conjecture is not true Wednesday, October 15, 14
8. 8. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. a. 7, 10, 13, 16 b. 3, 12, 48, 192 Wednesday, October 15, 14
9. 9. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. a. 7, 10, 13, 16 b. 3, 12, 48, 192 Conjecture: The nth term is found by adding 3 to the previous term Wednesday, October 15, 14
10. 10. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. a. 7, 10, 13, 16 b. 3, 12, 48, 192 Conjecture: The nth term is found by adding 3 to the previous term 19 Wednesday, October 15, 14
11. 11. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. a. 7, 10, 13, 16 b. 3, 12, 48, 192 Conjecture: The nth term is found by adding 3 to the previous term 19 Conjecture: The nth term is found by multiplying the previous term by 4 Wednesday, October 15, 14
12. 12. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. a. 7, 10, 13, 16 b. 3, 12, 48, 192 Conjecture: The nth term is found by adding 3 to the previous term 19 Conjecture: The nth term is found by multiplying the previous term by 4 768 Wednesday, October 15, 14
13. 13. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. c. Wednesday, October 15, 14
14. 14. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. c. Conjecture: The nth figure is found by rotating the previous figure 90° counterclockwise Wednesday, October 15, 14
15. 15. EXAMPLE 1 Write the conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. c. Conjecture: The nth figure is found by rotating the previous figure 90° counterclockwise Wednesday, October 15, 14
16. 16. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. a. The product of an odd and even number Wednesday, October 15, 14
17. 17. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. a. The product of an odd and even number Test out some examples to see what happens Wednesday, October 15, 14
18. 18. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. a. The product of an odd and even number Test out some examples to see what happens 3*2 = 6 Wednesday, October 15, 14
19. 19. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. a. The product of an odd and even number Test out some examples to see what happens 3*2 = 6 17*4 = 68 Wednesday, October 15, 14
20. 20. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. a. The product of an odd and even number Test out some examples to see what happens 3*2 = 6 17*4 = 68 5*10 = 50 Wednesday, October 15, 14
21. 21. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. a. The product of an odd and even number Test out some examples to see what happens 3*2 = 6 17*4 = 68 5*10 = 50 Conjecture: The product of an odd and even number will be even Wednesday, October 15, 14
22. 22. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. b. The radius and diameter of a circle Wednesday, October 15, 14
23. 23. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. b. The radius and diameter of a circle Test out some examples to see what happens Wednesday, October 15, 14
24. 24. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. b. The radius and diameter of a circle Test out some examples to see what happens Wednesday, October 15, 14
25. 25. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. b. The radius and diameter of a circle Test out some examples to see what happens Wednesday, October 15, 14
26. 26. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. b. The radius and diameter of a circle Test out some examples to see what happens Wednesday, October 15, 14
27. 27. EXAMPLE 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. b. The radius and diameter of a circle Test out some examples to see what happens Conjecture: The diameter is twice as long as the radius Wednesday, October 15, 14
28. 28. EXAMPLE 3 The table shows the total sales for the first three months that Matt Mitarnowski’s Wonderporium is open. Matt wants to predict the sales for the fourth month. Month Sales 1 2 3 \$400 \$800 \$1600 Make a conjecture about the sales in the fourth month and justify your claim. Wednesday, October 15, 14
29. 29. EXAMPLE 3 The table shows the total sales for the first three months that Matt Mitarnowski’s Wonderporium is open. Matt wants to predict the sales for the fourth month. Month Sales 1 2 3 \$400 \$800 \$1600 Make a conjecture about the sales in the fourth month and justify your claim. Conjecture: The sales double each month, so the fourth month should have \$3200 in sales Wednesday, October 15, 14
30. 30. EXAMPLE 4 Based on the table showing unemployment rates for various counties in Texas, find a counterexample for the following statement: The unemployment rate is highest in the cities with the most people. County Population Rate Armstrong Cameron El Paso Hopkins Maverick Mitchell 2,163 371,825 713,126 33,201 50,436 9,402 3.7% 7.2% 7.0% 4.3% 11.3% 6.1% Wednesday, October 15, 14
31. 31. EXAMPLE 4 Based on the table showing unemployment rates for various counties in Texas, find a counterexample for the following statement: The unemployment rate is highest in the cities with the most people. County Population Rate Armstrong Cameron El Paso Hopkins Maverick Mitchell 2,163 371,825 713,126 33,201 50,436 9,402 3.7% 7.2% 7.0% 4.3% 11.3% 6.1% Wednesday, October 15, 14
32. 32. EXAMPLE 4 Based on the table showing unemployment rates for various counties in Texas, find a counterexample for the following statement: The unemployment rate is highest in the cities with the most people. County Population Rate Armstrong Cameron El Paso Hopkins Maverick Mitchell 2,163 371,825 713,126 33,201 50,436 9,402 3.7% 7.2% 7.0% 4.3% 11.3% 6.1% Wednesday, October 15, 14
33. 33. PROBLEM SET Wednesday, October 15, 14
34. 34. PROBLEM SET p. 92 #1-45 odd, 50 “Success is the sum of small efforts, repeated day in and day out.” - Robert Collier Wednesday, October 15, 14