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Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 (For help, go the Skills Handbook, page 715.) Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . . Some are even and some are odd. 1. Make a list of the positive even numbers. 2. Make a list of the positive odd numbers. 3. Copy and extend this list to show the first 10 perfect squares. 12 = 1, 22 = 4, 32 = 9, 42 = 16, . . . 4. Which do you think describes the square of any odd number? It is odd. It is even. Make sure your name is in your book! 1-1 Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Solutions 1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . . 2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, . . . 3. 12 = (1)(1) = 1; 22 = (2)(2) = 4; 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 62 = (6)(6) = 36; 72 = (7)(7) = 49; 82 = (8)(8) = 64; 92 = (9)(9) = 81; 102 = (10)(10) = 100 4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd. 1-1 What is Geometry •Geometry is more than the study of shapes. It is the study of truths. •These truths are constant, no matter what the situation. •Geometry uses reason (logic) to prove truths and to build upon them to prove even more truths. •The study of geometry is a study of how to think logically. There are two types of logical strategies: 1. Inductive Reasoning 2. Deductive Reasoning Chapter 1 Section 1 • Goals: • Use inductive reasoning to make a conjecture. Vocabulary 1.1 •Inductive Reasoning •investigating using the observation of patterns •Conjecture •A conclusion reached based upon inductive observation •Counterexample •An example that shows the conjecture is not correct •Prime Number •A Positive number with no factors other than itself and 1. (The smallest prime number is 2.) Use Inductive Reasoning: GEOMETRY LESSON 1-1 Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. 384, 192, 96, 48, … # 1. Write the sequence 384 ÷2 2. What value is +,-,x, or ÷ each 192 ÷2 time? 96 ÷2 48 ÷2 24 ÷2 12 Each term is half the preceding term. So the next two terms are 48 ÷ 2 = 24 and 24 ÷ 2 = 12. 1-1 Use Inductive Reasoning GEOMETRY LESSON 1-1 Make a conjecture about the sum of the cubes of the first 25 counting numbers. Find the first few sums. 13 =1 = 12 = (1)2 13 + 23 =9 = 32 = (1+2)2 13 + 23 + 33 = 36 = 62 = (1+2+3)2 13 + 23 + 33 + 43 = 100 = 102 = (1+2+3+4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1+2+3+4+5)2 The sum of the cubes equals the square of the sum of the counting numbers. 1-1 Use Inductive Reasoning GEOMETRY LESSON 1-1 The first three odd prime numbers are 3, 5, and 7. Make and test a conjecture about the fourth odd prime number. One pattern of the sequence is that each term equals the preceding term plus 2. So a possible conjecture is that the fourth prime number is 7 + 2 = 9. However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false. By applying the assumed pattern and then testing the result against the initial directions, we have found a counterexample. A counterexample applies the presumed pattern and gives a false result. Only ONE counterexample is needed to prove a conjecture is false. Conjecture: odd prime numbers are found by adding 2 to each odd prime. Counterexample: 7 is odd prime, 7+2 = 9, 9 is not prime. Result: Conjecture is false. The fourth prime number is 11. 1-1 When points on a circle are joined, they produce unique regions within the circle: Points Regions 2 2 3 4 4 8 5 16 6 ?? Will the # of regions always be twice as many as the previous number? 6 points yields 30 regions. 30 is NOT 2x16! Conjecture is false. Use Inductive Reasoning GEOMETRY LESSON 1-1 The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003. year $$ Write the data in a table. Find a pattern. 2000 8.00 + 1.50 Each year the price increased by $1.50. 2001 9.50 + 1.50 A possible conjecture is that the 2002 11.00 + 1.50 price in 2003 will increase by $1.50. 2003 12.50 If so, the price in 2003 would be $11.00 + $1.50 = $12.50. Re-Cap •Inductive Reasoning •Based upon observation of patterns •Conjecture •A conclusion reached based upon inductive observation •Counterexample •An example that shows the conjecture is not correct •Prime Number •Number with no factors other than itself and 1. Tips for Inductive Reasoning: •Make a list •Make a table when comparing two sets of numbers •Look for simple numbers patterns Additional Practice GEOMETRY LESSON 1-1 Find a pattern for each sequence. Use the table and inductive reasoning. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 –2160; 15,120 3. Find the sum of the first 10 counting numbers. 55 4. Find the sum of the first 1000 2. counting numbers. 500,500 Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number. Sample: 2+3=5, and 5 is not even 1-1 Homework 1.1 Homework, due at the beginning of the NEXT class: page 6 Name Section # Page # Remember, this is an Honor Code Show your work Make sure your School! here IN PENCIL name is in your book! No Copying! I pledge that I have neither given nor received aid on this assignment Check in ink! GEOMETRY LESSON 1-1 Pages 6–9 Exercises 1. 80, 160 12. 1 , 1 19. The sum of the first 6 pos. 5 6 even numbers is 2. 33,333; 333,333 13. James, John 6 • 7, or 42. 3. –3, 4 14. Elizabeth, Louisa 4. 1, 1 15. Andrew, Ulysses 20. The sum of the first 30 pos. 16 32 even numbers is 5. 3, 0 16. Gemini, Cancer 30 • 31, or 930. 6. 1, 1 17. 3 7. N, T 21. The sum of the first 100 pos. even numbers is 8. J, J 18. 100 • 101, or 10,100. 9. 720, 5040 10. 64, 128 11. 1 , 1 36 49 1-1 Check in ink! GEOMETRY LESSON 1-1 22. The sum of the first 28. 1 ÷1 =3 and3 is 31. 31, 43 2 3 2 2 100 odd numbers is improper. 1002, or 10,000. 32. 10, 13 29. 75°F 33. 0.0001, 0.00001 23. 555,555,555 30. 40 push-ups; 24. 123,454,321 34. 201, 202 answers may vary. Sample: Not very 35. 63, 127 25–28. Answers may vary. Samples are given. confident, Dino may 36. 31 , 63 32 64 reach a limit to the 37. J, S 25. 8 + (–5 = 3) and 3 > 8 / number of push-ups 26. 1 • 1 > 1 and 1 • 1 > 1 he can do in his 38. CA, CO / / 2 3 2 3 3 2 27. –6 – (–4) < –6 and allotted time for 39. B, C exercises. –6 – (–4) < –4 1-1 Check in ink! GEOMETRY LESSON 1-1 40. Answers may vary. 42. 47. Answers may vary. Sample: In Exercise Samples are given. 31, each number a. Women may soon outrun increases by increasing men in running competitions. multiples of 2. In Exercise 43. b. The conclusion was based 33, to get the next term, on continuing the trend divide by 10. 44. shown in past records. c. The conclusions are 41. based on fairly recent records for women, 45. and those rates of improvement may not You would get a third line continue. The conclusion between and parallel to about the marathon is most the first two lines. 46. 102 cm suspect because records date only from 1955. 1-1 Check in ink! GEOMETRY LESSON 1-1 48. a. 50. His conjecture is 52. 21, 34, 55 probably false because most 53. a. Leap years are years people’s growth that are divisible by 4. slows by 18 until they stop growing b. 2020, 2100, and 2400 somewhere between b. about 12,000 radio 18 and 22 years. c. Leap years are years stations in 2010 divisible by 4, except c. Answers may vary. 51. a. the final year of a Sample: Confident; century which must the pattern has held be divisible by 400. for several decades. So, 2100 will not be a leap year, but 2400 49. Answers may vary. will be. Sample: 1, 3, 9, 27, b. H and I 81, . . . 1, 3, 5, 7, 9, . . . c. a circle 1-1 Check in ink! GEOMETRY LESSON 1-1 54. Answers may vary. 55. (continued) Sample: d. 100 + 99 + 98 + … + 3 + 2 + 1 1 + 2 + 3 + … + 98 + 99 + 100 101 + 101 + 101 + … + 101 + 101 + 101 56. B The sum of the first 100 numbers is 57. I 100 • 101 , or 5050. 2 The sum of the first n numbers is n(n+1) . 58. [2] a. 25, 36, 49 2 b. n2 55. a. 1, 3, 6, 10, 15, 21 [1] one part correct b. They are the same. c. The diagram shows the product of n and n + 1 divided by 2 when n = 3. The result is 6. 1-1 Check in ink! GEOMETRY LESSON 1-1 59. [4] a. The product of 11 59. (continued) 60-67. and a three-digit [3] minor error in number that begins explanation and ends in 1 is a four-digit number [2] incorrect description that begins and ends in part (a) in 1 and has middle digits that are each [1] correct products for one greater than the (151)(11), (161)(11), middle digit of the and (181)(11) three-digit number. (151)(11) = 1661 (161)(11) = 1771 68. B b. 1991 69. N c. No; (191)(11) = 2101 70. G 1-1 Coordinate Assignments Write your coordinate down. You will use this all year long! Put it on your math folder! (1,-1) (-1,2) (-1,-3) (1,4) (1,0) (2,-1) (-2,2) (-2,-3) (2,4) (0,5) (3,-1) (-3,2) (-3,-3) (3,4) (-3,0) (4,-1) (-4,2) (-4,-3) (4,4) (0,-5)