2. GEOMETRY LESSON 1-1 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. 1 2 = 1, 2 2 = 4, 3 2 = 9, 4 2 = 16,... 4. Which do you think describes the square of any odd number? It is odd. It is even. Patterns and Inductive Reasoning 1-1 Here is a list of the counting numbers: 1, 2, 3, 4, 5,... Some are even and some are odd. Make sure your name is in your book!

3. 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. 1 2 = (1)(1) = 1; 2 2 = (2)(2) = 4; 3 2 = (3)(3) = 9; 4 2 = (4)(4) = 16; 5 2 = (5)(5) = 25; 6 2 = (6)(6) = 36; 7 2 = (7)(7) = 49; 8 2 = (8)(8) = 64; 9 2 = (9)(9) = 81; 10 2 = (10)(10) = 100 4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd. Solutions GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

4. 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

5. Chapter 1 Section 1 Goals: Use inductive reasoning to make a conjecture.

6. Vocabulary 1.1 Inductive Reasoning Conjecture Counterexample Prime Number investigating using the observation of patterns A conclusion reached based upon inductive observation An example that shows the conjecture is not correct A Positive number with no factors other than itself and 1. (The smallest prime number is 2.)

7. Each term is half the preceding term. So the next two terms are 48 ÷ 2 = 24 and 24 ÷ 2 = 12. Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. 384, 192, 96, 48, … GEOMETRY LESSON 1-1 Use Inductive Reasoning: 1-1 1.Write the sequence 2.What value is +,-,x, or ÷ each time? # 384 192 96 48 ÷2 24 ÷2 12

8. Make a conjecture about the sum of the cubes of the first 25 counting numbers. Find the first few sums. 1 3 = 1 1 3 + 2 3 = 9 1 3 + 2 3 + 3 3 = 36 1 3 + 2 3 + 3 3 + 4 3 = 100 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 225 The sum of the cubes equals the square of the sum of the counting numbers. GEOMETRY LESSON 1-1 Use Inductive Reasoning 1-1 = 1 2 = 3 2 = 6 2 = 10 2 = 15 2 = (1) 2 = (1+2) 2 = (1+2+3) 2 = (1+2+3+4) 2 = (1+2+3+4+5) 2

9. The first three odd prime numbers are 3, 5, and 7. Make and test a conjecture about the fourth odd prime number. The fourth prime number is 11. 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. GEOMETRY LESSON 1-1 Use Inductive Reasoning 1-1 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.

10. When points on a circle are joined, they produce unique regions within the circle: Points Regions 2 2 3 4 4 8 Will the # of regions always be twice as many as the previous number? 5 16 6 ?? 6 points yields 30 regions. 30 is NOT 2x16! Conjecture is false.

11. 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. Write the data in a table. Find a pattern. Each year the price increased by $1.50. A possible conjecture is that the price in 2003 will increase by $1.50. If so, the price in 2003 would be $11.00 + $1.50 = $12.50. GEOMETRY LESSON 1-1 Use Inductive Reasoning year $$ 20008.00 20019.50 200211.00 2003 + 1.50 12.50

12. Re-Cap Inductive Reasoning Conjecture Counterexample Prime Number Based upon observation of patterns A conclusion reached based upon inductive observation An example that shows the conjecture is not correct 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

13. Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2. Use the table and inductive reasoning. 3. Find the sum of the first 10 counting numbers. 4. Find the sum of the first 1000 counting numbers. Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number. –2160; 15,120 55 500,500 Sample: 2+3=5, and 5 is not even GEOMETRY LESSON 1-1 Additional Practice 1-1