2-1: Use Inductive reasoning Geometry Chapter 2 2-1: Use Inductive reasoning

Warm-up Examine the 3 figures given below. Describe how to sketch the fourth figure in the pattern. Then sketch the fourth pattern on your whiteboards

2-1: Use inductive reasoning Objective: Students will be able to describe patterns and use inductive reasoning to make conjectures. Agenda Inductive Reasoning Conjecture Counterexample

Conjecture You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. A conjecture is an unproven statement that is based on observations. Example: Over the last four games, your favorite football team’s results have been win, loss, win, loss. What will be the result of their fifth game. If you said “win”, then you just used inductive reasoning.

Example 2 Example 2: Describe the pattern in the numbers −7, −21, −63, −189…. Next, give the next three numbers in the pattern Pattern: You multiply the previous number by 3 to get the next number. Next 3 Numbers: −567, −1701, −5103

Make and test a conjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers. Give Examples: Try adding some sets of consecutive integers Look for a Pattern: 12=4×3 3+4+5=12 15=5×3 4+5+6=15 24=8×3 7+8+9=24

Make and test a conjecture Make a Conjecture: The sum of consecutive numbers is always 3 times the value of the second of the three numbers added. Test your conjecture: Try it out on a few more examples 6+7+8=21 =7×3 21+22+23=66 =22×3 10+11+12=33 =11×3 25+26+27=78 =26×3 14+15+16=45 =15×3 32+33+34=99 =33×3

Counterexample To show that a conjecture is true, you must show that it is true for all cases. However, if you want to show that conjecture is false, you simply need to find one counterexample.

Find a Counterexample Find a counterexample to disprove the following conjecture: Conjecture: The sum of two numbers is always greater than the larger of the two numbers being added. Example: 2+3=5 In this example, 𝟑 is the larger number in the addition, 𝟓 is the sum, and 5>3. This demonstrates the conjecture.

Find a Counterexample To disprove this conjecture, find an example of a sum that is less than the larger number. Counterexample: −2+ −3 =−5 In this example, −𝟐 is the larger number in the addition, and −𝟓 is the sum. But −𝟐>−𝟓. This disproves the conjecture.

Make your own Patterns and Conjectures With a partner, come up with your own example of a statement that describes a pattern. It can be described in words, such as in example 1, or with a sequence of numbers, like example 2. I will select groups to share their statements with the class.

H.w. 2-1 Pgs. 67 – 68 #’s 1, 3, 5, 6-11, 13, 14-17, 18, 19, 22 (EC) – 23, 24, 26, 27