2-1 Patterns and Inductive Reasoning

1. 2-1Patterns and Inductive Reasoning Objective: Use inductive reasoning to make conjectures

2. Inductive Reasoning Reasoning based on patterns you observe. • Conjecture A conclusion you reach using inductive reasoning. Example A scientist dips a platinum wire into a solution containing salt (sodium chloride), passes the wire over a flame, and observes that it produces an orange-yellow flame. She does this with many other solutions that contain salt, finding that they all produce an orange-yellow flame. Conjecture If a solution contains sodium chloride, then in a flame test it produces an orange-yellow flame.

3. Example 2: Consider the sequence 2, 4, 7, 11, . . . Make a conjecture about the rule for generating the sequence. Then find the next three terms. Conjecture: You always add the next counting number to get the next term.

4. Patterns about 1st, 3rd, and 5th shape. • Half shaded circles, with the circle rotating a quarter turn counter clockwise from previous circle. • Patterns about 2nd, 4th, and 6th shape. • Polygons with consecutive odd numbered sides and a triangular pattern of dots, solid dot on top two hollow dots on bottom. • Draw next two shapes.

5. Counterexample An example that shows that a conjecture is incorrect. Find one counterexample to prove a conjecture is false. If the name of a month starts with the letter J, it is a summer month. Counterexample: January starts with the letter J and it is a winter month.

6. What is a counterexample for each conjecture? • If a flower is red, it is a rose. • When you multiply a number by 3, the product is divisible by 6. p. 85: 7-23 odd, 31-39 odd