2-12-1Patterns and Inductive Patterns and Inductive

ReasoningReasoning

Objective:Objective:

Use inductive reasoning to Use inductive reasoning to make conjecturesmake conjectures

• Inductive ReasoningReasoning based on patterns you observe.

• ConjectureA conclusion you reach using inductive reasoning.

ExampleA 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.

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.

• 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.

Find one counterexample to prove a conjecture is false.

CounterexampleAn example that shows that a conjecture is incorrect.

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.

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