Patterns and Inductive Reasoning

1. Patterns and Inductive Reasoning BIG IDEA: REASONING AND PROOF ESSENTIAL UNDERSTANDINGS: Patterns in some number sequences and some sequences of geometric figures can be used to discover relationships. Conjectures are not valid unless they are proven true. One counterexample can prove that a conjecture is false. MATHEMATICAL PRACTICE: Look for and make use of structure

2. Getting Ready • Fold a piece of paper in half. When you unfold it, the paper is divided into two rectangles. Refold the paper, and then fold it in half again. This time when you unfold it, there are four rectangles. How many rectangles would you get if you folded a piece of paper in half eight times? Explain

3. Recognize and Describe patterns • Inductive Reasoning: a type of reasoning that reaches ____________________ based on a _______________ of specific examples or past _____________. A process of identifying a ______________ based on limited data. • Deductive Reasoning: a process of reasoning _____________ from given ____________ to a conclusion. A process of concluding that a relationship is __________ because it is a __________________ case of a general principle that is known to be true. • Both types of reasoning are important to mathematical thinking.

4. EX 1: Sketch the next figure in the pattern • a) • b)

5. EX 2: Describe a pattern in the sequence of numbers. Predict the next two numbers. • c) • d) • e)

6. Recognize and Describe patterns • Mathematicians observe particular cases and then use inductive reasoning to make conjectures. Conjectures are proved to be true by means of deductive reasoning in a mathematical proof. You should test your conjecture multiple times. To prove that a conjecture is false, you need to provide a single counterexample. • Conjecture: a _____________________ reached by using inductive reasoning • Counterexample: an example showing that a statement is ____________

7. EX 3: Complete the conjecture • a) The product of any two consecutive positive integers is: • b) The product of 101 and a two digit number is: • c) The sum of the first 40 even numbers is:

8. EX 4: What is a counterexample for each conjecture? • a) If an animal is green, it is frog. • b) When you multiply a number by 2, the product is divisible by 4. • c) The difference of two positive numbers is always positive.

9. Examples • 5. Four adults attended a movie for $28. Three adults attended the same movie for $21 and six attended the movie for $42. Use inductive reasoning to make a conjecture about the price of each adult movie ticket. • 6. To get the next number in a sequence, you multiply the previous number by 2 and subtract 1. If the fourth number is 17, what is the first number in the sequence?

10. EXAMPLE • 7. • a) Sketch the next figure • b) Write the first four terms of a sequence of numbers that gives the number of toothpicks used to form each figure. Predict the next two numbers. • c) The number of toothpicks needed to make the nth figure is . Show that this conjecture is false by finding a counterexample.

11. 30 questions Patterns and Inductive Reasoning WS