Lesson 1-1 Patterns and Inductive Reasoning Applied Geometry Lesson 1-1 Patterns and Inductive Reasoning Objective: Learn and identify patterns and use inductive reasoning.

Inductive Reasoning - method used to make a conclusion based on a pattern of examples or past events Find the next three terms of the sequence 33, 39, 45 … 1, 3, 7, 13, 21… 51, 57, 63 + 6 + 6 31, 43, 57 +6 +10 +14 +2 +8 +12 +4

Your Turn Find the next 3 terms 1.25, 1.45, 1.65,… 13, 8, 3,… 1, 3, 9, … 32, 16, 8,… 10, 12, 15, 19… 1, 2, 6, 24… +0.2 each time / 2 each time 1.85, 2.05, 2.25 4, 2, 1 -5 each time +2, + 3, + 4 24, 30, 37 -2, -7, -12 *3 each time *2, *3, *4 27, 81, 243 120, 720, 5040

Geometric Pattern Draw the next figure in the pattern: Next figure:

Example Draw the next figure in the pattern. The next figure

Your turn Draw the next figure The next figure:

Conjecture A conjecture is a conclusion that you reach based on inductive reasoning.

The diagonals of a rectangle have the Hands-on Activity Draw 3 rectangles of different sizes Draw the diagonals of all the rectangles. Get a ruler and measure both diagonals of all 3 rectangles. What is your conjecture about the diagonals of a rectangle? The diagonals of a rectangle have the same measure

Counterexample Counterexample is an example that shows that a conjecture is false or not true.

Example Sample answer: ½ and 10 Said positive didn’t say had Akira studied the data in the table and made the following conjecture. The product of 2 positive numbers is always greater than either factor. Factors Product 2 8 16 5 15 75 20 38 760 54 62 3348 Find a counterexample for the conjecture. Sample answer: ½ and 10 Said positive didn’t say had to be a whole number

Example Provide a counterexample for the following conjecture: Multiplying a number by –1 produces a product that is less than –1. Sample answer: (-1)(-2) = 2 which is greater than –1

Real World example 2005 – 10.9 billion dollars The following graph shows the revenue from sale of waste management equipment in billions of dollars. Find a pattern in the graph and then make a conjecture about the revenue for 2005. Increase of 0.2 billion each year 2002 – 10.3 billion 2003 – 10.5 billion 2004 – 10.7 billion 2005 – 10.9 billion dollars

Homework Pg. 7 1 – 14 all, 16 – 38 Even