Chapter Using inductive reasoning to make conjectures

Objectives Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures.

Identifying Patterns  Find the next 2 items in the following pattern.  January, March, May,...  The next month is July. The next month is september  Alternating months of the year make up the pattern

Identifying patterns  Find the next 2 items in the following pattern.  1,8,27,64,…………….  1,1,2,3,5,8,…………………

Inductive reasoning  When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning

What is inductive reasoning ?  Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern.  Inductive reasoning is the process of observing, recognizing patterns and making conjectures about the observed patterns. Inductive reasoning is used commonly outside of the Geometry classroom; for example, if you touch a hot pan and burn yourself, you realize that touching another hot pan would produce a similar (undesired) effect.

What is conjecture?  A statement you believe to be true based on inductive reasoning is called a conjecture.

Making conjectures Ex#1  Complete the conjecture.  The sum of two positive numbers is ?.  List some examples and look for a pattern.  = = 3.15  3, ,000,017 = 1,003,917  The sum of two positive numbers is positive

Example #2  Complete the conjecture.  The number of lines formed by 4 points, no three of which are collinear, is ?.  Draw four points. Make sure no three points are collinear. Count the number of lines formed:  The number of lines formed by four points, no three of which are collinear, is 6.

Example #3  The sum of two odd numbers is __________

Example #4  Make a conjecture about the lengths of male and female whales based on the data.  Average length femaleAverage length male

Example #4 continue  In 5 of the 6 pairs of numbers above the female is longer.  Conjecture:  Female whales are longer than male whales.

Counter Example  To show that a conjecture is always true, you must prove it.  To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample.  A counterexample can be a drawing, a statement, or a number.

Counter example ex.#1  Show that the conjecture is false by finding a counterexample.  For every integer n, n 3 is positive.  Pick integers and substitute them into the expression to see if the conjecture holds.

Counterexample ex.#2  Show that the conjecture is false by finding a counterexample.  Two complementary angles are not congruent.  If the two congruent angles both measure 45°, the conjecture is false.

Counterexample ex.#3  Show that the conjecture is false by finding a counterexample.  For any real number x, x 2 ≥ x.

Counterexample ex.#4

How do inductive reasoning works  Inductive Reasoning  1. Look for a pattern.  2. Make a conjecture.  3. Prove the conjecture or find a counterexample.

Student guided practice  Lets do problems 2-10 on the book page 77

Homework !!!  Do problems in the book page 77

Closure  Today we saw about inductive reasoning and how to make conjectures and counterexamples.  Next class we are going to continue with conditional statements.