Chapter 22-1 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. 1 + 1 = 2 3.14 + 0.01 = 3.15 3,900 + 1,000,017 = 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 female

Average length male

49 47

51 45

50 44

48 46

51 48

47 48

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, n3 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, x2 ≥ x.

Counterexample ex.#4 Sow that each conjecture is false by

finding a counterexample For all positive numbers n,1/nn

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 11-23 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.