1.1 Patterns and Inductive Reasoning

GeometryMrs. Blanco

Objectives:• Find and describe patterns.• Use inductive reasoning to make real-life

conjectures.

Ex. 1: Describing a Visual Pattern

1)Sketch the next figure in the pattern.

1 2 3 54

Ex. 1: Describing a Visual Pattern - Solution

• The sixth figure in the pattern has 6 squares in the bottom row.

5 6

Ex. 1: Cont…2) Find the distance around each figure. Organize your results in a table.

3) Describe the patterns in the distances.

4) Predict the distance around the twentieth figure in this pattern.

Ex. 2: Describe a pattern in the sequence of numbers. Predict the next number.

a. 1, 4, 16, 64, …

b. 10, 5, 2.5, 1.25, …

c. 256, 16, 4, 2, …

d. 48, 16, , , …

16

3

16

9

256 (multiply previous number by 4)

0.625 (divide previous number by 2)

(take the square root of previous number)

16/27 (divide previous number by 3)

2

Using Inductive Reasoning

Much of the reasoning you need in geometry consists of 3 stages:

1. Look for a Pattern: Use diagrams and tables to help

2. Make a Conjecture. Unproven statement that is based on observations

3. Verify the conjecture—make sure that conjecture is true in all cases.

Ex. 3: Complete the ConjectureConjecture: The sum of the first n odd positive integers is

______.First odd positive integer:1 = 12 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1 + 3 + 5 + 7 = 16 = 42 The sum of the first n odd positive integers is n2.

Note:• To prove that a conjecture is true,

you need to prove it is true in all cases. • To prove that a conjecture is

false, you need to provide a single counterexample.

Ex. 4: Finding a counterexampleShow the conjecture is false by finding a

counterexample.1)For all real numbers x, the expressions x2 is

greater than or equal to x.

The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to 0.5. In fact, any number between 0 and 1 is a counterexample.

Another:Ex. 4: Finding a counterexample

Show the conjecture is false by finding a counterexample.

2) The difference of two positive numbers is always positive.

The conjecture is false. Here is a counterexample: 2-3 =-1

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