Lesson 2.1 Inductive Reasoning in Geometry

HOMEWORK: Lesson 2.1/1-15 odds, 22-24, 31-40, 42

EC: Due WednesdayPage 104 – “Improve Reasoning Skills” #1-8

Inductive reasoning: • make conclusions based on patterns you

observe

Conjecture: • conclusion reached by inductive reasoning

based on evidence

Geometric Pattern:• arrangement of geometric figures that repeat

Mathematicians use Inductive Reasoning to find patterns which will then allow

them to conjecture. We will be doing this ALOT this year!!

Objectives:

1) Use inductive reasoning to find the next term in a number or picture pattern

2) To use inductive reasoning to make conjectures.

ConjecturesA generalization made with inductive

reasoning (Drawing conclusions)EXAMPLES:• Bell rings M, T, W, TH at 7:40 am

Conjecture about Friday?

• Chemist puts NaCl on flame stick and puts into flame and sees an orange-yellow flame. Repeats for 5 other substances that also contain NaCl also producing the same color flame.Conjecture?

Ex. 1: Find the next term in the sequence:A) 3, 6, 12, 24, ___, ___B) 1, 2, 4, 7, 11, 16, 22, ___, ___

C) ,___, ___

Inductive Reasoning –reasoning that is based on

patterns you observe.

Solutions

Ex. 1: Find the next term in the sequence:A) 3, 6, 12, 24, ___, ___B) 1, 2, 4, 7, 11, 16, 22, ___, ___

C)

48 96 Rule: x229 37 Rule: +1, +2, +3, +4,

…

Rule: divide each section by half

Steps of Inductive Reasoning

1. Process of observing data

2. Recognizing patterns

3. Making generalizations based on those patterns

Identify the pattern and find the next 3 numbers:1) 1, 4, 9, 16, ____, ____, ____  2) 1, 3, 6, 10, ____, ____, ____

3) 1, 1, 2, 3, 5, 8, ____, ____, ____

Identify the pattern and find the next 3 numbers:1) 1, 4, 9, 16, ____, ____, ____   sequence of perfect squares

2) 1, 3, 6, 10, ____, ____, ____+2, +3, +4, +5, …

3) 1, 1, 2, 3, 5, 8, ____, ____, ____Fibonacci – add the 2 previous numbers to get the next.

25 36 49

15 21 28

13 21 34

Solutions

An example of inductive reasoning

Suppose your history teacher likes to give “surprise” quizzes. You notice that, for the first four chapters of the book, she gave a quiz the day after she covered the third lesson.

Based on the pattern in your observations, you might generalize …

Based on the pattern in your observations, you might generalize… that you will have a quiz after the third lesson of every chapter.

Solution

Identify the pattern and find the next item in the pattern.

Identifying a Pattern

January, March, May, ...

Observe the data..

Make a generalization

Identify the pattern..

The next month is July.

Alternating months of the year make up the pattern.(skip every other month)

January, March, May, ...

Solution

Identify the pattern and find the next item in the pattern.

Identifying a Pattern

7, 14, 21, 28, …

Observe the data..

Make a generalization

Identify the pattern..

The next multiple is 35.

Multiples of 7 make up the pattern.(add 7 to each term to get the next)

7, 14, 21, 28, …

Solution

Identify the pattern and find the next item in the pattern.

Identifying a Pattern

In this pattern, the figure rotates 90° counter-clockwise each time.

The next figure is .

Solution

Inductive reasoning can be used to make a conjecture about a number

sequence

Consider the sequence 10, 7, 9, 6, 8, 5, 7, . . .

Make a conjecture about the rule for generating the sequence.

Then find the next three terms.

Solution10, 7, 9, 6, 8, 5, 7, . .

Look at how the numbers change from term to termThe 1st term in the sequence is 10. You subtract 3 to get the 2nd term. Then you add 2 to get the 3rd term.

You continue alternating between subtracting 3 and adding 2 to generate the remaining terms. The next three terms are 4, 6, and 3.

10, 7, 9, 6, 8, 5, 7, . .

Find the next item in the pattern 0.4, 0.04, 0.004, …

Identifying a Pattern

Be very careful with the wording/terms you use to describe the pattern

Rules & descriptions can be stated in many different ways:Multiply each term by 0.1 to get the next. Divide each term by 10 to get the next.

The next item would be 0.0004.

0.4, 0.04, 0.004, … Solution

Arrangement of geometric figures that repeatUse inductive reasoning and make conjecture as to the next

figure in a pattern

Use inductive reasoning to describe the pattern and find the next two figures in the pattern.

Geometric Patterns

Following the pattern: blue L, red +, green T… the next figures would be

the red + and the green T

Solution

Use inductive reasoning to describe the pattern and find the next two figures in the pattern.

Geometric Patterns

Following the pattern: green triangle is moving CCW 120° (or rotating CCW every other side of

the hexagon)… the next figures would be Green triangle on the bottom and then two

sides CCW

Solution

Describe the figure that goes in the missing boxes.

Describe the next three figures in the pattern below.

Geometric Patterns

Solutions

Make a conjecture about the sum of the first 30 odd numbers.

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

..

1 + 3 + 5 +...+ 61 =

= 12

= 22

= 32

= 42

= 52

..

= 302900

cont.: Make a conjecture about the sum of the first 30 odd numbers.

Conjecture:Sum of the first 30 odd numbers =

= the amount of numbers added

Sum of the first odd numbers =

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.

Truth in Conjectures

Inductive Reasoning assumes that an observed pattern will continue. This may or may not be true.

Ex: x = x • xThis is true only for x = 0 and x = 1

Conjecture – A conclusion you reach using inductive reasoning.

Counter Example – To a conjecture is an example for which the conjecture is incorrect.

The first 3 odd prime numbers are 3, 5, 7. Make a conjecture about the 4th.

3, 5, 7, ___One would think that the rule is add 2, but that gives us 9 for the fourth prime number.

Is that true?What is the next odd prime number?

11

No

Show that the conjecture is false by finding a counterexample.

Finding a Counterexample

For every integer n, n3 is positive.

Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n3 = –27 and –27 0, the conjecture is false.n = –3 is a counterexample.

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.

45° + 45° = 90°

Finding a Counterexample

Show that the conjecture is false by finding a counterexample.

The monthly high temperature in Abilene is never below 90°F for two months in a row.

Monthly High Temperatures (ºF) in Abilene, TexasJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec88 89 97 99 107 109 110 107 106 103 92 89

The monthly high temperatures in January and February were 88°F and 89°F, so the conjecture is false.

Finding a Counterexample

The radius of every planet in the solar system is less than 50,000 km.

Show that the conjecture is false by finding a counterexample.

Planets’ Diameters (km)

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

4880 12,100 12,800 6790 143,000 121,000 51,100 49,500

Since the radius is half the diameter, the radius of Jupiter is 71,500 km and the radius of Saturn is 60,500 km. The conjecture is false.

Finding a Counterexample

Supplementary angles are adjacent.

Show that the conjecture is false by finding a counterexample.

The supplementary angles are not adjacent, so the conjecture is false.

23° 157°

Finding a Counterexample