C5S1 - Inductive and Deductive Reasoning

CYCLE 5

.

Synchronous Session 1

Inductive and

.

Deductive Reasoning

Giving Conjectures and Counterexample

Learning Targets

At the end of the lesson, I CAN:

1. illustrate inductive reasoning in a number of

situations.

2. state conjectures based on patterns observed.

3. distinguish between inductive and deductive

reasoning.

4. apply deductive reasoning in solving problems

and puzzles.

5. state a counterexample of the given conjecture.

Multiple Choice

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Jenna sleeps after playing. Jenna
sleeps after crying. Jenna sleeps

after studying. What can you
conclude that Jenna does after

taking a bath?

What kind of reasoning did you use to come up with your conjecture?

Conjecture

Ø
It is a conclusion made
from observing data. It
may or may not be true. It
is an “educated guess”.

Ø
It is the process of gathering
specific information, usually
through observation and
measurement and then
making a conjecture based
on the gathered data.

Inductive
Reasoning

Ø

“from specific to general”

Examples:

(Grade 8 Mathematics – Patterns and Practicalities, pages 295 and 296

a) Jason often comes to school late. His class starts at 7:30AM, so he

wakes up at 6AM and leaves the house at 7AM for the 30 minute

commute to school. Still he comes in late due to the unpredictable

traffic situation. One day, he wakes up at 5:30AM, leaves the house

at 6:30AM and is in school at 7:00AM. He does this for several days

and gets the same result. What can he conclude?

Examples:

(Grade 8 Mathematics – Patterns and Practicalities, pages 295 and 296

a) Jason often comes to school late. His class starts at 7:30AM, so he

wakes up at 6AM and leaves the house at 7AM for the 30 minute

commute to school. Still he comes in late due to the unpredictable

traffic situation. One day, he wakes up at 5:30AM, leaves the house

at 6:30AM and is in school at 7:00AM. He does this for several days

and gets the same result. What can he conclude?

Conclusion:
Jason will not be late in class if he wakes up at 5:30 AM.

Examples:

(Grade 8 Mathematics – Patterns and Practicalities, pages 295 and 296

b ) Write a conjecture that is based on the given information.
Given:

Tilapia and Bangus are fish and they can swim.

Tuna and Salmon are fish and they can swim.

Sharks are fish and they can swim.

Examples:

(Grade 8 Mathematics – Patterns and Practicalities, pages 295 and 296

b ) Write a conjecture that is based on the given information.
Given:

Tilapia and Bangus are fish and they can swim.

Tuna and Salmon are fish and they can swim.

Sharks are fish and they can swim.

Conclusion: All fish can swim.

Inductive Reasoning

Inductive reasoning is a practical, but not a foolproof way to make
conjectures.

For a conjecture to be ‘true’, it must be verified to be true for all possible
cases.

Example:

Premise: 2 is an even number and 22= 4 is an even number.

Premise: 4 is an even number and 42= 16 is an even number.

Inductive Reasoning

Inductive reasoning is a practical, but not a foolproof way to make
conjectures.

For a conjecture to be ‘true’, it must be verified to be true for all possible
cases.

Example:

Premise: 2 is an even number and 22= 4 is an even number.

Premise: 4 is an even number and 42= 16 is an even number.

Conclusion: An even number raised to 2 is also an even number.

Counterexample

An easy way to disprove a conjecture is to give a counterexample, that is,

an example that shows the statement to be false.

Example:

Premise: Parrot is a bird, it can fly.

Premise: Hawk is a bird, it can fly.

Conclusion: All birds can fly.

Counterexample:
Ostrich is a bird. It cannot fly.

Counterexample

An easy way to disprove a conjecture is to give a counterexample, that is,

an example that shows the statement to be false.

Example:

Premise: Parrot is a bird, it can fly.

Premise: Hawk is a bird, it can fly.

Conclusion: All birds can fly.

Give a counterexample to show that the given conjecture is false.

1. Given:

Premise: 16 is an even number and it is divisible by 4.

Premise: 32 is an even number and it is divisible by 4.

Conclusion: If a number is even, it is divisible by 4.

Let’s dothis!

Give a counterexample to show that the given conjecture is false.

1. Given:

Premise: 16 is an even number and it is divisible by 4.

Premise: 32 is an even number and it is divisible by 4.

Conclusion: If a number is even, it is divisible by 4.

Let’s dothis!

Counterexample:
10 is an even number.
It is not divisible by 4.

Give a counterexample to show that the given conjecture is false.

2. Given:
Premise: The ordered pair (2,3) consists of non-negative coordinates and
is located on Quadrant I.
Premise: The ordered pair (5,11) consists of non-negative coordinates
and is located on Quadrant I.

Conclusion: All ordered pairs with non-negative coordinates are located

on Quadrant I.

Let’s dothis!

Give a counterexample to show that the given conjecture is false.

2. Given:
Premise: The ordered pair (2,3) consists of non-negative coordinates and
is located on Quadrant I.
Premise: The ordered pair (5,11) consists of non-negative coordinates
and is located on Quadrant I.

Conclusion: All ordered pairs with non-negative coordinates are located

on Quadrant I.

Let’s dothis!

Counterexample:
(0,0) consists of non-negative
coordinates and it is not
located in Q1.

Practice Exercise 1

Practice Exercise 1

11

13

36

49

10 000 100 000

21

28

1
32

1
64

Open Ended

Give a counterexample that the given conditional statement is FALSE.

All aquatic creatures are fish.

Open Ended

Give a counterexample that the given conditional statement is FALSE.

All insects have eight legs.

Open Ended

Write a conjecture that is based on the given information:

A student who attains a perfect score in the first quiz will be given extra credit.

Kim attained a perfect score in the first quiz.

Open Ended

Write a conjecture that is based on the given information:

No foreigner can be elected as the president of our country.

Jin is a foreigner.

Open Ended

Write a conjecture that is based on the given information:

All the residents in Biñan, Laguna are exempted from paying taxes.

JK’s parents reside in Biñan, Laguna.

Deductive
Reasoning

Ø
It is the process of showing that certain

statements follow logically from agreed

upon assumptions and proven facts.

Ø
It is a type of logical reasoning that

uses accepted facts (undefined terms,

definitions, postulates, and theorems)

to reason in step – by – step manner

until arriving at the desired statement.

Ø

“from general to specific”

Examples:

Suppose that the given statements are true. Use deductive
reasoning to give another statement that must be true.

a) All snakes are cold-blooded animals.
A cobra is a snake.

b) Planets in our solar system revolve around the sun.
The planet Uranus belongs to the solar system.

c) The square of any real number is non-negative.
6 is a real number.

Examples:

Suppose that the given statements are true. Use deductive
reasoning to give another statement that must be true.

a) All snakes are cold-blooded animals.
A cobra is a snake.

b) Planets in our solar system revolve around the sun.
The planet Uranus belongs to the solar system.

c) The square of any real number is non-negative.
6 is a real number.

Therefore, a cobra is a cold-blooded animal.

Therefore, Uranus revolves around the sun.

Therefore, the square of 6 is non-negative.

Ø
In deductive reasoning, you must be able to justify any statement that
you make. You try to reason in an orderly way to convince/prove that
your conclusion is valid.

Ø
Example:

Ø
Prove that in 4 (3x - 8) + 5 = x − 5, x = 2.

12x − 32 + 5 = x − 5

Apply the Distributive Property

12x − 27 = x − 5

Combine like terms

11x − 27 = − 5

Addition Property of Equality

11x = 22

Addition Property of Equality

x = 2

Multiplication Property of Equality

Inductive Reasoning

from specific to a general

from general to specific

Deductive Reasoning

Andrei is a Filipino and he is
hospitable.
Nigel is a Filipino and he is
hospitable.
Therefore, all Filipinos are
hospitable.

All Filipinos are hospitable.
Andrei and Nigel are Filipinos.
Therefore, Andrei and Nigel are
hospitable.

Multiple Choice

Practice Exercise: Determine whether the reasoning used in the given argument is inductive or deductive.

A parallelogram has two pairs of parallel opposite sides.

A rectangle is a parallelogram.

Therefore, a rectangle has two pairs of parallel opposite sides.

• INDUCTIVE

• DEDUCTIVE

Multiple Choice

Practice Exercise: Determine whether the reasoning used in the given argument is inductive or deductive.

All rectangles have four right angles.

A square is a rectangle.

Therefore, a square has four right angles.

• INDUCTIVE

• DEDUCTIVE

Multiple Choice

Practice Exercise: Determine whether the reasoning used in the given argument is inductive or deductive.

100 is divisible by 10 and its last digit is 0.

250 is divisible by 10 and its last digit is 0.

Therefore, if a number is divisible by 10 then its last digit is 0.

• INDUCTIVE

• DEDUCTIVE

Practice Exercise 3

Asynchronous Task:

Open and answer CEREBRY 8.2.