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Mathematics
Quarter 2 - Module 10
Think Logically and Reason Out
Mathematics - Grade 8
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Quarter 2 – Module 10: Think Logically and Reason Out
First Edition, 2020
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8
Mathematics
Quarter 2 - Module 10
Think Logically and Reason Out
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Table of Contents
What This Module is About .................................................................................................... 1
What I Need to Know ............................................................................................................. 1
How to Learn from this Module ............................................................................................. .2
Icons of this Module .............................................................................................................. .2
What I Know ......................................................................................................................... ..3
Lesson 1
Converse, Inverse, and Contrapositive ...................................................................... 6
What I Need to Know ................................................................................................ 6
What’s In ……………………………………………………………………………….….6
What’s New: Study Show ..................................................................................... 7
What Is It …………….………………………………………………………………….…8
What’s More: Fill me Up…………………………………………………………………9
What I Have Learned: State the Truth ……………………………………………….9
What I Can Do: Relate ’n Table ……………………………………………………….10
Lesson 2
Inductive and Deductive Reasoning ........................................................................... 11
What I Need to Know ............................................................................................... 11
What’s In ................................................................................................................... 11
What’s New: Observe and take a Guess …………………………………………....11
What Is It: Define Me ............................................................................................. ..12
What’s More: Let’s Conclude ............................................................................... ..14
What I Have Learned: Draw It ………………………………………………………….14
What I Can Do: Show which Thumb ................................................................... ..15
Assessment: (Post-Test) ................................................................................................. ..16
Key to Answers ................................................................................................................. ..18
References ......................................................................................................................... ..21
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What This Module is About
Geometry deals with logical reasoning to prove a certain statement. Logic and
reasoning are tools in geometry to facilitate mathematical thinking for making valid
conclusions.
In this module, the learner will deal with statements/implications that are logically
equivalent which provide different methods of stating the same idea. And then, the learner
will study the kinds of reasoning in an argument.
What I Need to Know
At the end of this module, you should be able to:
1. Illustrate the equivalences of :
a. a statement and its contrapositive; and
b. the converse and inverse of the statement. (M8GE – IIg – 2)
2. Uses inductive or deductive reasoning in an argument. (M8GE – IIh – 1)
1
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
•
Take your time reading the lessons carefully.
•
Follow the directions and/or instructions in the activities and exercises diligently.
•
Answer all the given tests and exercises.
•
Use the internet if you need more information about the lesson.
Icons of this Module
What I Need to
This part contains learning objectives that
Know
are set for you to learn as you go along the
module.
What I know
This is an assessment as to your level of
knowledge to the subject matter at hand,
meant specifically to gauge prior related
knowledge
This part connects previous lesson with that
of the current one.
What’s In
What’s New
An introduction of the new lesson through
various activities, before it will be presented
to you
What is It
These are discussions of the activities as a
way to deepen your discovery and understanding of the concept.
What’s More
These are follow-up activities that are intended for you to practice further in order to
master the competencies.
What I Have
Learned
Activities designed to process what you
have learned from the lesson
What I can do
These are tasks that are designed to showcase your skills and knowledge gained, and
applied into real-life concerns and situations.
2
What I Know
Find out how much you already know about this module. Write the letter that you
think is the best answer to each question on a sheet of paper. Answer all items. After taking
and checking this short test, take note of the items that you were not able to answer correctly
and look for the right answer as you go through in this module.
1. What is the equivalent truth value of an inverse statement?
A. Conditional
B. Converse
C. Statement
D. Contrapositive
2. The statement, “If two angles are not congruent, then they do not have the same
measures.” is logically equivalent as ______.
A. If two angles do not have the same measures, then they are not congruent.
B. If two angles have the same measures, then they are congruent.
C. If two angles are not congruent, then they have the same measures.
D. If two angles are congruent, then they have the same measures.
3. Which statement has the same truth value as the following statement?
“If a figure is a segment, then it has exactly one midpoint.”
A. If a figure is not a segment, then it does not have exactly one midpoint.
B. If a figure is a segment, then it does not have exactly one midpoint.
C. If a figure has exactly one midpoint, then it is not a segment.
D. If a figure does not have exactly one midpoint, then it is not a segment
4. Which of the following best describes deductive reasoning?
A. Using logic to draw conclusions based on accepted statements.
B. Accepting the meaning of a term without definition.
C. Defining mathematical terms in relation to physical objects.
D. Inferring a general truth by examining a number of specific examples.
5. What conclusion can you draw from the following two statements?
If a person does not get enough sleep, that person will be tired.
Carl does not get enough sleep.
A. Carl will get enough sleep.
C. Carl will be tired.
B. Carl should get enough sleep.
D. Carl will not be tired.
6. What law of deductive reasoning is used in item #5?
A. Law of Syllogism
C. Modus Tollens
B. Modus Ponens
D. Law of Contrapositive
7. It uses specific examples to arrive at a general rule, generalizations, or conclusion.
A. Deductive reasoning
C. Inductive reasoning
B. Law of Syllogism
D. Law of Detachment
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8. What conclusion can you draw from the following two statements?
If you have a job, then you have an income.
If you have an income, then you must pay taxes.
A. If you have a job, then you must pay taxes.
B. If you don’t have a job, then you don’t pay taxes.
C. If you pay taxes then you have a job.
D. If you have a job, then you don’t have to pay taxes
9. For inductive reasoning: What is the next term in the sequence 1, 1, 2, 3, 5, 8?
A. 11
B. 12
C. 13
D. 14
10. Which of the arguments below use deductive reasoning?
I. Every multiple of 4 is even. 376 is a multiple of 4. Therefore, 376 is even.
II. A number can be written as a repeating decimal if it is rational. Pi cannot
be written as a repeating decimal. Therefore, pi is not rational.
A. I only
B. II only
C. both I and II
D. neither I nor II
11. Which of the following is an example of inductive reasoning?
A. Carlos learns that the measures of all acute angles are less than 90. He
conjectures that if he sees an acute angle, its measure will be less than 90.
B. Carlos reads in his textbook that the measure of all right angles is 90. He
conjectures that the measure of each right angle in a square equals 90.
C. Carlos measures the angles of several triangles and finds that their measures all
add up to 180. He conjectures that the sum of measures of the angles in any
triangle is always 180.
D. Carlos knows that the sum of the measures of the angles in a square is always
360. He conjectures that if he draws a square, the sum of the measures of the
angles will be 360.
12. Use deductive reasoning to complete the statement, “All right angles are congruent.
∠B and ∠C are both right angles. Therefore, ______.
A. ∠B and ∠C are congruent angles.
C. ∠B and ∠C have equal measures.
B.∠B and ∠C are right angles.
D.∠B and ∠C are not congruent.
For items 13-15, determine whether the reasoning is an example of deductive or inductive
reasoning. Choose the correct answer below.
A. The reasoning is deductive because general principles are being applied to
specific examples.
B. The reasoning is inductive because general principles are being applied to specific
examples.
C. The reasoning is inductive because a general conclusion is being made from
repeated observations of specific examples.
D. The reasoning is deductive because a general conclusion is being made from
repeated observations of specific examples.
13. In the sequence 9, 12, 15, 18, 21, ..., the most probable next term is 24.
14. It is a fact that every student who ever attended in a university was accepted into
graduate school. Because I am attending in a university, I can expect to be accepted
to graduate school, too.
15. If you build it, they will come. You build it. Therefore, they will come.
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5
Lesson
Converse, Inverse, and
Contrapositive
1
What I Need to Know
With aspects of the implication in our rear view mirror, we now want to form new
compound statement from that original implication. These new statements are called
converse, inverse, and contrapositive.
Logically equivalent statements are related conditional statements that have the
same truth value. Understanding the key concepts in dealing with if-then statement is much
of help in this lesson.
In this lesson, you will illustrate the equivalences of the statement and its
contrapositive, and the converse and inverse of a statement.
What’s In
Activity 1
“Give Me a Statement”
Study the table below to recall how you will convert the statement in terms of p and q.
If p, then q
If q, then p
If not p, then not q
If not q, then not p
Statement
Converse
Inverse
Contrapositive
Determine the inverse, converse, and contrapositive of the following if-then
statement.
Conditional: If a shape is a triangle, then it is a polygon.
Hypothesis: _______________________________________________
Conclusion: _______________________________________________
Converse: ______________________________________________________
Inverse: ________________________________________________________
Contrapositive: __________________________________________________
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In the next section, you will find out the equivalences of the given if-then statement
and its contrapositive, and the converse and inverse of a statement.
What’s New
How can you determine truth value of the conditional statement?
Recall that the implication (conditional statement) 𝒑 ⟶ 𝒒 is always true except in
the case that p is true and q is false. See the truth table for the implications below.
Conditional Statement
Truth Value
( p → q)
p
q
T
T
T
T
F
F
F
T
T
F
F
T
The truth value of conditional statement is either true or false. It is false only when
the hypothesis is true and the conclusion is false.
Illustrative Example: Determine the truth value of the following related conditionals.
Conditional: If a shape is a triangle, then it is a polygon.
The given statement is true since all triangles are polygons.
Converse: If a shape is a polygon, then it is a triangle.
Analyse the converse. Is it true? If not, give a counter example. A counterexample
is any example that shows the statement is false.
The converse is false because a square is also a polygon. It is not necessarily a
triangle. Square is what you call a counterexample.
Inverse: If a shape is not a triangle, then it is not a polygon.
The inverse is false because a square is not a triangle but it is a polygon.
Contrapositive: If a shape is not a polygon, then it is not a triangle.
The contrapositive is true since you cannot find a shape which is not a polygon but a
triangle.
Activity 2
“Study Show”
Study the table below and show the truth value of the statements.
Statement
An even
number is
divisible by
two.
If-then form
If a number is
even, then it is
divisible by 2.
Converse
If a number is
divisible by 2,
then it is even.
Truth Value
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Inverse
If a number is
not even, then it
is not divisible
by 2.
Contrapositive
If a number is
not divisible by
2, then it is not
even.
What Is It
From the example and the activity above, notice that the given statement and its
contrapositive have the same truth value. Therefore, the conditional statement is logically
equivalent to its contrapositive. Likewise with the converse of a statement is logically
equivalent to the inverse of a statement.
Logically equivalent statements are statements that have the same logical content,
i.e., truth value.
Illustrative examples:
Illustrate the equivalences of the statement and its contrapositive; and the converse
and inverse of a statement.
1. If a number is divisible by 2, then it is divisible by 4.
Solution:
Conditional: The given statement is false. Counterexample: 6 is divisible by 2 but not
divisible by 4.
Converse: If a number is divisible by 4, then it is divisible by 2. The converse is true.
Inverse: If a number is not divisible by 2, then it is not divisible by 4. The inverse is true.
Contrapositive: If a number is not divisible by 4, then it is not divisible by 2. The
contrapositive is false. Counterexample: 6 is not divisible by 4 but divisible by 2.
Therefore, the statement and its contrapositive are both false while the converse and
inverse of the statement are both true. Thus, the statement and its contrapositive; and the
converse and inverse of a statement are logically equivalent.
2. If a bird is an ostrich, then it cannot fly.
Solution:
Conditional: The given statement is true.
Converse: If a bird cannot fly, then it is an ostrich. The converse is false.
Counterexample: The bird could be a penguin.
Inverse: If a bird is not an ostrich, then it can fly. The inverse is false.
Counterexample: The bird could be a penguin.
Contrapositive: If a bird can fly, then the bird is not an ostrich. The contrapositive is true.
Therefore, the statement and its contrapositive are both false while the converse
and inverse of the statement are both true. Thus, the statement and its contrapositive; and
the converse and inverse of a statement are logically equivalent.
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What’s More
“Fill me Up”
Activity 3
Given the statements below, complete the following table to illustrate the
equivalences of a statement and its contrapositive; and the converse and inverse of the
statement.
TRUTH
VALUE
RELATED CONDITIONALS
Conditional
Converse
If two angles are right, then they are congruent.
If two angles are congruent, then they are right.
Inverse
Contrapositive
True
Counterexample
N/A
If two angles are not right, then they are not
congruent.
If two angles are not congruent, then they are
not right.
What I Have Learned
“State the Truth”
Activity 4
Direction: Illustrate the equivalences of the statement and its contrapositive; and the
converse and inverse of a statement by completing the following table and paragraph
below.
TRUTH
VALUE
RELATED CONDITIONALS
Conditional
Counterexample
If a number is a whole number, then
it is an integer.
Converse
Inverse
Contrapositive
Therefore, the ___________ and its ___________ are both _________while the
___________ and ____________ of the statement are both ____________. Thus, the
___________ and its ____________, and the ___________ and ______________of a
statement are _______________ equivalent.
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What I Can Do
Activity 5
“Relate ‘n Table”
Write the related conditional statements of the following conditional statement.
Determine its truth value and identify which statements are equivalent.
Table 1
TRUTH
VALUE
RELATED CONDITIONALS
Counterexample
If yesterday is Tuesday, then today
is Wednesday.
Conditional
Converse
Inverse
Contrapositive
Equivalent Statements: __________________________________________
__________________________________________
Table 2
TRUTH
VALUE
RELATED CONDITIONALS
Conditional
Counterexample
If an animal has stripes, then it is a
zebra.
Converse
Inverse
Contrapositive
Equivalent Statements: __________________________________________
__________________________________________
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Lesson
2
Inductive and Deductive
Reasoning
What I Need to Know
One of the tools used in proving is reasoning. The conclusion drawn from
observations, examples and pattern is called conjecture. The conjecture may or may not be
true.
In making conclusions, we can use either Inductive reasoning or Deductive
reasoning. A type of reasoning that allows you to reach conclusions based on a pattern of
specific examples or past events is inductive reasoning while a type of reasoning which
makes use of accepted rules of logic is deductive reasoning.
In this lesson, you will use inductive or deductive reasoning in an argument.
What’s In
The main focus in the study of geometry is to learn how to think logically. An
argument is a series of statements intended to determine the truth of another statement.
From lesson 1, you have learned that equivalent statements are related conditionals with the
same truth value. When the given conditional is a simple implication then we have two pairs
of equivalent statements, which are conditional-contrapositive and converse-inverse.
In the next section, you will find out one how these related conditionals with the same
truth value are used in an argument through inductive or deductive reasoning.
What’s New
Activity 1 “Observe and take a Guess”
Direction: Write the next term of the following numbered scenario:
1.
2. 3, 10, 17, 24, 31, …..
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3.
Activity 2 “Create Conclusion”
Complete the table.
Statement
1. Filipinos are hospitable.
Bonifacio is a Filipino.
2. If points are collinear, then they lie on the
same plane.
Points R, M, and N are collinear.
3. A quadrilateral is a polygon with four
sides. A parallelogram is a quadrilateral.
Conclusion
In the first activity, it’s the pattern that helps you guess and make judgment. And
because you use pattern to create conclusion, then that way of thinking or reasoning is
inductive. While in the second activity, it’s your comprehension and your common sense
that will drive you to a correct judgment. And because you use logic to create conclusion,
then your way of thinking/reasoning is deductive.
What Is It
“Define Me”
Inductive reasoning is a kind of reasoning where the conclusion is made based
upon current knowledge, observation, examples and patterns. It uses specific examples to
arrive at a general rule, generalizations or conclusions. It is judging by experience. It
involves uncertainty in making conclusions. Inductive Reasoning is a process of observing
data, recognizing patterns, and making generalizations from observations.
Illustrative examples:
Draw a conclusion from each given situation using inductive reasoning.
1. Look carefully at the figures, what is next?
2. Study the pattern and draw the next figure.
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3. My Math teacher is strict. My previous Math teacher was strict.
What can you say about all math teachers?
_All Math teachers are strict.
4. 1 × 10 = 10
2 × 10 = 20
3 × 10 = 30
24 × 10 = 240
2345 × 10 = _23450_
5. Every time Jackie visits her doctor she receives excellent services .With this she
believes that __her doctor gives excellent services__.
Deductive reasoning is a type of logical reasoning that uses accepted facts to
reason in a step-by-step manner until we arrive at the desired statements. From the given
statement, you are to make a sound judgment or a conclusion. For a clearer thought, let us
take some laws in logic that is vital in deduction.
Law of Detachment (Modus Ponens)
Major Premise: If p is true, then q is true.
Minor Premise: p is true.
Conclusion: Therefore, q is true.
Illustrative examples :
Draw a conclusion from each given situation using deductive reasoning.
1. Major Premise: If you are an 18-year old Filipino citizen, then you can vote.
Minor Premise: Pete is an 18-year old Filipino.
Conclusion: _Therefore, Pete can vote.
2. Major Premise: If a person has a driver’s license, the he is allowed to drive.
Minor Premise: Arturo has a driver’s license.
Conclusion: _Therefore, Arturo is allowed to drive.
Law of Syllogism (Chain Rule)
Major Premise: If p is true, then q is true.
Minor Premise: If q is true, then r is true.
Conclusion: If p, then r.
Illustrative examples:
Draw a conclusion from each given situation using deductive reasoning.
1. Major Premise: If it is May, then there are many flowers.
Minor Premise: If there are many flowers, then I am happy.
Conclusion: _If it is May, then I am happy.
2. Major Premise: If you drive a smaller car, then you will use less gasoline.
Minor Premise: If you use less gasoline, then you save money.
Conclusion: _If you drive a smaller car, then you save money.
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What’s More
Activity 3
“Let’s Conclude”
A. Draw conclusion from each given situation or find the next term of the sequence using
inductive reasoning.
1.
2. It has rained every day for the past six days, and it is raining today as well.
Conclusion: ______________________________________________________
B. Use the Law of Detachment to determine what you can conclude from the given
information, if possible.
3. If Eimon pass the final, then he passes the class. Eimon pass the final.
Conclusion: ______________________________________________________
4. If your uncle let you borrow the car, then you will go to the movies with your sister.
Your parents let you borrow the car.
Conclusion: ______________________________________________________
C. Use the Law of Syllogism to write a new conditional statement that follows from the pair of
true statements, if possible.
5. If
, then
. If
, then
.
Conclusion: ______________________________________________________
6. If a figure is a rhombus, then the figure is a parallelogram. If a figure is a
parallelogram, then the figure has two pairs of opposite sides that are parallel.
Conclusion: ______________________________________________________
What I have Learned
Activity 4
“Draw It”
Draw a conclusion from each given situation and identify Identify the following if the
argument uses an inductive or deductive reasoning.
1. 5, 10, 15, 20. The next number is _______________.
2. Coplanar points on the same plane. X, Y, Z are coplanar.
Therefore, __________________________________.
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3. A regular polygon is equilateral. A BELIN is a regular pentagon.
Therefore, __________________________________.
4. A child’s teacher in pre-school was a female. In his grade 1 and 2, his teachers
were both female. The child may say that, ___________________.
5. Filipinos are a peace-loving people. Julia is a Filipino.
Therefore, _______________________________________.
What I can Do
Activity 5
“Show which Thumb”
Instruction: Determine if each argument uses inductive or deductive reasoning. Write RIGHT
THUMB if the reasoning used in an item is inductive, and LEFT THUMB if deductive.
_________________1. Niku is Danica’s cousin. Since Donna is Danica’s twin sister, Niku is
also Donna’s cousin.
_________________2. The school librarian notices that many Grade 8 students are
requesting books about different countries in Africa. The librarian concludes that their social
studies class must be studying about Africa.
_________________3. Conrad notices that each term in the sequence 4, 8, 16, 32, …. is
found by multiplying the previous term by two. He concludes that the next two terms are 64
and 128.
_________________4. Given the sequence 13, 18, 23, and 28 you conclude that the next
term will be 33.
_________________5. All of the people that Ruby met in town are very strange. Ruby
conclude that everyone in town is very strange.
Summary
The conclusion drawn from observation, examples and pattern is called conjecture,
thus conjecture may or may not be true. That is why in inductive reasoning, it does not
guarantee a true result at all times. Law of Detachment (Modus Ponens) and Law of
Syllogism (Chain Rule) are some laws in logic that are vital in deductive reasoning.
Deductive reasoning is reasoning which begins using basic and general statements to
prove more complicated statements.
15
Assessment (Post-Test)
Find out how much you already know about this module. Write the letter that you
think is the best answer to each question on a sheet of paper. Answer all items. After taking
and checking this short test, take note of the items that you were not able to answer correctly
and look for the right answer as you go through in this module.
1. What is the equivalent truth value of a converse statement?
A. Conditional
B. Inverse
C. Statement
D. Contrapositive
2. The statement that has the same truth value as the given statement: “If a polygon is a
rectangle, then its pairs of opposite sides are parallel.” is _________.
A. If a polygon is a not a rectangle, then its pairs of opposite sides are parallel.
B. If the pairs of opposite sides of a polygon are not parallel, then the polygon is not
a rectangle.
C. If the opposite sides of a polygon are parallel, then the polygon is not a rectangle.
D. If a polygon is a not a rectangle, then its pairs of opposite sides are not parallel.
3. The equivalent statement of “If two angles are not complementary, then the sum of
their measures is not 90° is ______.
A. If two angles are complementary, then the sum of their measures is 90°.
B. If two angles are complementary, then the sum of their measures is not 90°.
C. If the sum of the measures of two angles is 90°, then the two angles are
complementary.
D. If two angles are not complementary, then the sum of their measures is not 90°.
4. Which of the following best describes inductive reasoning?
A. Using logic to draw conclusions based on accepted statements.
B. Accepting the meaning of a term without definition.
C. Defining mathematical terms in relation to physical objects.
D. Inferring a general truth by examining a number of specific examples.
5. What conclusion can you draw from the following two statements?
If a person does not get enough sleep, that person will be tired.
Marcos does not get enough sleep.
A. Marcos will be tired.
C. Marcos will get enough sleep.
B. Marcos should get enough sleep.
D. Marcos will not be tried.
6. What law of deductive reasoning is used in item #5?
A. Law of Syllogism
C. Modus Tollens
B. Modus Ponens
D. Law of Contrapositive
7. For inductive reasoning: What is the next term in the sequence 19, 23, 27, 31,…?
A. 33
B. 35
C. 37
D. 39
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8. Determine the right conclusion from the following statements:
I. A shape that has more than 2 sides is a polygon.
II. A regular polygon has both all sides and all angles congruent.
III. An equilateral triangle has 3 congruent sides and 3 congruent angles.
A. All triangles are polygons.
B. A rectangle with sides 2, 2, 4, and 4 is not a regular polygon.
C. An equilateral triangle is a regular polygon.
D. All of the above can be concluded.
9. It uses a general rule or fact to give a specific example.
A. Deductive reasoning
C. Inductive reasoning
B. Law of Syllogism
D. Law of Detachment
10. Determine the next number in the sequence: 1, 2, 4, 7, 11... Is this inductive or
deductive reasoning?
A. The next number is 22. This is inductive reasoning.
C. The next number is 16. This is inductive reasoning.
B. The next number is 22. This is deductive reasoning.
D. The next number is 16. This is deductive reasoning.
11. A conclusion which is arrived at by inductive reasoning is called a ________.
A. counterexample
B. proof
C. conjecture
D. theorem
12. Consider the following statements:
All piano players are musicians.
Fred is a piano player.
Which of the following is a conclusion which can be drawn from the statements?
A. All piano players are named Fred.
B. All musicians are piano players.
C. Fred is a musician.
D. Fred is not a musician.
For items 13-15, determine whether the reasoning is an example of deductive or inductive
reasoning. Choose the correct answer below.
A. The reasoning is deductive because general principles are being applied to
specific examples.
B. The reasoning is inductive because general principles are being applied to specific
examples.
C. The reasoning is inductive because a general conclusion is being made from
repeated observations of specific examples.
D. The reasoning is deductive because a general conclusion is being made from
repeated observations of specific examples.
13. It has rained every day for the past six days, and it is raining today as well. So it will
also rain tomorrow.
14. If the mechanic says that it will take seven days to repair your SUV, then it will
actually take ten days. The mechanic says, "I figure it'll take exactly one week to fix it,
ma'am." Then you can expect it to be ready ten days from now.
15. It is a fact that every student who ever attended in a university was accepted into
graduate school. Because I am attending in a university, I can expect to be accepted
to graduate school, too.
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Key to Answers
Pre-Test
1. B
2. B
3. D
4. A
5. C
6. B
7. C
8. A
9. C
10. C
11.C
12. A
13. C
14. A
15. A
Possible Answers
LESSON 1
Activity 1
Conditional
Hypothesis
Conclusion
Converse
Inverse
Contrapositive
Activity 2
Statement
An even
number is
divisible by
two.
Truth Value
If a shape is a triangle, then it is a polygon.
A shape is a triangle.
It is a polygon.
If a shape is a polygon, then it is a triangle.
If a shape is not a triangle, then it is not a polygon.
If a shape is not a polygon, then it is not a triangle.
If-then form
If a number is
even, then it is
divisible by 2.
true
Converse
If a number is
divisible by 2,
then it is even.
Inverse
If a number is
not even, then it
is not divisible
by 2.
Contrapositive
If a number is
not divisible by
2, then it is not
even.
true
true
true
Activity 3
RELATED CONDITIONALS
TRUTH
VALUE
Conditional
If two angles are right, then they True
are congruent.
Converse
If two angles are congruent, then
they are right.
False
Inverse
If two angles are not right, then False
they are not congruent.
Contrapositive If two angles are not congruent, True
then they are not right.
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Counterexample
N/A
Two angles can be congruent
but not necessarily right.
Two angles could not be right
but can be congruent.
N/A (There are no two angles
that are not congruent but are
right angles.)
Activity 4
RELATED CONDITIONALS
Conditional
If a number is a whole number,
then it is an integer.
If a number is an integer, then it
is a whole number.
If a number is not a whole
number, then it is not an integer.
Converse
Inverse
TRUTH VALUE
Counterexample
True
N/A
False
False
–1 or any negative
number
–1 or any negative
number
If a number is not an integer,
True
N/A
then it is not a whole number.
Therefore, the conditional and its contrapositive are both true while the converse and
Contrapositive
inverse of the statement are both false. Thus, the conditional and its converse, and the
inverse and contrapositive of a statement are logically equivalent.
Activity 5
Table 2
RELATED CONDITIONALS
If yesterday is Tuesday, then
Conditional
today is Wednesday.
If today is Wednesday, then
Converse
yesterday is Tuesday.
If yesterday is not Tuesday, then
Inverse
today is not Wednesday.
TRUTH VALUE
Counterexample
True
N/A
True
N/A
True
N/A
If today is not Wednesday, then
True
N/A
yesterday is not Tuesday.
EQUIVALENT STATEMENTS: conditional and its contrapositive; converse and inverse
Contrapositive
Table 2
TRUTH
VALUE
RELATED CONDITIONALS
Counterexample
If an animal has stripes, then it is
Tiger has stripes but it is
False
a zebra.
not a zebra.
If it is a zebra, then an animal
Converse
True
N/A
has stripes.
If an animal has no stripes, then
Inverse
True
N/A
it is not a zebra.
If it is not a zebra, then an
Tiger is not a zebra but it
Contrapositive
False
animal has no stripes.
has stripes.
EQUIVALENT STATEMENTS: conditional and its contrapositive; converse and inverse
Conditional
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LESSON 2
Activity 1
1.
2. 38
3.
Activity 2
Statement
Conclusion
1. Filipinos are hospitable.
Bonifacio is hospitable.
Bonifacio is a Filipino.
2. If points are collinear, then they lie on the
same plane.
Points R, M, and N lie on the same plane.
Points R, M, and N are collinear.
3. A quadrilateral is a polygon with four A parallelogram is a polygon with four sides.
sides. A parallelogram is a quadrilateral.
Activity 3
A.
1.
2. It will also rain tomorrow.
B. 3. Eimon pass the class.
4. You will go to the movies with your friends.
C. 5. If x < – 2, then |𝑥 | >2.
6. If a figure is a rhombus, then the figure has two pairs of opposite sides that are
parallel.
Activity 4
1. 25. Inductive reasoning
2. X, Y, Z are on the same plane. Deductive reasoning.
3. BELIN is equilateral. Deductive reasoning.
4. All teachers are female. Inductive reasoning
5. Julia is a peace-loving person. Deductive reasoning.
Activity 5
1. RIGHT THUMB
2. LEFT THUMB
3. RIGHT THUMB
4. RIGHT THUMB
5. RIGHT THUMB
Pre-Tes
1. B
2. B
3. C
4. D
5. A
6. B
7. B
8. C
9. A
10. C
11.C
12. C
13. C
14. C
15. A
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References
Abuzo, Emmanuel, Merden Bryant, Jem Boy Cabrella, Belen Caldez, Melvin Callanta,
Anastacia Proserfina Castro, Alicia Halabaso, Sonia Javier, Roger Nocom, and
Conception Ternida. Mathematics Learner’r Module 8. 1 st ed. Reprint,
Department of Education, 2013.
“Converse, Inverse, Contrapositive.”Tutors.com.Accessed June 16, 2020.
https://tutors.com/math-tutors/geometry-help/converse-inverse-contrapositive
“Converse, Inverse, Contrapositive.”Varsity Tutors. Accessed June 16, 2020.
https://www.varsitytrs.om/hotmath/hotmath_help/topics/converse-inversecontrapositive
Cuenco, Editha, Arnel Olofernes, Ralmond Roca, Rverie Vargas, and Mark Anthony Vidalgo.
Spiral Mathematics For Growth Mindset 8. Reprint, IEMI, 2020.
“Deductive and Inductive Arguments.”Internet Encyclopedia of Philosophy. Accessed June
16, 2020.https://www.iep.utm.edu/ded-ind/
Malaybalay Division. Subject Matter: Equivalent Statements, Inductive Reasoning, and
Deductive Reasoning. Math 8 DLL Exemplar, Quarter 2.
Miessler, Daniel.”The Difference Between Deductive and Inductive
Reasoning.”DANIELMIESSLER.August 29, 2018.
https://danielmiessler.com/blog/the-difference-between-deductive-and-inductivereasoning/
Taylor, Courtney.”What Are the Converse, Contrapositive, and Inverse?.”ThoughtCo.
Accesed June 16, 2020. https://www.thoughtco.com/converse-contrapositive-andinverse-3126458
Mathematics Teacher’s Guide 8 (pp. 341- 360)
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For inquiries and feedback, please write or call:
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Department of Education – Division of Gingoog City
Office Address:
Brgy. 23, National Highway,Gingoog City
Telefax:
088 328 0108/ 088328 0118
E-mail Address:
gingoog.city@deped.gov.ph
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