DIVISION OF NAVOTAS CITY

8
MATHEMATICS
Quarter 2

S.Y. 2021-2022
NAVOTAS CITY PHILIPPINES
Mathematics – Grade 8
Alternative Delivery Mode
Quarter 2
Second Edition, 2021

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Published by the Department of Education

Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer: Stephanie Joie S. Sanchez, Jennifer De Lara, Rosielyn P. Saavedra,Eugene P.

Santiago, Rhealiza F. San Miguel, Erros Josephus M. Gutierrez
Editors: Leonard S. Evangelista
Reviewers: Alberto J. Tiangco
Illustrator: Stephanie Joie S. Sanchez, Jennifer De Lara, Rosielyn P. Saavedra,Eugene
P. Santiago, Rhealiza F. San Miguel, Erros Josephus M. Gutierrez
Layout Artist: Kurt Russel M. Linao/Stephanie Joie S. Sanchez
Management Team: Alejandro G. Ibañez, OIC- Schools Division Superintendent
Isabelle S. Sibayan, OIC- Asst. Schools Division Superintendent
Loida O. Balasa, Chief, Curriculum Implementation Division
Alberto J. Tiangco, EPS in Mathematics
Grace R. Nieves, EPS In Charge of LRMS
Lorena J. Mutas, ADM Coordinator
Vergel Junior C. Eusebio, PDO II LRMS

Inilimbag sa Pilipinas ng ________________________

Department of Education – Navotas City

Office Address: BES Compound M. Naval St. Sipac-Almacen Navotas City
____________________________________________
Telefax: 02-8332-77-64
____________________________________________
E-mail Address: Navotas.city@deped.gov.ph
____________________________________________
 Table of Contents
What I Know .......................................................................... 1

Module 1 ............................................................................... 3

Module 2 ............................................................................... 8

Module 3 ............................................................................... 11

Module 4 ............................................................................... 17

Module 5 ............................................................................... 19

Module 6 ............................................................................... 21

Module 7 ............................................................................... 24

Module 8 ............................................................................... 28

Module 9 ............................................................................... 31

Assessment ........................................................................... 36

Answer Key ............................................................................ 38

Reference …………………………………………………………………46
Read and analyze each question carefully. Choose the letter of the correct
answer.

1. Which of the following is a linear inequality in two variables?

A. 4a – 3b = 9
B. 8f + 4 < 12
C. 3x < 16
D. 11 + 2t ≥ 3k

2. Which among the mathematical symbols best translate the verbal statement
Twice a number x and y is at most 24?

A. 2x + y < 24
B. 2x + y > 24
C. 2x + y ≥ 24
D. 2x + y ≤ 24

3. How many solutions does the following system have?

A. infinite
B. 2
C. 1
D. zero

4. Which of the following is an example of a function?

A. {(-3, 2), (-2, 3), (-2, 5)}

B. {(-1, 2), (0, 3), (0, 4)}
C. {(1, 3), (1, 2), (1, 4)}
D. {(0, 1), (1, 2), (-1, 4)}

5. Find the range of the function h(x) = x ² + 3.

A. All real
B. [3, )
C. (3, )
D. All of the above

1
6. Find the value of y if f(x) = 3x-2 and the value of x = 4.

A. -10
B. 10
C. 16
D. -16

7. Which of the following is the if-then form of the statement: “A prime number
has one and itself as factors.”?

A. If a number is prime, then its factors are 1 and itself.

B. If a number is prime, then its factors are not 1 and itself.
C. If a number is not prime, then its factors are 1 and itself.
D. If a number is not prime, then its factors are not 1 and itself.

8. Which of the following is the contrapositive of the statement, “If two angles
are vertical, then they are congruent.”?

A. If two angles are not vertical, then they are not congruent.
B. If two angles are congruent, then they are vertical.
C. If two angles are vertical, then they are congruent.
D. If two angles are not congruent, then they are not vertical.

9. The statement If <A and <B are supplementary, then the sum of their
measure is 180° is translated into If the sum of the measure of <A and <B is
180°, then they are supplementary. Which symbolization illustrates the
second statement?

A. p→q
B. q→p
C. ~p→~q
D. ~q→~p

10. Which appropriate conclusion can be drawn from the statement m∠J +
m∠S = 90?
A. ∠J≅∠S
B. ∠J and ∠S are right angles
C. ∠J and ∠S are complementary
D. ∠J and ∠S are supplementary

2
 MODULE 1

This module was designed and written with you in mind. It is here to help you
master illustrating linear inequalities in two variables. The scope of this module
permits it to be used in many different learning situations. The language used
recognizes the diverse vocabulary level of students. The lessons are arranged to
follow the standard sequence of the course. But the order in which you read them
can be changed to correspond with the textbook you are now using.
The module is divided into two lessons, namely:
Lesson 1.1 – Linear Inequalities in Two Variables
Lesson 1.2- Graph of Linear Inequalities in Two Variables
After going through this module, you are expected to:

1. Differentiates Linear Inequalities in two variables from linear

equations in two variables.
2. Illustrates and Graphs Linear Inequalities in Two Variables.
3. Solves Problems Involving Linear Inequalities in Two Variables.

Lesson Linear Inequalities in Two

1.1 Variables

Study the differences of linear inequalities from non-linear inequalities.

Linear Inequalities Not Linear

Inequalities
x>5 x=5
5y < 10 5y = 10
x+y>5 x + 5y = 10
5x – y ≤ 10 x2 – y < 5
x + 5y ≥ 10 y > x3 + 1
A linear inequality in two variables is an inequality that can be written in one of the
following forms:
Ax + By < C Ax + By ≤ C
Ax + By > C Ax + By ≥ C
where A, B, and C are real numbers, and A and B are not both equal to zero.

3
Linear equalities in two variables have a highest degree of 1. The solution can be
represented by sets of possible value that satisfies the given inequalities. The
solution of a linear inequality in two variables like Ax + By > C is an ordered pair (x,
y) that produces a true statement when the values of x and y are substituted into
the inequality.

To find the linear inequalities suited for the solution, let us substitute each ordered
pair to each inequality and check which one gives a true statement.
Examples:
Is the solution suited to x – 2y > -2?
x – 2y > -2 Given
(1) - 2(1) > -2 Substitution
1 - 2 > -2 Simplify
-1 > -2 True statement
Therefore, (1, 1) is a solution.

Is the solution suited to 2x + y > 4?

2x + y > 4 Given
2(1) + 1 > 4 Substitution
2+1>4 Simplify
3>4 False statement
Therefore, (1, 1) is not a solution.

Activity 1: Write "Inequality "if the given shows linear inequality in two
variables and "Not Inequality" if otherwise.
1. 3x – y ≥ 12 ________________________

2. 19 < y ________________________

3. y = 2/5 x ________________________

4. x ≤ 2y + 5 ________________________
5. 7(x – 3 ) < 4y ________________________

Activity 2: Encircle the point/s that represents as the solution/s of the given
linear inequalities in two variables.
1. x + y > 14 (10, 10) (-10, 10)
2. y – 2x < -1 (4, 9) (8, 9)
3. 2x + y ≥ 10 (-3, 1) (8, 10)
4. 4x + y < 11 (1,11) (1, -11)
5. 3x – y < 8 (-1, -2) (10, 2)

4
Read and analyze problem. Write an inequality statement about it.

1. Jazmine allowance for an online class is at most P200 a week. She bought
some school supplies in preparation for her portfolio. The price of a big
notebook is P30, while a long ordinary folder is P15.What are the possible
number of long folder and big notebook can she buy from her weekly
allowance?

Lesson Graphs of Linear Inequalities

1.2 in Two Variables

Graph of Linear Inequalities

The graph of linear inequalities in two variables is a region or a half-plane

where all points of the half-plane are the solution of the inequality. The graph of
an inequality in two variables is the set of points that represents all solutions
to inequality.

Steps in graphing linear inequalities in two variables:

1. Change the linear inequality to linear equation by changing the symbol >, <,
≥, ≤ to =.
2. Arranged the linear equation in the form y= mx + b.
3. Graph the linear equation. Use a broken line (dashed line) for > or < and a
solid line (heavy line) for ≥ or ≤. This will serve as the boundary line. This
line separates the Cartesian plane into two half-plane or region.
4. Use a test point like (0,0) to determine the half-plane. Use the original linear
inequality.
5. Shade the half-plane or region that contains the solution.
6.
Example 1: Graph x + y > 2
Step 1: x + y = 2
x 0 2
Step 2: y = -x + 2
y 2 0
Step 3: table of values
Step 4: Test for (0,0)
x + y> 2
0+0>2
0>2 (not true)

5
 Step 5:
Use broken line

Shade away from the point

(0,0) since the point (0,0) is not
part of the solution.

Therefore, any point that lies in the shaded part is a solution such as
(4,2), (0,6), and (3,1).

Example 2: Analyze the table below.

INEQUALITY PLANE SHADED SLOPE Y- INTERCEPT

DIVIDER REGION
1. y > 2x + 5 Broken line Upper region 2 5
2. 𝑦 <
3
𝑥 −3 Broken line Lower Region 3/2 -3
2
3. 𝑦 ≤
1
𝑥 −7 Solid Line Lower Region 1/2 -7
2
4. 2y ≥ 3x - 7 Solid Line Upper Region 3/2 -7/2

Given the inequality 𝒚 ≤ 𝟐𝒙 − 𝟓, determine the following and graph.

Slope: __________________
y-intercept: __________________
Plane divider: __________________
Shaded region: __________________

6
Read and analyze each problem. Write an inequality statement and solve it
using graphical method.

1. Emily bought two blouse and a pair of pants. The total amount she paid for
the items is not more than Php 980. How much is the possible cost of the
blouse and pants?

7
 MODULE 2

This module was designed and written with you in mind. It is here to help you
master system of linear inequalities in two variables. The scope of this module
permits it to be used in many different learning situations. The language used
recognizes the diverse vocabulary level of students. The lessons are arranged to
follow the standard sequence of the course. But the order in which you read them
can be changed to correspond with the textbook you are now using.

The module contains one lesson, namely:

Lesson 2 – System of Linear Inequalities in Two Variables

After going through this module, you are expected to:

1. Solves problems involving systems of linear inequalities in two variables.

Lesson System of Linear Inequalities

2 in Two Variables

A system of linear inequalities in two variables consists of at least two linear

inequalities in the same variables. A linear inequality solution is the ordered pair
that is a solution to all inequalities in the system. The graph of the linear inequality
is the graph of all solutions of the system.

The following is a system of linear inequalities in two variables.

x + y ≤ 6 Inequality 1
2x - y > 4 Inequality 2

A solution to a system of linear inequalities is an ordered pair that is a solution of

each inequality in the system. For example, (3, -1) is a solution of the system
above.

To check if (3, -1) is a solution, just substitute the value to the given
inequalities.
x+y≤6 2x - y > 4
3 + (-1) ≤ 6 2(3) – (-1) > 4
2 ≤ 6 TRUE 6+1>4
7 > 4 TRUE

8
The graph of a system of linear inequalities is the graph of all solutions of the
system.

To graph a system of linear inequalities, do the following for each inequality

in the system:
1. Graph the line that corresponds to the inequality. Use a dashed line for
inequality with < or > and a solid line for inequality with ≤ or ≥.
2. Lightly shade the half-plane that is the graph of the inequality. Use different
colors to distinguish different half-planes.
3. The double shaded region is the solution for the system of linear
inequalities.

Example: x + y ≤ 6 Inequality 1
2x - y > 4 Inequality 2

The solution set of a system of linear inequalities in two variables is the set
of all ordered pairs of real numbers (x, y) that simultaneously satisfy all the
inequalities in the system.
S = {(5, 1), (3, - 8), (4, -6), (5, -10)}. Example of solutions that satisfies the
inequalities x + y ≤ 6 and 2x - y > 4

A graph of all such ordered pairs is called the solution region for the
system. To solve a system of inequalities, graphically means to graph its solution
region.

9
Activity 1: Graph the given system of inequalities below. Determine five (5) ordered
pairs that are solution to the given system of linear inequalities.

Given: y < x + 4 Inequality 1

y ≥ -2x + 1 Inequality 2

Five (5) ordered pairs as

solution to the given system of
linear inequalities.

Read the problem carefully, then answer the questions below.

Leanne plans to cook fried chicken

and pork chop for her birthday. The
chicken price in the market is ₱ 160.00
per kilogram, while the pork is ₱320.00
per kilogram. How many kilograms of
each does she need to buy if the total
cost is at most ₱ 640.00?
Given:
Let x = the number of kilograms of
chicken
y = the number of kilograms of pork
Questions:
1. Write the inequalities that
represent the phrases below.
a. The total cost of a kilo of
chicken and pork is at most ₱ 640.00.
b. Kilo of chicken is not less than zero
c. Kilo of pork is not less than zero
2. Graph the linear inequalities that you get in number 1.
3. Give three ordered pairs that you can see in the double shaded region.

10
 MODULE 3

This module was designed and written with you in mind. It is here to help you
master illustrating relation and function. The scope of this module permits it to be
used in many different learning situations. The language used recognizes the
diverse vocabulary level of students. The lessons are arranged to follow the
standard sequence of the course. But the order in which you read them can be
changed to correspond with the textbook you are now using.

The module is divided into two lessons, namely:

Lesson 3.1 – Function and Relation

Lesson 3.2- Dependent and Independent Variables

After going through this module, you are expected to:

1. Illustrates a relation and a function.

2. Verifies if a given relation is a function.
3. Determines dependent and independent variables.

Lesson
Function and Relation
3.1

A relation is a set of ordered pairs (x, y), and { } is the symbol for a set. The
first coordinate set is the domain (x – values or the abscissas) of the relation, and
the second coordinates is the range (y – values or the ordinates) of the relation.
A function is a special type of a relation. It is a relation in which every
element in the domain is mapped to exactly one element in the range. Thus, a set
of ordered pairs is a function if no two distinct ordered pairs have equal abscissas.

All functions are relations, but not all relations are functions.

A relation may be represented by the following:

11
Ordered pairs Graph
1. {(1,5), (2, 10), (3,15), (4, 20)}

Table Mapping Diagram

x 1 2 3 4
y 5 10 15 20

Correspondence of Relation

1. One-to-one – If every element in the domain is mapped to a unique element

in the range.
Example:

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 2. Many-to-one – If any two or more elements of the domain are mapped to the
same element in the range.
Example:

3. One-to-many if each element in the domain is mapped to any two or more

elements in the range.
Example:

Activity 1: Determine whether each of the following describes a function or a mere

relation. Write your answer on the space provided before each number.

_________ 1. {(Navotas, Tiangco), (Manila, Moreno), (Quezon, Belmonte)}

_________ 2. {(Math, Saavedra), (Science, Gulmatico), (Filipino, Igcazensa)}
_________ 3.{(paper, pencil), (board, chalk), (paper, ballpen), (keyboard,
monitor)}
_________ 4.(student, i.d), (baby, birth certificate), (employee, employer
number)}
_________ 5. {(1, 4), (4, 5), (5, 7), (6, 5), (8, 7)}

13
Activity 2: Determine the correspondence of the relation. Write one-to-one, many-
to-one, or one-to-many, then identify if it is a FUNCTION or NOT FUNCTION. Write
the correct answer inside the box.

Use the mapping diagram below to answer the following questions.

Questions:
1. Describe the relation that shows in
the mapping diagram.
_______________________________

2. Write the set of ordered pairs that

defines the relation between the city
and its zip code.
O = {________, _________, ________,
________, ______}

3. From the diagram above, find the domain and range of the relation linking
the city and its zip code.
Domain : (_______, ________, ______, ________,)
Range: ( ________, _______, _______,________)

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 4. Is the given relation between the city and its zip code illustrates a function
or not? Why? ____________________________________________________
______________________________________________________________

Lesson Dependent and Independent

3.2 Variables

Variables may be dependent or independent. In an equation where y is

expressed in terms of x, the variable x is considered the independent variable
because any value could be assigned to it. However, the variable y is the
dependent variable because its value depends on the value of x.
In determining the dependent and independent variable, always remember
that x- values (independent variables) must come or happen first.

Independent variable – It is a variable that stands alone and is not changed by the
other variables you are trying to measure. (CAUSE)

Dependent variable – It is something that depends on other factors. (EFFECT)

The Independent variable causes a change in the Dependent variable.

Example: Time spent studying causes a change in Test Score

(Independent variable) (Dependent Variable)

It is not possible that Test Score could cause a change in time spent studying

Activity 1: Classify the variables as independent or dependent.

1. Drinking vitamins makes people healthier.

Independent Variable: ___________________________________
Dependent Variable: _____________________________________

2. Eating breakfast in the morning increases the energy to work.

Independent Variable: ___________________________________
Dependent Variable: _____________________________________

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 3. The number of bikes sold and the price of the bike.
Independent Variable: ___________________________________
Dependent Variable: _____________________________________

4. The weight of a person and the amount of food he takes.

Independent Variable: ___________________________________
Dependent Variable: _____________________________________

5. The amount of water bill to the water consumption every month.

Independent Variable: ___________________________________
Dependent Variable: _____________________________________

Write three (3) examples of dependent and independent variables. Write your
answer in the table below.
Independent Variable Dependent Variable

16
 MODULE 4

This module was designed and written with you in mind. It is here to help you
master graphing and illustrating a linear function. The scope of this module
permits it to be used in many different learning situations. The language used
recognizes the diverse vocabulary level of students. The lessons are arranged to
follow the standard sequence of the course. But the order in which you read them
can be changed to correspond with the textbook you are now using.

The module contains one lesson, namely:

Lesson 4 – Representation of Function and Relation

After going through this module, you are expected to:

1. Finds the domain and range of a function.

2. Graphs and illustrates a linear function and its (a) domain; (b) range; (c)
table of values; (d) intercepts; and (e) slope.

Lesson Representation of
4 Function and Relation

A relation is a set of ordered pairs. The domain of a relation is the set of

first coordinates. the range is the set of the second coordinates.
A function is a relation in which each element of the domain corresponds to
exactly one of the ranges.

Example 1:
Determine whether the following relations are function or not
a. {(1, 4), (2, 5), (3, 6), (4,7)} b. {(1, 2), (2, 5), (2, 0),(3,7)}

Solutions:
a. Each element in the domain, {(1), (2), (3), (4)} is assigned to no more than one
value in the range. 1 is assigned to 4, 2 is assigned to 5, 3 is assigned to 6
and 4 is assigned to 7. therefore, it is a function
b. Since the domain in element 2 is assigned to two different values in the
range, 5 and 0, it is not a function.

17
Example 2:
Solve for x and y intercepts
3x - 5y =15 3x - 5y =15
3(0) -5y = 15 3x - 5(0) =15
-5y= 15 3x = 15
-5 = -5 3 3
y = -3 x= 5

Activity 1: Determine the domain and the range of the relation given the set of
ordered pairs.

1. {(0, 2), (1, 3), (2, 4), (3, 5), (4, 6)}
2. {(0, 2), (0, 4), (0, 6), (0, 8), (0, 10)}
3. {(-5, -2), (-2, -2), (1, 0), (4, 2), (7, 2)}
4. {(0, 2), (-1, 3), (-2, 4), (-3, 5), (-4, 6)}
5. {(0, -2), (1, -3), (2, -4), (3, -5), (4, -6)}

Complete the table and graph the given linear function.

Given: f(x)= 2x+1, if x= 2 find the value of y

x -3 -2 -1 0 1 2 3

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 MODULE 5

This module was designed and written with you in mind. It is here to help
you master on solving problems involving linear function. The scope of this
module permits it to be used in many different learning situations. The
language used recognizes the diverse vocabulary level of students. The
lessons are arranged to follow the standard sequence of the course. But the
order in which you read them can be changed to correspond with the
textbook you are now using.

The module contains one lesson, namely:

Lesson 5 – Linear Function

After going through this module, you are expected to:

1. Describe the parts of a linear function.
2. Describe the Characteristic of a linear function.
3. Solves problems involving linear functions.

Lesson
Linear Function
5

Linear functions are those whose graph is a straight line. A linear function
has the following form. y = f(x) = a + bx. A linear function has one independent
variable and one dependent variable.
Example:
Determine the values of the function f if f(x) = 2x – 1 at x = -3, 0, and 2. Give
their meanings and ordered pairs.
Solution: If x = -3, then f(x) = 2x – 1 becomes f(-3) = 2(-3) – 1
f(-3) = -6 – 1
f(-3) = -7

19
which means the value of f at x = -3 is -7. or, if x = -3, then y = -7. This gives the
ordered pair (-3, -7).
Recall that an ordered pair can be written (x, y).
If x = 0, then f(x) = 2x – 1 becomes f(0) = 2(0) – 1
f(0) = 0 – 1
f(0) = -1
which means the value of f at x = 0 is -1. Or, if x = 0, then y = -1. This gives
another ordered pair (0, -1). If x = 2, then f(x) = 2x – 1 becomes f(2) = 2(2) – 1 f(2)
= 4 – 1 f(2) = 3, which means the value of f at x = 2 is 3. Or, if x = 2, then y = 3.
This gives the ordered pair (2, 3).
This implies that the graph of the function f will pass through the points (-3,
-7), (0, -1), and (2, 3).

Activity 1: Evaluate the following algebraic expressions. Match Column A with the
correct answer with Column B.

If x=2 y= -3 z=5

Column A Column B

1. 2xy A. 1
2. x-4y B. -37
3. x2+y C. -12
4. 4y-5z D. 27
5. 3x-7y E. 14

Directions: Complete the table below. Graph the linear function.

f(x)= 2-3x -6 -3 0 3 6

20
 MODULE 6

This module was designed and written with you in mind. It is here to help you
master on determining the relationship of between the hypothesis and conclusion
of an if -then statement. The scope of this module permits it to be used in many
different learning situations. The language used recognizes the diverse vocabulary
level of students. The lessons are arranged to follow the standard sequence of the
course. But the order in which you read them can be changed to correspond with
the textbook you are now using.

The module contains one lesson, namely:

Lesson 6 – If-Then Statement

After going through this module, you are expected to:

1. Identify the hypothesis and conclusion of an if-then statement

2. Transform a statement into an equivalent if-then statement.

Lesson
If-Then Statement
6

Read each statement below.

1. If students accomplish their homework on time, then they will get good
grades. Diane accomplished her homework on time; therefore, she got a good
grade.

2. If it rains, then the basketball game will be canceled.

It rains; therefore, the basketball game is canceled.

3. If the lines intersect, then they are not parallel.

Lines m and n are not parallel; therefore, they intersect.

21
 4. If two angles form a linear pair, then they are supplementary.
∠A and ∠B are supplementary; therefore, they form a linear pair.

If you will notice, the second statement is a valid consequence of the first
statement.

An If-then statement is called a conditional statement. It consists of two parts:

the hypothesis and the conclusion. The statement is written in the form, “If p then
q.” The “If” clause contains the hypothesis p and the “then” clause contains the
conclusion q.

Examples:
If students accomplish their homework on time, then they will get good
grades. (hypothesis) (conclusion)

If the lines intersect, then they are not parallel.

(hypothesis) (conclusion)

Activity 1: Identify the hypothesis and the conclusion for each of the following
conditional statements.

1. If you are kind and helpful, then you will have more friends.
Hypothesis: ___________________________________________
Conclusion: ___________________________________________

2. If you live your life to the fullest, then you will have no regrets.
Hypothesis: ___________________________________________
Conclusion: ___________________________________________

3. If you exercise regularly, then you will have a healthy mind and body.
Hypothesis: ___________________________________________
Conclusion: ___________________________________________

4. If you do your homework, then you will have a good grade.

Hypothesis: ___________________________________________
Conclusion: ___________________________________________

5. If the COVID-19 pandemic stops, then we will all be happy.

Hypothesis: ___________________________________________
Conclusion: ___________________________________________

22
Convert each statement to an if-then statement, then identify the hypothesis
and the conclusion.
1. Filipinos are God-fearing people.
If-Then Form: _____________________________________________________
Hypothesis: _______________________________________________________
Conclusion: _______________________________________________________

2. The hypotenuse is the longest side of a right triangle.

If-Then Form: _____________________________________________________
Hypothesis: _______________________________________________________
Conclusion: _______________________________________________________

3. The intersection of the coordinate axes is the origin.

If-Then Form: _____________________________________________________
Hypothesis: _______________________________________________________
Conclusion: _______________________________________________________

4. An even number is divisible by two.

If-Then Form: _____________________________________________________
Hypothesis: _______________________________________________________
Conclusion: _______________________________________________________

5. A triangle is a polygon with three sides.

If-Then Form: _____________________________________________________
Hypothesis: _______________________________________________________
Conclusion: _______________________________________________________

23
 MODULE 7

This module was designed and written with you in mind. It is here to help you
master on determining the inverse, converse, and contrapositive of n if-then
statement. The scope of this module permits it to be used in many different
learning situations. The language used recognizes the diverse vocabulary level of
students. The lessons are arranged to follow the standard sequence of the course.
But the order in which you read them can be changed to correspond with the
textbook you are now using.

The module contains one lesson, namely:

Lesson 7 – Inverse, Converse, and Contrapositive Statement

After going through this module, you are expected to:

1. Determine the inverse, converse, and contrapositive of an if-then statement.

Lesson Inverse, Converse, and

7 Contrapositive Statement

Study the table below and answer the questions.

Statement If-Then Form Converse Inverse Contrapositive

(Conditional)
𝒑→𝒒 𝒒→𝒑 ~𝒑 → ~𝒒 ~𝒒 → ~𝒑
A triangle is a If a polygon is If a polygon If a polygon is If a polygon
polygon with a triangle, has three not a triangle, does not have
three sides. then it has sides, then it then it does three sides,
three sides. is a triangle. not have three then it is not a
sides. triangle.

The converse of a statement is formed by interchanging the hypothesis and the

conclusion.

24
The inverse of a statement is formed by negating both the hypothesis and the
conclusion.

The contrapositive of a statement is formed by interchanging the hypothesis and

the conclusion and negating both.

We can summarize the process of converting a statement in terms of p and q

in this table.

Statement If p, then q
(p → q)
Converse If q, then p
(q → p)
Inverse If not p, then not q
(~p → ~q)
Contrapositive If not q, then not p
(~q → ~ p)

Activity 1: State the converse, inverse, and contrapositive of each conditional

statement. Then determine which are true and which are false.

1. If I can play the ukulele, then I am a musician.

Converse: __________________________________________________________

Inverse: ____________________________________________________________

Contrapositive: _____________________________________________________

2. If a triangle has no congruent sides, then it is scalene.

Converse: __________________________________________________________

Inverse: ____________________________________________________________

Contrapositive: _____________________________________________________

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 3. If a number is even, then it ends in the digit 0, 2, 4, 6, or 8.

Converse: __________________________________________________________

Inverse: ____________________________________________________________

Contrapositive: _____________________________________________________

4. If a shape is a triangle, then it is a polygon.

Converse: __________________________________________________________

Inverse: ____________________________________________________________

Contrapositive: _____________________________________________________

5. If you are a lawyer, then you passed the bar examination.

Converse: __________________________________________________________

Inverse: ____________________________________________________________

Contrapositive: _____________________________________________________

Complete the table by supplying the converse, inverse, and contrapositive of

the conditional statement.

Conditional Converse Inverse Contrapositive

If a man is a
teacher, then he is
a college graduate.

Conditional Converse Inverse Contrapositive

If an animal is a
bird, then it has
feathers.

Conditional Converse Inverse Contrapositive

If two lines are
parallel, then they
do not intersect.

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 Conditional Converse Inverse Contrapositive
If two points are
collinear, then they
lie on the same
line.

Conditional Converse Inverse Contrapositive

If a triangle is
isosceles, then it
has at least two
congruent sides.

27
 MODULE 8

This module was designed and written with you in mind. It is here to help
you master on illustrating the equivalences of a) the statement and its
contrapositive, and (b) the converse and inverse of a statement. The scope of
this module permits it to be used in many different learning situations. The
language used recognizes the diverse vocabulary level of students. The
lessons are arranged to follow the standard sequence of the course. But the
order in which you read them can be changed to correspond with the
textbook you are now using.

The module contains one lesson, namely:

Lesson 8 – Conditional Statement and Inverse, Converse, and Contrapositive

Statement

After going through this module, you are expected to:

1. Illustrate the equivalence of a statement, its contrapositive, converse,

and inverse.
2. Describe the truth value of contrapositive, converse, and inverse of a
statement.

Conditional Statement and its

Lesson
Contrapositive, Converse, and
8
Inverse

Two statements are logically equivalent if they have the same truth value for
all possible combinations of variables appearing in the two expressions. A
convenient and helpful way to organize truth values of various statements is in a
truth table. A truth table is a table whose columns are statements and whose rows
are possible scenarios. The table contains every possible scenario and the truth

28
values that would occur. The table below summarizes a conditional statement's
truth value, its converse, inverse, and contrapositive.

Conditional Converse Inverse Contrapositive

𝑝 𝑞 ~𝑝 ~𝑞 Statement
𝑝→𝑞 𝑞→𝑝 ~𝑝 → ~𝑞 ~𝑞 → ~𝑝

T T F F T T T T

T F F T F T T F

F T T F T F F T

F F T T T T T T

Activity 1: Rewrite the conditional statement into its contrapositive, converse and
inverse.

s = 4units

Area = 16 square units

Conditional Statement: ___________________________________________

Premise (p): ______________________________________________________

Conclusion (q):____________________________________________________

Converse (𝑞 → 𝑝): _________________________________________________

Inverse (~𝑝 → ~𝑞): ________________________________________________

Contrapositive (~𝑞 → ~𝑝): _________________________________________

29
Directions: Write the converse, inverse, and contrapositive of the statement.
Identify the truth value of the given statements. If the statement is false, provide a
counterexample.

If x = 3, then 𝐱 𝟐 = 9.

Types of Statement Truth Value Counter

Statement Example
Converse

Inverse

Contrapositive

30
 MODULE 9

This module was designed and written with you in mind. It is here to help you
master on using inductive and deductive reasoning, and writing direct and indirect
proof. The scope of this module permits it to be used in many different learning
situations. The language used recognizes the diverse vocabulary level of students.
The lessons are arranged to follow the standard sequence of the course. But the
order in which you read them can be changed to correspond with the textbook you
are now using.

The module is divided into two lessons, namely:

Lesson 9.1 – Inductive and Deductive Reasoning

Lesson 9.2- Direct and Indirect Proofs

After going through this module, you are expected to:

1. Define inductive and deductive reasoning.

2. Differentiate between direct and indirect proof.

3. Utilize different forms in representing proofs.

Lesson Inductive and Deductive

9.1 Reasoning

Reasoning means the process of thinking about something logically to form

a conclusion or judgment. There are two major types of reasoning; deductive and
inductive. They refer to the process by which someone creates a conclusion and
how they believe their conclusion to be true.

Inductive reasoning uses a set of specific observations to reach an

overarching conclusion. Particular premises create a pattern that gives way to a
broad idea that is likely true.

31
Examples:

1.

2. 1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
4 + 97 = 101
5 + 96 = 101

Deductive reasoning requires one to start with a few general ideas, called
premises, and apply them to a specific situation. Recognized rules, laws, theories,
and other widely accepted truths are used to prove that a conclusion is right.

Examples:

The Philippines is one of the top tourist destinations.

Palawan is in the Philippines.
Conclusion: Palawan is one of the top tourist destinations.
Filipinos are hardworking.
Juan is a Filipino.
Conclusion: Juan is hardworking.

We can illustrate inductive and deductive reasoning as:

Specific ideas General ideas

Deductive

Inductive

General ideas Specific ideas

Inductive reasoning Deductive reasoning is

can be shown using an often expressed visually using
inverted funnel that starts at a funnel that narrows a
the narrow premises and general idea into a specific
expands into a wider conclusion.
conclusion.

32
Activity 1: Observe the given statements. Provide a conclusion using the patterns
that you derived from the given statements.

1. Al got a high score in his Math test. He also loves to solve and compute. With
this, we can say that Al is _________________________________.

2. 5, 10, 15, 20, 25, ______

3. Collinear points are points on the same line.

Points A, B, and C are on the same line.
Conclusion: ________________________________________________

4. A right-angle measure exactly 90°.

Angle ABC is a right angle.
Conclusion: ________________________________________________

5. Equilateral triangles have equal measurements on sides.

Triangle XYZ is an equilateral triangle.
Conclusion: ________________________________________________

Draw a conclusion from the given situation and identify the reasoning being
used.
1. Acute angles measure less than 90. ∠ABC is an acute angle.
________________________________________________________________

2. 12, 24, 36, 48. The next number is ____________________________________.

3. Collinear points are points in the same line. Points X, Y and Z are on the
same line. ____________________________________________________________.
4. Filipinos are hospitable. Juan is a Filipino.
________________________________________________________________.

5. Every time I study hard, I get good grades. I believe that ___________________.

33
Lesson
Direct and Indirect Proofs
9.2

A proof is a collection of statements where the conclusion absolutely follows

from the premises of the arguments. It is a logical argument showing that the truth
of the premises guarantees the truth of the conclusion. There are two kinds of
proofs, namely:
Direct proof is a sequence of statements that are either givens or
deductions from previous statements and whose last statement is the conclusion to
be proved.
Indirect proof is a reasoning method where the opposite of the statement to
be proven is assumed true until the assumption leads to a contradiction.
A proof may be constructed in a two-column form or paragraph form. A two-
column proof is composed of a list of statements and reasons why these statements
are true. A proof in paragraph form is only a two-column form written in sentences.

34
 The given statements show the differences between direct and indirect proof.
Direct proof is a sequence of statements that are either givens or deductions from
previous statements and whose last statement is the conclusion to be proved. At
the same time, the indirect proof is a method of reasoning where the opposite of the
statement to be proven is assumed true until the assumption leads to a
contradiction.

Activity 1: Draw a conclusion based on the given statements and support it

with a reason.

Prove the given statement.

Given: 𝑚∠1 + 𝑚∠2 = 𝑚∠2 + 𝑚∠3

Prove: 𝑚∠1 = 𝑚∠3

35
Directions: Read and analyze ach question carefully. Choose the letter of the
correct answer.

1. Which of the following is a linear inequality in two variables?

A. 4a – 3b = 9
B. 8f + 4 < 12
C. 3x < 16
D. 11 + 2t ≥ 3k

2. Which among the mathematical symbols best translate the verbal statement
Twice a number x and y is at most 25?
A. 2x + y < 24
B. 2x + y > 24
C. 2x + y ≥ 24
D. 2x + y ≤ 24

3. How many solutions does the following system have?

A. infinite
B. 2
C. 1
D. zero

4. Which of the following is an example of a function?

A. {(-3, 2), (-2, 3), (-2, 5)}

B. {(-1, 2), (0, 3), (0, 4)}
C. {(1, 3), (1, 2), (1, 4)}
D. {(0, 1), (1, 2), (-1, 4)}

5. Find the range of the function h(x) = x ² + 3.

A. All real
B. [3, )
C. (3, )
D. All of the above

36
6. Find the value of y if f(x) = 3x-2 and the value of x = 4.

A. -10
B. 10
C. 16
D. -16

7. Which of the following is the if-then form of the statement: “A prime number
has one and itself as factors.”?

A. If a number is prime, then its factors are 1 and itself.

B. If a number is prime, then its factors are not 1 and itself.
C. If a number is not prime, then its factors are 1 and itself.
D. If a number is not prime, then its factors are not 1 and itself.

8. Which of the following is the contrapositive of the statement, “If two angles
are vertical, then they are congruent.”?

A. If two angles are not vertical, then they are not congruent.
B. If two angles are congruent, then they are vertical.
C. If two angles are vertical, then they are congruent.
D. If two angles are not congruent, then they are not vertical.

9. The statement If <A and <B are supplementary, then the sum of their
measure is 180° is translated into If the sum of the measure of <A and <B is
180°, then they are supplementary. Which symbolization illustrates the
second statement?

A. p→q
B. q→p
C. ~p→~q
D. ~q→~p

10. Which appropriate conclusion can be drawn from the statement m∠J + m∠S
= 90?

A. ∠J≅∠S
B. ∠J and ∠S are right angles
C. ∠J and ∠S are complementary
D. ∠J and ∠S are supplementary

37
 38
What I can Do What's More
Slope: 2
y-intercept: -
5
Plane Divider:
Solid Line
Shaded
Region: Lower
Region
Module 1
Module 2

Module 3

39
 40
What I can Do What’s More
x -3 -2 -1 0 1 2 3
y -5 -3 -1 1 3 5 7 1. Domain: {0, 1, 2, 3, 4}
Range: {2, 3, 4, 5, 6}
2. Domain: {0}
Range: {2, 4, 6, 8, 10}
3. Domain: {-5, -2, 1, 4,
7}
Range: {-2, 0, 2}
4. Domain: {0, -1, -2, -3,
-4}
Range: {2, 3, 4, 5, 6}
5. Domain: {0, 1, 2, 3, 4}
Range: {-2, -3, -4, -5, -
6}
Module 4
What I can Do
Note: Assist the students in
checking this part
Module 5

Module 6

41
Module 7

42
43
 44
What’s More
Conditional Statement: If the sides of the square measure 4 units,
then its area is 16 square units.
Premise: The sides of the square measures 4 units
Conclusion: The area of a square is 16 square units.
Converse: If the area of a square is 16 square units, then its sides
measure 4 units.
Inverse: If the sides of a square do not measure 4 units, then its area
is not 16 square units.
Contrapositive: If the are of the square in not 16 square units, then
its sides do not measure 4 units.
Module 8
Module 9

45
References
Antonio C. Coronel and Sr. Iluminada C. Coronel. (2013). Growing Up with Math 8.
Quezon City: FNB Educational, Inc.

Emmanuel P. Abuzo, M. L. (2013). Mathematics Learner's Module 8. Quezon City:

Book Media Press, Inc.

Orlando A. Oronce and Marilyn O. Mendoza. (2013). E-Math 8. Quezon City: Rex
Book Store, Inc.

46
For inquiries or feedback, please write or call:

Department of Education – Schools Division Office Navotas

Learning Resource Management Section

Bagumbayan Elementary School Compound

M, Naval St., Sipac Almacen, Navotas City

Telefax: 02-8332-77-64
Email Address: navotas.city@deped.gov.ph