7 math lm mod1

Learner’s Material
Module 1
7
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We value your feedback and recommendations.

ii
Department of Education
Republic of the Philippines

iii
Mathematics – Grade 7
Learner’s Material
First Edition, 2013
ISBN: 978-971-9990-60-4
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Published by the Department of Education
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Undersecretary: Yolanda S. Quijano, Ph.D.
Assistant Secretary: Elena R. Ruiz, Ph.D.
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Development Team of the Learner’s Material
Consultant: Ian June L. Garces, Ph.D.
Authors: Elizabeth R. Aseron, Angelo D. Armas, Allan M. Canonigo,
Ms. Jasmin T. Dullete, Flordeliza F. Francisco, Ph.D., Ian June L. Garces,
Ph.D., Eugenia V. Guerra, Phoebe V. Guerra, Almira D. Lacsina, Rhett
Anthony C. Latonio, Lambert G. Quesada, Ma. Christy R. Reyes, Rechilda P.
Villame, Debbie Marie B. Verzosa, Ph.D., and Catherine P. Vistro-Yu, Ph.D.
Editor: Catherine P. Vistro-Yu, Ph.D.
Reviewers: Melvin M. Callanta, Sonia Javier, and Corazon Lomibao

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Table of Contents
Number Sense
Lesson 1 Sets: An Introduction …………………………………... 1
Lesson 2.1: Union and Intersection of Sets ……………………….. 5
Lesson 2.2: Complement of a Set ………………………………….. 9
Lesson 3: Problems Involving Sets ………………………………. 13
Lesson 4.1: Fundamental Operation on Integers: Addition
of Integers ………………………………………………. 18
Lesson 4.2: Fundamental Operation on Integers: Subtraction
of Integers ………………………………………………. 22
Lesson 4.3: Fundamental Operation on Integers: Multiplication
of Integers ………………………………………………. 27
Lesson 4.4: Fundamental Operation on Integers: Division
of Integers ………………………...…………………….. 30
Lesson 5: Properties of the Operations on Integers ……………. 33
Lesson 6: Rational Numbers in the Number Line ………………. 41
Lesson 7: Forms of Rational Numbers and Addition
and Subtraction of Rational Numbers ……………….. 45
Lesson 8: Multiplication and Division of Rational Numbers ……. 53
Lesson 9: Properties of the Operations on Rational Numbers … 59
Lesson 10: Principal Roots and Irrational Numbers ……………… 64
Lesson 11: The Absolute Value of a Number …………………….. 70
Lesson 12: Subsets of Real Numbers …………………………….. 76
Lesson 13: Significant Digits and the Scientific Notation ………... 83
Lesson 14: More Problems Involving Real Numbers ……………. 87

1
Lesson 1: Sets: An Introduction
Pre-requisite Concepts: Whole numbers
About the Lesson:
This is an introductory lesson on sets. A clear understanding of the concepts in
this lesson will help you easily grasp number properties and enable you to quickly
identify multiple solutions involving sets of numbers.
Objectives:
In this lesson, you are expected to:
1. Describe and illustrate
a. well-defined sets;
b. subsets;
c. universal set, and;
d. the null set.
2. Use Venn Diagrams to represent sets and subsets.
Lesson Proper:
A.
I. Activity
Below are some objects. Group them as you see fit and label each group.
Answer the following questions:
a. How many groups are there?
b. Does each object belong to a group?
c. Is there an object that belongs to more than one group? Which one?
The groups are called sets for as long as the objects in the group share a
characteristic and are thus, well defined.

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Problem: Consider the set consisting of whole numbers from 1 to 200. Let this be
set U. Form smaller sets consisting of elements of U that share a different
characteristic. For example, let E be the set of all even numbers from 1 to 200.
Can you form three more such sets? How many elements are there in each
of these sets? Do any of these sets have any elements in common?
Did you think of a set with no element?
Important Terms to Remember
The following are terms that you must remember from this point on.
1. A set is a well-defined group of objects, called elements, that share a
common characteristic. For example, 3 of the objects belong to the set of
head covering or simply hats (ladies hat, baseball cap, hard hat).
2. Set F is a subset of set A if all elements of F are also elements of A.
For example, the even numbers 2, 4, and 12 all belong to the set of whole
numbers. Therefore, the even numbers 2, 4, and 12 form a subset of the set
of whole numbers. F is a proper subset of A if F does not contain all
elements of A.
3. The universal set U is the set that contains all objects under consideration.
4. The null set  is an empty set. The null set is a subset of any set.
5. The cardinality of a set A is the number of elements contained in A.
Notations and Symbols
In this section, you will learn some of the notations and symbols pertaining to sets.
1. Uppercase letters will be used to name sets and lowercase letters will be
used to refer to any element of a set. For example, let H be the set of all
objects on page 1 that cover or protect the head. We write
H = {ladies hat, baseball cap, hard hat}
This is the listing or roster method of naming the elements of a set.
Another way of writing the elements of a set is with the use of a
descriptor. This is the rule method. For example, H = {x|x covers and protects
the head}. This is read as “the set H contains the element x such that x
covers and protects the head.”
2. The symbol  or { } will be used to refer to an empty set.
3. If F is a subset of A, then we write . We also say that A contains the
set F and write it as . If F is a proper subset of A, then we write .
4. The cardinality of a set A is written as n(A).

F  A

AF

F  A

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II. Questions to Ponder (Post-Activity Discussion)
Let us answer the questions posed in the opening activity.
1. How many sets are there?
There is the set of head covers (hats), the set of trees, the set of even
numbers, and the set of polyhedrals. But, there is also a set of round objects
and a set of pointy objects. There are 6 well-defined sets.
2. Does each object belong to a set? Yes.
3. Is there an object that belongs to more than one set? Which ones?
All the hats belong to the set of round objects. The pine trees and two of the
polyhedrals belong to the set of pointy objects.
III. Exercises
Do the following exercises.
1. Give 3 examples of well-defined sets.
2. Name two subsets of the set of whole numbers using both the listing method
and the rule method.
3. Let B = [1, 3, 5, 7, 9}. List all the possible subsets of B.
4. Answer this question: How many subsets does a set of n elements have?
B. Venn Diagrams
Sets and subsets may be represented using Venn Diagrams. These are
diagrams that make use of geometric shapes to show relationships between sets.
Consider the Venn diagram below. Let the universal set U be all the elements in sets
A, B, C, and D.
D
A
B
C

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Each shape represents a set. Note that although there are no elements
shown inside each shape, we can surmise how the sets are related to each other.
Notice that set B is inside set A. This indicates that all elements in B are contained in
A. The same with set C. Set D, however, is separate from A, B, and C. What does it
mean?
Exercises
Draw a Venn diagram to show the relationships between the following pairs
or groups of sets:
1. E = {2, 4, 8, 16, 32}
F = {2, 32}
2. V is the set of all odd numbers
W = {5, 15, 25, 35, 45, 55 …}
3. R = {x| x is a factor of 24}
S = { }
T = {7, 9, 11}
Summary
In this lesson, you learned about sets, subsets, the universal set, the null set,
and the cardinality of a set. You also learned to use the Venn diagram to show
relationships between sets.

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Lesson 2.1: Union and Intersection of Sets
Prerequisite Concepts: Whole Numbers, Definition of Sets, Venn Diagrams
About the Lesson:
After learning some introductory concepts about sets, a lesson on set
operations follows. You will learn how to combine sets (union) and how to determine
the elements common to 2 or 3 sets (intersection).
Objectives:
1. Describe and define
a. union of sets
b. intersection of sets
2. Perform the set operations
a. union of sets
b. intersection of sets
3. Use Venn diagrams to represent the union and intersection of sets.
Lesson Proper:
I. Activities
A B
1. Which of the following shows the union of set A and set B? How many
elements are in the union of A and B?
1 2 3

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2. Which of the following shows the intersection of set A and set B?
How many elements are there in the intersection of A and B?
1 2 3
Here’s another activity:
Let
V = {2x | x  , 1  x  4}
W = {x2
| x  , -2  x  2}
What elements may be found in the intersection of V and W?
How many are there?
What elements may be found in the union of V and W?
How many are there?
Do you remember how to use Venn Diagrams? Based on the diagram below,
(1) determine the elements that belong to both A and B; (2) determine the elements
that belong to A or B or both. How many are there in each set?
Important Terms / Symbols to Remember
1. Let A and B be sets. The union of sets A and B, denoted by A ∪ B, is the set
that contains those elements that belong to A, B, or to both.
An element x belongs to the union of the sets A and B if and only if x
belongs to A or x belongs to B or to both. This tells us that
A  B = {x l x  A or x  B}
A B
10
20
1
12
25
36

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Using the Venn diagram, all of set A and B are shaded to show A  B.
2. Let A and B be sets. The intersection of the sets A and B, denoted by
A  B, is the set containing those elements that belong to both A and B.
An element x belongs to the intersection of the sets A and B if and only if x
belongs to A and x belongs to B. This tells us that
A  B = {x l x  A and x  B}
Using the Venn diagram, the set A  B consists of the shared regions
of A and B.
Sets whose intersection is an empty set are called disjoint sets.
3. The cardinality of the union of two sets is given by the following equation:
n (A  B) = n (A) + n (B) – n (A  B ).
1. Which of the following shows the union of set A and set B? Set 2. This is
because it contains all the elements that belong to A or B or both. There are
8 elements.
2. Which of the following shows the intersection of set A and set B? Set 3.
This is because it contains all elements that are in both A and B. There are
3 elements.
In the second activity:
V = {2, 4, 6, 8}
W = {0, 1, 4}
Therefore, V  W = {4} has 1 element and V  W = {0, 1, 2, 4, 6, 8} has 6
elements. Note that the element {4} is counted only once.
U
A B

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On the Venn Diagram: (1) The set that contains elements that belong to both A
and B consists of two elements {1, 12}; (2) The set that contains elements that
belong to A or B or both consists of 6 elements {1, 10, 12, 20, 25, 36}.
III. Exercises
1. Given sets A and B, determine which of the following shows (a) A  B;
and (b) A  B?
Set A
Students Who Play the
Guitar
Set B
Students Who Play the
Piano
Ethan Molina Mayumi Torres
Chris Clemente Janis Reyes
Angela Dominguez Chris Clemente
Mayumi Torres Ethan Molina
Joanna Cruz Nathan Santos
Set 1 Set 2 Set 3 Set 4
Ethan Molina
Chris Clemente
Angela Dominguez
Mayumi Torres
Joanna Cruz
Mayumi Torres
Ethan Molina
Chris Clemente
Mayumi Torres
Janis Reyes
Chris Clemente
Ethan Molina
Nathan Santos
Ethan Molina
Chris Clemente
Angela Dominguez
Mayumi Torres
Joanna Cruz
Janis Reyes
Nathan Santos
2. Do the following exercises. Write your answers on your paper.
A = {0, 1, 2, 3, 4} B = {0, 2, 4, 6, 8} C = {1, 3, 5, 7, 9}
Given the sets above, determine the elements and cardinality of:
a. A  B =
b. A  C =
c. A  B  C =
d. A  B =
e. B  C =
f. A  B  C =
g. (A  B)  C =
3. Let W = {x | 0 < x < 3}, Y = {x | x > 2}, and Z = {x | 0  x  4}.
Determine (a) (W  Y)  Z; (b) W  Y  Z.
Summary
In this lesson, you learned the definition of union and intersection of sets. You
also learned how use Venn diagram to represent the union and the intersection of
sets. You also learned how to determine the elements that belong to the union and
intersection of sets.

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Lesson 2.2: Complement of a Set
Prerequisite Concepts: Sets, Universal Set, Empty Set, Union and Intersection
of Sets, Cardinality of Sets, Venn Diagrams
About the Lesson:
The complement of a set is an important concept. There will be times when
you need to consider the elements not found in a particular set. This is when you
need the complement of a set.
Objectives:
1. Describe and define the complement of a set;
2. Find the complement of a given set; and
3. Use Venn diagrams to represent the complement of a set.
Lesson Proper:
I. Problem
In a population of 8000 students, 2100 are Freshmen, 2000 are Sophomores,
2050 are Juniors, and the remaining 1850 are either in their fourth or fifth year in
university. A student is selected from the 8000 students and it is not a sophomore,
how many possible choices are there?
Discussion
Definition: The complement of a set A, written as A’, is the set of all elements found
in the universal set, U, that are not found in set A. The cardinality n (A’) is given by
n (A’) = n (U) – n (A) .
Venn diagram:
Examples:
1. Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8}.
Then the elements of A’ are the elements from U that are not found in A.
Therefore, A’ = {1, 3, 5, 7, 9} and n (A’) = 5
2. Let U = {1, 2, 3, 4, 5}, A = {2, 4}, and B = {1, 5}. Then,
A’ = {1, 3, 5}
B’ = {2, 3, 4}
A’ ∪ B’ = {1, 2, 3, 4, 5} = U
3. Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 7, 8}. Then,
U
A’
A

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A’ = {5, 6, 7, 8}
B’ = {1, 2, 5, 6}
A’∩ B’ = {5, 6}
4. Let U = {1, 3, 5, 7, 9}, A = {5, 7, 9}, and B = {1, 5, 7, 9}. Then,
A ∩ B = {5, 7, 9}
(A ∩ B)’ = {1, 3}
5. Let U be the set of whole numbers. If A = {x | x is a whole number and x >
10}, then,
A’ = {x | x is a whole number and 0  x  10}.
The opening problem asks how many possible choices there are for a student
who was selected and known to be a non-Sophomore. Let U be the set of all
students and n (U) = 8000. Let A be the set of all Sophomores, then n (A) = 2000.
The set A’ consists of all students in U that are not Sophomores and n (A’) = n (U) –
n (A) = 6000. Therefore, there are 6000 possible choices for that selected student.
II. Activity
Shown in the table are names of students of a high school class by
sets according to the definition of each set.
A
Likes Singing
B
Likes Dancing
C
Likes Acting
D
Don’t Like Any
Jasper
Faith
Jacky
Miguel
Joel
Charmaine
Leby
Joel
Jezryl
Jacky
Jasper
Ben
Joel
Billy
Ethan
Camille
Tina
After the survey has been completed, find the following sets.
a. U = _________________________________________
b. A ∪ B’ = _____________________________________
c. A’ ∪ C = _____________________________________
d. (B ∪ D)’ = ____________________________________
e. A’ ∩ B = _____________________________________
f. A’ ∩ D’ = ____________________________________
g. (B ∩ C)’ = ____________________________________
The easier way to find the elements of the indicated sets is to use a Venn
diagram showing the relationships of U, sets A, B, C, and D. Set D does not share
any members with A, B, and C. However, these three sets share some members.

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The Venn diagram below is the correct picture:
Now, it is easier to identify the elements of the required sets.
a. U = {Ben, Billy, Camille, Charmaine, Ethan, Faith, Jacky, Jasper, Jezryl, Joel,
Leby, Miguel, Tina}
b. A ∪ B’ = {Faith, Miguel, Joel, Jacky, Jasper, Ben, Billy, Ethan, Camille, Tina}
c. A’ ∪ C = {Jasper, Jacky, Joel, Ben, Leby, Charmaine, Jezryl, Billy, Ethan,
Camille, Tina}
d. (B ∪ D)’ = {Faith, Miguel, Jacky, Jasper, Ben}
e. A’ ∩ B = {Leby, Charmaine, Jezryl}
f. A’ ∩ D’ = {Leby, Charmaine, Jezryl, Ben}
g. (B ∩ C)’ = {Ben, Billy, Camille, Charmaine, Ethan, Faith, Jacky, Jasper,
Jezryl, Leby, Miguel, Tina}
III. Exercises
1. True or False. If your answer is false, give the correct answer.
Let U = the set of the months of the year
X = {March, May, June, July, October}
Y = {January, June, July}
Z = {September, October, November, December}
a. Z’ = {January, February, March, April, May, June, July, August}
b. X’ ∩ Y’ = {June, July}
c. X’ ∪ Z’ = {January, February, March, April, May, June, July,
August, September, November, December}
d. (Y ∪ Z)’ = {February, March, April, May}
A
B
C
Joel
Jacky
Jasper
Ben
Leby
Charmaine
Jezryl
Faith
Miguel
Billy
Ethan
Camille
Tina
U

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2. Place the elements in their respective sets in the diagram below based on the
following elements assigned to each set:
U = {a, b, c, d, e, f, g, h, i, j}
A’ = {a, c, d, e, g, j}
B’ = {a, b, d, e, h, i}
C’ = {a, b, c, f, h, i, j}
3. Draw a Venn diagram to show the relationships between sets U, X, Y, and Z,
given the following information.
 U, the universal set, contains set X, set Y, and set Z.
 X  Y  Z = U
 Z is the complement of X.
 Y’ includes some elements of X and the set Z.
Summary
In this lesson, you learned about the complement of a given set. You learned
how to describe and define the complement of a set, and how it relates to the
universal set, U, and the given set.
A
B
C
U

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Lesson 3: Problems Involving Sets
Prerequisite Concepts: Operations on Sets and Venn Diagrams
About the Lesson:
This is an application of your past lessons about sets. You will appreciate
more the concepts and the use of Venn diagrams as you work through the different
word problems.
Objectives:
1. Solve word problems involving sets with the use of Venn diagrams
2. Apply set operations to solve a variety of word problems.
Lesson Proper:
I. Activity
Try solving the following problem:
In a class of 40 students, 17 have ridden an airplane, 28 have ridden a
boat. 10 have ridden a train, 12 have ridden both an airplane and a boat. Three
have ridden a train only and 4 have ridden an airplane only. Some students in
the class have not ridden any of the three modes of transportation and an equal
number have taken all three.
a. How many students have used all three modes of transportation?
b. How many students have taken only the boat?
II. Questions/Points to Ponder (Post-Activity Discussion)
Venn diagrams can be used to solve word problems involving union and
intersection of sets. Here are some worked out examples:
1. A group of 25 high school students were asked whether they use either
Facebook or Twitter or both. Fifteen of these students use Facebook and
twelve use Twitter.
a. How many use Facebook only?
b. How many use Twitter only?
c. How many use both social networking sites?
Solution:
Let S1 = set of students who use Facebook only
S2 = set of students who use both social networking sites
S3 = set of students who use Twitter only
The Venn diagram is shown below
Facebook Twitter
S1 S3
S2

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Finding the elements in each region:
The number of elements in each region is shown below
2. A group of 50 students went on a tour in Palawan province. Out of the 50
students, 24 joined the trip to Coron; 18 went to Tubbataha Reef; 20 visited
El Nido; 12 made a trip to Coron and Tubbataha Reef; 15 saw Tubbataha
Reef and El Nido; 11 made a trip to Coron and El Nido and 10 saw the
three tourist spots.
a. How many of the students went to Coron only?
b. How many of the students went to Tubbataha Reef only?
c. How many joined the El Nido trip only?
d. How many did not go to any of the tourist spots?
Solution:
To solve this problem, let
P1 = students who saw the three tourist spots
P2 = those who visited Coron only
P3 = those who saw Tubbataha Reef only
P4 = those who joined the El Nido trip only
P5 = those who visited Coron and Tubbataha Reef only
P6 = those who joined the Tubbataha Reef and El Nido trip only
P7 = those who saw Coron and El Nido only
P8 = those who did not see any of the three tourist spots
Draw the Venn diagram as shown below and identify the region where the
students went.
n(S1) + n(S2) + n(S3) = 25
n(S1) + n(S2) = 15
______________________
n(S3) = 10
But n(S2) + n(S3) = 12
________________
n(S2) = 2
n(S1) + n(S2) + n(S3) = 25
n(S2) + n(S3) = 12
_______________________
n(S1) = 13
U
Facebook
Twitter
13 2 10

15
Determine the elements in each region starting from P1.
P1 consists of students who went to all 3 tourist spots. Thus, n(P1) = 10.
P1  P5 consists of students who visited Coron and Tubbataha Reef but this
set includes those who also went to El Nido. Therefore,
n(P5) = 12 – 10 = 2 students visited Coron and Tubbatha Reef only.
P1  P6 consists of students who went to El Nido and Tubbataha Reef but
this set includes those who also went to Coron. Therefore,
n(P6) = 15 – 10 = 5 students visited El Nido and Tubbataha Reef only.
P1  P7 consists of students who went to Coron and El Nido but this set
includes those who also went to Tubbataha Reef. Therefore,
n(P7) = 11 – 10 = 1 student visited Coron and El Nido only.
From here, it follows that
n(P2) = 24 – n(P1) – n(P5) – n(P7) = 24 – 10 – 2 – 1 = 11 students visited
Coron only.
n(P3) = 18 – n(P1) – n(P5) – n(P6) = 18 – 10 – 2 – 5 = 1 student visited
Tubbataha Reef only
n(P4) = 20 – n(P1) – n(P6) – n(P7) = 20 – 10 – 5 – 1 = 4 students visited
Coron and El Nido only.
Therefore
n(P8) = 50 – n(P1) – n(P2) – n(P3) – n(P4) – n(P5) – n(P6) – n(P7) = 16
students did not visit any of the three spots.
The number of elements is shown below.
Now, what about the opening problem?
Coron El Nido
P8 Tubbataha
Reef
P5 P1
P6
P3
P2
P7
P4
Coron El Nido
5
16 Tubbataha Reef
11
1 4
2 10
1

16
Solution to the Opening Problem (Activity):
Can you explain the numbers?
III. Exercises
Do the following exercises. Represent the sets and draw a Venn diagram
when needed.
1. If A is a set, give two subsets of A.
2. (a) If 𝐴 and 𝐵 are finite sets and 𝐴 ⊂ 𝐵, what can you say about the
cardinalities of the two sets?
(b) If the cardinality of 𝐴 is less than the cardinality of 𝐵, does it follow
that 𝐴 ⊂ 𝐵?
3. If A and B have the same cardinality, does it follow that A = B? Explain.
4. If 𝐴 ⊂ 𝐵 and 𝐵 ⊂ 𝐶. Does it follow that 𝐴 ⊂ 𝐶? Illustrate your reasoning using
a Venn diagram.
5. Among the 70 kids in Barangay Magana, 53 like eating in Jollibee while 42
like eating in McDonalds. How many like eating both in Jollibee and in
McDonalds? in Jollibee only? in McDonalds only?
6. The following diagram shows how all the First Year students of Maningning
High School go to school.
a. How many students ride in a car, jeep, and the MRT going to their
school?
b. How many students ride in both a car and a jeep?
Walking 100 Jeep
Car
MRT
148
A B
3
4
21
4
4
T
19
55 15
76
17
20
67

17
c. How many students ride in both a car and the MRT?
d. How many students ride in both a jeep and the MRT?
e. How many students go to school
in a car only? a jeep only?
in the MRT only? walking?
f. How many First Year students of Maningning High School are there?
7. The blood-typing system is based on the presence of proteins called
antigens in the blood. A person with antigen A has blood type A. A person
with antigen B has blood type B, and a person with both antigens A and B
has blood type AB. If no antigen is present, the blood type is O. Draw a Venn
diagram representing the ABO System of blood typing.
A protein that coats the red blood cells of some persons was discovered in
1940. A person with the protein is classified as Rh positive (Rh+), and a
person whose blood cells lack the protein is Rh negative (Rh–). Draw a Venn
diagram illustrating all the blood types in the ABO System with the
corresponding Rh classifications.
Summary
In this lesson, you were able to apply what you have learned about sets, the
use of a Venn diagram and set operations in solving word problems.

18
1 2 3 4 5
123
Lesson 4.1: Fundamental Operation on Integers: Addition of Integers
Pre-requisite Concepts: Whole numbers, Exponents, Concept of Integers
About the Lesson:
This lesson focuses on addition of integers using different approaches. It is a
review of what the students learned in Grade 6.
Objectives:
1. Add integers using different approaches;
2. Solve word problems involving addition of integers.
Lesson Proper:
I. Activity
Study the following examples:
A. Addition Using Number Line
1. Use the number line to find the sum of 6 and 5.
On the number line, start with point 6 and count 5 units to the right. At what
point on the number line does it stop?
It stops at point 11; hence, 6 + 5 = 11.
2. Find the sum of 7 and (-3) .
On the number line, start from 7 and count 3 units going to the left since the
sign of 3 is negative.
At which point does it stop?
It stops at point 4; hence, (–3) + (7) = 4.
After the 2 examples, can you now try the next two problems?
a. (–5) + (–4) b. (–8) + (5)
We now have the following generalization:
Adding a positive integer 𝑛 to 𝑚 means moving along the real line a distance of 𝑛
units to the right from 𝑚. Adding a negative integer – 𝑛 to 𝑚 means moving along the
real line a distance of 𝑛 units to the left from 𝑚.
6 7 8 9 10 11 12 135
1 2 3 4 5 6 7 80

19
B. Addition Using Signed Tiles
This is another device that can be used to represent integers. The tile
represents integer 1, the tile represents –1, and the flexible + - represents
0.
Recall that a number and its negative cancel each other under the operation of
addition. This means
4 + (−4) = 0
15 + (−15) = 0
(−29) + 29 = 0
In general, 𝑛 + (−𝑛) = (−𝑛) + 𝑛 = 0.
Examples:
1. 4 + 5 ------ + +
four (+1) + five (+1)
hence, 4 + 5 = 9
2. 5 + (–3) -----
hence, 5 + (−3) = 2 + 3 + (−3) = 2 + 0 = 2
3. (−7) + (−6)
hence (−7) + (−6) = −13
Now, try these:
1. (–5) + (–11)
2. (6) + (–9)
+
-
+ + + + + + + + +
+ + + + + - - -
0
0
0
– – – – – – – – – – – – –+

20
II. Questions/Points to Ponder
Using the above model, we summarize the procedure for adding integers as follows:
1. If the integers have the same sign, just add the positive equivalents of the
integers and attach the common sign to the result.
a. 27 + 30 = + (/27/ + /30/)
= + (/57/)
= + 57
b. (–20) + (–15) = – (/20/ + /15/)
= –(20 + 15)
= –(35)
= –35
2. If the integers have different signs, get the difference of the positive
equivalents of the integers and attach the sign of the larger number to the
result.
a. (38) + (–20)
Get the difference between 38 and 20: 18
Since 38 is greater than 20, the sign of the sum is positive.
Hence 38 + (−20) = 18
b. (−42) + 16
Get the difference between 42 and 16: 26
Since 42 is greater than 16, the sum will have a negative sign.
Hence (−42) + 16 = −26
If there are more than two addends in the problem the first step to do is to combine
addends with same signs and then get the difference of their sums.
Examples:
1. (−14) + (22) + (8) + (−16) = −( 14 + 16) + ( 22 + 8)
= −30 + 30 = 0
2. 31 + 70 + 9 + (− 155) = ( 31 + 70 + 9) + (−155)
= 110 + (−155) = −45

21
III. Exercises
A. Who was the first English mathematician who first used the modern symbol of
equality in 1557?
(To get the answer, compute the sums of the given exercises below. Write the
letter of the problem corresponding to the answer found in each box at the
bottom).
A 25 + 95 C (30) + (–20) R 65 + 75
B 38 + (–15) D (110) + (–75) O (–120) + (–35)
O 45 + (–20) T (16) + (–38) R (165) + (–85)
R (–65) + (–20) R (–65) + (–40) E 47 + 98
E (78) + (–15) E (–75) + (20)
B. Add the following:
1. (18) + (–11) + (3)
2. (–9) + (–19) + (–6)
3. (–4) + (25) + (–15)
4. (50) + (–13) + (–12)
5. (–100) + (48) + (49)
C. Solve the following problems:
1. Mrs. Reyes charged Php3,752.00 worth of groceries on her credit card. Find
her balance after she made a payment of Php2,530.00.
2. In a game, Team Azcals lost 5 yards in one play but gained 7 yards in the
next play. What was the actual yardage gain of the team?
3. A vendor gained Php50.00 on the first day; lost Php28.00 on the second day,
and gained Php49.00 on the third day. How much profit did the vendor gain
in 3 days?
4. Ronnie had Php2,280 in his checking account at the beginning of the month.
He wrote checks for Php450, Php1,200, and Php900. He then made a
deposit of Php1,000. If at any time during the month the account is
overdrawn, a Php300 service charge is deducted. What was Ronnie’s
balance at the end of the month?
Summary
In this lesson, you learned how to add integers using two different methods.
The number line model is practical for small integers. For larger integers, the signed
tiles model provides a more useful tool.
–105 25 63 23 –85 –22
140 –55 10 –155 80 35 145

22
Lesson 4.2: Fundamental Operation on Integers: Subtraction of Integers
Prerequisite Concepts: Whole numbers, Exponents, Concept of Integers, Addition
of Integers
About the Lesson:
This lesson focuses on the subtraction of integers using different approaches.
It is a review of what the students learned in Grade 6.
Objectives:
1. Subtract integers using
a. Number line
b. Signed tiles
2. Solve problems involving subtraction of integers.
Lesson Proper:
I. Activity
Study the material below.
1. Subtraction as the reverse operation of addition
Recall how subtraction is defined. We have previously defined subtraction as
the reverse operation of addition. This means that when we ask “what is 5
minus 2?”, we are also asking “what number do we add to 2 in order to get
5?” Using this definition of subtraction, we can deduce how subtraction is
done using the number line.
a. Suppose you want to compute (−4) − 3. You ask “What number must be
added to 3 to get −4?”
To get from 3 to −4, you need to move 7 units to the left. This is equivalent to
adding −7 to 3. Hence in order to get −4, −7 must be added to 3. Therefore,
(−4) − 3 = −7
b. Compute (−8) − (−12)
What number must be added to −12 to get −8?
To go from −12 to −8, move 4 units to the right, or equivalently, add 4.
Therefore,
(−8) − (−12) = 4

23
2. Subtraction as the addition of the negative
Subtraction is also defined as the addition of the negative of the number. For
example, 5 − 3 = 5 + (−3). Keeping in mind that 𝑛 and −𝑛 are negatives of
each other, we can also have 5 − (−3) = 5 + 3. Hence the examples above
can be solved as follows:
(−4) − 3 = (−4) + (−3) = −7
(−8) − (−12) = (−8) + 12 = 4
This definition of subtraction allows the conversion of a subtraction problem
to an addition problem.
Problem:
Subtract (−45) from 39 using the two definitions of subtraction.
Can you draw your number line? Where do you start numbering it to make
the line shorter?
Solution:
1. 39 − (−45)
What number must be added to −45 in order to obtain 39?
39 − (−45) = 84
2. 39 − (−45) = 39 + 45 = 84
II. Questions / Points to Ponder
Rule in Subtracting Integers
In subtracting integers, add the negative of the subtrahend to the minuend,
𝑎 − 𝑏 = 𝑎 + (−𝑏)
𝑎 − (−𝑏) = 𝑎 + 𝑏
Using signed tiles or colored counters
Signed tiles or colored counters can also be used to model subtraction of integers.
In this model, the concept of subtraction as “taking away” is utilized.

24
Examples:
1. 10 − 6 means take away 6 from 10. Hence
10 − 6 = 4
2. −3 − (−2)
−3 − (−2) = −1
3. 4 − 9
Since there are not enough counters from which to take
away 9, we add 9 black counters and 9 white counters.
Remember that these added counters are equivalent to
zero.
We now take away
9 black counters.

25
4. 2 − (−4)
Hence 2 − (−4) = 6
The last two examples above illustrate the definition of subtraction as the addition
of the negative.
𝑚 − 𝑛 = 𝑚 − 𝑛 + [𝑛 + (−𝑛)] = [𝑚 − 𝑛 + 𝑛] + (−𝑛) = 𝑚 + (−𝑛)
III. Exercices
A. What is the name of the 4th highest mountain in the world?
Decode the answer by finding the difference of the following subtraction
problems. Write the letter to the answer corresponding to the item in the box
provided below:
O Subtract (–33) from 99
L Subtract (–30) from 49
H 18 less than (–77)
E Subtract (–99) from 0
T How much is 0 decreased by (–11)?
S (–42) – (–34) – (–9) – 18
79 –95 132 11 –17 99
Notice that this configuration is the
same configuration for 4 + (−9).
We proceed with the addition and
obtain the answer −5.

26
B. Mental Math
Give the difference:
1. 53 – 25 6. 25 – 43
2. (–6) – 123 7. (–30) – (–20)
3. (–4) – (–9) 8. (–19) – 2
4. 6 – 15 9. 30 – (–9)
5. 16 – (–20) 10. (–19) – (–15)
1. Maan deposited Php53,400.00 in her account and withdrew Php19,650.00
after a week. How much of her money was left in the bank?
2. Two trains start at the same station at the same time. Train A travels
92 km/h, while train B travels 82 km/h. If the two trains travel in opposite
directions, how far apart will they be after an hour? If the two trains travel
in the same direction, how far apart will they be in two hours?
3. During the Christmas season. The student gov’t association was able to
solicit 2356 grocery items and was able to distribute 2198 to one barangay.
If this group decided to distribute 1201 grocery items to the next barangay,
how many more grocery items did they need to solicit?
Summary
In this lesson, you learned how to subtract integers by reversing the process
of addition, and by converting subtraction to addition using the negative of the
subtrahend.

27
Lesson 4.3: Fundamental Operations on Integers: Multiplication
of Integers
Prerequisite Concepts: Operations on whole numbers, addition and subtraction of
integers
About the Lesson:
This is the third lesson on operations on integers. The intent of the lesson is
to deepen what students have learned in Grade 6, by expounding on the meaning of
multiplication of integers.
Objective:
1. Multiply integers
2. Apply multiplication of integers in solving problems
Lesson Proper:
I. Activity
Answer the following question.
How do we define multiplication?
We learned that with whole numbers, multiplication is repeated addition.
For example, 4 × 3 means three groups of 4. Or, putting it into a real context, 3 cars
with 4 passengers each, how many passenger in all? Thus
4 × 3 = 4 + 4 + 4 = 12.
But, if there are 4 cars with 3 passengers each, in counting the total number of
passengers, the equation is 3 × 4 = 3 + 3 + 3 + 3 = 12 . We can say then that
4 × 3 = 3 × 4 and
4 × 3 = 3 × 4 = 3 + 3 + 3 + 3 = 12.
We extend this definition to multiplication of a negative integer by a positive integer.
Consider the situation when a boy loses Php6 for 3 consecutive days. His total loss
for three days is (−6) × 3. Hence, we could have
(−6) × 3 = (−6) + (−6) + (−6) = −18.
II. Questions/Points to Ponder
The following examples illustrate further how integers are multiplied.
Example 1. Multiply: 5 x (−2)
However, 5 x (−2) = (−2) x (5)
Therefore: (−2) x (5) = (−2) + (−2) + (−2) + (−2) + (−2) = -10
The result shows that the product of a negative multiplier and a positive
multiplicand is a negative integer.

28
Generalization: Multiplying unlike signs
We know that adding negative numbers means adding their positive equivalents and
attaching the negative sign to the result, then
𝑎 × (−𝑏) = (−𝑏) × 𝑎 = (−𝑏) + (−𝑏) + ⋯ + (−𝑏)⏟
𝑎𝑑𝑑𝑒𝑛𝑑𝑠
= − (𝑏 + 𝑏 + ⋯ + 𝑏)⏟
𝑎𝑑𝑑𝑒𝑛𝑑𝑠
= −𝑎𝑏
for any positive integers 𝑎 and 𝑏.
We know that any whole number multiplied by 0 gives 0. Is this true for any integer
as well? The answer is YES. In fact, any number multiplied by 0 gives 0. This is
known as the Zero Property.
What do we get when we multiply two negative integers?
Example 2. Multiply: (−8) × (–3)
We know that (−8) × 3 = −24.
Therefore,
−24 + (−8) × (−3) = (−8) × 3 + (−8) × (−3)
= (−8) × [3 + (−3)] (Distributive Law)
= (−8) × 0 (3 and −3 are additive inverses)
= 0 (Zero Property)
The only number which when added to −24 gives 0 is the additive inverse of
−24. Therefore, (−8) × (−3) is the additive inverse of 24, or (−8) × (−3) = 24
The result shows that the product of two negative integers is a positive integer.
Generalization: Multiplying Two Negative Integers
If 𝑎 and 𝑏 are positive integers, then (−𝑎) × (−𝑏) = 𝑎𝑏.
Rules in Multiplying Integers:
In multiplying integers, find the product of their positive equivalents.
1. If the integers have the same signs, their product is positive.
2. If the integers have different signs their product is negative.
III. Exercises
A. Find the product of the following:
1. (5) (12)
2. (−8) (4)
3. (−5) (3) (2)
4. (−7) (4) (−2)
5. (3) (8) (−2)
6. (9) (−8) (−9)
7. (−9) (−4) (−6)

29
MATH DILEMMA
B. How can a person fairly divide 10 apples among 8 children so that each child
has the same share.
To solve the dilemma, match the letter in column II with the number that corresponds
to the numbers in column I.
Column I Column II
1. (6) (−12) C 270
2. (−13) (−13) P −72
3. (19) (−17) E 300
4. (−15) (29) K −323
5. (165) (0) A −435
6. (−18) (−15) M 0
7. (−15) (−20) L 16
8. (−5) (−5) (−5) J −125
9. (−2) (−2) (−2) (−2) U 169
10. (4) (6) (8) I 192
C. Problem Solving
1. Jof has twenty Php5 coins in her coin purse. If her niece took 5 of the coins,
how much has been taken away?
2. Mark can type 45 words per minute, how many words can Mark type in 30
minutes?
3. Give an arithmetic equation which will solve the following:
a. The messenger came and delivered 6 checks worth Php50 each.
Are you richer or poorer? By how much?
b. The messenger came and took away 3 checks worth Php120 each.
c. The messenger came and delivered 12 bills for Php86 each.
d. The messenger came and took away 15 bills for Php72 each.
Summary
This lesson emphasized the meaning of multiplication to set the rules for
multiplying integers. To multiply integers, first find the product of their positive
equivalents. If the integers have the same signs, their product is positive. If the
integers have different signs their product is negative.
_____
5
_____
4
_____
3
_____
7
_____
4
_____
1
_____
1
_____
9
_____
7
_____
8
_____
2
_____
10
_____
6
_____
7

30
Lesson 4.4: Fundamental Operations on Integers: Division of Integers
Prerequisite Concepts: Addition and subtraction of Integers, Multiplication of
Integers
About the Lesson: Like in the previous lessons, this lesson is meant to deepen
students’ understanding of the division operation on integers. The concept of division
used here relies on its relationship to multiplication.
Objective:
1. Find the quotient of two integers.
2. Solve problems involving division of integers.
Lesson Proper:
I. Activity
What is (−51) ÷ (−3)?
What is (−51) ÷ 3?
What is 51 ÷ (−3)?
What are the rules in dividing integers?
II. Questions / Points to Ponder
We have learned that Subtraction is the inverse operation of Addition,
In the same manner, Division is the inverse operation of Multiplication.
Example 1. Find the quotient of (−51) and (−3).
Solution:
Since division is the inverse of multiplication, determine what number
multiplied by (−3) produces (−51).
If we ignore the signs for the meantime, we know that
3 × 17 = 51
We also know that in order to get a negative product, the factors must have
different signs. Hence
(−3) × 17 = −51
Therefore
(−51) ÷ (−3) = 17
Example 2. What is (−57) ÷ 19?
Solution: 19 × 3 = 57
Hence
19 × (−3) = −57
Therefore
(−57) ÷ 19 = −3
Example 3. Show why 273 ÷ (–21) = –13.
Solution: (−13) × (−21) = 57
Therefore, 273 ÷ (–21) = –13

31
(-352) ÷ 22
(128) ÷ -16
(168) ÷ 6(144) ÷ -3
(108) ÷ 9
(-315) ÷ (-35)(-147) ÷ 7
(-120) ÷ 8
(-444) ÷ (-12)
Generalization
The quotient of two integers with the same signs is a positive integer and
the quotient of two integers having unlike signs is a negative integer. However,
division by zero is not possible.
When several operations have to be performed, the GEMDAS rule applies.
Example 4. Perform the indicated operations.
1. 2 − 3 × (−4)
2. 4 × 5 + 72 ÷ (−6)
3. 9 + 6 − (−3) × 12 ÷ (−9)
Solution:
1. 2 − 3 × (−4) = 2 − (−12) = 14
2. 4 × 5 + 72 ÷ (−6) = 20 + (−12) = 8
3. 9 + 6 − (−3) × 12 ÷ (−9) = 9 + 6 − (−36) ÷ (−9) = 9 + 6 − 4 = 11
III. Exercises:
A. Compute the following
1. (10 + 15) − 4 × 3 + 7 × (−2)
2. 22 × 9 ÷ (−6) − 5 × 8
3. 36 ÷ 12 + 53 + (−30)
4. (30 + 26) ÷ [(−2) × 7]
5. (124 − 5 × 12) ÷ 8
B. What was the original name for the butterfly?
To find the answer find the quotient of each of the following and write the letter of the
problems in the box corresponding to the quotient.
9 37 -15 -8 -8 28 -16 12 -48
R
L
E
B
T F
U
T
Y

32
1. Vergara’s store earned Php8,750 a week, How much is her average earning
in a day?
2. Russ worked in a factory and earned Php7,875.00 for 15 days. How much
is his earning in a day?
3. There are 336 oranges in 12 baskets. How many oranges are there in
3 baskets?
4. A teacher has to divide 280 pieces of graphing paper equally among his
35 students. How many pieces of graphing paper will each student recieve?
5. A father has 976 sq. meters lot, he has to divide it among his 4 children.
What is the share of each child?
D. Complete the three-by-three magic square (that is, the sums of the numbers in
each row, in each column and in each of the diagonals are the same) using the
numbers −10, −7, −4, −3, 0, 3, 4, 7, 10. What is the sum for each row, column
and diagonal?
Summary
Division is the reverse operation of multiplication. Using this definition, it is
easy to see that the quotient of two integers with the same signs is a positive integer
and the quotient of two integers having unlike signs is a negative integer.

33
Lesson 5: Properties of the Operations on Integers
Prerequisite Concepts: Addition, Subtraction, Multiplication, and Division
of Integers
About the Lesson:
This lesson will strengthen the skills of students in performing the
fundamental operations of integers. Knowledge of these will serve as an
axiom/guide in performing said operations. In addition, this will help students
solve problems including real life situations in algebra. This section also
discusses how an application of the properties of real numbers in real life
situations can be helpful in sustaining harmonious relationships among people.
Objectives
1. State and illustrate the different properties of the operations on integers
a. closure d. distributive
b. commutative e. identity
c. associative f. inverse
2. Rewrite given expressions according to the given property.
Lesson Proper:
I.
A. Activity 1: Try to reflect on these . . .
1. Give at least 5 words synonymous to the word “property”.
Activity 2: PICTIONARY GAME: DRAW AND TELL!
The following questions will be answered as you go along to the next activity.
Needed Materials:
5 strips of cartolina with adhesive tape
where each of the following words will
be written:
 Closure
 Commutative
 Associative
 Distributive
 Identity
 Inverse
Printed Description:
 Stays the same
 Swapping / Interchange
 Bracket Together/Group Together
 Share Out / Spread Out / Disseminate
 One and the Same / Alike
 Opposite / Contrary
Rules of the Game:
The mission of each player holding a
strip of cartolina is to let the
classmates guess the hidden word
by drawing symbols, figures or
images on the board without any
word.
If the hidden property is discovered,
a volunteer from the class will give
his/her own meaning of the identified
words. Then, from the printed
descriptions, he/she can choose the
appropriate definition of the disclosed
word and verify if his/her initial
description is correct.
The game ends when all the words
are revealed.

34
 What properties of real numbers were shown in the Pictionary Game?
Give one example and explain.
 How are said properties seen in real life?
Activity 3: SHOW AND TELL!
Determine what kind of property of real numbers is being illustrated in the
following images:
A. Fill in the blanks with the correct numerical values of the motorbike and
bicycle riders.
_______ _______
If a represents the number of motorbike riders and b represents the
number of bicycle riders, show the mathematical statement for the
diagram below.
_______ + _______ = _______ + _______
Guide Questions:
 What operation is used in illustrating the diagram?
 What happened to the terms in both sides of the equation?
 Based on the previous activity, what property is being applied?
 What if the operation is replaced by multiplication, will the same property
be applicable? Give an example to prove your answer.
 Define the property.
 Give a real life situation in which the commutative property can be
applied.
 Test the property on subtraction and division operations by using simple
examples. What did you discover?
+
Equals
+

35
B. Fill in the blanks with the correct numerical values of the set of cellphones,
ipods, and laptops.
_______ _______ _______
_______ _______ _______
If a represents the number of cellphones, b represents the ipods and c
represents the laptops, show the mathematical statement for the diagram
below.
(_______ + _______ ) +_______ = _______ + (_______ + _______ )
Guide Questions:
 What operation is used in illustrating the diagram?
 What happened to the groupings of the given sets that correspond to both
sides of the equation?
 Based on the previous activity, what property is being applied?
 What if the operation is replaced by multiplication, will the same property
be applicable? Give an example to prove your answer.
 Give a real life situation wherein associative property can be applied.
 Test the property on subtraction and division operations by using simple
examples. What did you discover?
equals
+ +
+ +

36
C. Fill in the blanks with the correct numerical values of the set of oranges and
set of strawberries.
_______ _______
_______ _______
If a represents the multiplier in front, b represents the set of oranges and
c represents the set of strawberries, show the mathematical statement for
the diagram below.
_______ (______ + _______) = ______ • _______ + _______ • ______
Guide Questions:
 Based on the previous activity, what property is being applied in the
images presented?
 In the said property can we add/subtract the numbers inside the
parentheses and then multiply or perform multiplication first and then
addition/subtraction? Give an example to prove your answer.
 Give a real life situation wherein distributive property can be applied.
+2 × 2 ×
equals
+2 ×

37
D. Fill in the blanks with the correct numerical representation of the given
illustration.
_______ _______ _______
Guide Questions:
images presented?
 What will be the result if you add something represented by any number
to nothing represented by zero?
 What do you call zero “0” in this case?
 Is there a number multiplied to any number that will result to that same
number? Give examples.
 What property is being illustrated? Define.
 What do you call one “1” in this case?
E. Give the correct mathematical statement of the given illustrations. To do this,
refer to the guide questions below.
PUT IN
PLUS
REMOVE
?
+ =

38
Guide Questions:
 How many cabbages are there in the crate?
 Using integers, represent “put in 14 cabbages” and “remove 14
cabbages?” What will be the result if you add these representations?
images presented?
 What will be the result if you add something to its negative?
 What do you call the opposite of a number in terms of sign? What is the
opposite of a number represented by a?
 What do you mean by reciprocal and what is the other term used for it?
 What if you multiply a number say 5 by its multiplicative inverse
1
5
, what
will be the result?
 What property is being illustrated? Define.
1. Closure Property
Two integers that are added and multiplied remain as integers. The set of
integers is closed under addition and multiplication.
2. Commutative Property
Changing the order of two numbers that are either being added or
multiplied does not change the value.
3. Associative Property
Changing the grouping of numbers that are either being added or
multiplied does not change its value.
4. Distributive Property
When two numbers have been added / subtracted and then multiplied by
a factor, the result will be the same when each number is multiplied by
the factor and the products are then added / subtracted.
5. Identity Property
Additive Identity
- states that the sum of any number and 0 is the given number.
Zero, “0” is the additive identity.
Multiplicative Identity
- states that the product of any number and 1 is the given number,
a • 1 = a. One, “1” is the multiplicative identity.
6. Inverse Property
In Addition
- states that the sum of any number and its additive inverse, is zero.
The additive inverse of the number a is –a.
In Multiplication
- states that the product of any number and its multiplicative inverse or
reciprocal, is 1.The multiplicative inverse of the number a is
a
1
.

39
III. Exercises
A. Complete the Table: Which property of real number justifies each
statement?
In this segment, you will learn some of the notations and symbols pertaining to
properties of real number applied in the operations of integers.
Closure Property under addition and
multiplication
a, b  I, then a+b  I,
a • b  I
Commutative property of addition
a + b = b + a
Commutative property of multiplication
ab = ba
Associative property of addition
(a + b) + c = a + (b + c)
Associative property of multiplication
(ab) c = a (bc)
Distributive property
a(b + c) = ab + ac
Additive identity property
a + 0 = a
Multiplicative identity property
a • 1 = a
Multiplicative inverse property
1
𝑎
•𝑎 = 1
Additive inverse property
a + (-a) = 0
Given Property
1. 0 + (−3) = −3
2. 2(3 − 5) = 2(3) − 2(5)
3. (−6) + (−7) = (−7) + (−6)
4. 1 x (−9) = −9
5. −4 x −
1
4
= 1
6. 2 x (3 x 7) = (2 x 3) x 7
7. 10 + (−10) = 0
8. 2(5) = 5(2)
9. 1 x (−
1
4
) = −
1
4
10. (−3)(4 + 9) = (−3)(4) + (−3)(9)

40
B. Rewrite the following expressions using the given property.
1. 12a – 5a Distributive Property
2. (7a) b Associative Property
3. 8 + 5 Commutative Property
4. –4 (1) Identity Property
5. 25 + (–25) Inverse Property
C. Fill in the blanks and determine what properties were used to solve the
equations.
1. 5 x ( ____ + 2) = 0
2. –4 + 4 = _____
3. –6 + 0 = _____
4. (–14 + 14) + 7 = _____
5. 7 x (____ + 7) = 49
Summary
The lesson on the properties or real numbers explains how numbers or
values are arranged or related in an equation. It further clarifies that no matter how
these numbers are arranged and what processes are used, the composition of the
equation and the final answer will still be the same. Our society is much like these
equations - composed of different numbers and operations, different people with
varied personalities, perspectives and experiences. We can choose to look at the
differences and forever highlight one's advantage or superiority over the others. Or
we can focus on the commonality among people and altogether, work for the
common good. A peaceful society and harmonious relationship starts with
recognizing, appreciating and fully maximizing the positive traits that we, as a
people, have in common.

41
Lesson 6: Rational Numbers in the Number Line
Prerequisite Concepts: Subsets of Real Numbers, Integers
About the lesson:
This lesson is a more in-depth discussion of the set of Rational Numbers and
focuses on where they are found in the real number line.
Objective:
In this lesson, the students are expected to
1. Define rational numbers;
2. Illustrate rational numbers on the number line;
3. Arrange rational numbers on the number line.
Lesson Proper
I. Activity
Determine whether the following numbers are rational numbers or not.
– 2, ,

1
11
,

43
,

16 , –1.89
Now, try to locate them on the real number line below by plotting:
II. Questions to Ponder
Consider the following examples and answer the questions that follow:
a. 7 ÷ 2 = 3 ½,
b. (–25) ÷ 4 = –6 ¼
c. (–6) ÷ (–12) = ½
1. Are quotients integers?
2. What kind of numbers are they?
3. Can you represent them on a number line?
Recall what rational numbers are ...
3 ½, –6 ¼, ½, are rational numbers. The word rational is derived from the word
“ratio” which means quotient. Rational numbers are numbers which can be written as
a quotient of two integers,
𝑎
𝑏
where b ≠ 0.
The following are more examples of rational numbers:
5 =

5
1 0.06 =

6
100 1.3 =
13
10
From the example, we can see that an integer is also a rational number and
therefore, integers are a subset of rational numbers. Why is that?
0-1-2-3 1 2 3 4

42
0 1
0 1
½
Let’s check on your work earlier. Among the numbers given, –2, ,

1
11
,

43
,

16 ,
–1.89, the numbers  and

43
are the only ones that are not rational numbers.
Neither can be expressed as a quotient of two integers. However, we can express
the remaining ones as a quotient of two integers:

2  2
1 , 16  4  4
1 , 1.89 
189
100
Of course,

1
11
is already a quotient by itself.
We can locate rational numbers on the real number line.
Example 1. Locate ½ on the number line.
a. Since 0 < ½ < 1, plot 0 and 1 on the number line.
b. Get the midpoint of the segment from 0 to 1. The midpoint now corresponds
to ½.
Example 2. Locate 1.75 on the number line.
a. The number 1.75 can be written as

7
4 and, 1 <

7
4 < 2. Divide the segment
from 0 to 2 into 8 equal parts.
0 1 2
b. The 7th
mark from 0 is the point 1.75.
0 1 21.75

43
Example 3. Locate the point −
5
3
on the number line.
Note that −2 < −
5
3
< −1. Dividing the segment from −2 to 0 into 6 equal parts,
it is easy to plot −
5
3
. The number −
5
3
is the 5th
mark from 0 to the left.
−2 −1 0
Go back to the opening activity. You were asked to locate the rational numbers and
plot them on the real number line. Before doing that, it is useful to arrange them in
order from least to greatest. To do this, express all numbers in the same form –
either as similar fractions or as decimals. Because integers are easy to locate, they
need not take any other form. It is easy to see that
− 2 < −1.89 <

1
11
<

16
Can you explain why?
Therefore, plotting them by approximating their location gives
-3 -2 -1 0 1 2 3 4
III. Exercises
1. Locate and plot the following on a number line (use only one number line).
a.

10
3
e. −0.01
b.

2.07 f.

7
1
9
c.

2
5
g. 0
d. 12 h.


1
6
−
5
3
−1.89
1
11

44
2. Name 10 rational numbers that are greater than −1 but less than 1 and arrange
them from least to greatest on the real number line?
3. Name one rational number x that satisfies the descriptions below:
a. 910  x
b.

1
10
 x 
1
2
c.

3  x  
d.

1
4
 x 
1
3
e.


1
8
 x  
1
9
Summary
In this lesson, you learned more about what rational numbers are and where
they can be found in the real number line. By changing all rational numbers to
equivalent forms, it is easy to arrange them in order, from least to greatest or vice
versa.

45
Lesson 7: Forms of Rational Numbers and Addition and Subtraction
of Rational Numbers
Prerequisite Concepts: definition of rational numbers, subsets of real numbers,
fractions, decimals
About the Lesson:
Like with any set of numbers, rational numbers can be added and subtracted.
In this lesson, you will learn techniques in adding and subtracting rational numbers.
Techniques include changing rational numbers into various forms convenient for the
operation as well as estimation and computation techniques.
Objectives:
1. Express rational numbers from fraction form to decimal form (terminating and
repeating and non-terminating) and vice versa;
2. Add and subtract rational numbers;
3. Solve problems involving addition and subtraction of rational numbers.
Lesson Proper:
A. Forms of Rational Numbers
I. Activity
1. Change the following rational numbers in fraction form or mixed number form
to decimal form:
a.


1
4
= ______ d.

5
2
= ______
b.

3
10
= ______ e.


17
10
= ______
c.

3
5
100
= _____ f.

2
1
5
= ______
2. Change the following rational numbers in decimal form to fraction form.
a. 1.8 = ______ d. −0.001 = ______
b. −3.5 = ______ e. 10.999 = ______
c. −2.2 = ______ f.

0.11 = ______

46
II. Discussion
Non-decimal Fractions
There is no doubt that most of the above exercises were easy for you. This is
because all except item 2f are what we call decimal fractions. These numbers are all
parts of powers of 10. For example,


1
4
=

25
100
which is easily convertible to a
decimal form, 0.25. Likewise, the number -3.5 =

3
5
10
 
35
10
.
What do you do when the rational number is not a decimal fraction? How do
you convert from one form to the other?
Remember that a rational number is a quotient of 2 integers. To change a
rational number in fraction form, you need only to divide the numerator by the
denominator.
Consider the number

1
8
. The smallest power of 10 that is divisible by 8 is
1000. But,

1
8
means you are dividing 1 whole unit into 8 equal parts. Therefore,
divide 1 whole unit first into 1000 equal parts and then take

1
8
of the thousandths
part. That is equal to

125
1000
or 0.125.
Example: Change

1
16
,
9
11
, and


1
3
to their decimal forms.
The smallest power of 10 that is divisible by 16 is 10 000. Divide 1 whole unit
into 10 000 equal parts and take

1
16
of the ten thousandths part. That is equal to
10000
625
or 0.625. You can obtain the same value if you perform the long division

1 16.
Do the same for

9
11
. Perform the long division

9 11 and you should obtain

0.81. Therefore,

9
11
=

0.81. Also,


1
3
 0.3. Note that both

9
11
and


1
3
are
non-terminating but repeating decimals.
To change rational numbers in decimal forms, express the decimal part of the
numbers as a fractional part of a power of 10. For example, −2.713 can be changed
initially to

2
713
1000
and then changed to


2173
1000
.

47
What about non-terminating but repeating decimal forms? How can they be
changed to fraction form? Study the following examples:
Example 1: Change

0.2 to its fraction form.
Solution: Let

r  0.222...
10r  2.222...
Then subtract the first equation from the second equation and obtain

9r  2.0
r 
2
9
Therefore,

0.2 =

2
9
.
Example 2. Change

1.35 to its fraction form.
Solution: Let

r  1.353535...
100r  135.353535...
Then subtract the first equation from the second equation and obtain

99r  134
r  
134
99
 1
35
99
Therefore,

1.35 =


135
99
.
B. Addition and Subtraction of Rational Numbers in Fraction Form
I. Activity
Recall that we added and subtracted whole numbers by using the number line or by
using objects in a set.
Using linear or area models, find the sum or difference.
a.
3
5
+
1
5
= _____ c.
10
11
−
3
11
= _____
b.
1
8
+
5
8
= _____ d. 3
6
7
− 1
2
7
= _____
Without using models, how would you get the sum or difference?
Since there is only 1 repeated digit,
multiply the first equation by 10.
Since there are 2 repeated digits,
multiply the first equation by 100. In
general, if there are n repeated digits,
multiply the first equation by

10n
.

48
Consider the following examples:
1.
1
6
+
1
2
=
1
6
+
3
6
=
4
6
𝑜𝑟
2
3
2.
6
7
+ (−
2
3
) =
18
21
+ (−
14
21
) =
4
21
3. −
4
3
+ (−
1
5
) = −
20
15
+ (−
3
15
) = −
23
15
𝑜𝑟 − 1
8
15
4.
14
5
−
4
7
=
98
35
−
20
35
=
78
35
𝑜𝑟 2
8
35
5. −
7
12
− (−
2
3
) = −
7
12
− (−
8
12
) =
−7 +8
12
=
1
12
6. −
1
6
− (−
11
20
) = −
10
60
− (−
33
60
) =
−10+33
60
=
23
60
1. Is the common denominator always the same as one of the denominators of
the given fractions?
2. Is the common denominator always the greater of the two denominators?
3. What is the least common denominator of the fractions in each example?
4. Is the resulting sum or difference the same when a pair of dissimilar
fractions is replaced by any pair of similar fractions?
Problem: Copy and complete the fraction magic square. The sum in each row,
column, and diagonal must be 2.
» What are the values of a, b, c, d, and e?
Important things to remember
To Add or Subtract Fraction
 With the same denominator,
If a, b, and c denote integers, and b ≠ 0, then
𝑎
𝑏
+
𝑐
𝑏
=
𝑎 + 𝑐
𝑏
and
𝑎
𝑏
−
𝑐
𝑏
=
𝑎 − 𝑐
𝑏
a 1
2
b
7
5
1
3
c
d e 2
5

49
 With different denominators,
𝑎
𝑏
𝑎𝑛𝑑
𝑐
𝑑
, where b ≠ 0 and d ≠ 0
If the fractions to be added or subtracted are dissimilar
» Rename the fractions to make them similar whose denominator is the
least common multiple of b and d.
» Add or subtract the numerators of the resulting fractions.
» Write the result as a fraction whose numerator is the sum or difference
of the numerators and whose denominator is the least common multiple
of b and d.
Examples:
To Add: To Subtract:
a.
3
7
+
2
7
=
3+2
7
=
2
7
a.
5
7
−
2
7
=
5−2
7
=
3
7
b.
2
5
+
1
4
b.
4
5
−
1
4
LCM/LCD of 5 and 4 is 20
4
5
−
1
4
=
16
20
−
5
20
=
16−5
20
=
11
20
2
5
+
1
4
=
8
20
+
5
20
=
8+5
20
=
13
20
II. Question to Ponder (Post - Activity Discussion)
Let us answer the questions posed in activity.
You were asked to find the sum or difference of the given fractions.
a.
3
5
+
1
5
=
4
5
c.
10
11
−
3
11
=
7
11
b.
1
8
+
5
8
=
6
8
𝑜𝑟
3
4
d. 3
6
7
− 1
2
7
= 2
4
7
Without using the models, how would you get the sum or difference?
You would have to apply the rule for adding or subtracting similar fractions.
1. Is the common denominator always the same as one of the denominators
of the given fractions?
Not always. Consider

2
5

3
4
. Their least common denominator is 20 not 5 or 4.
2. Is the common denominator always the greater of the two denominators?
Not always. The least common denominator is always greater than or equal
to one of the two denominators and it may not be the greater of the two
denominators.
3. What is the least common denominator of the fractions in each example?
(1) 6 (2) 21 (3) 15 (4) 35 (5) 12 (6) 60

50
4. Is the resulting sum or difference the same as when a pair of dissimilar
fractions is replaced by any pair of similar fractions?
Yes, for as long as the replacement fractions are equivalent to the original
fractions.
III. Exercises
Do the following exercises.
a. Perform the indicated operations and express your answer in simplest form.
1.
2
9
+
3
9
+
1
9
9.
7
9
−
1
12
2.
6
5
+
3
5
+
4
5
10. 11
5
9
− 7
5
6
3.
2
5
+
7
10
11.
1
4
+
2
3
−
1
2
4.
16
24
−
6
12
12. 10 − 3
5
11
5. 2
5
12
−
2
3
13.
7
20
+
3
8
+
2
5
6. 8
1
4
+
2
7
14.
5
12
+
4
9
−
1
4
7. 3
1
4
+ 6
2
3
15. 2
5
8
+
1
2
+ 7
3
4
8. 9
5
7
− 3
2
7
b. Give the number asked for.
1. What is three more than three and one-fourth?
2. Subtract from 15
1
2
the sum of 2
1
3
𝑎𝑛𝑑 4
2
5
. What is the result?
3. Increase the sum of 6
3
14
𝑎𝑛𝑑 2
2
7
𝑏𝑦 3
1
2
4. Decrease 21
3
8
𝑏𝑦 5
1
5
5. What is −8
4
5
𝑚𝑖𝑛𝑢𝑠 3
2
7
?
c. Solve each problem.
1. Michelle and Corazon are comparing their heights. If Michelle’s height
is 120
3
4
cm and Corazon’s height is 96
1
3
cm, what is the difference in their
heights?
2. Angel bought 6
3
4
meters of silk, 3
1
2
meters of satin and 8
2
5
meters
of velvet. How many meters of cloth did she buy?
3. Arah needs 10
1
4
kg of meat to serve 55 guests. If she has 3
1
2
kg
of chicken, a 2
3
4
kg of pork, and 4
1
4
kg of beef, is there enough meat
for 55 guests?

51
4. Mr. Tan has 13
2
5
liters of gasoline in his car. He wants to travel far so he
added 16
1
2
liters more. How many liters of gasoline is in the tank?
5. After boiling, the 17
3
4
liters of water was reduced to 9
2
3
liters.
How much water has evaporated?
C. Addition and Subtraction of Rational Numbers in Decimal Form
There are 2 ways of adding or subtracting decimals.
1. Express the decimal numbers in fractions then add or subtract as
described earlier.
Example:
Add: 2.3 + 7.21 Subtract: 9.6 – 3.25
2
3
10
+ 7
21
100
9
6
10
− 3
25
100
2
30
100
+ 7
21
100
9
60
100
− 3
25
100
(2 + 7) + (
30+21
100
) (9 – 3) +
60−25
100
9 +
51
100
= 9
51
100
or 9.51 6 +
35
100
= 6
35
100
or 6.35
2. Arrange the decimal numbers in a column such that the decimal
points are aligned, then add or subtract as with whole numbers.
Example:
Add: 2.3 + 7.21 Subtract: 9.6- 3.25
2.3 9.6
+ 7.21 - 3.25
9.51 6.35
Exercises:
1. Perform the indicated operation.
1. 1,902 + 21.36 + 8.7 6. 700 – 678.891
2. 45.08 + 9.2 + 30.545 7. 7.3 – 5.182
3. 900 + 676.34 + 78.003 8. 51.005 – 21.4591
4. 0.77 + 0.9768 + 0.05301 9. (2.45 + 7.89) – 4.56
5. 5.44 – 4.97 10. (10 – 5.891) + 7.99

52
2. Solve the following problems:
a. Helen had Php7,500 for shopping money. When she got home, she had
Php132.75 in her pocket. How much did she spend for shopping?
b. Ken contributed Php69.25, while John and Hanna gave Php56.25 each
for their gift to Teacher Daisy. How much were they able to gather
altogether?
c. Ryan said, “I’m thinking of a number N. If I subtract 10.34 from N, the
difference is 1.34.” What was Ryan’s number?
d. Agnes said, “I’m thinking of a number N. If I increase my number by 56.2, the
sum is 14.62.” What was Agnes number?
e. Kim ran the 100-meter race in 135.46 seconds. Tyron ran faster by 15.7
seconds. What was Tyron’s time for the 100-meter dash?
SUMMARY
This lesson began with some activities and instruction on how to change
rational numbers from one form to another and proceeded to discuss addition and
subtraction of rational numbers. The exercises given were not purely computational.
There were thought questions and problem solving activities that helped in
deepening one’s understanding of rational numbers.

53
Lesson 8: Multiplication and Division of Rational Numbers
Prerequisite Concepts: addition and subtraction of rational numbers, expressing
rational numbers in different forms
About the lesson:
In this lesson, you will learn how to multiply and divide rational numbers.
While there are rules and algorithms to remember, this lesson also shows why those
rules and algorithms work.
Objectives:
1. Multiply rational numbers;
2. Divide rational numbers;
3. Solve problems involving multiplication and division of rational numbers.
Lesson Proper
A. Models for the Multiplication and Division
I. Activity:
Make a model or a drawing to show the following:
1. A pizza is divided into 10 equal slices. Kim ate
𝟑
𝟓
of
1
2
of the pizza.
What part of the whole pizza did Kim eat?
2. Miriam made 8 chicken sandwiches for some street children. She cut up
each sandwich into 4 triangular pieces. If a child can only take a piece,
how many children can she feed?
Can you make a model or a drawing to help you solve these problems?
A model that we can use to illustrate multiplication and division of rational numbers
is the area model.
What is

1
4

1
3
? Suppose we have one bar of chocolate represent 1 unit.
Divide the bar first into 4 equal parts vertically. One part of it is

1
4
.

54
Then, divide each fourth into 3 equal parts, this time horizontally to make the
divisions easy to see. One part of the horizontal division is

1
3
.
There will be 12 equal-sized pieces and one piece is

1
12
. But, that one piece
is

1
3
of

1
4
, which we know from elementary mathematics to mean

1
3

1
4
.
What about a model for division of rational numbers?
Take the division problem:

4
5

1
2
. One unit is divided into 5 equal parts and 4
of them are shaded.
Each of the 4 parts now will be cut up in halves
Since there are 2 divisions per part (i.e.

1
5 ) and there are 4 of them (i.e.

4
5 ), then
there will be 8 pieces out of 5 original pieces or

4
5

1
2

8
5
.
1. A pizza is divided into 10 equal slices. Kim ate
𝟑
𝟓
of
1
2
of the pizza. What part
of the whole pizza did Kim eat?
½
3/5
// // //

1
3

1
4

1
12

3
5

1
2

3
10
Kim ate

3
10
of the whole pizza.

55
2. Miriam made 8 chicken sandwiches for some street children. She cut up
each sandwich into 4 triangular pieces. If a child can only take a piece, how
many children can she feed?
The equation is

8  1
4  32. Since there are 4 fourths in one sandwich, there
will be 4 x 8 = 32 triangular pieces and hence, 32 children will be fed.
How then can you multiply or divide rational numbers without using models or
drawings?
Important Rules to Remember
The following are rules that you must remember. From here on, the symbols to be
used for multiplication are any of the following:

, x,

, or x.
1. To multiply rational numbers in fraction form simply multiply the numerators and
multiply the denominators.
In symbol,
𝑎
𝑏
●
𝑐
𝑑
=
𝑎𝑐
𝑏𝑑
where b and d are NOT equal to zero, (b ≠ 0; d ≠ 0)
2. To divide rational numbers in fraction form, you take the reciprocal of the second
fraction (called the divisor) and multiply it by the first fraction.
In symbol,
𝑎
𝑏
÷
𝑐
𝑑
=
𝑎
𝑏
●
𝑑
𝑐
=
𝑎𝑑
𝑏𝑐
where b, c, and d are NOT equal to zero.
Example:
Multiply the following and write your answer in simplest form
a.
3
7
●
2
5
3
7
●
2
5
=
3 𝑥 2
7 𝑥 5
=
6
35
b. 4
1
3
● 2
1
4
13
3
●
9
4
=
13 ● 3 ● 3
3 ● 4
=
13 ●3
4
=
39
4
𝑜𝑟 9
3
4
The easiest way to solve for this
number is to change mixed
numbers to an improper fraction
and then multiply it. Or use
prime factors or the greatest
common factor, as part of the
multiplication process.

56
Divide:
8
11
÷
2
3
8
11
÷
2
3
=
8
11
●
3
2
=
2●4
11
●
3
2
4●3
11
=
12
11
𝑜𝑟 1
1
11
III. Exercises.
Do the following exercises. Write your answer on the spaces provided:
A. Find the products:
a.
5
6
●
2
3
f. 4
1
2
● 5
2
3
b. 7 ●
2
3
g.
2
15
●
3
4
c.
4
20
●
2
5
h.
1
6
●
2
3
●
1
4
d. 10
5
6
● 3
1
3
i. −
5
6
●
2
3
● (−
12
15
)
e. −
9
20
●
25
27
j.
9
16
●
4
15
● (−2)
B. Divide:
1. 20 ÷
2
3
6.
8
15
÷
12
25
2.
5
12
÷ (−
3
4
) 7. 13
1
6
÷ (−2)
3.
5
50
÷
20
35
8. −
5
6
÷ (−
10
14
)
4. 5
3
4
÷ 6
2
3
9. −
2
9
÷
11
15
5.
9
16
÷
3
4
÷
1
6
10.
15
6
÷
2
3
÷
5
8
C. Solve the following:
1. Julie spent 3
1
2
hours doing her assignment. Ken did his assignment for 1
2
3
times as many hours as Julie did. How many hours did Ken spend doing his
assignment?
2. How many thirds are there in six-fifths?
Take the reciprocal of
2
3
, which is
3
2
then multiply it
with the first fraction. Using prime factors, it is easy
to see that 2 can be factored out of the numerator then
cancelled out with the denominator, leaving 4 and 3
as the remaining factors in the numerator and 11 as
the remaining factors in the denominator.

57
3. Hanna donated
2
5
of her monthly allowance to the Iligan survivors. If her
monthly allowance is Php3,500, how much did she donate?
4. The enrolment for this school year is 2340. If
1
6
are sophomores and
1
4
are
seniors, how many are freshmen and juniors?
5. At the end of the day, a store had 2/5 of a cake leftover. The four employees
each took home the same amount of leftover cake. How much did each
employee take home?
B. Multiplication and Division of Rational Numbers in Decimal Form
This unit will draw upon your previous knowledge of multiplication and
division of whole numbers. Recall the strategies that you learned and developed
when working with whole numbers.
Activity:
1. Give students several examples of multiplication sentences with the
answers given. Place the decimal point in an incorrect spot and ask students
to explain why the decimal place does not go there and explain where it
should go and why.
Example:
215.2 x 3.2 = 68.864
2. Five students ordered buko pie and the total cost was Php135.75. How much
did each student have to pay if they shared the cost equally?
Questions and Points to Ponder:
1. In multiplying rational numbers in decimal form, note the importance of
knowing where to place the decimal point in a product of two decimal
numbers. Do you notice a pattern?
2. In dividing rational numbers in decimal form, how do you determine where to
place the decimal point in the quotient?
Rules in Multiplying Rational Numbers in Decimal Form
1. Arrange the numbers in a vertical column.
2. Multiply the numbers, as if you are multiplying whole numbers.
3. Starting from the rightmost end of the product, move the decimal point to the
left the same number of places as the sum of the decimal places in the
multiplicand and the multiplier.
Rules in Dividing Rational Numbers in Decimal Form
1. If the divisor is a whole number, divide the dividend by the divisor applying
the rules of a whole number. The position of the decimal point is the same as
that in the dividend.

58
2. If the divisor is not a whole number, make the divisor a whole number by
moving the decimal point in the divisor to the rightmost end, making the
number seem like a whole number.
3. Move the decimal point in the dividend to the right the same number of places
as the decimal point was moved to make the divisor a whole number.
4. Lastly divide the new dividend by the new divisor.
Exercises:
A. Perform the indicated operation
1. 3.5 ÷ 2 6. 27.3 x 2.5
2. 78 x 0.4 7. 9.7 x 4.1
3. 9.6 x 13 8. 3.415 ÷ 2.5
4. 3.24 ÷ 0.5 9. 53.61 x 1.02
5. 1.248 ÷ 0.024 10. 1948.324 ÷ 5.96
B. Finds the numbers that when multiplied give the products shown.
1. . 3. . 5. .
x_______ x______ x___________
1 0 . 6 2 1 . 6 2 1 . 9 8
2. . 4. .
x _______ x _______
1 6 . 8 9 . 5
Summary
In this lesson, you learned to use the area model to illustrate multiplication
and division of rational numbers. You also learned the rules for multiplying and
dividing rational numbers in both the fraction and decimal forms. You solved
problems involving multiplication and division of rational numbers.

59
Lesson 9: Properties of the Operations on Rational Numbers
Pre-requisite Concepts: Operations on rational numbers
About the Lesson:
The purpose of this lesson is to use properties of operations on rational
numbers when adding, subtracting, multiplying, and dividing rational numbers.
Objectives:
In this lesson, you are expected to
1. Describe and illustrate the different properties of the operations on rational
numbers.
2. Apply the properties in performing operations on rational numbers.
Lesson Proper:
I. Activity
Pick a Pair

2
14

3
5
0 1

13
40

13
12

1
3

3
20
From the box above, pick the correct rational number to be placed in the
spaces provided to make the equation true.
1.
3
14
+ ___ =
5
14
6. (
1
2
+
1
4
) +
1
3
= _____
2. ____ +
3
14
=
5
14
7.
1
2
+ (
1
4
+ ____) =
13
12
3.
1
3
𝑥 ____ = 0 8.

2
5
 ___ 
3
4






3
20
4. 1 x ____ =
3
5
9. (
2
5
𝑥
1
2
) 𝑥
3
4
= _____
5.
2
3
+ ____ =
2
3
10.
1
2
𝑥 (
2
5
+
1
4
) = (
1
2
𝑥
2
5
) + (
1
2
𝑥
1
4
) = ____
1. What is the missing number in item 1?
2. How do you compare the answers in items 1 and 2
3. What about item 3? What is the missing number?
4. In item 4, what number did you multiply with 1 to get
3
5
?
5. What number should be added to
2
3
in item 5 to get the same number?
6. What is the missing number in items 6 and 7?

60
7. What can you say about the grouping in items 6 and 7?
8. What do you think are the answers in items 8 and 9?
9. What operation did you apply in item 10?
Problem:
Consider the given expressions:
a.
1
4
+
1
8
+
1
2
+
2
3
=
1
4
+
1
2
+
2
3
+
1
8
b.
2
15
●
5
6
=
5
6
●
2
15
* Are the two expressions equal? If yes, state the property illustrated.
PROPERTIES OF RATIONAL NUMBERS (ADDITION AND MULTIPLICATION)
1. CLOSURE PROPERTY: For any two rational numbers.
𝑎
𝑏
𝑎𝑛𝑑
𝑐
𝑑
, their sum
𝑎
𝑏
+
𝑐
𝑑
and product
𝑎
𝑏
𝑥
𝑐
𝑑
is also rational.
For example:
a.
3
4
+
2
4
= (
3+2
4
) =
5
4
b.
3
4
●
2
4
=
6
16
𝑜𝑟
3
8
2. COMMUTATIVE PROPERTY: For any two rational numbers
𝑎
𝑏
𝑎𝑛𝑑
𝑐
𝑑
,
i.
𝑎
𝑏
+
𝑐
𝑑
=
𝑏
𝑑
+
𝑎
𝑏
ii.
𝑎
𝑏
●
𝑐
𝑑
=
𝑐
𝑑
𝑎𝑛𝑑
𝑎
𝑏
where a, b, c and d are integers and b and d are not equal to zero.
For example:
a.
2
7
+
1
3
=
1
3
+
1
7
b.
6
7
●
2
3
=
3
3
●
6
7
3. ASSOCIATIVE PROPERTY: For any three rational numbers
𝑎
𝑏
,
𝑐
𝑑
, 𝑎𝑛𝑑
𝑒
𝑓
i.
𝑎
𝑏
+ (
𝑐
𝑑
+
𝑒
𝑓
) = (
𝑎
𝑏
+
𝑐
𝑑
) +
𝑒
𝑓
ii.
𝑎
𝑏
● (
𝑐
𝑑
●
𝑒
𝑓
) = (
𝑎
𝑏
●
𝑐
𝑑
) ●
𝑒
𝑓
where a, b, c, d, e and f are integers and b, d and f are not equal to zero.

61
For example:
a.
3
5
+ (
2
3
+
1
4
) = (
3
5
+
2
3
) +
1
4
b.
1
4
● (
3
4
●
2
3
) = (
1
4
●
3
4
) ●
2
3
4. DISTRIBUTIVE PROPERTY of multiplication over addition for rational numbers.
If
𝑎
𝑏
,
𝑐
𝑑
, 𝑎𝑛𝑑
𝑒
𝑓
are any rational numbers, then
𝑎
𝑏
● (
𝑐
𝑑
+
𝑒
𝑓
) = (
𝑎
𝑏
●
𝑐
𝑑
) + (
𝑎
𝑏
●
𝑒
𝑓
)
For example:
3
7
● (
2
3
+
7
8
) = (
3
7
●
2
3
) + (
3
7
●
7
8
)
5. DISTRIBUTIVE PROPERTY of multiplication over subtraction for rational
numbers.
If
𝑎
𝑏
,
𝑐
𝑑
, 𝑎𝑛𝑑
𝑒
𝑓
are any rational numbers, then
𝑎
𝑏
● (
𝑐
𝑑
−
𝑒
𝑓
) = (
𝑎
𝑏
●
𝑐
𝑑
) − (
𝑎
𝑏
●
𝑒
𝑓
)
For example:
3
10
● (
2
3
−
2
8
) = (
3
10
●
2
3
) − (
3
10
●
2
8
)
6. IDENTITY PROPERTY
Addition: Adding 0 to a number will not change the identity or value of that
number.
𝑎
𝑏
+ 0 =
𝑎
𝑏
For example:
1
2
+ 0 =
1
2
Multiplication: Multiplying a number by 1 will not change the identity or value of
that number.
𝑎
𝑏
● 1 =
𝑎
𝑏
For example:
3
5
● 1 =
3
5
7. ZERO PROPERTY OF MULTIPLICATION: Any number multiplied by zero
equals 0, i. e.
𝑎
𝑏
● 0 = 0
For example:
2
7
● 0 = 0

62
II. Question to Ponder (Post-Activity Discussion)
1. What is the missing number in item1? »
2
14
2. How do you compare the answers in items 1 and 2? » The answer is the
same, the order of the numbers is not important.
3. What about item 3? What is the missing number? » The missing number is
0. When you multiply a number with zero the product is zero.
4. In item 4, what number did you multiply with 1 to get
3
5
? »
3
5
, When you
multiply a number by one the answer is the same.
5. What number should be added to
2
3
in item 5 to get the same number? » 0,
When you add zero to any number, the value of the number does not
change.
6. What do you think is the missing number in items 6 and 7?»
13
12
7. What can you say about the grouping in items 6 and 7? » The groupings are
different but they do not affect the sum.
8. What do you think are the answers in items 8 and 9? » The answer is the
same in both items,
3
20
.
8. What operation did you apply in item 10? » The Distributive Property
of Multiplication over Addition
III. Exercises:
Do the following exercises. Write your answer in the spaces provided.
A. State the property that justifies each of the following statements.
1.
2
3
+
5
8
=
5
8
+
2
3
________
2. 1 ●
9
35
=
9
35
________
3.
4
5
● (
3
4
+
2
3
) = (
4
5
●
3
4
) + (
4
5
●
2
3
) ________
4.
3
5
+ (
1
2
+
1
4
) = (
3
5
+
1
2
) +
1
4
________
5. (
2
7
+
1
5
+
2
3
) ● 1 = (
2
7
+
1
5
+
2
3
) _______________________
6. (
3
4
+ 0) =
3
4
________
7.
1
2
+
5
6
=
4
3
________________________________________
8.
3
8
●
1
4
●
2
3
●
1
2
=
3
8
●
2
3
●
1
4
●
1
2
________
9.
1
4
● (
5
7
−
2
3
) = (
1
4
●
5
7
) − (
1
4
●
2
3
) ________

63
10. (
2
15
●
5
7
) ● 0 = 0 _____
B. Find the value of N in each expression
1. N +
1
45
=
1
45
2. (
1
4
● 𝑁) ●
2
3
=
1
4
● (
6
7
●
2
3
)
3. (
2
15
+
12
30
) +
1
5
=
2
15
+ ( 𝑁 +
1
5
)
4. 0 + N =
5
18
5. N ● (
6
14
+
2
7
) = (
1
6
●
6
14
) + (
1
6
●
2
7
)
6.
8
23
● 1 = 𝑁
7.
2
9
+
2
3
= N
Summary
This lesson is about the properties of operations on rational numbers. The
properties are useful because they simplify computations on rational numbers. These
properties are true under the operations addition and multiplication. Note that for the
Distributive Property of Multiplication over Subtraction, subtraction is considered part
of addition. Think of subtraction as the addition of a negative rational number.

64
Lesson 10: Principal Roots and Irrational Numbers
Prerequisite Concepts: Set of rational numbers
About the Lesson:
This is an introductory lesson on irrational numbers, which may be daunting
to students at this level. The key is to introduce them by citing useful examples.
Objectives:
1. describe and define irrational numbers;
2. describe principal roots and tell whether they are rational or irrational;
3. determine between what two integers the square root of a number is;
4. estimate the square root of a number to the nearest tenth;
5. illustrate and graph irrational numbers (square roots) on a number line with
and without appropriate technology.
Lesson Proper:
I. Activities
A. Take a look at the unusual wristwatch and answer the questions below.
1. Can you tell the time?
2. What time is shown in the wristwatch?
3. What do you get when you take the √1 ? √4 ? √9 ? √16 ?
4. How will you describe the result?
5. Can you take the exact value of √130 ?
6. What value could you get?
Taking the square root of a number is like doing the reverse operation of
squaring a number. For example, both 7 and -7 are square roots of 49 since

72
 49
and

7 
2
 49 . Integers such as 1, 4, 9, 16, 25 and 36 are called perfect squares.
Rational numbers such as 0.16,

4
100
and 4.84 are also, perfect squares. Perfect
squares are numbers that have rational numbers as square roots. The square roots
of perfect squares are rational numbers while the square roots of numbers that are
not perfect squares are irrational numbers.
Any number that cannot be expressed as a quotient of two integers is
an irrational number. The numbers

2 , , and the special number e are all
irrational numbers. Decimal numbers that are non-repeating and non-terminating are
irrational numbers.

65
B. Activity
Use the

n button of a scientific calculator to find the following values:
1.

646
2.

164
3.

903
4.

31255
5.

24
II. Questions to Ponder (Post-Activity Discussions)
Let us answer the questions in the opening activity.
1. Can you tell the time? Yes
2. What time is it in the wristwatch? 11:45
3. What do you get when you take the √1 ? √4 ? √9 ? √16 ? 1, 2, 3, 4
4. How will you describe the result? They are all positive integers.
5. Can you take the exact value of √130 ? No.
6. What value could you get? Since the number is not a perfect square you could
estimate the value to be between

121 and

144, which is about 11.4.
Let us give the values asked for in Activity B. Using a scientific calculator,
you probably obtained the following:
1.

646
= 2
2.

164
Math Error, which means not defined
3.

903
= 4.481404747, which could mean non-terminating and non-repeating since
the calculator screen has a limited size
4.

31255
= -5
5.

24 = 4.898979486, which could mean non-terminating and non-repeating since
the calculator screen has a limited size
On Principal nth
Roots
Any number, say a, whose nth
power (n, a positive integer), is b is called the
nth
root of b. Consider the following:   497
2
 ,

24
16 and

10 
3
 1000 . This
means that −7 is a 2nd
or square root of 49, 2 is a 4th
root of 16 and -10 is a 3rd
or
cube root of -1000.
However, we are not simply interested in any nth
root of a number; we are
more concerned about the principal nth
root of a number. The principal nth
root of a
positive number is the positive nth
root. The principal nth
root of a negative
number is the negative nth
root if n is odd. If n is even and the number is
negative, the principal nth
root is not defined. The notation for the principal nth
root of a number b is

bn
. In this expression, n is the index and b is the
radicand. The nth
roots are also called radicals.

66
Classifying Principal nth
Roots as Rational or Irrational Numbers
To determine whether a principal root is a rational or irrational number,
determine if the radicand is a perfect nth
power or not. If it is, then the root is rational.
Otherwise, it is irrational.
Problem 1. Tell whether the principal root of each number is rational or irrational.
(a)

2253
(b)

0.04 (c)

1115
(d) √10000 (e)

6254
Answers:
a)

2253
is irrational
(b)

0.04 = 0.2 is rational
(c)

1115
is irrational
(d) √10000 = 100 is rational
(e)

6254
= 5 is rational
If a principal root is irrational, the best you can do for now is to give an
estimate of its value. Estimating is very important for all principal roots that are not
roots of perfect nth
powers.
Problem 2. The principal roots below are between two integers. Find the two closest
such integers.
(a) √19 (b)

1013
(c) √300
Solution:
(a) √19
16 is a perfect integer square and 4 is its principal square root. 25 is the next
perfect integer square and 5 is its principal square root. Therefore, √19 is between 4
and 5.
(b)

1013
64 is a perfect integer cube and 4 is its principal cube root. 125 is the next
perfect integer cube and 5 is its principal cube root. Therefore,

1013
is between 4
and 5.
(c) √300
289 is a perfect integer square and 17 is its principal square root. 324 is the
next perfect integer square and 18 is its principal square root. Therefore, √300 is
between 17 and 18.

67
Problem 3. Estimate each square root to the nearest tenth.
(a) √40 (b) √12 (c) √175
Solution:
(a) √40
The principal root √40 is between 6 and 7, principal roots of the two perfect
squares 36 and 49, respectively. Now, take the square of 6.5, midway between 6
and 7. Computing,

6.5 
2
 42.25 . Since 42.25 > 40 then √40 is closer to 6 than to
7. Now, compute for the squares of numbers between 6 and 6.5:

6.1 
2
 37.21,

6.2 
2
 38.44 ,

6.3 
2
 39.69, and

6.4 
2
 40.96. Since 40 is close to 39.69 than to
40.96, √40 is approximately 6.3.
(b) √12
The principal root √12 is between 3 and 4, principal roots of the two perfect
squares 9 and 16, respectively. Now take the square of 3.5, midway between 3 and
4. Computing

3.5 
2
12.25 . Since 12.25 > 12 then √12 is closer to 3 than to 4.
Compute for the squares of numbers between 3 and 3.5:

3.1 
2
 9.61,

3.2 
2
10.24 ,

3.3 
2
10.89, and

3.4 
2
11.56. Since 12 is closer to 12.25 than to
11.56, √12 is approximately 3.5.
(c) √175
The principal root √175 is between 13 and 14, principal roots of the two
perfect squares 169 and 196. The square of 13.5 is 182.25, which is greater than
175. Therefore, √175 is closer to 13 than to 14. Now:

13.1 
2
171.61,

13.2 
2
174.24 ,

13.3 
2
176.89 . Since 175 is closer to 174.24 than to 176.89
then, √175 is approximately 13.2.
Problem 4. Locate and plot each square root on a number line.
(a) √3 (b) √21 (c) √87
Solution: You may use a program like Geogebra to plot the square roots on a
number line.
(a) √3
This number is between 1 and 2, principal roots of 1 and 4. Since 3 is closer
to 4 than to 1, √3 is closer to 2. Plot √3 closer to 2.

68
(b) √21
This number is between 4 and 5, principal roots of 16 and 25. Since 21 is
closer to 25 than to 16, √21 is closer to 5 than to 4. Plot √21 closer to 5.
(c) √87
This number is between 9 and 10, principal roots of 81 and 100. Since 87 is
closer to 81, then √87 is closer to 9 than to 10. Plot √87 closer to 9.
III. Exercises
A. Tell whether the principal roots of each number is rational or irrational.
1. √400 6. √13,689
2. √64 7. √1000
3. √0.01 8. √2.25
4. √26 9. √39
5. √
1
49
10. √12.1
B. Between which two consecutive integers does the square root lie?
1. √77 6. √90
2. √700 7. √2045
3. √243 8. √903
4. √444 9. √1899
5. √48 10. √100000
C. Estimate each square root to the nearest tenth and plot on a number line.
1. √50 6. √250
2. √72 7. √5
3. √15 8. √85
4. √54 9. √38
5. √136 10. √101

69
D. Which point on the number line below corresponds to which square root?
A B C D E
0 1 2 3 4 5 6 7 8 9 10
1. √57
2. √6
3. √99
4. √38
5. √11
Summary
In this lesson, you learned about irrational numbers and principal nth
roots,
particularly square roots of numbers. You learned to find two consecutive integers
between which an irrational square root lies. You also learned how to estimate the
square roots of numbers to the nearest tenth and how to plot the estimated square
roots on a number line.

70
Lesson 11: The Absolute Value of a Number
Prerequisite Concepts: Set of real numbers
About the Lesson:
This lesson explains why a distance between two points, even if represented
on a number line cannot be expressed as a negative number. Intuitively, the absolute
value of a number may be thought of as the non-negative value of a number. The
concept of absolute value is important to designate the magnitude of a measure such
as the temperature dropped by 23 (the absolute value) degrees. A similar concept is
applied to profit vs loss, income against expense, and so on.
Objectives:
In this lesson, you are expected to describe and illustrate
a. the absolute value of a number on a number line.
b. the distance of the number from 0.
Lesson Proper:
I. Activity 1: THE METRO MANILA RAIL TRANSIT (MRT) TOUR
Suppose the MRT stations from Pasay City to Quezon City were on a straight
line and were 500 meters apart from each other.

71
1. How far would the North Avenue station be from Taft Avenue?
2. What if Elaine took the MRT from North Avenue and got off at the last
station? How far would she have travelled?
3. Suppose both Archie and Angelica rode the MRT at Shaw Boulevard and the
former got off in Ayala while the latter in Kamuning. How far would each have
travelled from the starting point to their destinations?
4. What can you say about the directions and the distances travelled by Archie
and Angelica?
Activity 2: THE BICYCLE JOY RIDE OF ARCHIEL AND ANGELICA
Problem: Archie and Angelica were at Aloys’ house. Angelica rode her bicycle
3 miles west of Aloys’ house and Archie rode his bicycle 3 miles east
of Aloys’ house. Who travelled a greater distance from Aloys’ house –
Archie or Angelica?
Questions To Ponder:
1. What subsets of real numbers are used in the problem? Represent the trip of
Archie and Angelica to the house of Aloys using a number line.
2. What are opposite numbers on the number line? Give examples and show on
the number line.
TaftAvenue
Magallanes
Ayala
Buendia
Guadalupe
Boni
ShawBoulevard
Ortigas
Santolan
AranetaCenter-Cubao
Kamuning
QuezonAvenue
NorthAvenue

72
3. What does it mean for the same distance travelled but in opposite directions?
How would you interpret using the numbers –3 and +3?
4. What can you say about the absolute value of opposite numbers say –5 and
+5?
5. How can we represent the absolute value of a number? What notation can
we use?
1. Absolute Value – of a number is the distance between that number
and zero on the number line.
2. Number Line – is best described as a straight line which is extended
in both directions as illustrated by arrowheads. A number line consists
of three elements:
a. set of positive numbers, and is located to the right of zero.
b. set of negative numbers, and is located to the left of zero; and
c. Zero.
The absolute value of a number is denoted by two bars ││.
Let's look at the number line:
The absolute value of a number, denoted "| |" is the distance of the number
from zero. This is why the absolute value of a number is never negative. In thinking
about the absolute value of a number, one only asks "how far?" not "in which
direction?" Therefore, the absolute value of 3 and of –3 is the same, which is 3
because both numbers have the same distance from zero.
Warning: The absolute-value notation is bars, not parentheses or brackets. Use the
proper notation; the other notations do not mean the same thing.
It is important to note that the absolute value bars do NOT work in the same way as
do parentheses. Whereas – (–3) = +3, this is NOT how it works for absolute value:
Problem: Simplify – | –3 |.
Solution: Given – | – 3 |, first find the absolute value of – 3.
– | –3 | = – (3)
Now take the negative of 3. Thus,
– | –3 | = – (3) = –3
This illustrates that if you take the negative of the absolute value of a number, you
will get a negative number for your answer.

73
Let us answer the questions posed in Activity 2.
1. What subsets of real numbers are used in the problem? Represent the trip of
Archie and Angelica to the house of Aloys using a number line.
The problem uses integers. Travelling 3 miles west can be represented by –3
(pronounced negative 3). Travelling 3 miles east can be represented by +3
(pronounced positive 3). Aloys’ house can be represented by the integer 0.
2. What are opposite numbers on the number line? Give examples and show on
the number line.
Two integers that are the same distance from zero in opposite directions
are called opposites. The integers +
3 and -
3 are opposites since they are
each 3 units from zero.
3. What does it mean for the same distance travelled but in opposite directions?
How would you interpret using the numbers –3 and +3?
The absolute value of a number is its distance from zero on the number line.
The absolute value of +3 is 3, and the absolute value of –3 is 3.
4. What can you say about the absolute value of opposite numbers say –5 and
+5?
Opposite numbers have the same absolute values.
5. How can we represent the absolute value of a number? What notation can
we use?
The symbol ││is used for the absolute value of a number.

74
III. Exercises
Carry out the following tasks. Write your answers on the spaces provided for
each number.
1. Find the absolute value of +
3, –3, +
7, –5, +
9, –8, +
4, –4. You may
refer to the number line below. What should you remember when we talk
about the absolute value of a number?
Solution: |+
3| = 3 |+
9| = 9
|–3| = 3 |–8| = 8
|+
7| = 7 |+
4| = 4
|–5| = 5 |–4| = 4
2. Find the absolute value of: +
11, –9, +
14, –10, +
17, –19, +
20, –20.
You may extend the number line below to help you solve this problem.
Solution: |+
11| = 11 |+
17| = 17
|–9| = 9 |–19| = 19
|+
14| = 14 |+
20| = 20
|–10| = 10 |–20| = 20
3. Use the number line below to find the value of N: |N| = 5.1
Solution
:
This problem asks us to find all numbers that are a distance of 5.1 units
from zero on the number line. We let N represent all integers that satisfy
this condition.
Remember that when we find the absolute value of a number, we are finding its
distance from 0 on the number line. Opposite numbers have the same absolute
value since they both have the same distance from 0. Also, you will notice that
taking the absolute value of a number automatically means taking the positive value
of that number.

75
The number +
5.1 is 5.1 units from zero on the number line, and the number
–5.1 is also 5.1 units from zero on the number line. Thus both +
5.1 and
–5.1 satisfy the given condition.
4.
Solution:
When is the absolute value of a number equal to itself?
When the value of the number is positive or zero.
5. Explain why the absolute value of a number is never negative. Give an
example that will support your answer.
Solution: Let │N │= –4. Think of a number that when you get the absolute value will
give you a negative answer. There will be no solution since the distance of
any number from 0 cannot be a negative quantity.
Enrichment Exercises:
A. Simplify the following.
1. │7.04 │
2. │0 │
3. │–

2
9 │
4. –│15 + 6 │
5. │–

2 2 │ – │–

3 2 │
B. List at least two integers that can replace N such that.
1. │N │= 4
2. │N │< 3
3. │N │> 5
4. │N │≤ 9
5. 0<│N │< 3
C. Answer the following.
1. Insert the correct relation symbol (>, =, <): │–7 │___ │–4 │.
2. If │x – 7│= 5, what are the possible values of x?
3. If │x │=
2
7
, what are the possible values of x?
4. Evaluate the expression, │x + y │ – │y – x │, if x = 4 and y = 7.
5. A submarine navigates at a depth of 50 meters below sea level while exactly
above it; an aircraft flies at an altitude of 185 meters. What is the distance
between the two carriers?
Summary
In this lesson you learned about the absolute value of a number, that it is a
distance from zero on the number line denoted by the notation |N|. This notation is
used for the absolute value of an unknown number that satisfies a given condition.
You also learned that a distance can never be a negative quantity and absolute
value pertains to the magnitude rather than the direction of a number.

76
Lesson 12: Subsets of Real Numbers
Prerequisite Concepts: whole numbers and operations, set of integers, rational
numbers, irrational numbers, sets and set operations, Venn diagrams
About the Lesson:
This lesson will intensify the study of mathematics since this requires a
good understanding of the sets of numbers for easier communication. Classifying
numbers is very helpful as it allows us to categorize what kind of numbers we are
dealing with every day.
Objectives:
1. Describe and illustrate the real number system.
2. Apply various procedures and manipulations on the different subsets
of the set of real numbers.
a. Describe, represent, and compare the different subsets of real number.
b. Find the union, intersection, and complement of the set of real numbers
and its subsets
Lesson Proper:
A. The Real Number System
I. Activity 1: Try to reflect on these …
It is difficult for us to realize that once upon a time there were no symbols or
names for numbers. In the early days, primitive man showed how many animals he
owned by placing an equal number of stones in a pile, or sticks in a row. Truly our
number system evolved over hundreds of years.
Sharing Ideas! What do you think?
1. In what ways do you think did primitive man need to use numbers?
2. Why do you think he needed names or words to tell “how many”?
3. How did number symbols come about?
4. What led man to invent numbers, words and symbols?

77
Activity 2: LOOK AROUND!
Fifteen different words/partitions of numbers are hidden in this puzzle. How many
can you find? Look up, down, across, backward, and diagonally. Figures are
scattered around that will serve as clues to help you locate the mystery words.
1. How many words in the puzzle were familiar to you?
2. What word/s have you encountered in your early years? Define and give
examples.
3. What word/s is/are still strange to you?
N A F R A C T I O N S I
S P B A C C D Z W N E L
T E O F T O G E H E R A
O R H S I U J R O G A M
I C R K I N R O L A T I
L E E L M T N A E T I C
A N A O P I I Q L I O E
R T L R S N T V U V N D
U I N T E G E R E E A A
T I R R A T I O N A L I
A N O N I N T E G E R S
N N U M N U M B E R S S
..., -3, -2, -1,
0, 1, 2, 3, ...
π, e, √ 𝟑
𝟐
𝟑
,
𝟕
𝟓
-4, -5, -6, ...
1, 2, 3, ...
0, 1, 2, 3, ...
0
-1, 0, 1, -
𝟏
𝟐
,
𝟒
𝟑
, 0.25,
0.1313...
100%, 15%,
25%
0.25,
0.33 ...

78
-1, 0, 1, -
𝟏
𝟐
,
𝟒
𝟑
,
0.25, 0.33...,
π, e, √ 𝟑, 10%,
15%, 25%
Activity 3: Determine what set of numbers will represent the following
situations:
1. Finding out how many cows there are in a barn
2. Corresponds to no more apples inside the basket
3. Describing the temperature in the North Pole
4. Representing the amount of money each member gets when Php200 prize
is divided among 3 members
5. Finding the ratio of the circumference to the diameter of a circle, denoted π
(read “pi”)
The set of numbers called the real number system consists of different partitions/
subsets that can be represented graphically on a number line.
II. Questions to Ponder
Consider the activities done earlier and recall the different terms you
encountered including the set of real numbers and together let us determine the
various subsets. Let us go back to the first time we encountered the numbers ...
Let's talk about the various subsets of real numbers.
Early Years...
1. What subset of real numbers do children learn at an early stage when they
were just starting to talk? Give examples.
One subset is the counting (or natural)
numbers. This subset includes all the numbers we
use to count starting with "1" and so on. The subset
would look like this: {1, 2, 3, 4, 5 ...}
In School at an Early Phase ...
2. What do you call the subset of real numbers that includes zero (the number
that represents nothing) and is combined with the subset of real numbers
learned in the early years? Give examples.
Another subset is the whole numbers. This subset is exactly like the subset
of counting numbers, with the addition of one extra number. This extra
number is "0". The subset would look like this: {0, 1, 2, 3, 4 ...}

79
In School at Middle Phase ...
3. What do you call the subset of real numbers that includes negative numbers
(that came from the concept of “opposites” and specifically used in describing
debt or below zero temperature) and is united with the whole numbers? Give
examples.
A third subset is the integers. This subset includes all the whole numbers
and their “opposites”. The subset would look like this: {... –4, –3, –2, –1, 0, 1,
2, 3, 4 ...}
Still in School at Middle Period ...
4. What do you call the subset of real numbers that includes fractions
(introduced first in elementary grades), decimals and percent?
Give examples.
A fourth subset is the non-integers. This subset includes all the fractions and
decimals. The subset would look like this: {…, –,
3
4
…, –0.5, –
1
4
, 0,
1
4
, 0.5,
3
4
…}
5. What do you call the subset of real numbers that includes integers and non-
integers and are useful in representing concepts like “half a gallon of milk”?
Give examples.
The next subset is the rational numbers. This subset includes all numbers
that "come to an end" or numbers that repeat and have a pattern. Examples
of rational numbers are: 5.34, 0.131313...,
6
7
,
2
3
, 9
6. What do you call the subset of real numbers that is not a rational number but
are physically represented like “the diagonal of a square”?
Lastly we have the set of irrational numbers. This subset includes numbers
that cannot be exactly written as a decimal or fraction. Irrational numbers
cannot be expressed as a ratio of two integers. Examples of irrational
numbers are:

2 ,

1013
, and π
1. Natural/Counting Numbers – are the numbers we use in counting
things, that is {1, 2, 3, 4, … }. The three dots, called ellipses, indicate
that the pattern continues indefinitely.
2. Whole Numbers – are numbers consisting of the set of natural or
counting numbers and zero.

80
3. Integers – are the result of the union of the set of whole numbers and
the negative of counting numbers.
4. Rational Numbers – are numbers that can be expressed as a
quotient
𝑎
𝑏
of two integers. The integer a is the numerator while the
integer b, which cannot be 0 is the denominator. This set includes
fractions and some decimal numbers.
5. Irrational Numbers – are numbers that cannot be expressed as a
quotient
𝑎
𝑏
of two integers. Every irrational number may be
represented by a decimal that neither repeats nor terminates.
6. Real Numbers – are any of the numbers from the preceding subsets.
They can be found on the real number line. The union of rational
numbers and irrational numbers is the set of real numbers.
7. Number Line – a straight line extended on both directions as
illustrated by arrowheads and is used to represent the set of real
numbers. On the real number line, there is a point for every real
number and there is a real number for every point.
III. Exercises
a. Locate the following numbers on the number line by naming the correct point.
b. Determine the subset of real numbers to which each number belongs. Use a tick
mark (√) to answer.
Number Whole
Number
Integer Rational Irrational
1. -86
2. 34.74
3.
4
7
4. √64
5. √11
6. -0.125
-2.66..., −𝟏
𝟏
𝟐
, -0.25,
𝟑
𝟒
, √ 𝟐 , √ 𝟏𝟏
𝟑
1 2 3 40-1-2-3-4

81
7. -√81
8. e
9. -45.37
10. -1.252525...
B. Points to Contemplate
It is interesting to note that the set of rational numbers and the set of irrational
numbers are disjoint sets; that is their intersection is empty. In fact, they are
complements of each other. The union of these two sets is the set of real numbers.
Exercises:
1. Based on the stated information, show the relationships among natural/counting
numbers, whole numbers, integers, rational numbers, irrational numbers and
real numbers using the Venn diagram. Fill each broken line with its corresponding
answer.

82
2. Answer the following questions.
a. Are all real numbers rational numbers? Prove your answer.
b. Are all rational numbers whole numbers? Prove your answer.
c. Are −
1
4
and −
2
5
negative integers? Prove your answer.
d. How is a rational number different from an irrational number?
e. How do natural numbers differ from whole numbers?
3. Complete the details in the Hierarchy Chart of the Set of Real Numbers.
Summary
In this lesson, you learned different subsets of real numbers that enable you
to name numbers in different ways. You also learned to determine the hierarchy and
relationship of one subset to another that leads to the composition of the real number
system using the Venn Diagram and Hierarchy Chart. You also learned that it was
because of necessity that led man to invent number, words and symbols.
The Real Number System

83
Lesson 13: Significant Digits and the Scientific Notation
Prerequisite Concepts: Rational numbers and powers of 10
About the Lesson:
This is a lesson on significant digits and the scientific notation combined. The
use of significant digits and the scientific notation is often in the area of measures
and in the natural sciences. The scientific notation simplifies the way we write very
large and very small numbers. On the other hand, numerical data become more
accurate when significant digits are taken into account.
Objectives:
1. determine the significant digits in a given situation.
2. write very large and very small numbers in scientific notation
Lesson Proper:
I. A. Activity
The following is a list of numbers. The number of significant digits in each
number is written in the parenthesis after the number.
234 (3) 0.0122 (3)
745.1 (4) 0.00430 (3)
6007 (4) 0.0003668 (4)
1.3 X 102
(2) 10000 (1)
7.50 X 10-7
(3) 1000. (4)
0.012300 (5) 2.222 X 10-3
(4)
100.0 (4) 8.004 X 105
(4)
100 (1) 6120. (4)
7890 (3) 120.0 (4)
4970.00 (6) 530 (2)
Describe what digits are not significant.
Significant digits are the digits in a number that express the precision of a
measurement rather than its magnitude. The number of significant digits in a given
measurement depends on the number of significant digits in the given data. In
calculations involving multiplication, division, trigonometric functions, for example,
the number of significant digits in the final answer is equal to the least number of
significant digits in any of the factors or data involved.
Rules for Determining Significant Digits
A. All digits that are not zeros are significant.
For example: 2781 has 4 significant digits
82.973 has 5 significant digits
B. Zeros may or may not be significant. Furthermore,
1. Zeros appearing between nonzero digits are significant.
For example: 20.1 has 3 significant digits
79002 has 5 significant digits

84
2. Zeros appearing in front of nonzero digits are not significant.
0.0000009 has 1 significant digit
3. Zeros at the end of a number and to the right of a decimal are significant
digits. Zeros between nonzero digits and significant zeros are also significant.
25000.00 has 7 significant digits
4. Zeros at the end of a number but to the left of a decimal may or may not be
significant. If such a zero has been measured or is the first estimated digit, it
is significant. On the other hand, if the zero has not been measured or
estimated but is just a place holder it is not significant. A decimal placed after
the zeros indicates that they are significant
For example: 560000 has 2 significant digits
560000 has 6 significant digits
Significant Figures in Calculations
1. When multiplying or dividing measured quantities, round the answer to as
many significant figures in the answer as there are in the measurement with
the least number of significant figures.
2. When adding or subtracting measured quantities, round the answer to the
same number of decimal places as there are in the measurement with the
least number of decimal places.
For example:
a. 3.0 x 20.536 = 61.608
Answer: 61 since the least number of significant digits is 2, coming
from 3.0
b. 3.0 + 20.536 = 23.536
Answer: 23.5 since the addend with the least number of decimal
places is 3.0
Describe what digits are not significant. The digits that are not significant are the
zeros before a non-zero digit and zeros at the end of numbers without the decimal
point.
Problem 1. Four students weigh an item using different scales. These are the values
they report:
a. 30.04 g
b. 30.0 g
c. 0.3004 kg
d. 30 g
How many significant digits are in each measurement?
Answer: 30.04 has 4 significant; 30.0 has 3 significant digits; 0.3004 has 4 significant
digits; 30 has 1 significant digit

85
Problem 2. Three students measure volumes of water with three different devices.
They report the following results:
Device Volume
Large graduated cylinder 175 mL
Small graduated cylinder 39.7 mL
Calibrated buret 18.16 mL
If the students pour all of the water into a single container, what is the total volume of
water in the container? How many digits should you keep in this answer?
Answer: The total volume is 232.86 mL. Based on the measures, the final answer
should be 232.9 mL.
On the Scientific Notation
The speed of light is 300 000 000 m/sec, quite a large number. It is cumbersome to
write this number in full. Another way to write it is 3.0 x 108
. How about a very small
number like 0.000 000 089? Like with a very large number, a very small number may
be written more efficiently. 0.000 000 089 may be written as 8.9 x 10-8
.
Writing a Number in Scientific Notation
1. Move the decimal point to the right or left until after the first significant digit
and copy the significant digits to the right of the first digit. If the number is a
whole number and has no decimal point, place a decimal point after the first
significant digit and copy the significant digits to its right.
For example, 300 000 000 has 1 significant digit, which is 3. Place a
decimal point after 3.0
The first significant digit in 0.000 000 089 is 8 and so place a decimal
point after 8, (8.9).
2. Multiply the adjusted number in step 1 by a power of 10, the exponent of
which is the number of digits that the decimal point moved, positive if moved
to the left and negative if moved to the right.
For example, 300 000 000 is written as 3.0 x 108
because the decimal
point was moved past 8 places.
0.0 000 089 is written as 8.9 x 10-8
because the decimal point was moved
8 places to the right past the first significant digit 8.
III. Exercises
A. Determine the number of significant digits in the following measurements.
Rewrite the numbers with at least 5 digits in scientific notation.
1. 0.0000056 L 6. 8207 mm
2. 4.003 kg 7. 0.83500 kg
3. 350 m 8. 50.800 km
4. 4113.000 cm 9. 0.0010003 m3
5. 700.0 mL 10. 8 000 L

86
B. a. Round off the following quantities to the specified number of significant
figures.
1. 5 487 129 m to three significant figures
2. 0.013 479 265 mL to six significant figures
3. 31 947.972 cm2
to four significant figures
4. 192.6739 m2
to five significant figures
5. 786.9164 cm to two significant figures
b. Rewrite the answers in (a) using the scientific notation
C. Write the answers to the correct number of significant figures
1. 4.5 X 6.3 ÷ 7.22
2. 5.567 X 3.0001 ÷ 3.45
3. (37 X 43) ÷ (4.2 X 6.0)
4. (112 X 20) ÷ (30 X 63)
5. 47.0 ÷ 2.2
D. Write the answers in the correct number of significant figures
1. 5.6713 + 0.31 + 8.123
2. 3.111 + 3.11 + 3.1
3. 1237.6 + 23 + 0.12
4. 43.65 – 23.7
5. 0.009 – 0.005 + 0.013
E. Answer the following.
1. A runner runs the last 45 m of a race in 6s. How many significant
figures will the runner's speed have?
2. A year is 356.25 days, and a decade has exactly 10 years in it.
How many significant figures should you use to express the number
of days in two decades?
3. Which of the following measurements was recorded to 3 significant
digits : 50 mL , 56 mL , 56.0 mL or 56.00 mL?
4. A rectangle measures 87.59 cm by 35.1 mm. Express its area with
the proper number of significant figures in the specified unit: a. in cm2
b. in mm2
5. A 125 mL sample of liquid has a mass of 0.16 kg. What is the density
of the liquid in g/mL?
Summary
In this lesson, you learned about significant digits and the scientific notation.
You learned the rules in determining the number of significant digits. You also
learned how to write very large and very small numbers using the scientific notation.

87
Lesson 14: More Problems Involving Real Numbers
Pre-requisite Concepts: Whole numbers, Integers, Rational Numbers, Real
Numbers, Sets
About the Lesson:
This is the culminating lesson on real numbers. It combines all the concepts
and skills learned in the past lessons on real numbers.
Objectives:
1. Apply the set operations and relations to sets of real numbers
2. Describe and represent real-life situations which involve integers, rational
numbers, square roots of rational numbers, and irrational numbers
3. Apply ordering and operations of real numbers in modeling and solving
real-life problems
Lesson Proper:
Recall how the set of real numbers was formed and how the operations are
performed. Numbers came about because people needed and learned to count. The
set of counting numbers was formed. To make the task of counting easier, addition
came about. Repeated addition then got simplified to multiplication. The set ℕ of
counting numbers is closed under both the operations of addition and multiplication.
When the need to represent zero arose, the set W of whole numbers was formed.
When the operation of subtraction began to be performed, the W was extended to
the set ℤ or integers. ℤ is closed under the operations of addition, multiplication and
subtraction. The introduction of division needed the expansion of ℤ to the set ℚ of
rational numbers. ℚ is closed under all the four arithmetic operations of addition,
multiplication, subtraction and division. When numbers are used to represent
measures of length, the set ℚ or rational numbers no longer sufficed. Hence, the set
ℝ of real numbers came to be the field where properties work.
The above is a short description of the way the set of real numbers was built
up to accommodate applications to counting and measurement and performance of
the four arithmetic operations. We can also explore the set of real numbers by
dissection – beginning from the big set, going into smaller subsets. We can say that
ℝ is the set of all decimals (positive, negative, and zero). The set ℚ includes all the
decimals which are repeating (we can think of terminating decimals as decimals in
which all the digits after a finite number of them are zero). The set ℤ comprises all
the decimals in which the digits to the right of the decimal point are all zero. This
view gives us a clearer picture of the relationship among the different subsets of ℝ in
terms of inclusion.

88
We know that the nth root of any number which is not the nth power of a rational
number is irrational. For instance, √2, √5, and √9
3
are irrational.
Example 1. Explain why 3√2 is irrational.
We use an argument called an indirect proof. This means that we will show why
3√2 becoming rational will lead to an absurd conclusion. What happens if 3√2 is
rational? Because ℚ is closed under multiplication and
1
3
is rational, then 3√2 ×
1
3
is rational. However,3√2 ×
1
3
= √2, which we know to be irrational. This is an
absurdity. Hence we have to conclude that 3√2 must be irrational.
Example 2. A deep-freeze compartment is maintained at a temperature of 12°C
below zero. If the room temperature is 31°C, how much warmer is the room
temperature than the temperature in the deep-freeze compartment.
Get the difference between room temperature and the temperature inside the
deep-freeze compartment
31 − (−12) = 43. Hence, room temperature is 43°C warmer than the
compartment.
Example 3. Hamming Code
A mathematician, Richard Hamming
developed an error detection code to
determine if the information sent
electronically is transmitted correctly.
Computers store information using bits
(binary digits, that is, a 0 or a 1). For
example, 1011 is a four-bit code.
ℝℚ′ ℚ
ℤ
ℕ
W
a
b
c
d
E
F
G

89
Hamming uses a Venn diagram with three “sets” as follows:
1. The digits of the four-bit code are placed in regions a, b, c, and d, in this
order.
2. Three additional digits of 0’s and 1’s are put in the regions E, F, and G so that
each “set” has an even number of 1’s.
3. The code is then extended to a 7-bit code using (in order) the digits in the
regions a, b, c, d, E, F, G.
For example, the code 1011 is encoded as follows:
Example 4. Two students are vying to represent their school in the regional chess
competition. Felix won 12 of the 17 games he played this year, while Rommel
won 11 of the 14 games he played this year. If you were the principal of the
school, which student would you choose? Explain.
The Principal will likely use fractions to get the winning ratio or percentage of
each player. Felix has a
12
17
winning ratio, while Rommel has a
11
14
winning ratio.
Since
11
14
>
12
17
, Rommel will be a logical choice.
Example 5. A class is having an election to decide whether they will go on a
fieldtrip. They will have a fieldtrip if more than 50% of the class will vote Yes.
Assume that every member of the class will vote. If 34% of the girls and 28% of
the boys will vote Yes, will the class go on a fieldtrip? Explain.
Although 38 + 28 = 64 > 50, less than half of the girls and less than half of the
boys voted Yes. This means that less than half all students voted Yes.
Example 6. A sale item was marked down by the same percentage for three years
in a row. After two years the item was 51% off the original price. By how much
was the price off the original price in the first year?
Since the price after 2 years is 51% off the original price, this means that the
price is then 49% of the original. Since the percentage ratio must be multiplied to
the original price twice (one per year), and 0.7 × 0.7 = 0.49, then the price per
year is 70% of the price in the preceding year. Hence the discount is 30% off the
original.
1
0
1
1
0
1
0
1
1
0 1
1011 1011001

90
Exercises:
1. The following table shows the mean temperature in Moscow by month from 2001
to 2011
January −6.4℃ May 13.7℃ September 12.6℃
February −7.6℃ June 16.9℃ October 6.0℃
March −0.8℃ July 21.0℃ November 0.5℃
April 6.8℃ August 18.4℃ December −4.9℃
Plot each temperature point on the number line and list from lowest to highest.
2. Below are the ingredients for chocolate oatmeal raisin cookies. The recipe yields
32 cookies. Make a list of ingredients for a batch of 2 dozen cookies.
3. In high-rise buildings, floors are numbered in increasing sequence from the
ground-level floor to second, third, etc, going up. The basement immediately
below the ground floor is usually labeled B1, the floor below it is B2, and so on.
How many floors does an elevator travel from the 39th
floor of a hotel to the
basement parking at level B6?
4. A piece of ribbon 25 m long is cut into pieces of equal length. Is it possible to get
a piece with irrational length? Explain.
5. Explain why 5 + √3 is irrational. (See Example 1.)
1 ½ cups all-purpose flour
1 tsp baking soda
1 tsp salt
1 cup unsalted butter
¾ cup light-brown sugar
¾ cup granulated sugar
2 large eggs
1 tsp vanilla extract
2 ½ cups rolled oats
1 ½ cups raisins
12 ounces semi-sweet chocolate
chips

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