Overview of the Handouts
A. Course Title:
Teaching Mathematics in the Intermediate Grades
B. Course Description:
This course emphasizes the integration of technological pedagogical content
knowledge that includes topics on rational numbers, measurement, geometric figures, prealgebra concepts, application of simple probability and data analysis. This course is capped
with microteaching that utilizes appropriate teaching strategies for the development of
critical and problem solving, reasoning, communicating, making corrections, representations
and decisions in real life situations.
B. Course Guide:
This handout in Teaching Mathematics in the Intermediate Grade is a continuation of
Teaching Mathematics in the Primary Grades.
This handout is divided into 11 Chapters which contain essential competencies in
Mathematics in the Intermediate Grades patterned from the Curriculum Guide of the
Department of Education to prepare you to become the best elementary mathematics
teacher in the future.
It consists of the following chapters to wit:
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Whole Numbers
Number Theory
Fractions
Decimals
Ratio and Proportion
Integers
Geometry
Measurement
Introduction to Algebra
Statistics and Probability for Primary Learners solving.
Each chapter could be learned in a self – paced or individual format and it is the
responsibility of the students to maximize their time and exert with determination in
completing the course with the best learning outcomes. However, contents presented in this
module may not be enough for their learning needs, so it is suggested to consider reading
other references related to the topics.
For every lesson of each chapter, there are Lesson Outcomes which enumerate the
expected objectives which you need to attain at the end of every lesson; Lesson Opener
which is a word problem that serves as a spring board for the lesson; Let’s Focus which
provided in-depth discussion of the topic utilizing appropriate strategies and techniques; Let’s
Wrap Up which summarizes important concepts learned; and Let’s Reflect which test
metacognitive ability as you devise appropriate motivational activity, teaching strategy and
mode of assessment that can capture the entire lesson.
Practice exercises and assessment for every lesson is given at the Google Classroom and
should be submitted at the Google Classroom. Students may contact her professor for any
concern or clarifications about the topics, tasks, and outputs to be done.
As to major exams, midterm examination will cover Chapter 1 to Chapter 6 of this
handouts and final examination will cover Chapter 7 to Chapter 11.
CLASS POLICIES UNDER THE NEW NORMAL (GENERIC POLICIES)
1. Take care of the handouts for the interest of the next users.
2. Return handouts during or after midterm examination.
3. Do not write on the handouts. Use the separate sheets for all exercises. Consider further
instructions from each of your professors/instructors as to how will you accomplish the
exercises whether printed or be sent online.
4. Read and/or refer to other sources or references on related topics for additional
learning contents.
5. Optimize your time and effort to complete this handout to achieve desired learning
outcomes for the semester.
6. Arrange with your course professor/instructor the schedule of submission of exercises
and/or requirements.
7. There are only three face-to-face sessions for this semester. All students must attend
the schedules of Onboarding or Orientation day, Midterm Examination day and Final
Examination day. Further information will be given as to the dates and venues of these
face-to-face sessions.
8. All exercises in the handout will also serve as your attendance. Hence, schedule of
submissions must be agreed upon between the professor/instructor and students to
secure attendance.
9. For smooth and fast communication, maintain your active contact number. Do not
be changing your mobile numbers if not necessary.
10. Username of your social media accounts like Facebook and Messenger must be your
real complete name. No pseudonyms, no aliases, no codenames.
11. Secure contact details of all your course professors. Feel free to contact or confer with
your course instructor/professor for any concern, clarification about the handout
content.
12. Above all, your safety and health are our concern. Always maintain the minimum
health guidelines provided by DOH and IATF. Wear face mask, bring your own
sanitizers, and maintain a 2-meter physical distancing.
C. Course Outcomes:
Having completed the handouts, the students are anticipated to meet the following
outcomes:
1. Demonstrate mastery of fundamental concepts of mathematical concepts in the
intermediate grades.
2. Gained pedagogical knowledge and utilized variety of effective strategies for
teaching and learning mathematics.
3. Utilized technology and online resources to deepen mathematics competency.
4. Applied problem solving and critical thinking skills in solving routine and nonroutine problems
5. Showed mastery of subject matter, applied appropriate strategies to facilitate
learning and gave appropriate assessment through microteaching.
D. Course Requirements:
Students are expected to submit the following requirements or outputs during major
exam.
Chapter
Chapter Title
Requirement/Output
1
Whole Number
2
Number Theory
3
Fractions
Chapter Exercises
4
Decimals
Midterm Examination
5
Ratio and Proportion
6
Percent
7
Integers
8
Geometry
9
Measurement
10
Introduction to Algebra
Virtual Demonstration
11
Statistics and Probability
Final Examination
Chapter Exercises
Lesson Planning
Chapter 1: WHOLE NUMBERS
LESSON 1
Multi- Digit Multiplication
Lesson Outcomes
At the end of the lesson, the student should be able to:


Multiply up to 3-digit numbers by up to 2-digit numbers without or with
regrouping; and
Solve word problems involving multi-digit multiplication
Lesson Opener
The Circle of Mathematicians of 123 Elementary Schools solicits notebooks to be
donated to the schools in the far-flung area of a certain municipality. They collected a total
of 232 packs of notebooks. If each pack contains 12 notebooks, how many notebooks did
the organization collect?
Let’s Focus
To visualize the problem, we can draw 232 packs with 12 notebooks each. But drawing
these will take us time. Instead, we represent them like below.
Based on our illustration, we can now visualize that there are 232 packs with 12
notebooks each. Thus, we can simply put this into multiplication sentence as 232 x 12 = N.
We can write the factors vertically to easily perform the operation as shown below
232
× 12
But how this work?
Let us first recall the terms involve in multiplication. Factors are the numbers which we
multiply. Product is the answer which we get in multiplication.
232
factors
× 12
N
product
In multiplying a three-digit by a two-digit number, we follow these steps.
Another way to multiply multi-digit numbers is through the use of Lattice Method. There
are three steps to follow in multiplying numbers using Lattice Method.
Using the Lattice method, we still got 2 784 as the answer.
Another method which we can use in multiplying multi-digit numbers is by applying
the Distributive Property of Multiplication over Addition (DPMA) which you have learned
when you were in the primary grades.
Our number sentence is 232 × 12 = 𝑁
12 can be renamed as 10 + 2.
So, our new equation is 232 × (10 + 2) = 𝑁
Working the equation, we get the results as follows:
(232 × 10) + (232 × 2)
2 320
464
2 784
Let’s Wrap up

There are many methods in multiplying multi-step numbers. Some of
them are the long method, the lattice method and the distributive
property of multiplication over addition.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 1: WHOLE NUMBERS
LESSON 2
Division of 3-to-4-Digit Numbers
b
By 1 – to 2-Digit Numbers
Lesson Outcomes
At the end of the lesson, the student should be able to:


Divide 3- to 4-digit numbers without and with remainder;
Solve word problems involving 3- to 4-digit numbers by 1- to 2-digit numbers.
Lesson Opener
A young organization in Brgy. #09 have raised 1 704 cans of sardines to be distributed
among the typhoon victims. If they will put 24 cans of sardines per box for transport, how
many boxes will they need?
Let’s Focus
To better understand the problem, we can illustrate this as follows:
With the aid of the illustration, the division sentence which we can form is
N
1 704 ÷ 24 = 𝑁 𝑜𝑟 24
1 704
1 704 is the dividend or the number to be divided by another number while 24 is the
divisor which divides another number. N represents the quotient or the answer in division.
To divide large numbers, we can use the Family mnemonic: Dad, Mom, Sister, Brother,
Rover. The initial letters stand for the steps in dividing large numbers as Divide, Multiply,
Subtract, Bring Down and Repeat or Remainder.
Since we do not have any more digit to bring down, we stop here. Therefore, the youth
organization needs 71 boxes to transparent the cans of sardines.
How can we be sure that our answer is correct?
To check, Quotient × Divisor = Dividend
71 × 24 = 1 704
Let’s Wrap up

To divide large numbers, we follow these steps:
Divide, Multiply, Subtract, Bring Down , and Repeat or Remainder
Let’s Reflect
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future
After learning this lesson, reflect on how it can be learned by your future pupils in
pupils in more effective, creative and meaningful way. Discuss you
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
1. Motivational Activity I will use
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
2. Teaching Strategy I will employ
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
3. Mode of Assessment I will administer
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
Chapter 1: WHOLE NUMBERS
LESSON 3
Order of Operations
B
Lesson Outcomes
At the end of the lesson, the student should be able to:


Simplify a series of operations on whole numbers involving more than two
operations using the PEMDAS or GEMDAS rule, and
Solve word problems involving order of operations
Lesson Opener
A pen cost P7.00 while a notebook costs thrice as the cost of the pen. Noel bought 5
notebooks and 2 pens. He gave the cashier a P200-bill. How much change did he got?
Let’s Focus
Let’s illustrate the problem in the Lesson Opener using models
pen
P7.00
pen
P7.00
notebook
notebook
notebook
Total cost of the
items that Noel
bought
notebook
notebook
1 unit = P7.00
17 units = P7.00 × 17 = 𝑃 119
total cost of the item Noel bought
P200 – P119 = P81
change of Noel
Another way to solve the problem is by translating the word problem is by translating
the word problem into an equation or number sentence. Since the cost of a notebook is
thrice as that of the pen, we know that a notebook costs P21. That is multiplying P7 by 3.
Putting it into a number sentence, we can come up with this: 200 − [(5 × 21) + (2 × 7)] = 𝑁.
Looking at the number sentence can be confusing but remembering the rules or how
to work on this will help us arrive at the correct answer. Remember the mnemonics GEMDAS.
This is the manner on how to solve the equation from left to right.
Grouping symbols, Exponents ,Multiplication or Division Addition or Subtraction
First, we will work on the operations inside the grouping symbols – parenthesis, then the
brackets.
200 − [(5 × 21) + (2 × 7)] = 𝑁.
200 −
(105
+
200 − 119
81
14)
= 𝑁.
=𝑁
= 𝑁
Let’s Wrap up

To solve an equation involving order of operation, remember the mnemonics
GEMDAS (Grouping symbols, Exponents, Multiplication or Division and
Addition of Subtraction.

Let’s Reflect
Let’s
Reflect
After
learning
this lesson, reflect on how it can be learned by your future pupils in
After learning
this
lesson,
reflect on
how
it canyou
be learned by your future
more effective,
creative
and
meaningful
way.
Discuss
pupils in more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
2._________________________________________________________________________________
Teaching Strategy I will employ
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
Teaching
Strategy I Iwill
3.2.Mode
of Assessment
willemploy
administer
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________
_________________________________________________________________________________
Chapter 1: WHOLE NUMBERS
LESSON 4
B
Exponential Notation
Lesson Outcomes
At the end of the lesson, the student should be able to:





Describe the exponents and the bases in a number expressed in exponential
notation.
Give the value of numbers expressed in exponential notation
Perform two or more different operations on whole numbers with
or without exponents and groupings symbols, and
Solve word problems involving exponential notation.
Lesson Opener
There are 28 cartons of milk in a grocery store. How many cartons of milk are there?
Let’s Focus
What does 28 mean?
28 is an example of a number written in exponential notation. It is read as ‘2 to the
eighth power”, “two to the eighth” or “2 raised to 8”.
2 is the base. It is a number that gets multiplied repeatedly a certain number of times.
8 is the exponent. The exponent tells the number of times the base is to be multiplied by itself.
When written in expanded form, 28 is equal to 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 .
standard form of 28 is 256.
Therefore, there are 256 cartons of milk in a grocery store.
Exponential Form
52
122
43
103
Read as
Five squared
Twelve squared
Four cubed
Ten cubed
Expanded Form
5x5
12 x 12
4x4x4
10 x 10 x 10
Standard Form
25
144
64
1000
The
Let’s Wrap up



Exponential notation is the shortest way of writing repeated
multiplication.
The base is the number that is being multiplies repeatedly.
Exponent tells us to multiply the number by itself certain number of
times.
Let’s Reflect
After learning
lesson, reflect on how it can be learned by your future pupils in
Let’s this
Reflect
more effective,
creative
andthis
meaningful
way.on
Discuss
you
After
learning
lesson, reflect
how it
can be learned by your future
pupils in more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
2. Teaching
Strategy I will employ
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
3. Mode
of Assessment I will administer
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 2: Number Theory
LESSON 1
Prime and Composite
B
Numbers
Lesson Outcomes
At the end of the lesson, the student should be able to:



Differentiate prime from composite numbers.
Write a given number as a product of its prime factors; and
Solve word problems involving prime and composite numbers.
Lesson Opener
Samuel writes the following numbers on a piece of paper, 48, 57, 37, 91 and 76. He
then asks Dave to identify the number which does not belong to the group. Dave gives the
correct answer. What is his answer?
Let’s Focus
What do you think is the basis of Dave in identifying the number which is different from
the rest? Let’s consider some possible ways:
1. All five numbers have two digits so it cannot be uses as basis.
2. Two of the numbers are even while three are odd, so again, this classification does
not make any one number different.
How about finding all the factors of each number?
48 – 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
57 – 1, 3, 19, 57
37 – 1, 37
91, 1, 7, 13, 91
76 – 1, 2, 4, 19, 38, 76
Observe that the number 37 has only two factors, 1 and 37 (the number itself) while
each of the rest has other factors aside from 1 and itself.
A number is called Prime if it has only two factors. Examples: 2, 13, 29 and 83.
On the other hand, numbers with more than two factors are called Composite. For
instance, 4, 15, 46 and 120 are composite.
Try this:
Tell whether each of the following numbers is prime or composite.
1. 2
2. 97
3. 138
4. 51
5. 1
Answer:
1. 2 is prime since it has only two factors – 1 and 2. In fact, 2 is the smallest prime
number.
2. 97 is prime. It has the biggest two-digit prime number.
3. 138 is obviously composite since it is even. Each of the even numbers greater than
2 has 2 as a factor other than 1 and itself which means all even numbers greater than 2 as
prime.
4. 51 is composite with factors of 1, 3, 17 and 51.
5. 1 is NEITHER prime nor composite because it has 1 factor only.
Every composite number can be expressed as a product of its prime factors. This
process is called prime factorization. For example, let use find the prime factorization of 24
using the Factor Tree Method. Let us then express the answer in exponential form.
Choose any pair of factors of 24. Aside from 12
and 2, we may also use 6 and 4 or 3 and 8.
The prime factorization of 24 is 2 x 2 x 2 x 3 or 23 x 3.
Give the prime factorization of 90.
Choose any pair of factors of 90. We may also
choose 2 & 45, 3 & 30, 5 & 18 and 6 & 15.
The prime factorization of 90 is 2 x 32 x 5.
Let’s Wrap up

A Prime Number is a whole number greater than 1 which has only two
factors – 1 and the number itself.


A Composite Number is a whole number with three or more factors.
Prime Factorization is a process of expressing a composite number as
a product of its prime factors.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future
Let’sinReflect
pupils
more effective, creative and meaningful way. Discuss you
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective,
creative
and meaningful
1. Motivational
Activity
I will use way. Discuss you
_________________________________________________________________________________
1. Motivational
Activity I will use
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
2. Teaching
Strategy, I will employ
________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________________
________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 2: Number Theory
LESSON 2
D
B
Divisibility Rules
Lesson Outcomes
At the end of the lesson, the student should be able to:


Identify the divisibility rules for one-digit numbers and selected two-digit
numbers, and
Solve routine and non-routine problems involving divisibility rules
Lesson Opener
What is the smallest number that can be exactly divided by all numbers from 1 to 10?
Let’s Focus
To answer the Lesson Opener, we need to know the divisibility rules for 1 to 10. It means
that, we need to know if when is a number divisible or can be exactly divided by 1,2, 3, …,
10.
For instance, we can say that 24 is divisible by 8 since 24 ÷ 8 = 3, while 24 is not divisible
by 7 because 24 ÷ 7 = 3 with a remainder of 3.
A Test for Divisibility by 1
Every number is divisible by 1
The Identity Property of Multiplication states that when a number is multiplied by 1, the
product is the number itself. Inversely, when a number is divided by 1, the quotient is the
number itself. This property supports that any number is divisible by 1
A Test for Divisibility by 2
Every number is divisible by 1.
A Test for Divisibility by 4, 8, 16 and other Powers of 2
Let n be a natural number. Then 4 divides n if and only f, and only if, 4 divides the
number named by the last two digits of n. Similarly, 8 divides n if, and only if, 8 divides
the number named by the last three digits of n.
A Test for Divisibility by 3 and 9
A natural number is divisible by 3 if, and only if, the sum of its digits is divisible by 3.
Similarly, a natural number is divisible by 9, if and only if, the sum of the digits is divisible
by 9.
Example: Is 423 divisible by 3?
To find out, we add 4 + 2 + 3; the answer is 9. Since 9 is divisible by 3, then 423 is divisible
by 3. Similarly, 423 is also divisible by 9 since, obviously, the sum which is 9 is divisible by 9.
A Test for Divisibility by 10
Let n be a natural number. Then n is divisible by 10 if, and only if, its unit digit is 0.
A Test for Divisibility by 5
Let n be a natural number. Then n is divisible by 5 if, and only if, its unit digit is 0 and
5.
A Test for Divisibility by 7
A natural number is divisible by 7 if the last digit is doubled and subtracted from the
number formed by remaining digits, the result is divisible by 7.
Example: Find out if 203 is divisible by 7



Double the last digit: 3 x 2 = 6
Subtract that from the rest of the number 20 – 6 = 14
Check to see if the difference is divisible by 7.
14 is divisible by 7, therefore 203 is divisible by 7.
A Test for Divisibility by 11
A natural number is divisible by 11 if, and only if, the difference of the sums of the
digits in the even and odd positions in the number is divisible by 11.
Example: Find out if 41, 019 is divisible by 11.
In 41, 019, the digits in the odd position (first, third and fifth) are 4, 0 and 9 are their sum
is 13. On the other hand, the digits in the even position (second and fourth) are 1 and 1 and
their sum is 2. The difference between the sums of the odd-placed and even-placed digits 13
– 2 = 11, is divisible by 11. Therefore, 41, 019 is divisible by 11.
Let’s Wrap up










Every number is divisible by 1.
A number is divisible by 2 if it is even.
A number is divisible by 3 if the sum of its digits is a multiple of 3.
A number is divisible by 4 if its last two digits are divisible by 4.
A number is divisible by 5 if its last digit is either 0 or 5.
A number is divisible by 6 if it is divisible by both 2 and 3.
A number is divisible by 7 if the difference twice the units digit and the
remaining digit is divisible by 7.
A number is divisible by 8 if the last three digits are divisible by 8.
A number is divisible by 9 if the sum of its digits is divisible by 9.
A number is divisible by 10 if the units digit is zero.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 2: Number Theory
LESSON 3
D
Greatest Common Factor
B
Lesson Outcomes
At the end of the lesson, the student should be able to:



Find the factors of a number.
Find the common factors and the greatest common factor (GCF) of two to
three numbers using the following methods: listing, prime factorization and
continuous division.
Solve real-life problems involving GCF of 2-3 given numbers.
Lesson Opener
Ivan has three pieces of string with lengths of 48 m, 80 m and 96 m. He wishes to cut
the three pieces of string into smaller whole meter pieces of equal length with no remainders.
What is the greatest possible length of each of the smaller pieces of string?
Let’s Focus
Listing Method
One way to answer the problem above is to simply list down all the possible whole
meter pieces which each string can be cut into exactly. The list is arranged in rows below:
48 – 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
80 – 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
96 – 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
To know if in which lengths of smaller pieces can the three strings be cut into, we take
the pieces common to all three strings, and these are: 1, 2, 4, 8 and 16. It means that the
longest piece in which the three strings can be cut into is 16 m.
The numbers 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 are factors of 48. These are numbers that
can exactly divide 48. The factors or 80 are 1, 2, 4, 5, 8. 10. 16, 20, 40, 80. On the other hand,
the factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
The common factors of 48, 80 and 96 ate 1, 2, 4, 8 and 16. The greatest among the
common factors which is 16 is called the Greatest Common Factor (GCF) or Greatest
Common Divisor (GCD) of 48, 80 and 96.
The process used above to find the GCF of 48, 80 and 96 is called Listing Method.
Prime Factorization Method
To find the GCF of 48, 80 and 96 using the prime factorization method, simply find the
prime factorization of the given numbers- the product of the prime factors common to all
given numbers is their GCF. That is,
48 = 2 x 2 x 2 x 2 x 3
80 = 2 x 2 x 2 x 2 x
5
96 = 2 x 2 x 2 x 2 x 3 x 2
Common Prime Factors: 2 x 2 x 2 x 2 = 16
Therefore, the GCF of 48, 80 and 96 is 16
Continuous Division Method
True to its name, in using the continuous method, we continue dividing the given
numbers by a common prime number until the quotients are relatively prime. Let’s take a look
at the process using the same numbers below.
2
48
80
96
Divide each number by 2. Write the quotient below
the numbers
2
24
40
48
Divide by 2.
2
12
20
24
Divide by 2.
2
3
10
12
3
5
6
Divide by 2.
Since 3, 5, 6 are relative prime, then the GCF of 48, 80 and 96 is the product of the
prime factors used as divisors which is 2 x 2 x 2 x 2 or 16.
Let’s Wrap up



Factors are numbers being multiplied to get the product.
Greatest Common Factor (GCF) is the biggest number that can
exactly divide the given numbers.
The methods of finding the GCF of two or more numbers are Listing,
Prime Factorization, Continuous Division.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 2: Number Theory
LESSON 4
D
B
Least Common Multiple
Lesson Outcomes
At the end of the lesson, the student should be able to:
Find the multiples of a number.
Find the common multiples and least common multiple (LCM) of two or more
numbers using the following methods: listing, prime factorization, and
continuous division; and
Solve real-life problems involving given numbers.



Lesson Opener
ESSU-Salcedo Campus has three bells. Bell A rings every 60 minutes, Bell B every 90
minutes and Bell C every 45 minutes. They all ring together at 7:00 a.m. When is the next time
that they will all ring together again?
Let’s Focus
The most logical way to solve the problem is by listing the time from 7:00 and adding
successively 60 minutes or 1 hour for Bell A 90 minutes for Bell B, and 45 minutes for Bell C until
the first common time emerges. This, however, might take too long to do. The best option is
to solve by finding the Least Common Multiple of the numbers (in minutes) and convert them
to hours, then add to 7:00.
Let us first define “multiple”. What is multiple or what are multiples of a number?
Multiples are products of the natural numbers and the given number. For instance, the
multiples of 8 are 8, 16, 24, 32, 40 and so on. These are derived by multiplying 8 by 1, 2, 3, 4, 5,
and so on.
Let us now solve the problem using the Listing Method
Listing Method
Step 1: List the multiples of each number
60 – 60, 120, 180, 240, 300…
90 – 90, 180, 270, 360 …
45 – 45, 90, 135, 180…
Step 2: Find the common multiple of the numbers.
Since 180 is the first multiple common to all three numbers, then it is the LCM of
the numbers.
Prime Factorization
Step 1: Find the prime factorization of the numbers
60 – 2 x 2 x 3 x 5
90 – 2
x3x5x3
45 -
3x5x3
Step 2: Multiply the common multiples
2 x 2 x 3 x 5 x 3 = 180
Observe that unlike in GCF where prime factor has to be common to all given
numbers, for LCM, even if a prime factor is common to only two numbers, it can still be
considered as a common prime factor.
Continuous Division Method
True to its name, in using the continuous method, we continue dividing the given
numbers by a common prime factor until the quotients are different relatively prime numbers.
Let’s take a look at the process using the same numbers above.
Step 1: Divide the numbers by their common prime factor.
3
60
90
45
Divide each number by 3. Write the quotient below
the numbers
Step 2: Since there is no more common prime factor for all three numbers, then find a
common prime factor for any two numbers. Bring down the number that is nor divisible by the
prime divisor.
Divide by 2. Bring down 15 since it is not divisible by
2 20 30 15
2.
3
10
15
15
5
10
5
5
2
1
1
Divide by 3
Divide by 5
Step 3: The remaining numbers 2, 1, and 1 are now relatively prime. Therefore, the
prime factorization of 60, 90 and 45 is the product of the prime divisors and the remaining
quotient 3 x 2 x 3 x 5 x 2 = 180.
Let’s Wrap up



Multiples are products of a given number and the natural/counting
numbers.
Least Common Multiples (LCM) is the least number that can exactly
be divided by the given numbers.
The three methods of finding the LCM f 2 or more numbers are Listing,
Prime Factorization and Continuous Division.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 3: Fractions
LESSON 1
D
Changing Improper Fraction to
B
Mixed Number and Vice Versa
Lesson Outcomes
At the end of the lesson, the student should be able to:



Visualize and recognize mixed number and improper fraction;
Change improper fraction to mixed number and vice versa;
Solve word problems involving changing improper fraction to mixed number
and vice versa.
Lesson Opener
1
1
Mother brought home 2 2 plates of buko pie. Illustrate 2 2 and change it is improper
fraction.
Let’s Focus
Two and one-half can be illustrated as follows:
2
1
2
can also be written as
5
2
which is an improper fraction. That is by multiplying the
whole number by the denominator (number found at the bottom of the fraction bar) and
adding the numerator (number found at the top of the fraction bar). The answer will become
the new numerator. Then, copy the same denominator.
How about if an improper fraction is given, and you are tasked to change it into a
mixed number? For example, change
15
4
to a mixed number.
To change an improper fraction to a mixed number, we will divide the numerator by
the denominator. The quotient will become the whole number and the remainder will
become the numerator of our fraction. Finally, copy the same denominator.
15 divide by 4 is 3 r 3, which gives us 3
3
15
as the improper form of 4
4
.
Let’s Wrap up



A mixed number or mixed fraction is composed of a whole number
and fraction while an improper fraction has a numerator which is
greater than or equal to the denominator.
To change a mixed number to an improper fraction, multiply the whole
number by the denominator, then add the numerator. The answer will
become the new numerator. Finally, copy the same denominator.
To change an improper fraction to a mixed number, divide the
numerator by the denominator. The quotient will become the whole
number, while the remainder will become the numerator. Lastly, copy
the same denominator.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 3: Fractions
LESSON 2
D
Fractions in Simplest Form
B
Lesson Outcomes
At the end of the lesson, the student should be able to:


Express fractions to lowest forms.
Solve word problems involving fractions in simplest forms.
Lesson Opener
Jayson brought home
9
12
plates of pizza. Express
9
12
as fraction in its simplest form.
Let’s Focus
9
12
can be illustrated as follows:
Minimizing the number of divisions, we can
have the illustration below which is
3
4
is the lowest term or the simplest form of
3
4
9
12
We need not illustrate the fraction and minimize the number of divisions in order to
express it in simplest form. The easiest way to do this is by dividing both the numerator and
denominator by their greatest common factor (GCF).
As a recall, we can find the GCF of two r more numbers by the continuous division
method, listing method or through prime factorization method. Using any of these methods,
we arrive at the same GCF of 9 and 12 which is 3.
Performing the operations, we have
9
12
÷
3
3
=
3
4
Now, the only common factor of 3 and 4 is 1. This means that 3 and 4 are relatively
prime which indicates that
3
4
is the simplest form already.
Let’s Wrap up

To express a fraction in its lowest term or in its simplest form, we divide
both the numerator and the denominator by their greatest common
factor.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 3: Fractions
D
Addition
of Dissimilar Fractions
LESSON 3
B
and Mixed Numbers
Lesson Outcomes
At the end of the lesson, the student should be able to:



Perform addition of dissimilar fractions;
Add fractions and mixed fractions without and with regrouping
Solve word problems involving addition of dissimilar fractions and mixed
numbers.
Lesson Opener
Lola Titay bought
1
2
kilogram of onions. Lolo Isko bought
many kilograms of onion did Lola Titay and Lolo Isko buy in all?
1
4
kilograms of onions. How
Let’s Focus
Based on the problem given in the Lesson Opener, we can form the number sentence
1
4
+
1
2
= 𝑁. We can illustrate this as follows:
1
1
2
4
+
+
=N
= N
+
Therefore ,
1
4
+
1
2
=
=
3
4
We can also find the sum of
1
4
+
1
2
by changing both fractions into similar
fractions. We cannot add these fractions immediately because they have different
denominators. To do this, we find the least common multiple of 4 and 2 and that is 4. This will
serve as the the least common denominator (LCD) of
1
4
+
1
2
=
1
4
+
2
4
=
1
2
𝑎𝑛𝑑
1
4
3
4
Let’s Wrap up


To add dissimilar fractions, change them first to similar fractions, then
add.
To add mixed numbers:
1. add the whole number parts.
2. change the fractional parts to like fractions;
3. add the fractional parts;
4. write the answer in the simplest form.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 3: Fractions
LESSON 4
D
Subtraction
of Dissimilar
B
Fractions and Mixed Numbers
Lesson Outcomes
At the end of the lesson, the student should be able to:



Perform subtraction of dissimilar fractions;
subtract fractions and mixed fractions without and with regrouping
Solve word problems involving subtraction of dissimilar fractions and mixed
numbers.
Lesson Opener
Camille used
5
6
meters of ribbon in her project. Joan used
project. Who used a longer ribbon? By how much?
2
3
meters of ribbon in her
Let’s Focus
Based on the problem presented in the lesson opener, we need to compare first the
fractions before performing any operations. Since both fractions have the same whole
number, we only have to compare the fractional parts.
15
12
5
2
6
3
Comparing the fractions
5
6
is greater than
2
3
which means that Camille used a longer
ribbon than Joan.
To know how much longer Camiile’s ribbon is than Joan’s, subtract the fractions as
illustrated below. Since the fractions are dissimilar, we will rename them as similar fractions
before performing subtration.
Therefore, Camille used a longer ribbon than Jaon by
1
.
6
Let’s Wrap up


To subtract dissimilar fractions, change them first to similar fractions,
then subtract the numerators and copy the common denominator.
To subtract mixed numbers:
1. subtract the whole number parts.
2. change the fractional parts to like fractions;
3. subtract the fractional parts;
4. write the answer in the simplest form.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 3: Fractions
LESSON 5
D
B
Multiplication of Fractions
Lesson Outcomes
At the end of the lesson, the student should be able to:



Visualize multiplication of fractions using models;
Multiply simple fractions, whole numbers and mixed numbers; and
Solve word problems involving multiplication without or with subtraction of
fractions and whole numbers using appropriate problems solving strategies
and tools.
Lesson Opener
One recipe of maja blanca needs
2
1
2
cups of milk. If Anglelica will make 5 recipes of
maja blanca, how many cups of mils will she need?
Let’s Focus
We can illustrate the problem as follows:
Based on the illustration, how many wholes and halves do we have in all?
We have 10 wholes and 5 halves which are equal to 12
1
2
.
We can easily solve this by multiplying the fractions. That is 2
or
1
12 .
2
1
2
×5=
5
2
5
25
1
2
× =
Remember that in multiplying fractions we multiply the numerators and the
denominators and express the answer in simplest form. In multiplying mixed numbers, express
the mixed numbers as an improper fraction and follow the same rule. Meanwhile, whole
numbers have denominators of 1.
Let’s Wrap up

To multiply fractions, we follow these rules:
1. In multiplying a fraction by a fraction, multiply the numerators as well as
the denominators and express the answer in simplest form.
2. In multiplying a fraction by a whole number, express the whole number
as a fraction with a denominator and express the answer in simplest form.
3. In multiplying mixed numbers, express the mixed numbers as improper
fractions. Then, multiply the numerators and the denominators and express
the answer in simplest form
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 3: Fractions
LESSON 6
D
B
Division of Fractions
Lesson Outcomes
At the end of the lesson, the student should be able to:



Visualize division of fractions using models;
Divide simple fractions, whole numbers and mixed numbers; and
Solve word problems involving division without or with any it the other
operations of fractions and whole numbers using appropriate problems solving
strategies and tools.
Lesson Opener
Josam brought home 2 plates of banana cake. She sliced them into fourths. How
many pieces did Josam slice the cake into?
Let’s Focus
Number of pieces of cake = 2
wholes?
÷
1
means how many quarters or fourths are there in 2
4
Based from the illustration we can see that:
Number of quarters or fourths in 1 cake = 4
Number of quarters or fourths in 2 cakes = 2 x 4
So, 2
1
÷ =2 ×4=8
4
We can say that dividing by
1
4
is the same as multiplying by 4.
Thus, Jose sliced the cake into 8 pieces.
Let’s Wrap up

To divide fractions, we follow these steps:
1. get the reciprocal or the divisor;
2. change the “ ÷ " (division sign) to “×” (multiplication sign)
3. multiply the numerators;
4. multiply the denominators;
5. express the answer in its simplest form, if needed.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
LESSON 1
D
B
Visualization of Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:



Visualize decimal numbers using models like blocks, grids, number lines and
money to show the relationship to fractions.
Read and write decimal numbers through ten thousandths; and
Solve word problems involving visualization of decimals.
Lesson Opener
Teacher Marriane asked her pupils: “What is the smlalles number?
Mirel answered, “One”,
Myrine replied, “ Zero”,
How about you, Melchor, whose answer do you think is correct?
Melchor answered: Neither is correct, Ma’am.
Is Melchor correct?
Let’s Focus
In the story above, yes Melchor is correct. Mirel’s answer is correct for the set of
counting or natural numbers which are 1,2, 3, . . . On the other hand, Myrine’s answer is correct
for the set of whole numbers which are 0, 1, 2, …
However, there are numbers which are less than zero. These are the negative numbers
which we will learn in the later part of this learning material. Also, there are numbers greater
than zero but less than 1. Examples of this set of number are
on.
1 3
, , 0.25, 0.01, 0.085 and so
2 4
Fractions can be expressed as decimals. A proper fraction, when converted to a
decimal has a zero in the whole number part. For instance,
3
4
is equal to
0.75.
Decimal is a rational number whose denominator is a power of 10 such as 100 or 1, 102
or 100, and so on.
Let’s take a look at the illustrations below then we will practice how to read decimals
by looking at the corresponding fraction and decimal being represented by the illustrations.
The fractions and decimals that represent the shaded parts in the illustrations.
Fraction
1.
Decimal
0.1
1
10
2. .
0.5
5
10
3. 1 10
1.3
7
2.7
3
4. . 2
10
To read decimal number:
1. read the whole number part first (if the whole number part is zero, then simply read
the decimal part).
2. read the decimal point as “and”, then
3. read the decimal place like whole number and affixing the place value of the
right-most digit.
Let us place the decimals above in the place value hacrt and learn how to read
them.
Fraction
1.
0
1
0
5
1
3
Decimal
Read as
0.1
One-tenth
0.5
Five-tenths
3
10
1.3
One and three- tenths
7
2.7
Two and seven-tenths
1
10
2. .
5
10
3. 1
4. . 2
10
Practice reading the following decimals:
Decimal
1. 8.007
2. 0.0005
3. 12.0231
4. 9. 6109
5. 0.0023
Read as
8 and 7 thousandths
5 ten-thousandths
12 and 231 ten-thousandths
9 and 6 thousand, 109 ten-thousandths
23 ten-thousandths
Let’s Wrap up

To divide fractions, we follow these steps:
1. get the reciprocal or the divisor;
2. change the “ ÷ " (division sign) to “×” (multiplication sign)
3. multiply the numerators;
4. multiply the denominators;
5. express the answer in its simplest form, if needed.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
D
LESSON 2
B
Changing Decimals to
Fractions and Vice Versa
Lesson Outcomes
At the end of the lesson, the student should be able to:


Rename decimal numbers to fractions and vice versa;
Solve problems involving changing decimals to fractions and vice versa.
Lesson Opener
𝜋 (pi) is the sixteenth letter of the Greek alphabet. It is always introduced as a value of
3.14 or 3.14159. while there is no exact value of pi, many mathematicians and math fans are
interested in calculating it to as many digits as possible. The Guinness World Record for reciting
the most digits of pi belongs to Rajveer Meenaof of India, who recited pi to 70,000 decimal
places (while blindfolded) in 2015. Meanwhile, some computer programmers have
calculated the value of pi to more than 22 trillion digits. The first 100 digits of pi are:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230
78164 06286 20899 86280 34825 34211 7067
(source:https://www.livescience.com/29197-what-si-pi.html)
Let’s Focus
Though 𝜋 is an irrational number, some use rational expressions to estimate it, like
22
𝑜𝑟
7
333
106
. This show that decimals may be expressed as fractions and vice versa.
For instance, the decimal number 0.5 is read as five-tenths which may also be written
as
5
10
. Thereore, 0.5 =
5
10
1
𝑜𝑟 .
2
How is decimal written in fraction?
Decimals are written in fractions the way they are read. Let’s take a look at the
following examples.
1.
2.
3.
4.
Decimal
0.2
0.75
0.009
3.05
Read as
Two-tenths
Seventy-five hundredths
Nine-thousandths
Three and five-hundredths
Fraction
2/10 or 1/5
75/100 or ¾
9/1000
3 5/100 or 3 1/20
All rational numbers can be written as wither terminating or repeating decimals.
Examples of terminating decimals are 0.5, 2.84 and 0.875 while examples of repeating
decimals are 0.333… and 0.727272… Non-terminating decimals that do not repeat such as
0.1010010001…,are called irrational nubers. Examples of irrational numbers are 0.1372…, 𝜋
and √2.
To express a fraction as decimal, simply divide the numerator by the denominator.
Let’s Try this:
Convert the following fractions into decimals:
1.
2.
3.
4
5
3
8
7
100
4
4. 8
25
1
5. 3
4
Let’s Check:
1.
2.
3.
4
5
3
= 0.8
= 0.375
8
7
= 0.07
100
4
4. 8 25 = 8.16
1
5. 3 4 = 3.25
Let’s Wrap up



A decimal is expressed as a fraction the way it is read.
Decimals are written as fractions the way they are read.
To convert a fraction to decimal, divide the numerator by the
denominator.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
D
LESSON 3
B
Place Value and Value
of Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:


Give the place value and the value of a digit of a given decimal number
through ten thousandths; and
Solve word problems involving place value of decimals.
Lesson Opener
In a decimal number, how many times as large is the digit 7 in the hundreds place
than the digit in the hundredths place?
Let’s Focus
If we write a number with 7 in both the hundreds and hundredths places we will have
something like the one below.
Hundreds
(100)
Tens
(10)
PLACE VALUE CHART
Ones
Decimal
(1)
Point
Tenths
( 0.1 or
1
Hundredths
)
10
7
( 0.01 or
1
)
100
7
To answer the problem posed above, we need not know what digits are in the tens,
ones and tenths places. However, we will just assign digits in these place values to have a
complete picture of the decimal number and the values and place values of each digit.
Hundreds
(100)
Tens
(10)
7
4
PLACE VALUE CHART
Ones
Decimal
(1)
Point
6
.
Tenths
1
Hundredths
1
( 0.1 or 10)
( 0.01 or 100)
9
7
Let us now give the place value and value of each digit in the number 746.97
Digit
Place Value
(in numeral)
100
10
1
0.1
0.01
7
4
6
9
7
Value
(Digit x Place Value
700
40
6
0.9
0.07
The value of digit 7 in the hundreds place is 700 while that in the hundredths place is
0.07. the chart clearly shows tht the place value of a digit in a number is always ten times as
large as the one in its immediate right. Therefore, the value of digit 7 in the hundreds place is
10,000 times as large as that of the hundredths place.
If we continue the place value chart thorugh ten-thousands, it will look like this.
PLACE VALUE CHART
Word
Decimal
Fraction
Hundreds
Tens
Ones
100
100
10
10
1
1
3
Decimal
Point
Tenths
Hundredths
Thousandths
Tenthousandths
0.1
1
10
1
0.01
1
100
2
0.001
1
1000
0
0.0001
1
10000
5
Take 3.1205 as example. What is the place value an value of each digit? Let us place
3.1205, read as three and one thousand thwo hundred five ten-thousandths, in the place
value chart.
Now, it is easier for us to determine the place value and value of each digit of the
numeral. Let’s use a table to show that.
Digit
3
1
2
0
5
Place Value
(in numeral)
1
0.1
0.01
0.001
0.0001
Value
(Digit x Place Value
3x1=3
1 x 0.1 = 0.1
2 x 0.01 = 0.02
0 x 0.001 = 0
5 x 0.0001 = 0.0005
Let’s Wrap up


The place value is the value of the location of a digit in a number.
The place values are determined by how many places the digit lies
to the right or the left of the decimal point.
The value of a digit is the product of that digit and it place value.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
LESSON 4
D
B
Rounding Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:


Round decimal numbers to the nearest hundredth and thousandth; and
Solve word problems involving rounding decimals.
Lesson Opener
What are the decimal numbers with two decimal places that can be rounded of 0.8?
Let’s Focus
We usually round off when we don’t need exact values or we only want an estimare
or appropriate of values. Moreover, no mathematician likes working with a long, awkward
string of decimals, so they often use a technique called “rounding” to make these numbers
easier to work with.
Rounding a decimal is much like rounding a whole number. First is to find the place
value needed to be rounded and look at the digit to the right. If is is 5 or bigger, round up. If
is less than 5, round down.
Let’s do sme examples step by step.
Round 9.4738 to the nearest thousandth
Step 1: Find the decimal place you need to round to.
The problem above tells us to round the decimal number 9.4738 to the nearest
thousandth- the digit in the thousandths place is “3”.
Step 2: Determine the digit to the right of the place you are rounding to. The digit to the rigt
of “3” is “8”.
Step 3. If the digit to the right of the place you are rounding to is less than 5, round down-that
is- copy all digits from the left up to the digit you are rounding to. However, if the digit to the
right of the place you are rounding to is equal to or greater than 5, round up or add 1 to the
digit in the place you are rounding to and drop all digits to its right.
Since 8 > 5, then we add 1 to 3. All digits to the left of the thousandths place are
retained. Thus, our answer is 9.474.
Let us have more examples.
Round the following decimals to the nearest place value indicated.
1. 0.749 (tenth)
2. 2.457 ( hundredth)
3. 0.8195 (thousandth)
4. 37.806 (hundredth)
5. 0.9631 (greatest place value
Let’s show our answers using a table.
1.
2.
3.
4.
5.
Given
Decimal
Place
Rounding
to
Digit in the
Place
Rounding to
0.749
2.457
0.8195
37.806
0.9631
Tenth
Hundredth
Thousandth
Hundredths
Greatest
place vale
7
5
9
0
9
If digit to the right of the
place rounding to is
equal to or greater than
5, round up. If less than 5,
round down
4 < 5, retain 7
7 > 5, add 1 to 5
5 = 5; add 1 tp 9
6 > 5; add 1 to 0
6 > 5, add 1 to 9
Answer
0.7
2.46
0.820
37.81
1.0
To answer the Lesson Opener, we may use a number line showing the hundredth.
The number linshows that all numbers starting from 0.75 until 0.84 can be rounded to
0.8. the 0.74 is rounded down to 0.7 since the digit in the hundedths place is less than 5 while
0.85 is to be rounded up to 0.9. it means that there are 10 decimals that can be rounded off
to 0.8
Let’s Wrap up

To round off a decimal:
1. find the decimal place you need to round to;
2. determine the digit to its right; and
3. if the digit is less than 5, round down- that is to copy all the digits from
the left up to the digit you are rounding to. If the digit to the right of the
place you are rounding to is equal to or greater than 5, round up or
add 1 to the digit in the place you are rounding to and drop all digits
to its right.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
LESSON 5
D
B
Comparing Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:


Compare and arrange decimal numbers; and
Solve word problems involving comparing and ordering decimals
Lesson Opener
During the West Visayas State University-Himamaylan City Campus Sports Festival, the
sprinters in the 100-m dash have the following time.
Name
Josiah Bonsico
Ralph Barreto
Russel Banico
Ivan Tanate
James Enesio
Jairyll Prava
Time
12.2
11.75
12.18
9.98
10.05
11.8
Let’s Focus
Decimals are compared in exactly the same way as whole numbers. We start by
comparing the different place values from left to right-that is-from the largest place. Writing
the numbers in colums in a place value table would be of great help.
Let us place the decimals on the table below.
Josiah Bonsico
Ralph Barreto
Russel Banico
Ivan Tanate
James Enesio
Jairyll Prava
Tens
1
1
1
1
1
Ones
2
1
2
9
0
1
.
.
.
.
.
.
Tenths
2
7
1
9
0
8
Hundredths
5
8
8
5
We may then add 0s to the beginning and/or end of th numbers so they have the
same number of digits.
Tens
1
1
1
0
1
1
Josiah Bonsico
Ralph Barreto
Russel Banico
Ivan Tanate
James Enesio
Jairyll Prava
Ones
2
1
2
9
0
1
.
.
.
.
.
.
Tenths
2
7
1
9
0
8
Hundredths
0
5
8
8
5
0
Now, let’s compare, starting from the highest place value which is the tens place. The
smaller the number means the faster the sprinter. So we arrange the numbers fro least to
greatest.
So far, only Ivan Tanate has a zero in the tens place which means 9,98 is the least
number. Therefore, Ivan is the fastest so he wins first place.
Josiah Bonsico
Ralph Barreto
Russel Banico
Ivan Tanate
James Enesio
Jairyll Prava
Tens
1
1
1
0
1
1
Ones
2
1
2
9
0
1
.
.
.
.
.
.
Tenths
2
7
1
9
0
8
Hundredths
0
5
8
8
5
0
Rank
1st
All other athletes have “1” in the tens place, so we proceed to the ones place. The
least digit in the ones place is zero, that is James Enesio. So his time, 10.05 is second least,
therefore he is the second fastest sprinter.
Josiah Bonsico
Ralph Barreto
Russel Banico
Ivan Tanate
James Enesio
Jairyll Prava
Tens
1
1
1
0
1
1
Ones
2
1
2
9
0
1
.
.
.
.
.
.
Tenths
2
7
1
9
0
8
Hundredths
0
5
8
8
5
0
Rank
1st
2nd
Next, there are two 1s in the ones place-the time of Ralph Barret – 11.75 and that of
Jairyll Prava – 11.8. We now compare the tenths digits of the two decimals. Since, 7 is less than
8, so 11.75 is less tha 11.8
Josiah Bonsico
Ralph Barreto
Russel Banico
Ivan Tanate
James Enesio
Jairyll Prava
Tens
1
1
1
0
1
1
Ones
2
1
2
9
0
1
.
.
.
.
.
.
Tenths
2
7
1
9
0
8
Hundredths
0
5
8
8
5
0
Rank
6th
3rd
5th
1st
2nd
4th
Let’s Wrap up

To compare decimal numbers, arrange the numbers in column, with
digits of the same place value properly aligned. Start comparing from
the leftmost digits, that is, the digits with the greatest place value. The
number with the greater digit in the higher place value is the bigger
number.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
LESSON 6
D
B
Addition and Subtraction of
Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Add and subtract decimal numbers; and
 Solve word problems involving addition and subtraction of decimals.
Lesson Opener
Mrs. Rivera bought the following items from a grocery store: a kilo of sugar ar P56..50,
2 kg of rice at P48 per kilo, a pack of milk at P96.75 and a pack of match sticks at P6. How
much change would she get from her P1000.00
Let’s Focus
To solve the problem, we first sum up the amounts of the items Mrs. Rivera bought and
subtract it from her P1000.00 to determine the amount of change she will get.
How ae decimals added and subtracted?
Let us learn the steps in adding and subtracting decimals by using the given data in
the problem.
Let us add P56.60, P48, P48, P96.75 and P6
To add decimals:
Step 1: Arrange the decimal points of the numbers being added in a column. This way, we
are adding the digits of the same place value.
Hundreds
Tens
5
4
4
9
Ones
6
8
8
6
6
.
.
.
.
.
Tenths
5
Hundredths
0
7
5
Step 2: Extend numbers with zeroes if necessary to line them up. Add 0s to the beginning
and/or end of the numbers so they have the same nuber of digits. This doesn’t change the
value of the number.
Hundreds
Tens
5
4
4
9
0
Ones
6
8
8
6
6
.
.
.
.
.
Tenths
5
0
0
7
0
Hundredths
0
0
0
5
0
Step 3. Write the decimal point on the answer line, directly below the decimal points lined up
in the decimal numbers being added.
Hundreds
Tens
5
4
4
9
0
Ones
6
8
8
6
6
.
.
.
.
.
.
Tenths
5
0
0
7
0
Hundredths
0
0
0
5
0
Step 4. Add up the numbers in the colum starting farthest to the right, moving to the left, and
write the answer on the answer line, directly beneath them.
Hundreds
2
Tens
5
4
4
9
0
5
Ones
6
8
8
6
6
5
.
.
.
.
.
.
Tenths
5
0
0
7
0
2
Hundredths
0
0
0
5
0
5
The sum we gor is 255.25. it means that the total amount of the purchases of Mrs.
Rivera is P255.25
Now, to determine the amount of her change, we subtract P255.25 fro her money
which is P1000.00
Subtraction of decimals is very much like subtraction of whole numbers, only that, like
in addition of decimals, we arrange the numbers in column base on the decimal point.
Step 1: Arrange the decimal points of the numbers to be subtracte in a column. This way,
we are subtracting the digits of the same place value.
Thousands Hundreds
Tens
Ones
Tenths
1
0
0
0
.
0
2
5
5
.
2
Step 2: Extend numbers with zeroes if necessary to line them up.
Hundredths
50
Thousands
1
0
Hundreds
0
2
Tens
0
5
Ones
0
5
.
.
Tenths
0
2
Hundredths
0
5
Step 3: Write the decimal point on the answer line, directly below the decimal pints lined up
in the numbers being subtracted.
Thousands
1
0
Hundreds
0
2
Tens
0
5
Ones
0
5
.
.
.
Tenths
0
2
Hundredths
0
5
Step 4: This time,it’s just like doing subtraction of whole numbers. Subtract the numbers in the
column starting farthest to the right, moving to the left, and write the answer on the answer
line, directly beneath them.
Thousands
1
0
Hundreds
0
2
7
Tens
0
5
4
Ones
0
5
4
.
.
.
Tenths
0
2
7
Hundredths
0
5
5
The difference is 744.75. Therefore, the amount of change Mrs. Rivera received was
P744.75
Let’s Wrap up

To add decimals, the following are the steps:
Step 1: Arrange the decimal points of the numbers being added in a column. This way, we
are adding the digits of the same place value.
Step 2: Extend numbers with zeroes if necessary to line them up. Add 0s to the beginning
and/or end of the numbers so they have the same nuber of digits. This doesn’t change the
value of the number.
Step 3. Write the decimal point on the answer line, directly below the decimal points lined
up in the decimal numbers being added.
Step 4. Add up the numbers in the colum starting farthest to the right, moving to t
he left, and write the answer on the answer line, directly beneath them.

To subtract decimals, ere are the steps:
Step 1: Arrange the decimal points of the numbers to be subtracte in a column. This way,
we are subtracting the digits of the same place value.
Step 2: Extend numbers with zeroes if necessary to line them up.
Step 3: Write the decimal point on the answer line, directly below the decimal pints lined
up in the numbers being subtracted.
Step 4: This time,it’s just like doing subtraction of whole numbers. Subtract the numbers in
the column starting farthest to the right, moving to the left, and write the answer on the
answer line, directly beneath them.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
LESSON 7
D
B
Multiplication of Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Multiply decimals and mixed with factors up to 2 decimal places.
 Multiply mentally decimals up to 2 decimals places by 0.1, 0.01, 10 and 100;
and
 Solve routine and non-routine problems involving multiplication of decimals
including money using appropriate problem-solving strategies and tools.
Lesson Opener
Danilo is thrice as heavy as Claud. If Claud weighs 9.75 kg, then what is the weight of
Danilo?
Let’s Focus
The word “thrice” in the problem above is used to signify three times or multiplication
by 3. It means that since Claud is 9.75 kg heavy, then the weight of Danilo is 9.75 x 3.
Let’s solve the problem by following the steps on how to multiply decimals.
Step 1: Line up the numbers on top of each other above the multiplication bar.
9
7
X
5
3
Step 2: Multiply the numbers as you usually would in ordinary multiplication, just ignore the
decimal point first.
9
7
X
2
9
5
3
2
5
Step 3: Count up all of the decimal places or the number of digits that are located at the
right side of the decimal points of the factors. In 9.75, there are 2 digits-7 and 5, while in 3,
therer is none for a total of 2. The same number f digits must be to the right of the decimal
point in the product.
9
7
X
2
9
5
3
2
5
The product of 9.75 and 3 is 29.25. Therefore, Danilo weighs 29.25 kg.
Apply what you have just learned. Give the product of 0.238 and 0.35.
Solution:
0.0
0
2
3
8
X
0
3
5
1
1
9
0
7
1
4_______
8
3
3
0
There are 5 decimal places in all in the factors
So there must be 5 also in the product.
The product of 0.238 and 0.35 is 0.0833.
Let’s Wrap up

To multiply decimals, multiply the numbers as you usually would in
ordinary multiplication, place the decimal point in the product so that
its number of decimal places corresponds with the number of decimal
places in the factors.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 4: Decimals
LESSON 8
D
B
Division of Decimals
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Divide decimals/mixed decimals up to 2 decimal places.
 Divide decimals up to 4 decimal places by 0.1, 0.01 and 0.001
 Divide decimals up to 2 decimal places by 10, 100, and 1000 mentally, and
 Solve routine and non-routine problems involving division without or with any
of the other operations of decimals and whole numbers including money using
appropriate problem-solving strategies and tools.
Lesson Opener
Donessa has P32. She wants to buy bread that costs p2.50 each. How many pieces
can she buy from her money.
Let’s Focus
We can find out how many pieces of bread Donessa bought by dividing 32 by 2.50
You already learned how to divide whole numbers. Division of decimals is very much
similar with division of whole numbers except for the placement of the decimal points which
we will learn first.
To make division of decimals easy, we may first make the decimal divisor a whole
number by multiplying it by a power of 10, 100, 1 000 and so on. However, if we multiply the
divisor by a power of 10, we also need to multiply the dividend by the same number.
Let’s have some practice.
1. 350 ÷ 0.8 To make the divisor, which is 0.8, a whole number, multiply it by 10 so it
becomes 8, also multiply 350 by 10 which makes it 3 500.
So, the new numbers to be divided are 3 500 and 8
2.4 ÷ 0.004 Multiply both numbers by 1000 to make 0.004 equal to 4.
The new numbers are 4 000 ÷ 4.
Let’s Try This:
Give the new dividend and divisor.
1. 1.2 ÷ 0.05
2. 0.5 ÷ 1.2
3. 1 000 ÷ 0.009
Let’s Check:
1. 1.2 ÷ 0.05
Multiply by 100 to make 0.05 a whole number.
2. 0.5 ÷ 1.2
Multiply by 10 to make 1.2 equal to 12
3. 1 000 ÷ 0.009
Multiply by 1 000 to make 0.009 as 9
Let’s Wrap up

To divide decimal numbers:
1. make the divisor a whole number by multiplying it by a power of 10; use the
same power of ten to multiply to the divided;
2. divide the way you do with whole numbers;
3. place the decimal point in the quotient right above the one in the dividend;
and
4.if there is still a remainder, zero(s) may be affixed at the right of the decimal
point of the dividend. Division may be continued until completed or until
desired number of decimal places in the quotient is attained in the case of
non-terminating decimals.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 5: Ratio and Proportion
LESSON 1
D
B
Ratio
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Visualize the ratio of 2 given numbers;
 Express ration using either the colon (:) or fraction;
 Define and illustrate the meaning of ratio and proportion using concrete or
pictorial models; and
 Solve problems involving ratio
Lesson Opener
Jay bought 3 ballpens and 5 pencils in a bookstore. What is the ratio of the number of
ballpens to the number of pencils?
Let’s Focus
The ratio of ballpens to pencils is 3:5. It is read as “3 to 5”.
Ratio can be expressed in 3 ways
Form
Word Form
Colon Form
Fraction Form
Symbol
3 to 5
3:5
3
5
The ratio of cupes to saucers is 3:2
The ratio of saucers to cups is 2:3
The ratio of cups to the total number of objects is 3:5
What conclusion can you formulate from these statements?
Let’s Wrap up

To multiply decimals, multiply the numbers as you usually would in
ordinary multiplication, place the decimal point in the product so that
its number of decimal places corresponds with the number of decimal
places in the factors.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 5: Ratio and Proportion
LESSON 2
D
Equivalent
and Simplest Form
B
of Ratios
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Identify and write equivalent rations;
 Express ratios in their simplest forms;
 Find the missing term in a pair of equivalent rations; and
 Solve problems involving equivalent and simplest form of ratio.
Lesson Opener
Maureen has 4 books and 8 notebooks in her bag
Books
The ratio of
books to
notebooks
is 4:8
Notebooks
Let’s Focus
The ratio of
books to
notebooks
2:4
The ratio of
books to
notebooks
1:2
The ratio of books to notebooks is 1:2
4:8. 2:4 and 1:2 are equivalent ratios, 1:2 is written in the simplest form.
Let’s Wrap up


Equivalent ratios are ratios that are in proportion. They are formed by
multiplying or dividing all their terms by the same number.
A ration is in its simplest form if the common factor of all its terns is 1. It is
obtained by dividing the terms by their GCF.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 5: Ratio and Proportion
LESSON 3
D
B
Kinds of Proportion
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Find a missing term in a proportion (direct, inverse and partitive); and
 Solve problems involving direct proportion, partitive proportion, and inverse
proportion in different contexts such as distance, rate and time using
appropriate strategies and tools.
Lesson Opener
A recipe calls for 3 cups of milk foe every 5 cups of flour. How many cups of milk are
needed for the same recipe which has 20 cups of flour?
Let’s Focus
Direct Proportion
Every 3 cups of milk requires 5 cups of flour can be illustrated as follows.
If there are 20 cups of flour, there will be 12 cups of milk.
Alternative Solution
Partitive Proportion
Three siblings, aldrin, Benjie and Cherry, will divide P 1 000 in the ration of 2;3:5
respectively. How much eill Cherry get?
Solution:
Aldrin
Benjie
Cherry
10 units = P 1000
1 unit
= P 100
5 units = P100 x 5 = P500
Therefore, Cherry will receive P500
Alternative Solution:
2 + 3 + 5 = 10
P1 000 ÷ 10 = P100
P100 x 5 = P500
Inverse Proportion
It took 20 minutes for three pipes to fill a pool. How long will it take 4 pipes to fill the
same pool?
No. of Pipes
Time
3
20
4
𝑁
3
4
=
𝑁
20
4𝑁 = 60
𝑁 = 15
Therefore, it will take 15 minutes for 4 pipes to fill the pool.
Let’s Wrap up



Two quantities are in direct proportion if an increase/decrease in one
quantity results to increase/decrease in another quantity.
Partitive proportion is used if a whole is divided into two or more
unequal parts.
In an inverse proportion, as the value of one quantity increases the
other decreases.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 6: Percent
LESSON 1
D
B
Visualization of Percent
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Visualize percent, and its relationship to fractions, ratios and decimal numbers
using models; and
 Solve word problems involving visualization of percent.
Lesson Opener
In the recent elections for the University Student Council chairperson, the candidates
garnered the following percent of the total number of votes cast.
Nikki De Guzam
– 30%
Vincent Lamis
- 35%
Jeriel Mahinay
- 20%
Mark Lim
- 15%
If 1500 students cast their votes, then how many students voted for the winner?
Let’s Focus
The word percent comes from the Latin per centum, which means per hundred or
for every one hundred. It is one of the most common rations, second term of which is 100.
The percent symbol “%” has a value of
1
100
For example, 5% is equivalent to 5
fraction
5
100
x
1
100
𝑜𝑟
5
100
. Its equivalent in ratio is 5:100. The
is read as 5 hundredth which when written in decimal is 0.05
It means that 5% may be written as:
1. Fraction:
5
100
𝑜𝑟
1
20
2. Decimal: 0.05
3. Ratio : 5:100 or 1:20
We will discuss in detailes and give example on how percent is converte to decimal,
fraction and ratio and vice versa.
Meanwhile, let’s take a look at the 10 x 10 grid and see how 5% is represented.
The grid shows a total of 100 1 x 1 squares. Out of these, 5 squares are shaded. We
can, then, say that 5 out or 100 or 5% of the big square is shaded.
Converting Percent to Fraction
To convert a percent to a fraction, multiply the number of percent to the value of
the percent sig which is
1
100
then express in simplest form.
Express each percent of the total votes that each candidate received as a fractiion.
Candidate
Nikki De Guzam
Percent
30%
Fraction
Vincent Lamis
35%
35
7
𝑜𝑟
100
25
Jeriel Mahinay
20%
20
1
𝑜𝑟
100
5
Mark Lim
15%
15
3
𝑜𝑟
100
20
3
10
Converting Fraction to Percent
One method to convert a fraction to percent is to first express it as a decimal using
the following steps.
1. Divide the numerator by the denominator.
2. Multiply by 100.
3. Affix the percent sign.
Example: Express
7
8
in percent.
Step 1. 7 ÷ 8 = 0.875
Step 2. 0.875 𝑥 100 = 87.5
Step 3. 87.5%
Converting Percent to Decimal
To convert percent to decimal, divide the number by 100 and drop the percent
symbol. For instance, 10% is equal to
10
100
which is 0.10 or 0.1 in decimal. In our lesson on
decimals, the short cut in dividing by 100 is to simply move the decimal point two places to
the left.
Convert to decimal the percent of votes garnered by each candidate.
Candidate
Percent
Decimal
Nikki De Guzam
30%
0.3
Vincent Lamis
35%
0.35
Jeriel Mahinay
20%
0.2
Mark Lim
15%
0.15
Converting Decimal to Percent
To express a percent as decimal, we divide by 100. To express a decimal as percent,
we therefore do the inverse operation which is multiplication by 100.
For example, 0.8 is equal to 0.8 x 100 or 80%; 0.125 x 100 or 12.5%
Converting Percent to Ratio
To express a percent as ratio, the number of percent becomes the first term of the
ration while the second term is always 100. It means that we may also first find the equivalent
in simple fraction, the express it as a ratio-that is the numerator becomes the first term of the
ration and the denominator becomes the second term.
Let us convert to ratio the percent of votes garnered by each candidate.
Candidate
Nikki De Guzam
Percent
30%
Ratio
30: 100 𝑜𝑟 3: 10
Vincent Lamis
35%
35: 100 𝑜𝑟 7: 20
Jeriel Mahinay
20%
20: 100 𝑜𝑟 1: 5
Mark Lim
15%
15: 100 𝑜𝑟 3; 20
Converting a Ratio to Percent
A ratio may be expressed as percent by first converting it to a decimal, then
multiplying the result by 100.
For instance, to express 4:5 as percent.
4 ÷ 5 = 0.8 𝑎𝑛𝑑
0.8 × 100 = 80
So 4:5 is equal to 80%
To summaruze the percent of votes received by the candidates, we have:
Percent
Decimal
Fraction
Ratio
30%
0.3
3: 10
35%
0.35
20%
0.2
15%
0.15
3
10
7
20
1
5
3
20
7: 20
1: 5
3; 20
Let’s now answer the Lesson Opener. The winner is Vincent Lamis who garnered 35%
of the total votes which is 1 500. To find the number of votes he received, we may use the
decimal, fraction or ratio form of 35%. To use the fraction form, we have:
7
𝑥 1500 = 525
20
Therefore, Vincent Lamis who garnered 525.
Let’s Wrap up

The word percent comes from the Latin per centum, which means per
hundred or for every one hundred. The percent symbol “%” has a value
of 1/100
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 6: Percent
LESSON 2
D
Percentage, Base and Rate
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Define percentage, rate or percent and base.
 Identify the base, percentage, and rate in a problem; and
 Solve routine and non-routine problems involving finding the percentage, rate
and base using appropriate strategies and tools.
Lesson Opener
A test has 20 questions. If Divina gets 80% correct, how many questions did she miss?
Let’s Focus
We will start by translating the bove problem into a number sentence.
Let n + the number of items missed.
Since the problem is asking if how many items were missed, then instead of using 80%,
we use 20% because if 80% of the test was answered correctly, then 100% - 80% or 20% was
answered incorrectly or was missed.
So 20% of 20 = n or 20% x 20 = n
In the number sentence, 20 is the total number of items in the test or it is the perfect
score. It is equal to a 100%. It is called base (b). On the other hand, the answer which is still
unkown and is represented by n, is called the percentage. It is a part of the base. In our
problems, it is 20% of the base. Finally, 20% is called the rate (r). Rate is the ratio between the
percentage and the base.
Rate
20% x 20 = n
percentage
Base
From this formula, we can then derive the formula finding the base and the rate, which
are as follows:
If percentage (p) = rate (r) x base (b), then
𝑝
r=
b=
𝑏
𝑝
𝑟
To solve our problem, we use the first formula since we are solving for the percentage.
𝑝=𝑟 ×𝑏
= 20% × 20
= 0.2 × 20
=4
It means that Divina missed 4 items.
Let’s Try this:
What percent of 56 is 21?
Let’s Check:
Number Sentence: 𝑛% × 56 = 21
Given Facts: 𝑏 = 56 𝑎𝑛𝑑 𝑝 = 21
Unknown: rate
Formula: 𝑟
=
𝑝
𝑏
21
= 56
=
0.375 𝑜𝑟 37.5%
Let’s try another problem:
Teacher Shiela Mae has 24 boys in her class. She said that the boys comprise
60% of the class. How many pupils does Teacer Shiela Mae have?
Solution:
If n – total number of pupils in the class, then the number sentence would be:
60% × 𝑛 = 24
Given facts: 𝑟 − 60% 𝑜𝑟 0.6;
𝑝 = 24
Unknown: rate
Formula: 𝑏
=
=
𝑝
𝑟
24
0.6
= 40
Answer: Teacher Shiela Mae has 40 pupils in her class.
Let’s Wrap up



In percentage problems, the base is equal to 100%.
The percentage is a part of the base
Rate is the ratio between the percentage and base.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 6: Percent
LESSON 3
Discount,
Simple Interest and
D
Commission
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Calculate discount, simple interest and commission; and
 Solve routine and non-routine problems involving finding the discount, simple
interest and commission.
Lesson Opener
The marked price of a rice cooker is P1 250. The shopkeeper offers an off-season
discount offers an off season discount of 20% on it. Find its selling price.
Let’s Focus
Discount Formula
In this lesson, three important applications of percent will be discussed. These are
discount, simple interest and commission. If we can relate these three with percentage
formula, the rest would be easy.
Let us discuss each one by starting with the discount formula.
Discount or mark down is the amount deducted from the original price.
The table below shows the given facts in the Lesson Opener and how they are used
in the problem. It also presents the euivalent of each term in Percentage Formula and their
corresponding formula.
Given
Information
P 1, 250
20%
20% × 1250
Description
Marked price (MP)/Original
Price (OP)/ List Price (LP)
Ratio
between
the
percentage and the base
Rate of Discount
Discount
Formula
𝑂𝑃 =
𝑅=
𝐷
𝑅
𝐷
𝑂𝑃
Percentage
Percentage
𝑃 = 𝑅 × 𝑂𝑃
Discount Rate multiplied by
the base
Equivalent in
Percentage Formula
Base
80% x 1250
Net or Selling Price
NP = OP – D
Amount to be paid
𝑁𝑒𝑡 𝑃𝑟𝑖𝑐𝑒 (𝑁𝑃)
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑 𝑃𝑟𝑖𝑐𝑒
Solution:
Find the amount of discount.
D = R x OP
= 0.2 x 1 250
= P250
Subtract the amount of discount from the original price
NP = OP – D
= 1 250 – 250
= P 1000
The selling or net price of the rice cooker is P1 000.
Another way of solving this problem, is of cours, by multiplying 80% by 1 250 which will
give an answer which is already the net price.
To derive the formulla for Original Price and Rate of Discount, we may just relate them
with their equivalent in the Percentage Formula. Please refer to the traingle below.
Percentage Formula
Discount Formula
P
D
RxB
R x OP
D = R x OP
R = D ÷ OP
OP = D ÷ R
Simple Inteterest
Problem: Beverly deposited an amount of P25 000 in a bank that offers a simple interest
rate of 10% per annum. How much will her total money be in 18 months?
Interest is an amount charge for the use of money. The amount of interest depends on
how big the principal and rate of interest are and the length of time by which the amount
stays in the bank or is paid by the borrower. Again, ler us present the given facts in tabular
form so we can see the equivalent of each term in Percentage and Discount Formula.
Given
Information
P 25 000
Description
10% per annum
18 months or 1.5 y
Unknown
P 25 000 + Interest
Formula
Amount deposited, invested,
loaned
100%
Rate of Interest ( R ) for every
year
Length of time which the
money is borrowed or
loaned
Amount paid or changed for
the use of money over a
certain period of time
Principal + Interest
Percentage Formula
Term
𝑃=
𝐼
𝑅 ∗𝑇
𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝐵𝑎𝑠𝑒
𝑅=
𝐼
𝑃 ∗𝑇
Rate
𝑇=
𝐼
𝑃 ∗𝑅
Time
I = P*R*T
Interest
A=P+I
A = P + (P*R*T)
Total Amount
Interest Formula
P
I
RxB
P× 𝑅 × 𝑇
I=PxRxT
𝐼
𝑃=
𝑅 ×𝑇
𝐼
𝑅=
𝑃 ×𝑇
𝑇=
Let us now present the solution to our problem
Solution: I = P x R x T
𝐼
𝑃 ×𝑅
= 25 000 x 0.5 x 1.5 (convert 18 months to 1.5 years since the rate is er
annum/year)
= 3 750
The interest for 18 months is P 3 750. We add this to the principal to find the total
amount.
A=P+I
= 25 000 + 3 750
= 28 750
Beverly’s total money in the bank after 18 months is P 28, 750.
Commisision
Problem: Mrs Tamon, a real estate agent, sold a house and lot for P 1 750 000 and
received a 5% commission from the sale. How much was her commission?
Commission is paid to a person for the services rendered. In our problem, it is an amount
of money paid to an agent for selling the house and lot. Commission is generally a
percentage of sales. If the agent has a regular salary, the commission is added to that to
get the total take home pay.
Let’s now present the given facts and their corresponding formula and description in a
table.
Given
Information
5% of 1 750
P 1 750 000
5%
Description
Formula
Commission
Gross Proceeds (GP) or Total
Sales (TS)
- 100% of the sales
Rate of Commission (R)
𝐶 = 𝑅 × 𝐺𝑃
Take Home Pay (T)
Percentage Formula
𝐶
𝑅
Base
𝐶
𝐺𝑃
Rate
𝐺𝑃 =
𝑅=
Equivalent in
Percentage Formula
Percentage
T
=
Regular
Salary
+
Commission
Commission Formula
P
C
RxB
R x GP
C = R x GP
𝐶
𝐺𝑃 =
𝑅
𝐶
𝑅=
𝐺𝑃
Solution
C = R x GP
= 0.05 × 1 750 000
= 87 500
It means that Mrs. Tamon gets a commission of P87 500 for the sale of the
house and lot.
Let’s Wrap up








Discount is the amount deducted from the original price.
Original Price is the price of an item before a discount is deducted or
before the mark-up is added.
Net Price is the selling price of an item.
Interest is the amount earned for the use of money over a certain
period of time.
Principal is the amount deposited or loaned.
Amount is the result after the interest is added to the principal
Commission is the amount paid for the service rendered.
Gross Proceeds or Total Sales is the total amount earned for selling
products or services.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 7: Integers
LESSON 1
D
Concepts of Integers
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Identify real-life situations that make use of integers.
 Describe the set of integers; and
 Solve problems involving concepts of integers.
Lesson Opener
Alex is now 1 kilogram lighter than last week. How do you represeent Alex weight drop
as an integer?
Let’s Focus
1-kilogram lighter means 1 kilogram less than last week. We can represent this weight
drop as -1.
Integers are part of the number system. They are a combination of natural numbers or
counting numbers (1, 2, 3, …) and their opposites ( -1, -2 -3, …) including zero (0). We can
visualize this further with the aid of a number line.
Zero is neutral or has no value because it is neither positive nor negative. To the right of
0 are positive integers. Positive integers can be denoted by a plus sigh (+). However, if the
sign of a number is not indicated, it is understood that we are referring to a positive integers.
The farther you are to the right of zero, the greater the value of the number. Meanwhile,
negative numbers are found at the left of zero. They are denoted by a minus sign (-). The
farther you are to the left from zero, the lesser is the value of the number.
Let’s Wrap up

Integers are composed of positive and negative whole numbers
including zero.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 7: Integers
LESSON 2
Comparing and Arranging
Integers
D
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Compare and arrange integers with other numbers such as whole numbers,
fractions, and decimals;
 Represent integers on the number line and
 Solve problems involving comparing and arranging integers.
Lesson Opener
Here is a table showing the recorded coldest temperatures of a certain city in 6
months.
Month
Temperature
1
2°C
2
-1°C
3
-2°C
4
5°C
5
4°C
6
6°C
Let’s Focus
How do you compare the temperature in month 1 with the temperature in month 3?
What about the temperature in month 5 and in month 6?
Arrange the months according to their temperature in ascending order.
One way to know the answer is by visualizing the integers in a number line.
Plotting the integers in a numer line gives us a clearer picture of the position of the
integers. We have already known that integers are composed of positive whole numbers
and their opposites including zero. Zero serves as our reference point. Positive integers are
found to the right of zero while negative integers are on the left of zero. The farther the
integer ot the right of zero, the greater is its value. Meanwhile, integers have lesser value if
they lie farther to the left of zero.
Comparing integers is also the same as comparing natural numbers. We use the signs
less than ( < ), greater than ( > ), or equal ( = ).
Based on the given table presented in the Lesson Opener, the temperature of month
1 is 2°C while the recorded temperature in month 3 is -2°C. this means that month 1 has a
warmer temperature than month 3. We can put them in symol as 2°C . -2°C.
Month 5 has colder temperature than month 6. In symbols, we have 4 < 6.
Since 2=we have already plotted the temperatures in the number line, we can easily
arrange them in ascending order. We have month 3, month 2, month 1, month 5, month 4
and month 6. Their numerical arrangement in ascending order are as follows: -2, -1, 2, 4, 5,
6.
Let’s Wrap up



In comparing integers, we use the symbols <, > or =.
The farther the integer to the right of zero, the greater is its value.
Integers have lesser value if they lie farther to the left of zero.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 7: Integers
LESSON 3
D
Adding and Subtracting Integers
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Describe the addition and subtraction of integers using materials such as
algebra tiles, counters, chips and cards;
 Perform the addition and subtraction of integers; and
 Solve word problems involving addition and subtraction of integers using
appropriate strategies and tools.
Lesson Opener
Yesterday, the temperature in a certain area was -4°C. Today, its temperature
increased by 5°C. what is today’s temperature?
Let’s Focus
We can put this the number sentence as -4 + 5 = n
T better understand how addition of integers work, we can use a number line to
visualize this.
We mark the first integer on the number line. In our example, we mark – 4.
Then, we add 5 to it. This means that we move 5 places to the right since 5 is a positive
integer.
The number line shows that ( -4 ) + 5 = 1
Can you find the sum of the following expressions using a number line?
1. 4 + -7 =
2. 3 + 2 +
3. -2 + -5 =
4. 3 + -3
We are adding integers with the same signs in examples 2 and 3. Take note that we
added the numbers and copied their sign.
In example 4, we added numbers with opposite signs and got zero as an answer.
Subtraction of Integers
When Neo took the height f the mongo seedlings, which he planted for his Science
experiment, it was 2 cm tall. During the second measurement, it wa 5 cm tall. How many
centimeters is the height increase of the plant?
Looking at the number line, how many steps did we move from 2 to 5? Taking away 2
from 5 is like adding 5 by the additive inverse of 2 which is -2. That is 5 + (-2) = 3.
Find the difference using the number line
1. (-3) – 1 =
In addition sentence, we can rewrite this as (-3) + (-1) =
2. (-4) – (-2) =
In addition sentence, we can rewrite this as (-4) + 2 =
Let’s Wrap up




To add integers with the same signs, add the numbers and copy the
sign.
To add integers with different signs, ignore the signs then subtract the
smaller number form the bigger numbers and copy the sign of the
bigger number.
The sum of two the same numbers having opposite signs is equal to
zero
To subtract an integer, add its additive inverse. E.g. x – y = x + (-y) or
3-2 = 3 + (2)
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 7: Integers
LESSON 4
D
Multiplying and Dividing Integers
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Describe and interpret the multiplication and division of integers using
materials such as algebra tiles, counters, chips and card.
 Perform the multiplication and division of integers; and
 Solve routine and non-routine problems involving multiplication and division of
integers using appropriate strategies and tools.
Lesson Opener
Study the examples below. Show how these number sentences work using algebra
tiles, chips, counters or cards.
3 x 4 = 12
12 ÷ 4 = 3
3 x -4 = -12
-12 ÷ 4 = -3
-3 x 4 = -12
-12 ÷ -4 = 3
-3 x – 4 = 12
12 ÷ -4 = -3
-3 x 0 = 0
0 ÷ -4 = 0
Let’s Focus
In visualizing multiplication and division of integers, we may use algebra tiles, chips,
counters or cards.
Chips
Tiles
1
-Positive integer
-1
-Negative integer
-Positive integer
-Positive integer
-Negative integer
-Negative integer
Multiplication of Integers
Multiplication is a repeated addition. The numbers that we multiply are called factors
while the answer in multiplication is called the product.
Observe that -3 x -4 = -( 3 x -4) = 12
Flip 3 groups of 4
Division of Integers
Recall that division is the inverse of multipliation. In a division sentence, the first number
is the dividend which tells us the total number of elements. The second number is the divisor
which may be referred to as number of groups or number of memebers in a group. In this
lesson, we refer the divisor as the number of groups. Take note that the number of groups
cannot be a negative number, we will flip the chips.
1. 12 ÷ 4 = 3 means “How many members are there in each group if 12 is divided into 4
groups?”
Therefore, each group has 3 members.
2. -12 ÷ 4 = 3 means “How many members are there in each group if -12 is divided into
f4 groups?
Therefore, each group has -3 members.
3. -12 ÷ -4 = 3 means “How many members are there in each group if -12 is divided
into -4 groups.
Since our second number is negative, we flip the tiles because we cannot have a
negative number of groups.
Therefore, each group has 3 members.
4. 12 ÷ -4 = -3 means “How many members are there in each group if 12 is divided into
-4 groups?
Since our second number is negative, we filip the tiles because we cannot have a
negative number of groups.
Let’s Wrap up

The product and quotient of same signed integers is a positive while
those of different integers is negative.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 8: Geometry
LESSON 1
Angles
D
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Illustrate the different angles (right, acute, and obtuse) using models
 Measure and construct angles;
 Find complement and supplement of an angle; and
 Solve word problems involving angles.
Lesson Opener
Study the objects below.
What do you notice in these figures?
Let’s Focus
Kinds of Angles
Sample figure
Kinds of Angles
Description
Acute Angle
Measures between 0° and
90°
Right Angle
Measures exactly 90%
Obtuse Angle
Mesures between 90° and
180°
Angle Construction and Measurement
A protractor is a tool which is used in
constructing and measuring an angle. Degree (°) is
the unit of measure for angles.
To measure an angle, align the vertex of the
angle to the center mark of the protractor and place
the 0° mark on one side of the angle and then read
the scale where the other side of the angle
intersects.
Center mark
Zero line or
baseline
A
C
B
Complementary and Supplementary Angles
Sample Figure
Pair of Angles
Description
Complementary Angles
Two angles which have sum
of 90°
Supplementary Angles
Two angles which have a
sum of 180°
Let’s Wrap up





Angles can be classified as acute (measure between 0° and 90°), right
angles (measures exactly 90°) or obtuse angle (measures between 90°
and 180°).
In a straight line, the sum of the angles is equal to 180°.
The sum of all angles at a point is equal to 360°.
Complementary angles are two angles which have a sum of 90°.
Supplementary angles are two angles whose sum is equal to 180°.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 8: Geometry
LESSON 2
Triangles
D
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Describe the attributes/properties of triangles using concrete objects or
models;
 Identify and describe triangle according to sides and angles; and
 Solve word problems involving triangles.
Lesson Opener
Study the figure and find out what is common among them
Let’s Focus
What is common among the figures above is their shape which is triangle. A triangle is
a plane figure made up of 3 sides and 3 angles.
Kinds of Triangles According to Sides
Sample Figure
Kinds of Triangles
Description
Scalene
No sides are equal.
Isosceles
At least two sides are equal
Equaliateral
All sides are equal.
Kinds of Triangles According to Angles
Sample Figure
Kinds of Triangles
Description
Acute
All angles are acute
Right
One angle measures 90°
Obtuse
One angle is obtuse
When all angles in a triangle are equal, it is called equiangular. In an equiangular
triangle, each of the interior angles measures 60°.
Angles in a Triangle
The sum of all the interior angles in a triangle is 180°
In ABC, m∠𝑎 + 𝑚 ∠𝑏 + 𝑚∠𝑐 = 180°
Let’s Wrap up




A triangle is a three-sided polygon.
Triangles can be classified according to sides (scalene, isosceles,
equilateral) and angles (acute, right, obtuse)
The total measure of the interior angles of a triangle is 180°.
The exterior angle of a triangle is equal to the sum of its two opposite
interior angles.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 8: Geometry
LESSON 3
D
Quadrilaterals
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Describe the attributes/properties of quadrilaterals using concrete objects or
models;
 Describe the different kinds of quadrilaterals. Square, rectangle,
parallelogram, trapezoid and rhombus; and
 Solve word problems involving quadrilaterals.
Lesson Opener
Study the diagram of the quadrilateral. Can you describe each of them? What
relationship among them can you form?
Let’s Focus
A quadrilateral is a polygon with four sides. “Quadrilateral” came from Latin words
which means a variant of four andlatus which means “side”.
Quadrilaterals have special types such as parallelogram, rectangle, rhombus, square,
kite and trapezoid.
In quadrilateral, the total measure of its interior angles is 360°. The sum of the interior
angles of a polygon can be solved using the formula (n – 2) x 180° where n refers to the
number of sides of a polygon.
Let’s Wrap up




A quadrilateral is a four-sided polygon.
Special types of quadrilateral include parallelogram, rectangle,
rhombus, square, kite and trapezoid.
The sum of the interior angles of a quadrilateral is 360°.
The formula in finding the total measure of interior angles in a polygon
is (n-2) x 180° where n is the number of sides.

Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 8: Geometry
LESSON 4
D
Circles
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Describe a circle;
 Identify the terms related to a circle;
 Draw circles with different radii using a compass; and
 Solve word problems involving circles.
Lesson Opener
Teacher A stood at the center of the classroom. She asked 10 learners to hold hands
around her maintaining an even distance from her as well as between their classmates. What
shape did the learners form?
Let’s Focus
The learners formed a circle just like the figure below.
A
A circle is a closed plane figure made up a set of points that are equidistant (same
distance) from a fixed point called the center. In this illustration, A is the center of the
circle.
Aside from the center, other parts of the circle includes radius, diameter and chord.
Basic Terms Related to Circle
Figure
D
C
Term
Description
Center
A point equidistant to any apoint on the circle. It
is named by its capital letter. In this example
A is the center of the circle.
Radius
A segment drawn from the center to any point
on the circle. It is half of the diameter. It is named
by two capital letters such as AB.
Chord
A segment joining on the circle. It is named by
two capital letters. In this example, DE is the
chord.
E
Diameter
B
Arc
Semi-circle
Circumference
A chord passing through the center of the circle.
It is the longest chord. It is twice the radius. It is
named by two capital letters. In this example CB
is the diameter.
A portion of the circumference of a circle AB is
an example of the arc. When an arc is less than
half of the rotation, it is called as the minor arc. If
it is more than half of the rotation, it is called as
the major arc
Half of the circumference of the circle. It is
measure 180°.
Distance around the circle
𝜋 (pi) is the ratio between the distance around the circle and the distance across the
circle. In other words, dividing the circumference of the circle by its diameter will result to 𝜋.
22
Its value is 7 or approximately equal to 3.14.
Constructing a Circle
A compass is an instrument which aids in construction a circle.
In constructing a circle follow these steps:
1. Draw a line representing the radius of the circle with the aid of a ruler on a piece
of paper.
2. Attach the pencil to the drawing compass.
3. point the needel of the compass on one end of the line and the pencil lead to the
other point. The point where the needle points to represents the center of the
circle.
4. rotate the compass withour moving the needle until it reaches a full turn.
Let’s Wrap up



A circle is a plane figure wherein all points are equidistant from the
center.
Parts of the circle include center, radius, diameter and chord.
Pi (𝜋) is the ration between the circumference and the diameter of the
circle.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 9: Measurement
LESSON 1
D
Speed, Distance, Time
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Calculate speed distance, and time; and
 Solve problems involving speed.
Lesson Opener
Alkie drives his car at a constant speed of 75kph. How far will he be from his starting
point after 3.5 hours?
Let’s Focus
Surely you have already been asked questions such as :”How far is your house from
school?” or “How long does it take you to arrive in school?”
Most of us, if not all, travel to and from school everyday. It means that we already have
at least an estiate of the distance between our house and school and the length of time it
takes to travel.
Distance, speed/rate and time are three important concepts in mathematics which
we usually encounter as word problems.
Distance is the length measured between two points or the length of space travelled
by a moving object. It is usually denoted by d in math problems. The most common unit being
used to measure distance is kilometers or meters. So one would usually say, my house is
around 5 km from school.
Time is the measured or measurable period during which a condition, process or action
exists or takes place. It is usually expressed in hours or minutes. On the other hand, speed or
rate is the speed at which an object or person travels. We usually say, the bus runs 60
kilometers per hour (kph).
For example, a car runds 60 kph. It means that in one hour, the car covers 60 km;
therefore, in 2 hours, it covers 60 x 2 or 120 km. This clearly shows that if the rate and time are
multiplied, the resul is the distance. That is,
Distance = Speed x Time
Speed =
Time =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑠𝑝𝑒𝑒𝑑
Knowing the formula in solving distance problems, we are now ready to solve our
Lesson Opener.
Given Facts:
S = 75 kph
t = 3.5 h
d=sxt
= 75 x 3.5
= 262.5 km
Simple problems like these may also be solved by using a drawing such as the one
below.
75 km
1h
75 km
75 km
1h
1h
75
2
km
0.5 h
It shows that to find the total distance the car travels at a speed of 75 kph in 3.5 h, we
may simply add the numbers inside the rectanges, that is:
d = 75 + 75 + 75 +
75
2
= 75 + 75 + 75 + 37.5
= 262.2 km
Problem:
Rhyne travels at a speed of 60 kph. At the same rate, how long will it take him to cover
a 240 km distance?
Solution:
Given Facts:
s = 60 kph
d = 240 km
Unknown time:
t=
𝑑
=
240
𝑠
60
=4h
It tkes Rhyne 4 h to cover a distance of 240 km.
Problem:
Mr. Argie Palomo left his apartment at 10:30 a.m. and arrived in Dumaguete City, 245
km away, at 2:00 p.m. What was his speed in kilometers per hour?
Solution:
Given Fracts:
Time of Arrival : 2:00 pm.
Time of Departure: 10;30 a.m.
D = 245 km
Unknown: speed
First, find our the time by subtracting Time of Departure from Time of Arrival.
Time Arrived
Time Departed
Time Elapsed
Hour
2
10
Minute
00
30
Hour
14
10
Minute
00
30
Hour
13
10
3
Minute
60
30
30
It took Mr. Palomo 3 h 30 min or 3.5 h to reach Dumaguete City. Mr. Palomo’s speed is
70kph.
s=
=
𝑑
𝑡
245
35
= 70 kph
Let’s Wrap up



Distance is the length measured between two points or the length of
space travelled by a moving object.
Time is the measured or measurable period during which a condition,
process or action exist, takes place or continues.
Speed or rate is the speed at which an object or person travels.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 9: Measurement
D
LESSON 2
Circumference
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Visualize circumference of a circle;
 Derive a formula in finding the circumference of a circle;
 Find the circumference of a circle; and
 Solve routine and non-routine problems involving circumference of a circle
Lesson Opener
A circular lagoon has a diameter of 56 m. if you walk around it times, how many
meters will that be?
Let’s Focus
The concept of circumference can be best be understood once you have a good
grasp of some terms related to it.
The ratio between the circumference and the diameter of a circle is called pi, a Greek
22
letter whose symbor is 𝜋. The value of pi is always 7 𝑜𝑟 3.14 … As we have discussed earlier, its
value is constant. It means that regardless of the size of the circle, the ratio between the
circumference and the diameter is the same.
Using pi as a ratio, we can now derive the formula for circumference and diameter. If
𝑐
𝑟 = 𝑑, then c = 𝜋𝑑 𝑎𝑛𝑑 𝑑 =
𝑐
𝜋
Using the formula for circumference which is 𝜋𝑑, we can now solve the Lesson Opener.
Given Facts:
d = 56
Unknown: circumference (times 5)
Solution:
C = 𝜋𝑑
=
22
7
x 56
= 176
The distance around the circular lagood is 176 m. Since you will walk around it 5 times,
then multiply 176 by 5.
= 176 x 5
= 880
Answer: 880 m
Let’s Wrap up


Circumference is the distance around a circle.
Formula for Circumference: 2𝜋 𝑜𝑟 𝜋𝑑
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
, if an increa
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 9: Measurement
LESSON 3
D
Surface Area
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Visualize and describe surface area and names the unit of measure used for
measuring the surface area of solid/space figures
 Derive a formula for finding the surface area of cubes, prisms, pyramids,
cylinders, cones and spheres;
 Find the surface area of cubes, prisms, pyramids, cylinders, cones and spheres;
and
 Solve word problems involving measurement of surface area
Lesson Opener
A closed sewing box, 28 cm by 6 cm, is to be completely covered. What area is to be
covered?
Let’s Focus
The surface area of a solid object is a measure of the total area that the surface of the
object occupies. So the first step to do in the study of surface area is to visualize a threedimensional idea in a two-dimensional drawing. To get there, you nedd to see surface area
in the 3-D realm and then transfer everything to paper. It might be a bit difficult at first, but
with a lot of practice it will sure becoe easier eventually.
Another way to make the study of surface area more concrete is to visualize or draw
the net of the solid figures. Let us have some examples:
Net of Solids
Cube
Rectangular Prism
Square Pyramid
Cylinder
Cone
Sphere
Surface Area of a Cube
It is easy to visualize the surface are of a cube since we already know that it has six
square faces which are all congrurent. It means that we simply solve the area of one face
and multiply the result by 6.
Surface Area of a Cube = 6s2
6 cm
6 cm
If the side or edge of the cube above is 4 cm, then its surface area (SA) is:
SA of a Cube = 6s2
= 6 x 42
= 96
Surface Area of a Rectangular Prism
To find the surface area of a rectangular prism, we need the measure of the length,
width and height.
4 cm
3 cm
6 cm
If we took a look at the net of the rectangular prism, we see three parts of faces:

Front and back faces - 2 (length x height) or 2/h


Right and left faces – 2 (width x height) or 2w/h
Top and bottom face – 2 (length x width) or 2/w
To sum up, the surface area of a rectangular prism is:
SA = front & back faces + right & left faces + top & bottom faces
= 2/h + 2wh + 2/w or 2( lw + wh + lw)
Using the formula, the surface are of the rectangular prism above, then
Sa = 2lh + 2wh + 2lw
= (2 x 6 x 4) + (2 x 3 x 4) + (2 x 6 x 3)
= 48 + 24 + 36
= 108 cm2
Surface Area of a Square Pyramid
To find the surface area of a square pyramid, we only need the measure of the side of
the square base and the height of the triangular lateral face (not the height of the pyramid)
h
l
8 mm
12 mm
Surface Area of a Square Pyramid
= area of the square base + are of the 4 triangular lateral faces
1
SA = s2 + 4( 𝑏ℎ)
2
1
= 82 + 4(2 𝑥 8 𝑥 12)
= 64 + 192
= 256 mm2
Surface Area of a Cylinder
Looking at the net of a cylinder, it is easy for us to see its surface area which is
compose of the 2 circular bases + the curved lateral side.
What we need, therefore, are the following:



Height of the cylinder which becomes the width or height of the rectangular
lateral side;
Radius of the circular base, and
Circumference of the circular base which is the length of the rectangular
lateral side.
h
h = 20m
r = 14 m
SA of the cylinder = 2 circular bases + the curved lateral side
= 2𝜋𝑟2 + (circumference height)
= 2𝜋𝑟2 + 2𝜋𝑟 x h
= 2𝜋𝑟(r + h)
Using DPMA
The SA of the cylinder above is:
= 2𝜋𝑟(r + h)
=2x
22
7
× (14 + 20)
= 88(34)
= 2 992 m2
Surface Area of a Cone
The surface area of a cone is the sum of the lateral surface are and the circular base.
If you know the radius of the base and the slant height of the cone, you can easily find the
total surface area using a standard formula.
Slant h = 15 mm
r= 8 mm
Slant h
SA of a Cone = lateral area + circular base
If the giveen data are:
 Slant height (x) – the diagonal distance from the top vertex of the cone to the
edge of the base (Not the height of the cone which is perpendicular to the
base), and
 Radius of the circular base ((𝜋)(𝑟)(𝑠)
SA of a Cone = lateral area + circular base
= 𝜋𝑟 (𝑠 + 𝑟)
= 3.14 x 8 (15 + 8)
= 577.76 mm2
using DPMA
Sometimes, however, you might have the radius and some other measurement, such
as the height or volume of the cone. In these instances, you can use the Pythagoren Theorem
and the volume formula to derive the slant height, and thus the surface area of the cone.
Surface Area of a Sphere
The surface area of a sphere is the number of units (cm, inches, feet – whatever your
measurement) that cover the outside of a spherical object. The formular was discovered by
the Greek philosopher and mathematician Aristotle thousand years ago.
r
SA of a Sphere = 4𝜋𝑟2
Problem:
Given a radius of 4.5 dm, find surface area of the sphere.
Solution:
SA = 4𝜋𝑟2
= 4 x 3.14 x 4.52
= 254.34 dm2
Let us now solve the problem given in the Lesson Opener. The solid figure being referred
to is a rectangular prism. Therefore, the formula to find its surface are is SA = 2lh + 2wh + 2lw.
Solution:
SA = 2lh + 2wh + 2lw or 2 (lh + wh+ lw)
= 2 (28 x 6 + 20 x 6 + 28 x 20)
= 2 ( 168 + 120+ 560)
= 2 (848)
= 1 696 cm2
The total surface area of the sewing kit to be covered is 1 696 cm 2
Let’s Wrap up






SA of a Cube = 6s2
SA of a Rectangular Prism = 2lh + 2wh + 2lw
1
SA of a Square Pyramid = s2 + 4(2 𝑏ℎ)
SA of a Cylinder = 2𝜋𝑟(r + h)
SA of a Cone = 𝜋𝑟 (𝑠 + 𝑟)
SA of a Sphere = 4𝜋𝑟2
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 9: Measurement
LESSON 3
D
Volume
Lesson Outcomes
At the end of the lesson, the student should be able to:
 Visualize the volume of solid figures in different situation using non-standard
(e.g. marbles, etc.) and standard units
 Find the volume of a given solid;
 Determine the relationship of the volume between
 A rectangular prism and a pyramid
 A cylinder and a cone
 And a cylinder and sphere; and
 Solve routine and non-routine problems involving volumes of solids.
Lesson Opener
How many cubes of side 2 cm can be made from a block of wood 28 cm by 20 cm
by 14 cm?
Let’s Focus
Volume is the amount of space taken up by a three dimensional object. You can
measure the vollume of solid figures using non-standard or standard units. For instance, you
can measure the volume of a marble in different ways. One is by direct measurement of the
diameter and using a formula. Amother is by water displacement method. The latter may be
more appropriate if you want to find the volume of an irregularly-shaped object or a large
number of marbles at once.
Finding Volume Using Water Displacement Method
By dropping the object into a measuring container of water, where the volume of the
water is known, the object’s volume can then be calculated by subtracting the volume of
the water from the volume of the water and object combined.
For example, the water level in a container is initially at 100 ml. after a marble is
dropped into the container, the water level rises to 150 mL. When you subtract 100 mL from
150 mL, the difference is 50 mL which is obviously the volume of the marble.
Volume of a Rectangular Prism
The figure at the bottom shows a rectangular prism with a length of 6 units, width of 4
units and the height of 5 units and a cube which is 1 unit x 1 unit x 1 unit or 1 cubic unit. The
volume of the prism is the number of 1 x 1 x 1 cubes that it contains.
Let’sfind the volume of the prism by counting the unit cubes starting from the bottom
layer:
Length – cubes
Width – cubes
It means, there are 4 rows of 6 cubes or 24 cubes in the first layer from the bottom. It
follows that there are also 24 cubes in each of the 2nd, 3rd, 4th and 5th layers. To sum up,
therer are 24 x 5 or 120 cubes in all.
So to find the volume of a rectangular prism:
1. Find the area of the base: length x width;
2. multiply the area of the base by the height.
Volume of a Rectangular Prism = lwh
Volume of a Cube
In the figure below, how many cubic ( 1 x 1 x 1 smaller cubes) are there in the bigger
cube?
The edge of the bigger cube is 3 units. It means, the base is 3 units by 3 units and the
height is also 3 units. In the first layer at the bottom, there are 3 rows of 3 cubes each or 9
cubes. Since there are 3 layers, then there is a total of 9 x 3 or 27 smaller cubes in the bigger
cube.
The formula in finding the volume of a cube is very closely similar with that of the
rectangular prism. In fact, it is easier to memorize since the dimensions of a cube are the
same-that is-all edges are congruent.
Volume of a Cube = edge x edge x edge or e3
Relationship between the Volume of Rectangular Prism and Pyramid
The pictures below show the relationship between the volume of a rectangular prism
with that of a pyramid with the same base and height.
1
It shows that if the pyramid is filled with water and the water is poured into the prism, 3
of the prism is filled. It means that this process is to be done three times before the prism is
filled to the brim.
1
Therefore, we can say that the volume of a pyramid is that of a prism with the same
3
base and height.
The base of a rectangular pyramid is of course a rectangle. So to find its area, multiply
the length and the width. If you multiply the result by the height and then divide by 3, you ge
the volume of the pyraid.
Since the formula for finding the volume of a prism is lwh , therefore, the formula in
finding the volume of a pyramid is
1
Volume of a Rectangular Pyramid:
1
lwh.
3
3
(length x width x height) or
𝑙𝑤ℎ
3
Example:
The dimensions of a rectangular prism are 48 mm, 32 mm and 20 mm. What is the
volume of a pyramid with the same base and height?
Solution:
Volume of the pyramid =
=
1
3
1
3
=
1
=
1
3
3
(volume of the prism)
( 𝑙𝑤ℎ)
(48 x 32 x 20)
(30 720)
= 10 240 mm3
Relationship between the Volume of Cylinder and Cone
The volume of a cylinder is related with that of a cone is similar with that of a prism and
a pyramid. It means that the volume of a cone is
and heights are equal.
1
3
the volume of a cyllinder. It their bases
The figures above clearly show that the volume of a cone in on-third that of a cylinder
with the same base and height.
Volume of a Cylinder = Area of the circular base x Height
= 𝜋𝑟 2 ℎ
Volume of a Cone
=
1
3
𝜋𝑟2 ℎ
Example:
The radius of the circular base of a cylindrical container is 28 cm and its height is 30
cm. What is its volume? What is the volume of a cone with the same base and height as the
container?
Solution:
Volume of a Cylinder = 𝜋𝑟 2 ℎ
=
22
7
× 282 × 30
= 73 920 cm3
Volume of a Cone =
=
=
1
3
(Volume of a Cylinder)
𝜋𝑟 2 ℎ
3
73 720
3
= 24 640 cm2
Relationship between the Volume of Cylinder and Sphere
In the figure below, the radius of the sphere is equal to the radius of the base of the
cylinder becomes
a cylinder.
2
3
full. It shows, then that the volume of a sphere is
2
3
that of the volume of
Volume of a Cylinder = Area of the circular Base x Height = 𝜋𝑟 2 ℎ
Volume of a Sphere =
2
3
𝜋𝑟2 ℎ or
2𝜋𝑟2 ℎ
3
Example:
The radius of s sphere is 2.5 dm. what is its volume? What is the volume of the
cylinder whose base has a radius of 2.5 dm and whose height is 5 dm?
Solution:
In the problem, the radius of the cylinder and sphere are equal. Also the height of
the cylinder is twice the radius of the sphere. Therefore, you can already tell that the
volume of the sphere is
first.
2
3
that of the cylinder. Hence, we solve for the volume of the cylinder
Volume of the Cyllinder = 𝜋𝑟 2 ℎ
= 3.14. x 2.52 x 5
= 98.125 dm3
Volume of the Sphere =
=
=
2
3
𝜋𝑟2 ℎ
2 ( 3.14 × 2.52 × 5
3
2 ( 98.125)
3
= 65.42 dm3
Before we proceed to more activities, let us first solve the problem in our Lesson Opener.
The problem may have been difficult for you when we started the lesso. Now that you learned
more about volume, you are for sure ready to solve the problem.
Solution:
Making a drawing of 28 cm x 20 cm by 14 cm blok and cut into 2-cm cubes woud
clearly show the answer to the problem.
However, another way is to simply divide 28 x 20 x 14 by 2 x 2 x 2.
=
28 ×20 ×14
2× 2 × 2
= 980
Let’s Wrap up



Volume is the amount of space occupied by a three-dimensional
object.
Volume of a Rectangular Prism = 𝑙𝑤ℎ
Volume of a Cube = edge x edge x edge or e3
Volume of a Cylinder = 𝜋𝑟 2 ℎ

Volume of a Cone =


1
3
𝜋𝑟2 ℎ
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 10: Introduction to Algebra
LESSON 1
Sequence
D
Lesson Outcomes
At the end of the lesson, the student should be able to:




Determine the missing term/s in a sequence of numbers (e.g. odd numbers,
even numbers, multiples of a number, factors of a number, etc.)
Formulate the rule in finding the next term in a sequence;
Formulate the rule in finding the nth term using different strategies (looking for
a pattern, guessing and checking, working backwards); and
Solve word problems involving sequence.
Lesson Opener
Janeen arranges stamps in her album. She arranges 4 stamps in the first page, 7
stamps in the secnd page, 10 stamps in the third page, and so on. If her album has 10 pages,
how many stamps are there on the last page?
Let’s Focus
One way that we can solve this problem is by plotting the data in a table.
Page Number
Number of Stamps
1
4
2
7
3
10
10
n
Study the data. What do you notice with the number of stamps in relation to the page
number? The number of stamps in the current page is 3 more than the previous one.
Page Number
Number of Stamps
Rule
1
4
2
7
+3
3
10
+3
If we continue to fill in the table, we come up with these data
10
n
Page Number
Number
of
Stamps
Rule
1
4
2
7
+3
3
10
+3
4
13
+3
5
16
+3
6
19
+3
7
22
+3
8
25
+3
9
28
+3
10
31
+3
Making a table to compe up with the answer helps us in visualizing the given data but
listing many numbers may consume huge amount of our time if we are asked to find the
number of stamps on the 100th page.
Based on the problem, we have this sequence.
4, 7, 10 …
First term
second
third
three dots
term
term
(ellipsis)
Each term of the number that composes a sequence is called a term.
The three consecutive dots or the ellipsis tells that the sequence is infinite which means
that it goes on. We cannot identify the last tem of an infinite sequence. On the other hand,
a sequence is finite if the last term in known. It stops on a specific number.
A sequence has a rule or formula in order to find the value of each term. From the
previous example, we know that the rule is adding 3 to the first term to get the second term.
The same is true in finding the third term which is by adding 3 to the second term.
Let us find the 100th term in the sequence 4, 7, 10… We may use a table to determine
the pattern and formula a rule.
Term
Value
Difference
Formula
1
4
(1 x 3) + 1
2
7
(2 x 3) + 1
3
3
10
(3 x 3) + 1
…
…
…
100
N
(100 x 3) + 1
Based on the table, we can find the 100th term by multiplying the term by the common
difference and then adding 1. Finding the nth term will give us a formula 3n + 1.
The sequence 4, 7, 10 . . . , is an example of an arithmetic sequence. The difference
between one term and the next term is constant. In an arithmetic sequence, we can use the
formula: 𝑎𝑛 = 𝑎1 + ( 𝑛 − 1)𝑑 , wherein, 𝒂𝒏 is the nth term, 𝒂𝟏 is the first term, n is the term
position and d is the common difference.
study the next example below.
3, 6, 12, … What is the 8th term?
Plotting on a table, we have this.
Term
Value
1
3
2
6
3
12
...
...
8
n
Notice that the preceding term is multiplie by 2 to get the next term.
Term
Value
Rule
1
4
2
7
3
10
x2
...
...
8
n
x2
If we input the data, we come up with these.
Term
Value
Rule
1
3
2
6
x2
3
12
4
24
x2
x2
5
48
x2
6
96
x2
7
192
x2
8
384
x2
Here is another way to plot the data on the table.
Term
1
2
3
…
100
Value
3
6
12
…
N
Ratio
2
Formula
3x
3x
3x
…
3 x 28-1
Based on the table, we cand find the 8th term by working on the equation.
3 x 28-1 = 3 x 72
= 3 x 128
= 384
We can now generalize that the formula in finding the nth term is a geometric
sequence is 𝑎𝑛 = 𝑎𝑟 𝑛−1 , where 𝒂𝒏 is the nth term in a sequence, 𝒂 is the first term, and 𝒓 is the
common ratio
Let’s Wrap up




A sequence is a list of things (usually numbers) that are in order.
A sequence may be classified ad finite or infinite.
Arithmetic sequence has a formula 𝑎𝑛 = 𝑎1 + ( 𝑛 − 1)𝑑
We can use formula 𝑎𝑛 = 𝑎𝑟 𝑛−1 , for geometric sequence.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 10: Introduction to Algebra
LESSON 2
Number Sentence
D
Lesson Outcomes
At the end of the lesson, the student should be able to:



Find the missing number in an equation involving properties of operations;
Use different strategies (looking for a pattern, working backgrounds, etc.) to
solve for the unknown in simple equations involving one or more operations on
whole numbers and fractions; and
Solve word problems involving number sentence.
Lesson Opener
Ember is 9 kilograms heavier than Maryann. Their total weight is 77 kilograms. Find the
wieght of Ember.
Let’s Focus
We can solve this by using blocks.
Ember
9
Maryann
77
2 units + 9 = 77
2 units = 68
1 unit = 34
Ember’s weight = 34 + 9
= 43
Therefore, Ember’s weight if 443 kilograms
We can also assign letters or variables to stand for the missing values in a number
sentence.
Let r be the weight of Maryann.
Let r + 9 be the weight of Ember.
R + r + 9 = 77
2r + 9 = 77
2r
= 68
r
= 34
Ember’s weight = r + 9
= 34 + 9
= 43
Therefore,Ember’s weight is 43 kilograms.
Let’s Wrap up



A number sentence or mathematical sentence is made up of
numbers and/or variables (letters) and signs ( +, -, x, ÷, <, =, >)
An equation is a special type of number sentence. It has an equal
sign (=) to show that expression on both sides are equal (e.g. 1 + 2 =
3)
The missing value in a number sentence can be represented by a
symbol such as shape or letter.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 10: Introduction to Algebra
LESSON 3
D
Expression and Equation
Lesson Outcomes
At the end of the lesson, the student should be able to:




Differentiate expression from equation’
Give translation of real-life verbal expression and equations into letters or
symbols and vice versa
Define a variable in an algebraic expression and equation
Solve word problems involving expression and equation
Lesson Opener
Jose is 9 years old. How old will Jose be 5 years from now?
Let’s Focus
Five years from now, Jose will be 9 + 5 years old.
Notice that 9 + 5 is composed of the numbers 9 and 5 which are joined by a plus sign
(+). 9 + 5 is an example of a mathematical expression.
How old will Jose be in h years?
We can represent this as 9 + h. we have already known that 9 + h is an example of
expression. However, it consists of a number 9 and a variable h. This is called an algebraic
expression.
Here are other examples of expressions and algebraic expression
Examples of Expressions
Symbols
8+6
7–3
5x2
9÷3
Phrase
The sum of 8 and 5
The difference of 7 and 3
The product of 5 and 2
The qoutient of 9 and 3
Examples of Algebraic Expressions
Symbols
Phrase
Y+6
The sum of y and 6
X–3
The difference of x and 3
5m
The product of 5 and m
h÷3
The qoutient of h and 3
To solve for Jose’s age, we will add 9 and 5. That is 9 + 5 = 14 years old.
After how many years will Jose be at 14 years old?
We can represent this as 9 + h = 14. Notice that the equation is composed of number
9 and variable h joined by a plus sign in the left side of the equation while 14 in on the right
side of the equation. The expressions are separated by an equal sign. This is an example of
algebraic equation.
Let’s Wrap up




An expression is a number or a combination of numbers and
operation symbols.
An equation is made up of two expressions connected by an equal
sign (=),
An algebraic expression is an expression that contains one or more
variables.
An algebraic equation is a combination of algebraic expressions
connected by and equal sign.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 11: Statistics and Probability
LESSON 1
D
Double Bar Graph
Lesson Outcomes
At the end of the lesson, the student should be able to:




Organize data in tabular form and present them in a single/double horizontal
or vertical bar graph;
Interpret data presented in different kinds of bar graphs;
Construct bar graphs; and
Solve routine and non-routine problems using data presented in a single or
double-bar graph.
Lesson Opener
Teacher Jonalyn made a survey among 100 first year students as to their favorite
sports. The results are as follows: Volleyball: male – 12, female – 30; Basketball: male – 25,
female -10; Football: male – 15, female – 8.
Construct a double bar graph for more organized and attractive presentation of data.
Let’s Focus
You are surely familiar with single bar graphs by this time. However, this might probably
be your first encounter with double bar graph. A double bar graph is a graphical display of
information using two bars besides each other at various heights. The bars can be arranged
vertically or horizontally – the length of which represent frequency. Double bar graphs
compare two data sets ot once. For instance, if you want to gauge the favorite sport of the
students in your school, a single bar graph may be used. To gain even more insight as to the
favorite sport of the boys as compared to that of the girls, then a double bar graph would be
a perfect for this. Hence, when an item has two different measurable categories, a double
bar graph is needed to accurately represent the data.
An example of a vertical double bar graph with its parts is shown below. The graph not
only show the average scores of students on five subjects, but it also presents the average
scores of the boys and the girls separately.
Parts of a Double Bar Graph
Title
The title of the double bar graph provides a general overview to the learner of what is
bieng measured and compared. It helps the learners identify what the are about to look at.
In the example, the title tells you that the graph contains data on the average scores of the
100 students in five subjects. Specifically, it offers the average scores of boys and girls in each
of the five subjects.
Bars
Bar graph show information with bars or rectangles, which may be vertical or horizontal.
The length of the bar represents the frequency which means that the longer or taller bars
show greater numbers than the short ones do.
Horizontal Axis ( X-Axis)
The Horizontal – Axis of a double bar graph show the categories being compard. It runs
left to right. It has labels such as numbers representing different time periods or names of
things being compard. In the graph above, the horizontal axis has names of five subjects.
Vertical Axis ( Y – Axis)
The Vertical – Axis represents the scale. A scale is a set of numbers that represents the
data organized into equal intervals. In most bar graphs, like the one above, the y-axis runs
vertically (up and down). Typically, the labels of the vertical-axis are numbers for the amount
of items being measured. It usually start from 0 and is divided into as many equal parts as
necessay. In our example, the vertical-axis shows the possible average scores of students.
Legend or Key
A double bar graph also includes a legend or key. The key for a double bar graph
represents the groups being compared with two separates colors. The legend tells us what
each bar represents. Just like on a map, the legend helps the reader understand what they
are looking at. In our example, the legend tells us that the black bars represent the boys while
the gray represents the girls.
Let us now construct a double bar graph using data in our Lesson Opener.
Steps in constructing double bar graphs:
1. Write the title at the top center.
2. Draw the horizontal and vertical axes.
3. Write the labels for both axes.
4. Draw the double bars. The bars for the same group being compared must be beside
each other but of different colors. The number of bars corresponds to the number of
things being compared. The height or length of the bars must also correspond to the
value of the thins they represent.
5. Make a legend or key.
The double bar graph may be interpreted as follows:
The double bar graph above present the favorite sports of first year students. The key tells
us that the black color represent the number of male students while the gray one represent
the number of female students who like the given sports. We can also tell the number of male
and female students who like each sport. At a glance, we can clearly see the most female
students like volleyball while most male students like basketball. Also,the sport which is leastliked by male students is volleyball whil that of female students is football.
Let’s Wrap up


A double bar graph is a graphical display of information using two
bars beside each other at a various height. The bars can be arranged
vertically or horizontally. Double bar graphs compare two data sets at
one.
The parts of a double bar graph are the following: 1. Title; 2. Horizontal
axix; 3. Vertical axis; 4. Bars, and 5. Legend
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Chapter 11: Statistics and Probability
LESSON 2
D
Line Graph
Lesson Outcomes
At the end of the lesson, the student should be able to:




Interpret data presented in different kinds of line graphs (single to double-line
graph);
Construct a line graph based on a given set of data.
Draw inferences based on data presented in a line graph; and
Solve word problems using data presented in a line graph.
Lesson Opener
Two friends, Alexinne and Naomi, reviewed their grade in Math for the School Year
2018-2019. Their grades are presented in the table below.
Pupil
1st Qrtr. Grade
2nd Qrtr. Grade
3rd Qrtr. Grade
4th Qrtr Grade
Alexinne
95
97
94
96
Naomi
94
96
95
94
Construct a double-line graph to make the given data more presentable.
Let’s Focus
One of the best ways to present data the creative way is by using graphs. At this moment,
you might have this in-depth knowledge on how to interpret graphical data presentations or
even how to construct graphs. I’m sure that you have already seen graphs that use lines to
presents the behavior of data being interpreted. Line graph or line chart is a graphical
presentation of data, with set of points connected by straight lines, which shows how
something changes over time. For instance, the Philippine Statistics Authority (PSA) uses a line
graph to show the change in population of the Philippines over the years. Line graphs have
two kinds, the single-line and double-line graphs. For example, we can use the single-line
graph if we just only have to graph the temperature levels over a week-long period. This
means, there is only one category being observed over a given-period of time. In constrast,
a double-line graph could be utilized if there are two categories that are being compared
over some time intervals.
The basic parts of a well-constructed line graph include accurate title for the given set of
data, properly labelled vertical and horizontal axes, vividly place connected data points,
appropriately placed condition labels and a descriptive figure caption.
An example of a single-line graph with its parts is shown below. The graph presents the
change in the temperature. In degrees Celsius, in Himalayan City over six-day period.
Title of the Line Graph
The title provides the key words to tell what the graph is all about. In other words, it
summarizes the data in the graph. It helps the readers identify what they are about to look
at. In the given example above, the graph speaks about the temperature in Himalayan City
for the six days.
Scale
The scale shows the units used on the vertical axis or the y-axis. The numbers fused on the
scale must be of equal intervals. In the given example, the scale ranges from 0 to 40 with
equal intervals of 10 degrees Celsius between two temperatures.
Point
The points represent the amount of something being measured. Using the given data in
the example, the points represent the temperature in Himalayan City over the period of 6
days.
Line
The lines connect the data points. These help us clearly visualize the amount of increase
or decrease in the amount of something being measured. When the line goes upward, that
means there is an increase in the amount of something. On the other hand, when the line
goes downward, that means a decrease in the amount of stuff being measured. However, if
the line moves in a constant horizontal direction, that means there is no change in the value
between two consecutive data points.
Labels
The labels help us identify what kind of data is shown on the vertical and horizontal axes.
Horizontal Axis
It is also called the independent axis because its values do not depend on something. In
the given example, the horizontal axis represents the days which the temperatures in
Himalayan City are being monitored.
Vertical Axis
It is also called the dependent axis because its values depend on those of the horizontal
axis. In the given example, the vertical axis represents the amount of temperature meausured
in degrees Celsius.
At this time, let’s construct a double-line graph using the given data in the Lesson Opener.
Steps in Creating a Double-line Graph
1. Create a table. Draw the horizontal and vertical axes. On the top of the page, place an
appropriate title that briefly describes the data.
2. Label the horizontal and vertical axes. Make sure that the vertical and horizonatal lines are
equally spaced depending on the amount of something being measured. It does not
necessarily mean that you should always start the scale of numbers with zero.
3. Plot the data points. This time, you may have multiple sets of points depending on how may
items are being compared.
4. Connect the data points using straight lines. Make sure to connect the points that belong
on the same item only.
5. Create a key or a legend if you are comparing multiple items.
Grades in Math for the School Year 2018-2019
97,5
97
97
96,5
96
96
Grades
96
95,5
95
95
95
94,5
94
94
94
93,5
93
92,5
1st Quarter
2nd Quarter
3rd Quarter
4th quarter
Grading Period
Alexinne's Grade
Naomi's Grade
The graph shows the grades in Math of Naomi and Alexinne for the school year 2018-2019.
The legend tells us that the blue color represent the grades of Alexinne while the orange color
represents Noami’s grades. Based on the graph presented, we can say that both Naomi and
Alexinne acquired their highest grades in math on the 2nd Quarter of the school year 20182019. It could be inferred that both Naomi and Alexinne had spent more time in studying their
lessons in Math for the 2nd Quarter or maybe the lessons are quite easy for them to
comprehend.
Let’s Wrap up


Line graph or line chart is a graphical presentation of data, with set of
points connected by straight lines, which shows how something
change over time.
The major parts of a line graph are the following: 1. Title ; 2. Horizontal
Axis; 3. Vertical axis; 4. Points; 5. Lines and 6. Legend
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
_________________________________________________________________________________________
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2. Teaching Strategy I will employ
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3. Mode of Assessment I will administer
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Chapter 11: Statistics and Probability
LESSON 3
D
Pie Graph
Lesson Outcomes
At the end of the lesson, the student should be able to:



Construct a pie graph based on a given set of data;
Interpret data presented in a pie graph; and
Solve word problems using data presented in a pie graph.
Lesson Opener
Francis, a Bachelor in Elementary Education student, recieves a monthly allowance of
P10 000. He spends this amount on the following: boarding house- P1 000; food – P3 500;
transportation – P2 000; school supplies and project – P2 500. He then saves the remaining
amount.
How will you present these data in a pie graph?
Let’s Focus
You have already learned that data can be presented in tablular or graphical for. The
graphical form is, of course, more visually appealing aside from its being abel to display data
or information in an easy-to-read and interpret format. After learning about double bar and
line graphs, it’s now time to learn another kind of graph which is called a pie or circle graph.
The main use of a pie graph is to show comparison. When items are presented on a
pie graph, you can easily see which item constitutes the biggest and least part or amount.
Using different colors for the slices to denote the different categories makes the pie graph
more attractive and easy to interpret. One limitation when using pie graphs, however, is that
they do not give you the exact values of data. Usually, uou can only tell which category is
least of biggest. To solve this problem, values in terms of percent are included in the pie graph
for readers to see outright. Another limitation is inaccuracy on how each slice in the pie is
drawn as some values may contain percents with decimal places. However, with the use of
Microsoft Excel Software, it is now a lot easier to construct more accurate an colorful pie
graphs.
Let us now answer the problem given in the Lesson Opener by constructing a pie graph
using the data given. It is to be notied that in constructing a pie graph, one must assure that
all the slices are of equal proportions, such that all parts make up the whole data when
summed up. When amount for categories are expressed in percent, all percent values for the
parts must total 100%.
Steps in Constructing a Pie Graph
1. Find the Total
Add the values of all categories
In our problem, the total is already given which is P10 000.
The breakdown is as follows:
Category
Boarding House
Food
Transportation
School Supplies/Projects
Savings
Total
Value
P1 000
P4 000
P 500
P3 500
P1 000
P10 000
2. Convert the Categories to Percentages
Divide the value for each category by the total and multiply by 100.
Example:
For boardng house :
1 000 ×100
10 000
= 10%
Category
Value
Boarding House
P1 000
Food
P4 000
Transportation
P 500
School Supplies/Projects
P3 500
Savings
P1 000
Total
P10 000
Observe that the total of all percentage values is always 100%
Percentage
10%
40%
5%
35%
10%
100%
3. Calculate the Degrees
Multiply the decimal forms from Step 2 by 360° to get the number of degrees each slice
of pie should take up in the pie graph.
Example:
For food: 40% x 360 = 0.4 x 360 = 144°
Category
Value
Percentage
Boarding House
P1 000
10%
Food
P4 000
40%
Transportation
P 500
5%
School Supplies/Projects
P3 500
35%
Savings
P1 000
10%
Total
P10 000
100%
Again, remember that the total of all degree values must be 360°.
Degree
36°
144°
18°
126°
36°
360°
4. Construct the Pie Graph
a. Draw a circle.
b. Put a dot in the exact middle of the circle.
c. Draw a radius or one straight line from the center to one of the edges of the circle.
d.Measure out the first slice of the pie with a protractor.
e. Draw a second line from the center of the circle at the angle you just measured.
f. Repeat steps d and e until the whole circle is divided up and all the slices are
represented.
g. Label each slice of the circle. Include the legend.
Boarding House;
1000
Savings; 3500
Food; 4000
school Supplies ;
3500
So this is how our pie looks like.
Transportation;
500
Let’s Wrap up

A pie graph is in the form of a circle or a big pie that represents the
whole data. It is divided into varying slice sizes telling you how much of
one data part exists. Each part of the pie represents a subcategory or
component of the whole data which is proportional to the quantity of
data it represents.
Let’s Reflect
After learning this lesson, reflect on how it can be learned by your future pupils in
more effective, creative and meaningful way. Discuss you
1. Motivational Activity I will use
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_________________________________________________________________________________________
, if an increaSA
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
2. Teaching Strategy I will employ
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
3. Mode of Assessment I will administer
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
References:
1. Camarista, Genesis G. & Oranio Ian B. (2020), Teaching Mathematics in the Intermediate
Grades, Lorimar Pbublishing Inc.
2. Camarista, Genesis G. & Oranio Ian B. (2019), Teaching Mathematics in the Primary Grades,
Lorimar Pbublishing Inc.
3. Kiong, T. S. (2018), Mathematics Heuristics and Strategies to Problem Solving Primary 5,
Farfield Book Publsihers
4. Leong S (2017), Conquer Thinking Skills & Heuristics Workbook 4, Singapore Asia Publishers
Group Pte Ltd.
5. Tan J (2014), Maths World Problems Daily Practice, Educational Publishing House Pte Ltd.