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(Modular Approach Teaching Horizons)

Diversified M.A.T.H.

For Gen Z Learners

Prepared by:

Marie Christine A. Libetario, MAIE

Ma. Victoria P. Amado, MAT

Ladylen M. Vidal, MAEd

Jherosam M. Samonte, MAT

Randy L. Pepito, MEd

Philippine School Doha Intermediate Department S.Y. 2021 – 2022

Revised by:

Marie Christine A. Libetario, MAIE

Revised with:

Ma. Petsusan F. Moran, MAEd

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PREFACE

Welcome to Diversified MATH Module for Gen Z Learners!

Mathematics today is considered an indispensable subject. It plays a very

important role in one’s daily living as this is used throughout as an essential tool

in many fields of study. Moreover, it serves a special function to prepare the new

generation of today for them to become technically competent users of their gained

knowledge and skills.

Diversified MATH Module for Gen Z Learners is designed to help students

like you to gain comprehensive knowledge which will strengthen your Math skills

and competencies.

The components of the lessons in this module are as follows:

A pretest activity is a quick check of your retention from the

previous lessons.

These are activities that serve as springboards of the lesson and

short discussion of what the lesson is all about.

These are practice exercises crafted in varied forms, which are

provided in this section.

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These are activities that develop critical thinking skills that

provide you with opportunities to communicate and practice

sound reasoning using mathematical ideas.

These are learning outputs to connect you to the practical

applications of Math in different situations.

These are activities that will allow you to work independently or

collaboratively in exploring Math concepts and skills learned.

Here are some icons that will guide you around this module.

Take notes Complete solutions in the math n.b.

Answer book exercises

Important terms, definitions, etc., so…

Chill because you completed a part of the module

Task Icons

This module will likewise assist you to discover and prepare yourself to the

changing times and trends in Math.

May your journey to real world of Mathematics be as fulfilling as we have

planned for you!

Math 5 Teachers

Journal Drawing Survey Investigation Pair share Individual Group Writing Art Act. Activity Activity

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ROLE OF PARENTS

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At the end of this module, you will be able to use different methods of

finding the GCF & LCM of 2 or more numbers and use these concepts to justify

your solutions to problems related to real life. Specifically, you will be able

to:

1. define factors, multiple, GCF and LCM;

2. identify the best or easiest method to use in finding GCF and LCM;

3. find the GCF of two or more numbers using different methods;

4. find the LCM of two or more numbers using different methods;

5. make conjectures using concepts and techniques in finding GCF and LCM

in solving problems related to real life; and

6. create an artwork incorporating the learned concepts on factors,

multiples, GCF and LCM.

These will help you to become an adaptable and creative thinker.

FIRST QUARTER

MODULE NO. 1: NUMBER THEORY

Lesson 1: Factors and Multiples

(160 MINUTES)

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DIRECTIONS: Answer this pretest in your Math notebook. Don’t forget to read

and follow the questions carefully. (10 minutes)

Pretest: Number Theory

1. Which of the following is a true statement?

A. 39 is an example of a composite number.

B. The only even prime number is 2.

C. 1 is both prime and composite.

D. All even numbers are prime.

2. What is the smallest two-digit number that has only three factors?

A. 10 B. 25

C. 15 D. 35

3. Which of the following are the first five multiples of 5?

A. 3, 6, 9, 12, 15 B. 4, 8, 12, 16, 20

C. 5, 10, 15, 20, 25 D. 6, 12, 18, 24, 30

4. Which of the following sets of even numbers are all multiples of 6?

A. 6, 12, 30, 56 B. 0, 8, 16, 24

C. 6, 12, 24, 36 D. 48, 56, 64, 72

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5. What is the LCM of 9 and 4?

A. 1 B. 16 C. 36 D. 49

6. Which number is both an even number and a prime number?

A. 2 B. 4 C. 5 D. 8

7. Which of the following sets of odd numbers are all multiples of 7?

A. 7, 14, 32, 70 B. 7, 21, 35, 61

C. 15, 21, 29, 31 D. 7, 21, 35, 49

8. What is the GCF of 24 and 40?

A. 2 B. 5 C. 6 D. 8

9. Which of the ff. pairs are prime numbers and has a difference equal to 1?

A. 3, 4 B. 9, 10

C. 2, 3 D. 101, 102

10. Which pair of factors of 32 has a sum equal to 12?

A. 16, 2 B. 32, 1

C. 8, 4 D. 12, 3

End of Pretest

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From grades 1 to 4, you learned many things about whole numbers. These

included reading and writing numerals and performing mathematical operations

such as addition, subtraction, multiplication and division of whole numbers.

This lesson will introduce you to concepts that define or describe subsets of

whole numbers and some of its properties that we can sort of play with.

FACTORS OF A NUMBER

Recall the properties of multiplication. Can you define the word factors

based on the example below? Compare your definition with the one given below.

8 × 3 = 24

factors product

The factors of a number are the numbers which when multiplied will result

to the given number as their product.

How will you know if a whole number is a factor of another whole number?

To answer this question, read and study the following examples:

Example No. 1: Two factors of 24 are 6 and 4. Is 5 also a factor of 24?

5 × 𝑛 = 24

factors product

To find one of the factors, divide the product by the given factor.

24 ÷ 5 = 𝑛

product factors

Note: If there exists a

remainder (r), there

is no whole number

solution to n. Solving

24 ÷ 5 will give a

remainder of 4.

Therefore, 5 is not a factor of 24.

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Example No. 2: How many factors does 36 have?

36 = 1 × 36

36 = 2 × 18

36 = 3 × 12

36 = 4 × 9

36 = 6 × 6

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. Therefore, 36 has 9 factors.

Try this: 1. Find the set of factors of 40.

40 = 1 × 40

40 = ____ × _____

40 = ____ × _____

40 = ____ × _____

The factors of 40 are 1, ____, ____, ____, ____, ____, ____, 40.

How many factors does 40 have? ___________________

Example No. 3: What are the factors of 13?

The factors of 13 are 1 and 13.

To make sure that you have listed the factors completely, start with the number 1 and multiplied by the given number (1 × 36). Work your way up to the next factor of the given number that is 2, (2× 18). Until you have reached the factor listed in the previous pair. List the factors starting with the smallest factor 1 up to the highest factor of the number, that is 36.

Prime numbers – numbers that have only 2 factors, one and itself. Examples of prime nos. 2, 3, 5, 11, 13, 17, 19, 23…. 13 is a prime number. Therefore, its factors are 1 and 13 only.

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Try this: 2. What are the factors of the following numbers?

a. 47 → _____, _____

b. 13 → _____, _____

c. 7→ _____, ______

What do you call numbers that has exactly 2 factors? ______________

• Factors are numbers we multiply to get another number.

• A factor is also a number’s divisor

• There is an exact number of factors for any given number.

• The factors of a number always include 1 and the number itself.

• Factors are less than or equal to the given number.

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MULTIPLES OF A NUMBER

Multiples of a number refer to a set of numbers obtained when they are

multiplied by the digits 1, 2, 3, 4, 5, …

Example No. 4: Find the first five multiples of 6

6 × 1 6 × 2 6 × 3 6 × 4 6 × 5

6 12 18 24 30

Therefore, the first five multiples of 6 are 6, 12, 18, 24, 30.

Try this: 3. List the first five multiples of the following numbers.

a. 3 → ____________________________

b. 5 → ____________________________

c. 10 → __________________________

d. 9 → ____________________________

Skip counting – the simplest method used to find the multiples of a given number. In this exercise, start with the given number and work your way up to the fifth multiple of the number using skip counting. Ex. To find the first five multiples of 2, skip count by 2s. 2, 4, 6, 8, 10

• Multiples are numbers we get when we multiply a number by

1, 2, 3,… and so on.

• There is an infinite number of multiples for any given

number.

• Multiples of a number are equal to or more than the number.

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Activity 1.1

The concept on factors can be used in grouping people, objects, and other

things around us the we can count. While multiples most of the time includes

situations that call for succession and increase in number by multiplication. Solve the

given situations using the concept on factors and multiples. Write your organized

solutions in your Math notebook.

1. Isabel has 36 new angel figurines. In how many ways can she group the

figurines in such a way that each group will have the same number of figurines?

2. This year, Aldy bought 24 balls for a new Christmas tree. He decided to

increase the number of balls on the tree by 24 each year. How many Christmas

balls will the tree have after four years?

Clue: No. of ways = No. of factors of 36

Clue: In this exercise, use the concept on multiples of a number.

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3. Answer the following questions coupled with explanations. Then

ask the same set of questions (only yes and no part) to 10 people in

your household or your friends online. Prepare a table of responses.

Make a feedback on how many correct and wrong answers you got

from your respondents.

a. Is 1 a factor of any whole number? Why?

b. Is 0 a factor of 25? Why?

c. Are all the multiples of an even number also even? Give examples

to support your answer.

Work on these exercises to check and to sharpen your understanding on the

concepts of factors and multiples.

Exercise 1.1

Answer the exercises found in your textbook

Phoenix Math for the 21st Century Learners Gr.5

Towards Better Understanding: A,B,C pp. 53-54

Follow-up Practice: A,B,C pp. 54-55

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Exercise 1.2

A. Find the factors of each number

B. Give the first ten nonzero multiples of each number.

1. 18 → ________________________

2. 42 → ________________________

3. 65 → ________________________

4. 96 → ________________________

5. 120 → ________________________

6. 8 → ________________________________________________

7. 11→ ________________________________________________

8. 20→ ________________________________________________

9. 15→ ________________________________________________

10. 25→ ________________________________________________

End of Lesson 1

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As you’ve learned previously, factors are the numbers you multiply to give a

product and every whole number has a set of factors corresponding to it. You may

use this knowledge to solve the given problem below.

Example No. 5: The Intermediate department is holding a sportsfest for all grade

5 students. Two classes form their teams within their sections. Section A has 42

students while section B has 48 students. All students must participate and be part

of the team. If each team for both sections must have the same number of players,

what is the greatest number of players that can be formed per team?

FIRST QUARTER

MODULE NO. 1: NUMBER THEORY

Lesson 2: Greatest Common Factor (GCF)

(160 MINUTES)

Section A Section B

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Section A (42 students) Section B (48 students)

Number of Teams

Number of Players in each team

Number of

Teams

Number of Players in each team

1 42 1 48 2 21 2 24 3 14 3 16 6 7 4 12 7 6 6 8

14 3 8 6 21 2 12 4 42 1 16 3

24 2 48 1

All the numbers that you see in each column are called factors.

42 = 1 × 42

42 = 2 × 21

42 = 3 × 14

42 = 6 × 7

48 = 1 × 48

48 = 2 × 24

48 = 3 × 14

48 = 4 × 12

48 = 6 × 8

42 teams will not be possible because it means a student will play alone. Same with 48 teams

is not a team.

The yellow arrow is a guide on how you are going to list the factors of the given no. a.k.a

U TURN

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DIFFERENT METHODS IN FINDING THE GCF

Using Listing of Factors method (LF), the factors of the given number of

students… 42 → 1, 2, 3, 6, 7, 14, 21, 42

48 → 1, 2, 3, 4, 6, 8, 12, 14, 24, 48

The common factors of two or more numbers are the common numbers that

can divide exactly the given numbers, thus the common factors of 42 and 48 are

1,2,3, and 6. The greatest common factor or GCF is 6.

Section A 42 ÷ 6 = 7 𝑡𝑒𝑎𝑚𝑠

Section B 48 ÷ 6 = 8 𝑡𝑒𝑎𝑚𝑠

Therefore, we form 7 teams for section A and 8 teams for section B. Each with the same number of players.

Another method that we can use to get the greatest common factor is

Prime Factorization (PF)

Section A 42

6 × 7

2 × 3

Section B 48 2 × 24 3 × 8 2 × 4 2 × 2

PF: 42 = 2 × 3 × 7

48 = 2 × 3 × 2 × 2 × 2

𝐺𝐶𝐹 = 2 × 3 = 𝟔

A factor tree shows that every composite number can be written as a product of its prime factors. This process is call prime factorization.

The common factors are 2 and 3 and their product is 6. Hence, the greatest common factor of 42 and 48 is 6. Therefore, each team has 6 players.

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Continuous Division Method (CD)

Example No. 6: Find the greatest common factor of 48, 56, and 64 using

continuous division.

2 48, 56, 64

2 24, 28, 32

2 12, 14, 16

6, 7, 8

𝐺𝐶𝐹 = 2 × 2 × 2 = 𝟖

If there is no common factor except 1 just like in the case of 6,7,8… STOP dividing. The product of the divisors found in the left is the resulting GCF of the given nos. 48, 56 and 64.

• The GCF of two or more numbers is the largest whole

number that is a factor of each of the numbers.

• To find the GCF of a set of numbers, you can make use of any

one of these methods that will work best for you.

1. Listing of Factors → list the factors completely using the

U-Turn pattern/rainbow method

2. Prime Factorization →write the given nos. as product of

prime nos. Start with a factor tree as part of your solution.

3. Continuous Division → write the nos. horizontally then

perform continuous division. Use prime divisors. Continue

dividing until there is no common divisor.

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Activity 1.2

Get ready to show how well you’ve understood the concept on greatest common

factor. Work on the following by writing your organized solutions in your Math

notebook.

1. Henry has a collection of white clouds in fishbowls. He also has 16 black pebbles

and 12 white pebbles for the fishbowls. He wants to put these pebbles in fishbowls

in such a way that each fishbowl will have the same number of black and white

pebbles. Find the greatest number of fishbowls where he can put the pebbles.

2. Patrick buys 32 basketball cards, 24 baseball cards and 40 football cards. He

wants to give the same number of each kind of card to some of his friends. He also

wants to give the cards to the greatest possible number of friends. To how many

friends can he give all the cards?

Use Listing of Factors to solve this problem.

Use either Prime Factorization or Continuous Division to solve this problem.

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3. Hazel will bake cookies for her teachers. That will include 25 chocolate chip, 50

red velvet and 100 peanut butter cookies. She will put the cookies in tin cans in

such a way that each can will have the same number of each kind of cookie.

a. If she wants to use the greatest possible number of tin cans,

how many cans will she need?

b. How many of each kind of cookie will each tin can contain?

c. Make an illustration of the given problem and show it in our

next virtual class.

Work on these exercises to check and to sharpen your understanding on the

concepts of greatest common factor.

Exercise 1.3

Answer the exercises found in your textbook.

Phoenix Math for the 21st Century Learners Gr.5

Towards Better Understanding: A, B p. 74

Follow-up Practice: p. 74

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Exercise 1.4

A. Find the GCF of each set of number using listing method (LF)

B. Find the GCF using the continuous division method (CD).

1. 16 and 20

16 → ________________________

20 → ________________________

GCF = ________

2. 10, 25 and 30

10 → ________________________

25 → ________________________

30 → ________________________

GCF = ________

3. 36 and 81

4. 12, 16 and 20

5. 33, 44 and 77

End of Lesson 2

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It may sound funny or redundant but the simplest way to define/describe

the least common multiple is that it is the least of the common multiples of two or

more numbers. Study the given examples…

Example No. 7: Luisa waters her plants every 2 days, while Mercy waters her

plants every 3 days. If both of them water their plants on the same day, how many

days will it take before they water their plants on the same day again?

To solve the problem, let us list down some of the multiples of 2 and 3.

DIFFERENT METHODS IN FINDING THE LCM

Using Listing of Multiples method (LM), the multiples of the given number

will be listed as shown. Start with at least 4 multiples for each number until you

find a first common multiple, that is the least common multiple.

Multiples of 2 → 2, 4, 6, 8, …

Multiples of 3 → 3, 6, 9, 12,…

FIRST QUARTER

MODULE NO. 1: NUMBER THEORY

Lesson 3: Least Common Multiple (LCM)

(160 MINUTES)

Therefore, the least common multiple (LCM) of 2 and 3 is 6.

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Numbers may have several common multiples. Some of the common

multiples of 2 and 3 are 6, 12, 18 and so on but the least common multiple or LCM

is the least multiple common to the given numbers. In this case, 6 is the LCM of 2

and 3.

Therefore, it will take 6 days before the girls water their plants on the

same day again.

Example No. 8: Find the LCM of 8 and 12 using…

Prime Factorization (PF)

8

4 × 2 2 × 2

12 2 × 6 2 × 3

PF: 8 = 2 × 2 × 2

12 = 2 × 2 × 3

𝐿𝐶𝑀 = 2 × 2 × 2 × 3 = 𝟐𝟒

A factor tree shows that every composite number can be written as a product of its prime factors. This process is call prime factorization.

Encircle each prime factor that is common to both nos. Then, multiply the common factors with he other factors (those that were not encircled). Therefore, the LCM (8,12) = 24

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Continuous Division Method (CD)

Similar in finding the GCF, the LCM can also be obtained using the continuous

division method.

Example No. 9: Find the LCM of 12, 20, 32 using continuous division.

2 12, 20, 32

2 6, 10, 16

3, 5, 8

𝐿𝐶𝑀 = 2 × 2 × 3 × 5 × 8 = 𝟒𝟖𝟎

Let’s try another example.

Example No. 10: Find the LCM of 36, 54, 72 using continuous division.

2 36, 54, 72

3 18, 27, 36

3 6, 9, 12

2 2, 3, 4

1, 3, 2

𝐿𝐶𝑀 = 2 × 3 × 3 × 2 × 1 × 3 × 2 = 𝟐𝟏𝟔

Think of a prime number that can exactly divide at least two of the given numbers. Perform continuous division until the quotients obtained are already relatively prime.

Multiply all the numbers outside of the inverted division symbol.

In the case of the quotients, 2,3,4, 2 and 4 are divisible by 2 Since 3 is not divisible by 2, in the CD method, bring down no. 3 since it is the only one not divisible by 2.

Multiply all the numbers outside of the inverted division symbol.

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Activity 1.3

Hurray! You are almost done with our lesson 3 on Number Theory. Get ready

for more challenges on how you can make use of the concepts on least common

multiple. Work on the following problems by writing your organized solutions in

your Math notebook.

• The LCM of two or more numbers is the smallest nonzero

number that is a multiple of each of the given numbers.

• To find the LCM of two or more numbers, you can use the

following methods:

1. Listing of Multiples → list the first 4 multiples of the nos.

then work your way up until the least common multiple appears

in the list.

2. Prime Factorization →write the given nos. as product of

prime nos. Start with a factor tree as part of your solution.

3. Continuous Division → write the nos. horizontally then

perform continuous division. Use prime divisors. Continue

dividing until there is no common divisor. Get the product of the

nos. outside the continuous division symbol including the

remaining quotients at the bottom.

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1. Kit, Kokoy and Karl help in washing the dishes at home. Kit helps every 3 days,

Kokoy helps every 4 days, and Karl helps every 5 days. If the three boys help in

washing the dishes today, after how many days will they all help wash the dishes

again on the same day?

2. Three girls regularly visit an orphanage on the first Sunday of a certain month

to have storytelling sessions with the children. Charlene visits the orphanage every

2 months, Caryl every 5 months, and Dory every 6 months. If the three girls visit

the orphanage together on the first Sunday of June, after how many months will

the three girls visit the orphanage together again? In what month will that be?

3. Find the GCF and LCM of the given set of numbers. Use any method and show

your solutions in your Math notebook.

Given GCF LCM a. 4, 8, 12 b. 6,8, 9 c. 25, 75 d. 5, 11

Use Listing of Multiples to solve this problem.

Use either Prime Factorization or Continuous Division to solve this problem

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4. When finding the LCM of two or more numbers, which among

the methods discussed is easiest to use? Explain why you prefer

this method over the other methods. Share with your

classmates how you were able to find GCF and LCM of 2 or more

numbers and which method works best for you.

Work on these exercises to check and to sharpen your understanding on the

concepts of least common multiple.

Exercise 1.5

Answer the exercises found in your textbook

Phoenix Math for the 21st Century Learners Gr.5

Towards Better Understanding: A, B p. 79

Follow-up Practice: B nos. 2 to 5 p. 81

End of Lesson 3

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Congratulations! You are now ready to apply these learned concepts on

number theory. Complete the task given below following the guidelines indicated.

(Learning Output No. 1)

1st Quarter Learning Output No. 1

WORD PROBLEM STORY ON GCF/LCM

Individual Task: Create a word problem story in real-life setting that will allow you to solve the problem in the situation using either GCF or LCM.

Output format: Google Classroom Question (LO1 Word Problem Story) This will be posted during our online class. Your classmates will try to answer your word problem and from there we will be able to determine if the problem is solvable.

Rubric for Scoring:

Learning Output 1: WORD PROBLEM STORY ON GCF/LCM 10 pts. Concept & Word problem is related to either GCF or LCM 10 pts. There is logic in the situation & problem is solvable. 10 pts. Problem is original & submitted on time 30 points in total

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Exploring the world of number theory is more enjoyable when you

creatively interpret your own word problem as a work of art. The next task will be

individually.

1st Quarter Product No. 1

DRAW YOUR PROBLEM & SOLVE IT!

Individual Task: Creatively interpret your own word problem as a work of art.

Output format: Google Classroom Assignment Attachment (PR1 Draw Your Problem) Handmade artwork in A4 bond paper. Scanned

images or pictures of your actual output will be

sent as an attachment in the Google classroom

assignment. Best works will be presented in

our online class.

Rubric for Scoring:

1st Quarter Product No. 1. PROBLEM DRAWING Points 10 Concept & Word problem is related to either GCF or LCM 10 Solutions were presented logically & with correct answer 15 Drawings were completed creatively, has visual appeal & related to the given problem 5 Output was submitted on time 40 points in total

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Here are some additional learning resources that will help you understand

the other lessons related to Number Theory. Check out the listed materials during

your free study period.

In your Math textbook:

Bandejas, J. A., Reyes, D. E., Sagusay, E. L., & Dela Cruz, E. O. (2016). Phoenix Math for the

21st Century Learners Grade 5 K to 12. Phoenix Publishing House.

• Read and answer the exercises

Chapter 2 Lesson 3 Prime Factorization pp. 62-69

Webpage:

• Watch Math Antics. (2012, April 16). Prime Factorization [Video].

YouTube. https://www.youtube.com/watch?v=XGbOiYhHY2c

REFERENCES

Grade 5 Textbook: Bandejas, J. A., Reyes, D. E., Sagusay, E. L., & Dela Cruz, E. O. (2016). Phoenix Math

for the 21st Century Learners Grade 5 K to 12. Phoenix Publishing House. Note: All references will be reflected after the final completion of the last module for

this subject.

End of First Quarter Module 1