(m a teaching horizons for gen z learners

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Diversified M.A.T.H. For Gen Z Learners (Modular Approach Teaching Horizons) Philippine School Doha Intermediate Department S.Y. 2021 -2022 Authors: Ladylen M. Vidal, MAEd Ma. Victoria P. Amado, MAT Marie Christine A. Libetario, MAIE Jherosam M. Samonte, MAT Jocelyn Q. Gimpayan, MAEd Randy L. Pepito, MEd > >

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Page 1: (M A Teaching Horizons For Gen Z Learners

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Diversified M.A.T.H.

For Gen Z Learners

(Modular Approach Teaching Horizons)

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

Authors:

Ladylen M. Vidal, MAEd

Ma. Victoria P. Amado, MAT

Marie Christine A. Libetario, MAIE

Jherosam M. Samonte, MAT

Jocelyn Q. Gimpayan, MAEd

Randy L. Pepito, MEd

> >

Page 2: (M A Teaching Horizons For Gen Z Learners

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This module will focus on Number theory. It is

a branch of pure mathematics devoted to the study

of the natural numbers and the integers. It is the

study of the set of positive whole numbers which

are usually called the set of natural numbers. As it

holds the foundational place in the discipline,

Number theory is also called "The Queen of

Mathematics".

Are you ready to know more about Number

Theory?

THIRD QUARTER

MODULE NO. 1: NUMBER THEORY

Lessons: Odd and Even Numbers,

Prime and Composite Numbers,

Factors and Multiples, GCF and LCM

All the activities in the

and will be answered in

your notebook. Checking will be done during our

face-to-face class.

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At the end of this module, you will be able to state the key concepts, importance and techniques involve in theory of numbers that will aid you to discern and structure solution. Specifically, you are expected to:

1. identify odd and even numbers;

2. differentiate factors and multiples;

3. differentiate prime and composite numbers;

4. write a given number as a product of its prime factors using

prime factorization;

5. identify the Greatest Common Factor (GCF) and Least

Common Multiple (LCM) of two given numbers using; and

6. solve real-life problems involving GCF and LCM of two given

numbers.

These will help you become mathematically proficient

problem solver.

\

ROLE OF THE PARENTS

Monitor the schedule of the students for

synchronous online learning.

Guide the students to fully understand the lessons.

Supervise the students in performing/accomplishing

the given home-based activity.

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Read each item carefully and write the letter of the

correct answer in your notebook.

A. Write the letter of the correct answer on the space provided

before each number.

______1. What do you call to two or more numbers that are being multiplied?

A. divisor B. factors C. multiplier D. product

______2. What do you call to the greatest number among the common factors of two or more numbers?

A. factors B. Greatest Common Factors C. common factors D. Greatest Common Multiple

______3. Which of the following is a set of even numbers?

A. (2, 3, 4, 5, 6) B. (3, 5, 7, 9, 11) C. (2, 4, 6, 8, 10) D. (4, 8, 12, 15, 16)

______4. Which of the following are factors of 12?

A. 1, 3, 4, 12 B. 1, 2, 6, 12 B. 1, 2, 3, 4, 12 D. 1, 2, 3, 4, 6, 12

______5. Which of the following is NOT a factor of 20?

A. 2 B. 4 B. 5 D. 6

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.

______6. How many factors do 10 and 20 have in common? A. 3 B. 5 B. 4 D. 6

_______7. What is the Greatest Common Factor of 12 and 24?

A. 12 B. 6

C. 4 D. 2 ______8. What is Least Common Multiple of 8 and 32?

A. 8 B. 16 C. 24 D. 32

______9. Which number is a factor of 16, but NOT a multiple of 4?

A. 2 B. 4 C. 8 D. 16

______10. Which factors of 36 has a sum of 13? A. 1 and 36 B. 2 and 18 C. 4 and 9 D. 6 and 6

______11. Which of the following is a factor 28?

A. 7 B. 35 C. 21 D. 42

______12. What is the smallest prime number?

A. 0 B. 2 C. 1 D. 3

______13. Which number is neither prime nor composite?

A. 1 B. 5 C. 13 D. 37

_____14. Which of the following numbers is prime? A. 12 B. 9 C. 7 D. 6

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_____15. How many prime numbers are there between 1 - 10? A. 1 B. 2 C. 3 D. 4

_____16. What is the prime factorization of 18?

A. 2 x 3 x 3 B. 2 x 2 x 3

C. 2 x 9 D. 3 x 6 _____17. What is the prime factorization of 24?

A. 2 x 2 x 4 x 6 B. 2 x 2 x 3 x 3 C. 2 x 2 x 2 x 3 D. 2 x 3 x 4

_____18. Alexis has a soccer game every 3rd day, Paul has one every 4th day. When will they have a game on the same

day? A. Day 3 C. Day 4

C. Day 7 D. Day 12 ______19. Danny has 10 baseballs and 20 basketballs. If she wants to divide them into identical groups without any balls left over, what is the greatest number of groups Danny can make?

A. 2 C. 10 B. 5 D. 20

_____20. While performing a piece of music, Alice strikes the cymbals

every 8 beats and the triangle every 6 beats. If she just struck both at the same time, how many beats will pass before she again strikes them at the same time?

A. 6 C. 24 B. 8 D. 48

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Before we start our lesson, let us

review odd and even numbers. Do you

still remember what is these numbers?

Let us try if you can answer this.

Look at each number, then identify if it is odd and

even number.

a. 15 - ____ d. 10 - ____

b. 28 - ____ e. 60 - ____

c. 3 - ____ f. 49 - ____

You are correct!

a. 15 - odd d. 10 - even

b. 28 - even e. 60 - even

c. 3 - odd f. 49 - odd

An even number is a number that can be divided

into two equal groups. (Ex. 2, 4, 6, 8, 10, …)

An odd number is a number that cannot be

divided into two equal groups. (Ex.1, 3, 5, 7, 9, …)

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Lesson 1.1 Factors

Kimberly arranges 6 pots of flowers in different ways:

Arrangement 1: 1 group of 6

or 1 x 6 = 6

Arrangement 2: 2 groups of 3 or 2 x 3 = 6

Arrangement 3: 3 groups of 2 or 3 x 2 = 6

Arrangement 4: 6 groups of 1

or 6 x 1 = 6

As shown, the arrangement are by 1, 2, 3 and 6. So, we can say

that 6 pots of flowers can be arranged in 4 different ways because 6

has 4 factors.

1, 2, 3 and 6 are called the factors of 6.

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Let us find the factors of some numbers.

16

1. 16 = 1 x 16 1 16

= 2 x 8 or 2 8

= 4 x 4 4 4

36

2. 36 = 1 x 36 1 36

= 2 x 18 or 2 18

= 3 x 12 3 12

= 4 x 9 4 9

80

3. 80 = 1 x 80 1 80

= 2 x 40 2 40

= 4 x 20 or 4 20

= 5 x 16 5 16

= 8 x 10 8 10

Note: 1 is always a factor of any number. If 2 factors are the same

such as 4 x 4 = 16, consider it as 1 factor only.

The factors of 16 are 1, 2,

4, 8, and 16.

It has 5 factors.

The factors of 36 are 1, 2,

3, 4, 9, 12, 18 and 36.

It has 8 factors.

The factors of 80 are 1, 2, 4,

5, 8, 10, 16, 20, 40 and 80.

It has 10 factors.

Factors are numbers that are multiplied to give a

product.

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A. List all the factors of the following numbers.

Let’s get started!

ACTIVITY NO.1A

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You are doing great! Let’s keep

moving.

ACTIVITY NO.1B

A. Is the first number a factor of the second? Write YES or NO in

the blank.

______1. 3; 15 ______4. 5; 24

______2. 9; 63 ______5. 4; 32

______3. 4; 21

B. Write the set of all factors of each number then, identify the

total number of factors do each given have.

1. 12: _________________________ No. of Factors: ______

2. 28: _________________________ No. of Factors: ______

2. 36: _________________________ No. of Factors: ______

2. 40: _________________________ No. of Factors: ______

2. 56: _________________________ No. of Factors: ______

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Lesson 1.2 Prime and Composite Numbers

Study these numbers and their list of factors.

Factors Factors Factors

2 – 1, 2 6 – 1, 2, 3, 6 9 – 1, 3, 9

3 – 1, 3 7 – 1, 7 10 – 1, 2, 5, 10

4 – 1, 2, 4 8 – 1, 2, 4, 8 11 – 1, 11

5 – 1, 5

Which of these

numbers are

composite?

Which of them

are prime

numbers?

Which numbers have two factors? Which nubers have more

than tow factors?

Counting numbers that have exactly two factors are prime

numbers. Numbers 2, 3, 5, 7 and 11 are prime numbers. Each of

them has exactly two factors which ae 1 and the number itself.

Composite numbers have more than two factors. Numbers

4, 6, 8, 9 and 10 are composite numbers, because they have more

that two factors.

Number 1 is neither prime nor composite,

because it has only factors.

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Number Factors Number

of Factors

Prime or

Composite

8

18

23

43

100

Let’s get started!

ACTIVITY NO.2A

A. Fill in the table with the correct answers.

B. Write C on the blank if the number is composite and P if it

is prime.

_____ 1. 17 _____ 6. 35

_____2. 20 _____ 7. 48

_____ 3. 25 _____ 8. 90

_____ 4. 32 _____ 9. 71

_____ 5. 47 _____ 10. 37

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You are doing great! Let’s keep

moving.

ACTIVITY NO.2B

A. Name two primes whose product is -

1. 15 - ____ ____ 4. 39 - ____ ____

2. 21 - ____ ____ 5. 77 - ____ ____

3. 34 - ____ ____

B. Give 3 different primes whose sum is -

1. 10 - ____ ____ ____ 4. 31 - ____ ____ ____

2. 15 - ____ ____ ____ 5. 47 - ____ ____ ____

3. 23 - ____ ____ ____

C. Answer the following questions.

1. What is the smallest prime number? ________

2. What is the largest 2-digit prime number? ________

3. What is the largest 2-digit composite number? ________

4. What is the sum of all the prime numbers between 5 and 20?

________

5. What is the product of the prime numbers immediately before

and after 20? ________

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Lesson 1.3 Prime Factorization

What kind of numbes are 2 and 3?

All the factors of 12 in the expression, 2 x 2 x 3 = 12, are

prime numbers. That is why we say that 2 x 2 x 3 is the prime

factorization of 12.

The prime

factorization of

12 is 2 x 2 x 3.

What is prime

factorization?

Study the two ways on how to get the prime factorization of 12.

x

x

12 = 2 x 2 x 3

1. Think of two numbers which, when multiplied

together, will give 12. (2 x 6 = 12) 2 is prime

and 6 is not.

2. Think of two factors that will give 6. (2 x 3 = 6)

2 is prime and 3 is also prime.

3. Get all the prime factors ( 2 x 2 x 3).

A.

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Prime factorization is a way of expressing a

number as a product of prime factors. To get the

prime factorization of a number, build a factor tree

starting with primes.

12 = 2 x 2 x 3

B.

1. Think of two numbers which, when multiplied

together, will give 12. (3 x 4 = 12) 3 is prime

and 4 is not.

2. Think of two factors that will give 4. (2 x 2 = 4)

2 is prime.

3. Get all the prime factors ( 2 x 2 x 3).

Here is another example:

56 = 2 x 2 x 2 x 7

1. Think of two factors that will give 56.

(7 x 8 = 56) 7 is prime and 8 is not.

2. Think of two factors that will give 8.

(2 x 4 = 8) 2 is prime and 4 is not.

3. Think of two factors that will give 4.

(2 x 2 = 4) 2 is prime.

4. Get all the prime factors (2 x 2 x 2 x 7).

x

x

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1.

5. 4.

3. 2.

27 = ___________ 63 = ___________ 30 = ___________

100 = _______________ 56 = ________________

Let’s get started!

ACTIVITY NO.3A

A. Complete the factor tree by filiing in the missing

factors. Write the prime factorization of each number in

the given blank.

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You are doing great! Let’s keep

moving.

ACTIVITY NO.3B

A. Build a factor tree and then write the prime factorization

of each number below.

1. 30 2. 60 3. 72

30= ___________ 60= __________ 72=___________

4. 150 5. 200

150= _____________ 200= ______________

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Method A: Listing Method

1. List all the factors of 12 and 18.

12 – 1, 2, 3, 4, 6, 12

18 – 1, 2, 3, 6, 9, 18

2. List the common factors of 12 and 18. Common

factors: 1, 2, 3, 6

3. Select the greatest number among the common

factors.

The greatest number is 6.

Therefore, the greatest common factor (GCF) is 6.

Let us find out if the Greatest Common Factor of 12 and 18

is really 6. There are different ways of finding the GCF.

Lesson 1.4 Common Factors and Greatest Common Factors

(GCF)

The Greatest Common

Factor of 12 and 18 is 6.

How did you know that the

Greatest Common

Factor of 12 and 18 is 6?

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2 12 18

3 6 9

2 3

1. Divide by the least prime number which is 2.

2. Continue dividing until no prime number can divide

both numbers.

3. Then, multiply the numbers at the left or the prime-

number divisor.

4. 2 x 3 = 6 (GCF)

Method B: Prime Factorization

1. Get the prime factorization of 12 and 18.

2. Find the common prime factors.

12 = 2 x 2 x 3

18 = 2 x 3 x 3

3. Get the product of the common prime factors.

The greatest common factor (GCF) is 2 x 3 = 6.

Method C: Continuous Division

12 = 2 x 2 x 3 18 = 2 x 3 x 3

x

x

x

x

The number pairs whose GCF is 1 are called

relatively prime numbers.

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Let’s get started!

ACTIVITY NO.4A

A. Fill in the blank to find the greatest common factor of the

following pair of numbers.

1. 10 and 12 10: _____, ______, ______, ______ 12: _____, ______, ______, ______, ______, ______

Common Factors: __________________ GCF: _____ 2. 45 and 50 45:_____, ______, ______, ______, ______, ______

50: _____, ______, ______, ______, ______, ______ Common Factors: __________________ GCF: _____

3. 15 and 18

15:_____, ______, ______, ______, 18: _____, ______, ______, ______, ______, ______

Common Factors: __________________ GCF: _____ 4. 16 and 28 16: _____, ______, ______, ______, ______

28: _____, ______, ______, ______, ______, ______ Common Factors: __________________ GCF: _____

5. 30 and 100 30: _____, ______, ______, ______, ______, ______ 100: _____, ______, ______, ______, ______, ______

______, ______, ______ Common Factors: __________________ GCF: _____

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You are doing great! Let’s keep

moving.

ACTIVITY NO.4B

A. Find the product of the factors in the prime factorization.

Encircle the common factors, then give the GCF. Number 1

is done for you.

1. 2 x 3 x 3 = 18 3. 2 x 2 x 5 = ___

3 x 3 x 3 = 27 2 x 2 x 3 = ___

GCF of 18 and 27 = 9 GCF of ___ and ___ = ____

2. 2 x 3 x 3 = ___ 4. 2 x 3 x 7 = ___

2 x 3 x 5 = ___ 2 x 2 x 3 x 5 = ___

GCF of ___ and ___ = ____ GCF of ___ and ___ = ____

B. Give the prime factorization of each, then write the GCF

of the numbers. (Write your solution in your math

notebook.)

1. 20 = ________________ 2. 12 = ________________

50 = ________________ 36 = ________________

GCF: _______ GCF: _______

C. Use the continuous division to get the GCF of each set of

numbers. (Write your solution in your math notebook.)

1. 15 and 45 – GCF = _______

2. 20 and 36 – GCF = _______

3. 50 and 90 – GCF = _______

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Method A: Listing Method

1. List the multiples of 8 and 24 in ascending order.

8 – 8, 16, 24, 32, 40, 48, 56, 64, 72,…

12 – 12, 24, 36, 48, 60, 72, 84,…

2. The common multiples of 8 and 12 in the list are 24,

48, 72,…

3. The least common multiple or LCM of 8 and 12 is

24.

Lesson 1.5 Common Multiples and Least Common

Muliple (LCM)

There are different ways of finding the least common multiple of

numbers.

Multiples are like products in

multiplication. In division,

multiples are like dividends.

The Least Common

Multiple of 8 and 12 is 24. Can you tell

why?

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1. Divide by the least prime number which is 2.

2. Continue dividing until no prime number can divide

both numbers.

3. Then, multiply the numbers at the left and the last

numbers at the bottom.

4. 2 x 2 x 2 x 3 = 24 (LCM)

Method B: Prime Factorization

1. Get the prime factorization of 8 and 12.

2. Find the common factors.

8 = 2 x 2 x 2

12 = 2 x 2 x 3

3. Multiply the common factors by the rest of the factors.

Common factors are 2 x 2. The rest of the factors are 3

and 2. So the LCM of 8 and 12 is 2 x 2 x 2 x 3 = 24.

Method C: Continuous Division

x

x

12 = 2 x 2 x 3

x

x

8 = 2 x 2 x 2

2 8 12

2 4 6

2 3

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Let’s get started!

ACTIVITY NO.5A

A. List down the first 10 multiples of each number.

Then find the common multiples (CM) and the least

common multiple (LCM).

1. 4 = _____________________________________

6 = _____________________________________

CM: ____________________ LCM: ________

2. 8 = _____________________________________

10 = _____________________________________

CM: ____________________ LCM: ________

3. 3 = ______________________________________

7 = ______________________________________

CM: ____________________ LCM: ________

4. 5 = _____________________________________

20 = _____________________________________

CM: ____________________ LCM: ________

5. 3 = _____________________________________

5 = _____________________________________

6 = _____________________________________

CM: ____________________ LCM: ________

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ACTIVITY NO.5B

You are doing great! Let’s keep

moving.

A. Give the prime factorization of each number. Then find

the LCM. (Write your solution in your math notebook.)

1. 9 = ________________ 4. 6: ________________

27 = ________________ 15: = ________________

LCM: _______ LCM: _______

2. 4 = ________________ 5. 12: ________________

8 = ________________ 30 = ________________

LCM: _______ LCM: _______

3. 10= ________________

20 = ________________

LCM: _______

B. Use the continuous division to get the LCM of each set of

numbers. (Write your solution in your math notebook.)

1. 4 and 6 – LCM = _______ 3. 32 and 40 – LCM = ______

2. 8 and 12 – LCM = _______ 3. 50 and 75 – LCM = ______

3. 15 and 20 – LCM = _______

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The radio station DZLV pauses for

commercials every 10 minutes. DZAA

pauses for commercials every 15

minutes. After how many minutes

will they pause for commercials

at the same time?

How will you solve for the answer to the problem?

You can use the 4-step plan in solving for the answer.

A. Understand:

What does the problem ask for?

After how many minutes will they pause for commercials at the same time?

What facts are given? 10 minutes and 15 minutes

B. Plan:

How will you solve the problem?

By finding the Least Common Multiple (LCM)

C. Solve:

How is the solution done?

By Listing Method: 10 = 10, 20, 30, 40, … 15 = 15, 30, 45, 60, … LCM: 30 By Prime Factorization: 10 = 2 x 5 15 = 3 x 5 LCM: 2 x 3 x 5 = 30

Lesson 1.5. Solving Real-Life Problems Involving GCF and

LCM of Given Numbers

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The same steps will be applied in solving real-life

problems involving GCF using any of the methods in finding

the GCF of two numbers discussed in the previous lessons.

By Continuous Division: LCM: 2 x 3 x 5 = 30

D. Check and Look Back:

What is the answer to

the problem?

The two radio stations will pause

for commercials at the same time after 30 minutes.

5 10 15

2 3

In solving word problems, the keywords will help you to

decide how to solve it. Always look for the keywords.

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Let’s get started!

ACTIVITY NO.5A

A. Read each problem and answer the questions that follow.

1. David is going to put eggs in trays of 3 and 15 eggs. What is

the smallest number of eggs that David can put using the trays?

a. What is asked in the problem? ______________________

b. What facts are given? ____________________________

c. How will you solve the problem? ____________________

d. What is the answer to the problem? _________________

2. There are 16 boys and 24 girls. If they will be grouped

separately in teams with the same number, what is the biggest

number of children in a group?

a. What is asked in the problem? ______________________

b. What facts are given? ____________________________

c. How will you solve the problem? ____________________

d. What is the answer to the problem? _________________

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ACTIVITY NO.5B

You are doing great! Let’s keep

moving.

A. Read and solve each problem.

1. Samuel would like to put and equal number of mangoes and

apples in baskets. If there were 18 mangoes and 21 apples,

what is the biggest number of baskets that he could use? How

many mangoes and apples are in each basket?

2. Randy and Henry both began travelling around a circular track.

Randy is riding his bike, and Henry is walking. It takes Randy 5

minutes to make it all the way around, and Henry takes 20

minutes. How much time will pass until they meet at the

starting line?

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