(m a teaching horizons for gen z learners
TRANSCRIPT
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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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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.
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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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