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UNIT 1: INTRODUCTION
TO ALGEBRA
Algebra (from Arabic: ‫الجبر‬ (al-jabr, meaning "reunion of broken
parts" and "bonesetting") is one of the broad parts of mathematics, together
with number theory, geometry and analysis.
It is the study of mathematical symbols and the rules for manipulating
these symbols; it is a unifying thread of almost all of mathematics. It also
includes everything from elementary equation solving to the study of
abstractions such as groups, rings, and fields.
In this module, it will focus on simplifying algebraic expression by using
order of operations. It also includes solving different word problems involving
ratio and proportion.
Having basic skills in algebra will be a great help in solving different
complex and complicated word problems in different areas of mathematics
such as geometry, probability, statistics and calculus.
LESSON 1: ORDER OF OPERATIONS
OBJECTIVES:
 Define what order of operation means,
 Demonstrate their understanding of the order of operations on word
problems, and
 Simplify the mathematical expressions using the order of operation.
LESSON PROPER:
"Operations" mean things like add, subtract, multiply, divide, squaring,
etc. If it isn't a number it is probably an operation.
But, when you see something like ...

2
7 + (6 × 52
+ 3)
... what part should you calculate first?
Start at the left and go to the right?
Or go from right to left?
Warning: Calculate them in the wrong order, and you can get a wrong answer!
So, long ago people agreed to follow rules when doing calculations, and they
are:
Do things in Parentheses First
4 × (5 + 3) = 4 × 8 = 32
4 × (5 + 3) = 20 + 3 = 23 (wrong)
Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract
5 × 22
= 5 × 4 = 20
5 × 22
= 102
= 100 (wrong)
Multiply or Divide before you Add or Subtract
2 + 5 × 3 = 2 + 15 = 17
2 + 5 × 3 = 7 × 3 = 21 (wrong)
Otherwise just go left to right
30 ÷ 5 × 3 = 6 × 3 = 18
30 ÷ 5 × 3 = 30 ÷ 15 = 2 (wrong)
How Do I Remember It All...? PEMDAS!
P Parentheses first
E Exponents (i.e. Powers and Square Roots, etc.)
MD Multiplication and Division (left-to-right)
AS Addition and Subtraction (left-to-right)
Example 1: How do you work out 3 + 6 × 2?

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Multiplication before Addition:
First 6 × 2 = 12, then 3 + 12 = 15
Example 2: How do you work out (3 + 6) × 2?
Parentheses first then Multiplication:
First (3 + 6) = 9, then 9 × 2 = 18
Example 3: Sam threw a ball straight up at 20
meters per second, how far did it go in 2 seconds?
Sam uses this special formula that includes the effects of
gravity:
Height = velocity × time − (1/2) × 9.8 × time2
Sam puts in the velocity of 20 meters per second and time of 2 seconds:
Height = 20 × 2 − (1/2) × 9.8 × 22
Now for the calculations!
Start with: 20 × 2 − (1/2) × 9.8 × 22
Parentheses first: 20 × 2 − 0.5 × 9.8 × 22
Then Exponents: 20 × 2 − 0.5 × 9.8 × 4
Then the Multiplies: 40 − 19.6
Subtract and DONE! 40 − 19.6=20.4
Therefore, the ball reaches 20.4 meters after 2 seconds.
ACTIVITY: Simplify the following mathematical expression using
the order of operation. Show your complete solution.
1) 21 ÷ 3 + (3 × 9) × 9 + 5
2) 18 ÷ 6 × (4 - 3) + 6
3) 14 - 8 + 3 + 8 × (24 ÷ 8)
4) 4 × 5 + (14 + 8) - 36 ÷ 9
5) (17 - 7) × 6 + 2 + 56 - 8

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LESSON 2: RATIO
OBJECTIVES:
 Identify the concept of ratio,
 Solve different word problems involving ratio, and
 Develop honesty, confidence and accuracy in solving ratio problems.
LESSON PROPER:
A Ratio compares values.
A ratio says how much of one thing there is compared to another thing.
There are 3 blue squares to 1 yellow square
Ratios can be shown in different ways:
Use the ":" to separate the values: 3 : 1
Or we can use the word "to": 3 to 1
Or write it like a fraction: 3/1
A ratio can be scaled up:
Here the ratio is also 3 blue squares to 1 yellow square,
even though there are more squares.
The trick with ratios is to always multiply or divide the numbers by the
same value.

5
Example 1:
4 : 5 is the same as 4×2 : 5×2 = 8 : 10
Example 2: There are 5 pups; 2 are boys, and 3 are girls
Part-to-Part:
The ratio of boys to girls is 2:3 or 2
/3
The ratio of girls to boys is 3:2 or 3
/2
Part-to-Whole:
The ratio of boys to all pups is 2:5 or 2
/5
The ratio of girls to all pups is 3:5 or 3
/5
Ratios can have more than two numbers!
For example, concrete is made by mixing cement, sand, stones and
water.
A typical mix of cement, sand and stones is written as a ratio, such
as 1:2:6.
We can multiply all values by the same amount and still have the same
ratio.
10:20:60 is the same as 1:2:6

6
x 2
So when we use 10 buckets of cement, we should use 20 of sand and 60
of stones.
You have 12 buckets of stones but the ratio says 6.
That is OK; you simply have twice as many stones as the number in the
ratio ... so you need twice as much of everything to keep the ratio.
Here is the solution:
Cement Sand Stones
Ratio Needed: 1 2 6
You Have: 12
And the ratio 2:4:12 is the same as 1:2:6 (because they show the
same relative sizes)
So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You
will also need water and a lot of stirring....)
Why are they the same ratio? Well, the 1:2:6 ratios say to have:
 twice as much Sand as Cement (1:2:6)
 6 times as much Stones as Cement (1:2:6)
In our mix we have:
 twice as much Sand as Cement (2:4:12)
 6 times as much Stones as Cement (2:4:12)
ACTIVITY: COUNT ME IN!
1. There are 10 animals: 5 are chicken, 2 are dogs and the rest are cats.
a) What is the ratio of chicken to dogs?
b) What is the ratio of cats to chicken?
c) What is the ratio of total number of animals to cats?
d) What is the ratio of dogs to cats?
2. In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. If
the bag contains 120 green sweets, how many red sweets are there?
3. A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. If a
bag of the mixture contains 12 pounds of rice, how much corn does it contain?
4. Clothing store A sells T-shirts in only three colours: red, blue and green. The
colours are in the ratio of 3 to 4 to 5. If the store has 20 blue T-shirts, how many
T-shirts does it have altogether?
5. Mr. Fenuyat D. Jenuwa handles a class of 3232 students, of which 2020 are girls.
Write the ratio of girls to boys.

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LESSON 3: PROPORTION
OBJECTIVES:
 Understand the concept of proportion,
 Illustrate problems involving proportion, and
 Develop honesty, confidence and accuracy in solving word problems
involving proportion.
LESSON PROPER:
Proportion says that two ratios (or fractions) are equal.
Example 1:
So 1-out-of-3 is equal to 2-out-of-6
The ratios are the same, so they are in proportion.
Example 2: Rope
A rope's length and weight are in proportion.
When 20m of rope weighs 1kg, then:
 40m of that rope weighs 2kg
 200m of that rope weighs 10kg
 etc.

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So:
20:1 = 40:2
When shapes are "in proportion" their relative
sizes are the same. Here we see that the
ratios of head length to body length are the
same in both drawings.
So they are proportional.
Making the head too long or short would look
bad!
Example 3:
International paper sizes (like A3, A4, A5, etc.) all have the
same proportions:
Working With Proportions
NOW, how do we use this?
Example: you want to draw the dog's head ... how long should
it be?
Let us write the proportion with the help of the 10/20 ratio from above:
?:42 = 10:20
Now we solve it using a special method:
Multiply across the known corners,
then divide by the third number

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And we get this:
? = (42 × 10) / 20
= 420 / 20
= 21
So you should draw the head 21 long.
Using Proportions to Solve Percent
A percent is actually a ratio! Saying "25%" is actually saying "25 per
100":
25% = 25/100
We can use proportions to solve questions involving percent.
The trick is to put what we know into this form:
Part : Whole = Percent : 100
Example 4: What is 25% of 160?
The percent is 25, the whole is 160, and we want to find the "part":
Part : 160 = 25 : 100
Multiply across the known corners, then divide by the third number:
Part = (160 × 25) / 100
= 4000 / 100
= 40
Example 5: What is $12 as a percent of $80 ?
Fill in what we know:
$12 : $80 = Percent : 100

10
Multiply across the known corners, then divide by
the third number. This time the known corners are top
left and bottom right:
Percent = ($12 × 100) / $80
= 1200 / 80
= 15%
Example 6: The sale price of a phone was $150, which was only
80% of normal price. What was the normal price?
Fill in what we know:
$150 : Whole = 80 : 100
Whole = ($150 × 100) / 80
= 15000 / 80
= 187.50
Example 7: How tall is the Tree?
Sam tried using a ladder, tape measure, ropes and various other things,
but still couldn't work out how tall the tree was.
But then Sam has a clever idea ... similar triangles!
Sam measures a stick and its shadow (in meters), and also the shadow of
the tree, and this is what he gets:
Now Sam makes a sketch of the triangles, and writes down the "Height to
Length" ratio for both triangles:

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Height: Shadow Length: h2.9 m = 2.4 m1.3 m
h = (2.9 × 2.4) / 1.3
= 6.96 / 1.3
= 5.4 m (to nearest 0.1)
ACTIVITY: Illustrate the following word problems involving
proportion. Show your solution.
1. David paints 3 rooms in 7 hours. At the same pace, how long would it
take him to paint 15 rooms?
2. A monsoon dumped rain on a coastal area. In twelve hours 20 inches
of rain had fallen. How much rain will fall over a period of 2 days, if it continues
at this rate?
3. Sarah Beth makes cookies at the rate of 2 dozen per hour. She has to
make a total of 216 cookies for a wedding party. How long will it take her?
4. When juggling a ball travels in a complete circle every 2 seconds. How
many circles does it make in a minute?
5. Baseball cards come in packs of a dozen (12) cards. Matt has 132
baseball cards. How many packs of baseball cards did he buy?

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UNIT 2:
NUMBER THEORY
We have used the natural numbers to solve problems. This was the right set of
numbers to work with in discrete mathematics because we always dealt with a whole
number of things. The natural numbers have been a tool. Let's take a moment now to
inspect that tool. What mathematical discoveries can we make about the natural numbers
themselves? This is the main question of number theory: a huge, ancient, complex, and
above all, beautiful branch of mathematics.
Historically, NUMBER THEORY was known as the Queen of Mathematics and was
very much a branch of pure mathematics, studied for its own sake instead of as a means to
understanding real world applications. This has changed in recent years however, as
applications of number theory has been unearthed. Probably the most well-known example
of this is RSA cryptography, one of the methods used in encrypt data on the internet. It is
number theory that makes this possible.
In this unit, you must have differentiated prime from composite numbers; expressed
a given number as a product of its factors; identified the divisibility rules of one-digit
numbers and selected two-digit numbers; identified factors and multiples of numbers; and
solved for GCF and LCM of two or more numbers using various methods.
LESSON 1: PRIME AND COMPOSITE
NUMBERS
OBJECTIVES:
At the end of the lesson, you must 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 PROPER:
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?
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:

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1. All five numbers have two digits so it cannot be used as basis.
2. Two of the numbers are even while three are odd, so again, this classification
does not make any one number different.
3. 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
 Numbers with more than two factors are called COMPOSITE.
Examples:
4, 15, 46, and 120
Direction: 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 is the biggest two-digit prime number.
3. 138 is COMPOSITE since it is even. Each of the even numbers greater than 2 has 2
as factor other than 1 and itself which makes 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.
PRIME FACTORIZATION
 It is a process by which every composite number can be expressed as a product of
its prime factors.

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For example, let us factorization of 24 using the Factor Tree Method. Let us then
express the answer in exponential form.
24
4 6
2 2 2 3
 The prime factorization of 24 is 2 x 2 x 2 x 3 or .
 To check if the prime factorization of a given number is correct, we ask
ourselves two questions:
1. Are all factors PRIME already?
2. Is the product of prime factors the SAME with the given
number?
 In our given number,
1. The factors are 2 and 3 which are both PRIMES.
2. The product of 2 x 2 x 2 x 3 is 24.
 Therefore, our prime factorization of 24 is CORRECT.
Give the prime factorization of 90.
90
10 9
2 5 3 3
 The prime factorization of 90 is .
d
Choose any pair of factors of 24.
Aside from 4 and 6, we may also use 2 and
12 or 3 and 8.
Choose any pair of factors of 24.
We may also use choose from 2 & 45, 3 &
30, 5 & 18, and 6 & 15.

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In summary, we have
 PRIME NUMBER – a whole number greater than 1 which has only two
factors: – 1 and the number itself.
 COMPOSITE NUMBER – a whole number with three or more factors.
 PRIME FACTORIZATION – process of expressing a composite number as a
product of its primes.
ACTIVITY: Solve the following word problem involving prime and
composite numbers. Show your complete solution.
1. Fenuyat D. Jenuwa is thinking of a composite number between 60 and 70. The
number has prime factors that have a sum of 12. What is Odessa’s number?
2. The area of Mr. Paeffal’s house and lot in Cagayan City is 342 square meters. If
the length is a prime less than 25, then what is the width of the lot?
3. Use prime factorization to find the prime factors of 939.
4. Can a prime number be a negative number? Justify your answer.
5. Are all even numbers considered to be composite numbers? Explain.
LESSON 2: GREATEST COMMON FACTOR
(GCF)
OBJECTIVES:
 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 and
 Solve real-life problems involving GCF of 2-3 given numbers (M5NS-Ie-70.2).
LESSON PROPER:
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 piece length with no remainders.
What is the greatest possible length of each of the smallest pieces of string?
1. LISTING METHOD

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One way to answer the problem above is to simply LISTING down all the possible
whole meter pieces which each string can be cut into exactly.
For instance, the 48 m string can be cut into 1 m, 2 m, 3 m, 4 m, 6m, 8 m, 12 m, 16 m,
24 m, and 48 m pieces. The second string, 80 m, can be cut into 1 m, 2 m, 4 m, 5 m, 8 m, 10
m, 16 m,20 m, 40 m, 'and 80 m pieces. Finally, the 96 m string can be cut into 1 m, 2 m, 3
m,4m, 6 m, 8 m, 12 m, 16 m, 24 m, 32 m, 48 m, and so m pieces.
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 the three strings can 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 of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80. On the
other hand, the factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.
The common factors of 48, 80 and 96 are 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.
2. 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 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 2x 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.

17
3. CONTINUOUS DIVISION METHOD
True to its name, in using the continuous division method, we continue dividing the
given numbers by a common prime number until the quotients are relatively prime .
Let's take a look the process using the same numbers above.
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 | 6 10 12__________ Divide by 2.
3 5 6
Since 3, 5, 6 are already relatively prime, therefore 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.
ACTIVITY: Find the GCF of the following numbers using the method
indicated.
1. 15, 20, 36 – Listing Method
2. 24, 36, 60 – Prime Factorization
3. 42, 72, 90 – Continuous Division
4. Mr. Demaca Moobon wishes to distribute 84 balls and 108 bats equally
among the number of boys. Find the greatest number of boys who will receive the gift in
this way.
5. I am a single-digit number. If I divide 39, 85, and 113, there will be
remainders of 3, 4, and 5, respectively. What is the greatest number I could possibly be?

18
LESSON 3: LEAST COMMON MULTIPLE
(LCM)
OBJECTIVES:
 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 (M4NS-IIc-68.1 and 69.1); and
 Solve real-life problems involving LCM of 2-3 given numbers (M5NS-Ie-70.2).
LESSON PROPER:
WVSU-Himalayan City 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?
1. LISTING METHOD
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 above.
STEP 1: List the multiples of each number.
60 - 60, 120, 180, 240, 300, …
90 - 90, 180, 270, 360, …
45 - 45, 90, 135, 180, …

19
STEP 2: Find the FIRST COMMON MULTIPLE of the numbers.
Since 180 is the first multiple common to all three numbers, then it is LCM of the
numbers.
2. PRIME FACTORIZATION METHOD
STEP 1: Find the prime factorization of the numbers.
60 - 2 x 2 x 3 x 5
90 - 2 x 3 x 5 x 3
45 - 3 x 5 x 3
2 x 2 x 3 x 5 x 3
STEP 2: Multiply the common multiples.
2 x 2 x 3 x 5 x 3 = 180
Observe that unlike in GCF where a 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.
3. CONTINUOUS DIVISION METHOD
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
not divisible by the prime divisor.
2 |20 30 15 Divide by 2. Bring down 15 since it is not
divisible by 2.
 Two or more numbers are
said to be relatively prime if
they have no more common
divisors except 1.

20
3 |10 15 15 Divide by 3.
5 |10 5 5 Divide by 5.
2 1 1
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
We can say that the next time they will all ring together again is 10:00 a.m.
ACTIVITY: Find the LCM of the following numbers using the given method.
1. 48, 64 – Listing Method
2. 24, 36, 42 – Prime Factorization
3. 2, 4, 6, 8, 10 – Continuous Division
4. A lighthouse flashes its light every 12 minutes. Another lighthouse flashes
every 18 minutes. If the two lighthouse flash together at 12:00 noon, at what time will they
next flash together?
5. I have 3 numbers. They are consecutive multiples of 3. Their sum is 27.
What is the LCM of these 3 numbers?

21
REFERENCES
Lopez-Mariano, Norma D. Business Mathematics. REX Bookstore. 2016
Daligdig, Romeo M. Mathematics in the Modern World. LORIMAR Publishing, Inc. 2019
Earnhart & Adina. Mathematics in the Modern World. C & E Publishing, Inc. 2018
Camarista & Oranio. Teaching Mathematics in the Intermediate Grades. Lorimar Publishing Inc.
2020
https://www.mathsisfun.com/operation-order-pemdas.html
https://www.mathsisfun.com/numbers/ratio.html
https://www.mathsisfun.com/algebra/proportions.html
https://www.onlinemathlearning.com/proportion-problems.html
https://www.chilimath.com/lessons/introductory-algebra/greatest-common-factor/
https://www.brighthubeducation.com/homework-math-help/31394-two-methods-of-
finding-the-greatest-common-factor/
https://www.mathsisfun.com/least-common-multiple.html

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