8

Mathematics
Quarter 1 – Module 3:
Illustrating Rational
Algebraic Expressions
Mathematics – Grade 8
Self-Learning Module (SLM)
Quarter 1 – Module 3: Illustrating Rational Algebraic Expressions
First Edition, 2020

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Writers: John Rey P. Taberna, Ly Harvey A. Campos
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 8

Mathematics
Quarter 1 – Module 3:
Illustrating Rational
Algebraic Expressions
Introductory Message
For the facilitator:

Welcome to the Mathematics 8 Self-Learning Module (SLM) on Illustrating Rational

Algebraic Expressions!

This module was collaboratively designed, developed and reviewed by educators both
from public and private institutions to assist you, the teacher or facilitator in helping
the learners meet the standards set by the K to 12 Curriculum while overcoming
their personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help
learners acquire the needed 21st century skills while taking into consideration their
needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of
the module:

Notes to the Teacher

This contains helpful tips or strategies that
will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to manage
their own learning. Furthermore, you are expected to encourage and assist the
learners as they do the tasks included in the module.
For the learner:

Welcome to the Mathematics 8 Self-Learning Module (SLM) on Illustrating Rational

Algebraic Expressions!

The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a learner
is capable and empowered to successfully achieve the relevant competencies and
skills at your own pace and time. Your academic success lies in your own hands!
This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at your own pace and time. You will be enabled to
process the contents of the learning resource while being an active learner.
This module has the following parts and corresponding icons:

What I Need to Know This will give you an idea of the skills or
competencies you are expected to learn in the
module.

What I Know This part includes an activity that aims to

check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.

What’s In This is a brief drill or review to help you link

the current lesson with the previous one.

What’s New In this portion, the new lesson will be

introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.

What is It This section provides a brief discussion of the

lesson. This aims to help you discover and
understand new concepts and skills.

What’s More This comprises activities for independent

practice to solidify your understanding and
skills of the topic. You may check the answers
to the exercises using the Answer Key at the
end of the module.

What I Have Learned This includes questions or blank

sentence/paragraph to be filled in to process
what you learned from the lesson.

What I Can Do This section provides an activity which will

help you transfer your new knowledge or skill
into real life situations or concerns.
 Assessment This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.

Additional Activities In this portion, another activity will be given

to you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.

Answer Key This contains answers to all activities in the

module.

At the end of this module you will also find:

References This is a list of all sources used in developing

this module.

The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.

We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
 What I Need to Know

This module was designed and written to help you master how to illustrate
rational algebraic expressions. The scope of this module permits it to be used in
many different learning situations. The language used recognizes the diverse
vocabulary level of learners. The lessons are arranged to follow the standard
sequence of the course. But the order in which you read them can be changed to
correspond with the textbook you are now using. After going through this module,
you are expected to:

In this module, you will be able to:

 illustrate rational algebraic expressions.
M8AL-Ic-1

Specifically, you are expected to:

1. illustrate rational algebraic expressions; and

2. differentiate rational algebraic expressions from non-rational algebraic
expressions.
 What I Know

Before we start our lesson, let me know about your prior knowledge on
illustrating rational algebraic expressions.

Direction: Encircle the letter of the correct answer.

1. What expression contains operations, numbers, and one or more variables?

a. Basic Expressions
b. Numerical Phrases
c. Algebraic Expressions
d. Operational Coefficients

2. What is the constant in the expression 4x2 – 2x + 5?

a. 4
b. 2
c. 1
d. 5

3. How many terms are there in the expression 3x + 1?

a. 1
b. 2
c. 3
d. 4
4. What are the operations used in the expression 2x + y – 1?
a. + only
b. – only
c. Both + and –
d. Neither + nor –

5. What are the literal coefficients in the expression 2x + y + 3z?

a. x and y
b. x and z
c. y and z
d. x, y and z

6. What expression refers to the ratio of both polynomial numerator and

denominator?
a. Algebraic Expressions
b. Basic Rational Expressions
c. Rational Numerical Phrases
d. Rational Algebraic Expressions
 𝑃(𝑥)
7. What is the excluded value of the rational function, ?
𝑄(𝑥)
a. Q(x) ≠0
b. Q(x) =0
c. Q(x) ≥0
d. Q(x) ≤0

8. Which of the following is an algebraic expression?

a. 3x + 2y = 2
b. x + 1
c. 3 + 4 = 7
d. 4 + 2

9. What is the name of a term with no variable in an algebraic expression?

a. Coefficient
b. Variable
c. Constant
d. Factor

10. What do you call of the numerical factor of a term that contains a variable?
a. Coefficient
b. Term
c. Constant
d. Factor

11. Why is it important to know the excluded value/s of a rational algebraic

expression?
a. To prevent the expression to become define
b. To prevent the expression to become 1
c. To prevent the expression to become undefined
d. To prevent the expression to become greater than or equal to zero.

12. Which of the following expressions is a rational algebraic expression?

𝑥
a.
√3𝑦

3𝐶 −3
b.
√(𝑎+1)0

c. 4𝑦 2 + 𝑧 −3
𝑎−𝑏
d.
𝑎+𝑏
13. Which of the following expressions is NOT a rational algebraic expression?
𝑥+𝑦
a.
𝑥−𝑦

𝑦 2 +1
b.
𝑥𝑦 −3
𝑥
c.
√25𝑦 2
𝑎+𝑏+𝑐
d.
𝑎−𝑏+𝑐
14. Identify the terms in the expression, 2x 2 + x + 1.
a. 2, x, and 1
b. 2, 1, and 1
c. 2x2, x, and 1
d. x2, x, and 1

15. Which of the following expressions could be considered as rational algebraic

expression?
a. √50𝑥
1
b. 5𝑥 2
c. 4𝑦 2 − 9𝑧 −2
𝑏−𝑎
d. 𝑏+𝑎
 Lesson
Illustrating Rational
3 Algebraic Expressions
It is a good day to start this module. This module will help you illustrate
rational algebraic expressions. Moreover, this will focus on illustrating rational
algebraic expressions, and differentiating rational algebraic expressions from non-
rational algebraic expressions.

What’s In

Before the discussion of the new topic, let us review first your knowledge about
the following: translating the verbal phrases to mathematical phrases, the laws of
exponents, and identifying polynomials from non-polynomials.

Activity 1. Let us Match

A. Match the symbol with their correct implications.

Symbol Implication of Mathematical Symbols

______ 1. ( ) a. ratio of, the quotient of, divided by
______ 2. / b. equals, is equal to
______ 3. + c. the sum of, more than, increased by,
______ 4.- d. multiplied by, times
______ 5. = e. less than, decreased by, subtracted from,
the difference

B. Match the laws exponents with appropriate expressions.

Laws of exponent Rule

______1. Product Rule a. (ax)y = ax∙y

______2. Quotient Rule b. ax / ay = ax-y
______3. Power Rule c. ax ∙ ay = ax+y
______4. Inverse Rule d. a0 = 1
______5. Zero Exponent e. a-1 = 1/a
C. Match the name of polynomials with appropriate expressions.

Names of Polynomial Expression

______1. Binomial a. 2x
______2. Monomial b. 2x + 1
______3. Polynomial c. 2x2+2x+1
______4. Trinomial d. 2x3+2x2+2x+1
What are Non-polynomials?
1
1. √𝑥 or 𝑥 2 , a variable that has a rational power is not a polynomial.
1
2. , a variable found in the denominator is not a polynomial.
𝑥

3. 𝑥 −1 , a variable with a negative power is also not a polynomial.

What’s New

Good Job! We are done with our review. Let’s go to our new topic the Rational
Algebraic Expression. First, let us discuss what an algebraic expression is.

Activity 2. Can You Break It Down?

Directions: Identify the terms, numerical coefficient/s, literal coefficients and

constant of the following algebraic expression below.

Algebraic Terms Numerical Literal Constants

Expressions Coefficients Coefficients
Ex. 3b2 + 2c -5 3b2, 2c and -5 3 and 2 b2 and c -5

1. 4x + 2

2. 3c2 + 5x

3. 5b + 5

4. 3a2 + 2b2 + 1

5. 3r3 + 7s2
 What is It

A Algebraic Expression is an expression that contains operations, numbers,

and one or more variables.

Example 1: 5x – 2

In the above example, we can determine the parts of an algebraic expression.

5x and 2 are the two terms in the expression.
5 is the numerical coefficient or the numerical value of the term 5x.
x is the literal coefficient or variable written using a letter.
- is the operation used.
-2 is the constant or a term with no variable.

Example 2: 6y3 + 2z - 5

We can also easily identify the parts of the algebraic expression in the above
example.
6y3, 2z, and -5 are the three terms in the expression.
6, 2 are numerical coefficients.
y3, z are the literal coefficients.
- 5 is the constant.

In this section, you will answer the activity 1 to apply your knowledge in
identifying parts of algebraic expressions.

A Rational algebraic expression is an expression containing a numerator

and denominator that are both polynomials. It is an expression of P(x) and Q(x) in
𝑃(𝑥)
the form of , where Q(x) ≠ 0.
𝑄(𝑥)

𝑃(𝑥) 𝑥 2 +3𝑥+2 numerator

Example,
𝑄(𝑥) 𝑥 2 −1 denominator

where 𝑥 ≠ 1 and − 1.

You have learned in the past lesson about polynomials that if there is a radical
sign and a negative exponent, then the expression is not an algebraic expression.

Example:
 2√𝑥
the numerator has a radical sign ( ),so, this is
3
not a rational algebraic expression.
Except: √4 , √9 , √16 or any number that has a square root.

𝑥 −5 +2
the numerator and also the denominator has a
𝑦 −2
negative exponent, so, this is not a rational
algebraic expression

After this, try your ability to know if the expressions are rational or not.

What’s More

Activity 3. Can You Recognize Me?

Direction: Identify which of the following are rational algebraic expressions and
which are not. Write R if it is a rational algebraic expression and N if it
is a non- rational algebraic expression.

𝑘
_____1.
𝑚2 +3𝑘

𝑥+2
_____2.
𝑥−2

3𝑥
_____3.
𝑥 2 +𝑥

2
_____4.
𝑚−5

𝑏
_____5.
√2𝑥
 What I Have Learned

To summarize what you have learned. Fill in the blank with the appropriate
words.

In a rational algebraic, expression, the numerator and denominator should

both be _______________. Furthermore, the numerator with a variable inside radical
sign ( ) is ___ a rational algebraic expression. Moreover, the numerator and the
denominator with a _______ or ______ exponent, so, this is not a rational algebraic
expression.

What I Can Do

This activity will apply your knowledge on illustrating rational algebraic

expression in real-life situation.

Activity 4. Can you help John?

John has given a task to determine if the given expression is a rational
algebraic expression or not. Write H if it is a rational algebraic expression and A if it
is a non- rational algebraic expression.

64𝑟 3𝑎𝑏𝑐
_____1. _____4.
𝑚3 +5𝑛−17𝑜 𝑚−5

3𝑒𝑥 −2 25𝑚𝑛
_____2. _____5.
𝑓𝑥 2 +𝑔𝑥 2

𝑦+𝑟+7 3𝑎𝑏𝑐
_____3. _____6.
𝑦−𝑠−9 √𝑎𝑏𝑐𝑑𝑤𝑥𝑦𝑧

I hope that the pattern of the answers makes you laugh.

 Assessment

Good Job! Let us test what you have learned from the very start of our lesson.
Now, I want you to read carefully and answer the questions below.

Direction: Encircle the letter of the correct answer.

1. How many terms are there in the expression 5x + 4?

a. 1
b. 2
c. 3
d. 4

2. What are the literal coefficients in the expression 5x + 3y +2z?

a. x and y
b. x and z
c. y and z
d. x, y and z

3. What are the operations used in the expression 5x + 8y – 9?

a. + only
b. – only
c. Both + and –
d. Neither + nor –

4. What is the constant in the expression 5x2 – 3x + 9?

a. 2
b. 3
c. 5
d. 9

5. What expression contains operations, numbers, and one or more variables?

a. Basic Expressions
b. Numerical Phrases
c. Algebraic Expressions
d. Operational Coefficients

6. What expression refers to the ratio of both polynomial numerator and

denominator?
a. Algebraic Expression
b. Basic Rational Expression
c. Rational Numerical Phrase
d. Rational Algebraic Expression
7. Why is it important to know the excluded value/s of a rational algebraic
expression?
a. Can determined the values that can make the denominator equal to one
b. Can determined the values that can make the denominator equal to zero
c. Can determined the values that can make the denominator equal to
positive numbers
d. Can determined the values that can make the denominator equal to
negative numbers

𝑃(𝑥)
8. What is the excluded value of the rational function, ?
𝑄(𝑥)
a. Q(x) ≠ 0
b. Q(x) = 0
c. Q(x) ≥ 0
d. Q(x) ≤ 0

9. Which of the following is an algebraic expression?

a. 5x2 + 4y = 8
b. 2x + 4
c. 4 + 5 = 9
d. 7 + 2

10. What is the name of a term with no variable in an algebraic expression?

a. Coefficient
b. Variable
c. Constant
d. Factor

11. What do you call that number before variable in a term?

a. Numerical coefficient
b. Term
c. Constant
d. Literal coefficient

12. What are the terms in the expression, 4x2 + 2x + 3?

a. 4, x2, and 3
b. 4, 2, and 3
c. 4x2, 2x, and 3
d. 4x2, x, and 3
13. Which of the following expressions is NOT a rational algebraic expression?
2𝑥+𝑦
a.
𝑥−2𝑦

𝑦 2 +12
b.
4𝑥𝑦 −3
5𝑥
c.
√25𝑦 2
2𝑎+𝑏+𝑐
d.
𝑎−𝑏+2𝑐

14. Which of the following expressions could be considered as rational algebraic

expression?

a. √110𝑥
1
b. 5𝑥 2
c. 4𝑦 2 − 9𝑧 −2
2𝑏−𝑎
d.
𝑏+2𝑎

15. Which of the following expressions is a rational algebraic expression?

3𝑥
a.
√3𝑦

3𝐶−3
b.
√(𝑎+1)0

c. 4𝑦 2 + 𝑧 −3
4𝑎−𝑏+𝑐
d.
𝑎+𝑏
 Additional Activities

Activity 3. Can you differentiate?

Identify which of the following are rational algebraic expression and which are
not. Write Y if it is a rational algebraic expression and N if it is a non- rational
algebraic expression.

6𝑝
_____1.
𝑚4 +5𝑛

3𝑦+6
_____2.
3𝑦−8

4𝑥 −8
_____3.
𝑥 4 +𝑥

50𝑥𝑦
_____4.
𝑥 −5

5𝑎𝑏𝑐
_____5.
√5𝑎
 Assessment
What’s What’s What I Additional
1. B
In Activities
More Can Do 2. D
A.
1. d 3. C 1. Y
1. R 1. H
2. a 4. D 2. Y
3. c 2. R 2. A
5. C 3. N
4. e 3. R 3. H
6. D 4. N
5. b 4. N 4. A
7. B 5. N
5. N 5. H
B. 6. A 8. A
1. c 9. A
2. b
3. a
10. C
4. e 11. A
5.d 12. C
13. B
C.
1. b 14. D
2. a 15. D
3. d
4. c
Answer Key
 K to 10 Grade 8 Math Teachers Guide pp. 82-86
K to 10 Grade 8 Math Teachers Guide pp. 60-63
References
What I Know
What New
1. C
2. D Algebraic terms numerical literal constants
3. B Expressions coefficients coefficients
4. C Ex. 3b2 + 2c 3b2 3 and 2 b2 and c -3
5. D -3 and 2c
6. D 1. 4x + 2 4x 4 x 2
and 2
7. A
1. 3c2 + 5x 3c2 3 and 5 c2 and x None
8. B and
9. C 5x
10. A 2. 5b + 5 5b 5 b 5
11. C and 5
12. D 3. 3a2 + 2b2 3a2 3 and 2 a2 and b2 1
13. B +1 ,2b2 ,
14. B and 1
4. 3r3 + s2 3r3 3 and 1 r3 and s2 none
15. C
and s2
16. D
 DISCLAIMER
This Self-Learning Module (SLM) was developed by DepEd SOCCSKSARGEN
with the primary objective of preparing for and addressing the new normal.
Contents of this module were based on DepEd’s Most Essential Learning
Competencies (MELC). This is a supplementary material to be used by all
learners of Region XII in all public schools beginning SY 2020-2021. The
process of LR development was observed in the production of this module.
This is version 1.0. We highly encourage feedback, comments, and
recommendations.

For inquiries or feedback, please write or call:

Department of Education – SOCCSKSARGEN

Learning Resource Management System (LRMS)

Regional Center, Brgy. Carpenter Hill, City of Koronadal

Telefax No.: (083) 2288825/ (083) 2281893

Email Address: region12@deped.gov.ph