Republic of the Philippines

Visayas State University

Visca, Baybay, City Leyte
College of Education
Department of Teacher Education

Practice Teachers: Date and Time:

Val Edmar C. Devocion October 24-25, 2022 (3:00-5:00)
Resyl Bei S. Morales October 27-28, 2022 (1:00-3:00)

Cooperating Teacher:
Dr. Leo A. Mamolo

DETAILED LESSON PLAN FOR GENERAL MATHEMATICS

Subject: General Mathematics Grade Level: Grade XI

Content Area: Rational Functions
Module/Lesson: Module 2 – Lesson 2.1 and 2.2

Title of the Lessons:

Rational Functions, Equations and Inequalities
Solving Rational Equations

I. Intended Learning Outcomes:

At the end of the lesson, the Grade XI students should be able:
1. Define rational functions confidently
2. Represent real-life situations of rational functions easily
3. Differentiate rational functions, rational equations, and rational inequalities
confidently
4. Solve rational equations

Values Integration:
 Cooperation and engagement in learning deeper on rational functions by
means of the activities/seatwork given.
 Accuracy – students will be able to represent real-life situations of rational
functions accurately.
 Mastery – students will be able to differentiate rational functions, rational
equations, and rational inequalities. They will also be able to master the
topic at hand.
 Perseverance – students will keep trying to solve rational equations.

II. Content and Materials:

 Subject: General Mathematics
Topic: Rational Functions, Rational Inequalities
Solving Rational Equations
References: Mamolo, L. (2020). General Mathematics: Functions, Business
Mathematics and Logic. Module 2 pp 9-17.
Polynomials: Their Terms, Names, and Rules Explained |
Purplemath

Instructional Materials:
 Modules
 Laptop
 PowerPoint Presentation
 Projector
 Paper and Pen

III. Teaching-Learning Experiences

Planned Teacher Behavior Expected Student Behavior

A. Routinary Activities:

(Resyl’s part)

a. Prayer
 Requesting everyone to please stand for the
prayer. Who will lead the prayer today?  Student A will lead the prayer.

b. Energizer
 Please remain standing for the energizer. *The student will execute the energizer*

c. Greetings
 Good morning, everyone!  Good morning and mabuhay
ma’am Resyl
 You may take your seats.  Thank you, ma’am.

d. Checking of attendance
 Who is absent today class?  No one is absent ma’am.
 Okay very good! We have a perfect attendance.

e. Checking of Learning Task/Check

 Are you done with your learning task and your
learning check that was uploaded to our VSU E-  Yes ma’am.
Learning?
 Wow very good!
 Now, before we proceed, I just want to tell
everyone that every time I give instructions or
during my discussion, if I will ask you “Am I
 clear?” you have to respond to me “crystal
ma’am” and if I will ask “Am I crystal?”, you have
to respond “clear ma’am”.

 Am I clear class?  Crystal ma’am.

B. Preparatory Activity

a. Review
 In our previous lesson, we have tackled about
Multiplying and Dividing Functions as well as
Composite Functions.
*The students will raise their hands and
 Now, in order to verify if you really understood answer the given questions. *
the previous topic, let us have a short review.
1.
1. Let f and g be functions. a. ( f ∙ g ) ( x )=f ( x ) ∙ g ( x )

a. How do we compute their product?

b. How do we compute their quotient? b. ( fg ) ( x )= gf (( xx))
c. How about their composite?
c. ( f ᴏ g )( x )=f ( g ( x ) )
2. In multiplying functions, the product should be
in what form? 2. Reduced or simplest form

3. What are the important steps in dividing 3. Change the operation into
functions? multiplication and get the reciprocal of
the divisor
4. Let f ( x )=2 x +1 and g ( x )=5 x+3 . Find ( g ᴏ f )( x ) .
4. ( g ᴏ f )( x )=10 x +8
 Okay very good! Based on your answer you have
already learned multiplying and dividing functions
as well as composite function and I think you’re
all ready for our next lesson.

b. Motivation

 But before anything else, I know all of you find

Mathematics difficult, but always keep in mind
that math is significantly essential to our life. As
what sir Leo always say “Learning mathematics
is a matter of mindset” If you set your mind with
positive thoughts like “I can do this” then “You
can do it”

(Val’s Part)
C. Developmental Activities

a. Activating Prior Knowledge

 Now let us have a group activity. I will group you
into four groups. Please count 1 to 4 class.
*The student will count 1 to 4*

 Each group will identify if the given mathematical

expression is a polynomial or not. You will raise *The class forms a circle*
the word YES if it is polynomial, otherwise raise
the word NO. Each member should participate
because the group who will get the highest points
will be given rewards.

 Are you ready class?

 Yes, we are ready sir.
Mathematical Expressions:
Answer:
3 2
1. x −2 x +5 x +10
1. YES
2. 5 x+ 1
2. YES
−2
3. x +5
3. NO
2
4. 4. NO
x+5
5. NO
5.
√x
5
6. YES
6. 10
7. NO
1
7. 3 x 2 +2

 Thank you, class, for your participation. Great

job!

(Resyl’s part)

b. Analysis
 Now, who can share with us their observation in
the previous activity?
 It is all about Polynomials and Not
Polynomials ma’am.
  Yes student 3?
*Students’ answer may vary*
 Okay thank you. Now, how did you identify that
the given mathematical expression is a
polynomial or not.

 Okay very good class!

 Well done, everyone. And along the discussion

we will learn deeper and will encounter more of
these.

c. Abstraction

 Our topic for today is all about Rational

Functions. At the end of the lesson, students are
expected to:
1. Define rational functions confidently
2. Represent real-life situations of rational
functions easily
3. Differentiate rational functions, rational
equations, and rational inequalities
confidently
4. Solve rational equations

 Now, let us define first some important

definitions.

Definitions:
 Polynomial function – is a function in the form
n n−1
f ( x )=an x + an−1 x +…+ a1 x +a 0 where a 0 , a 1 , … a n
are real numbers, a n ≠ 0, and n is a positive
integer. Each addend of the sum is a term of the
polynomials. The constants a 0 , a 1 , a2 , … , an are
coefficients. The leading coefficient is a n . The
n
leading term is a n x , and constant term is a 0.

 To better understand this, let us have an

example

Example:
 Signs/Indication that the given expression is
not a polynomial:

1. Variables with negative exponent - x−2 +5

5 x +1
2. Variables in the denominator -
3x
 Crystal ma’am.
3. Variables with radical sign -
√x
5
1
4. Variables with fractional exponent - 3 x 2 +2

 Am I clear class?

 Rational Function - a function of the form

p(x )
f ( x )= w here p(x ) and q (x) are polynomial Answer:
q (x)
functions and q (x) is not zero function (i.e., 1. Rational Function (Clap 2 times)
q (x) ≠ 0 ¿.
2. Not (Stamp left foot 2 times)
 Try this! Identify if the given expression is a
rational function or not. Clap 2 times if the given 3. Rational Function (Clap 2 times)
expression is a RATIONAL FUNCTION. Stamp
your left foot 2 times if it is NOT. 4. Not (Stamp left foot 2 times)

4 x+ 2 5. Not (Stamp left foot 2 times)

1. f ( x )= 2
x + 10

2. f ( x )=
√x
2  Yes ma’am.

3. f ( x )=4
−3
( ) x +1
4. f x =
4

x+2
5. f ( x )=
x−x
 Can you now easily distinguish rational functions
from not?

 Okay very good class! So now, let us proceed.

 Now, before we proceed, I have one question.

Why do you think studying rational functions is
important?

 Rational functions can model a number of real-

life situations. and can find answers to real
problems. For example:

Work-time-rate problems ( w=rt )

Distance-speed-time problems (d =st )

 Now, if you wonder how rational functions can

help in real-life situation, let us proceed to
representing real-life situations of rational
function.

 Example 1: A truck that delivers essentials in

remote areas can travel 85 kilometers. Express
the velocity v as a function of travel time t in
d
hours? Use the formula of velocity v=
t
85
Solution: The function v (t)= can represent v
t
as a function of t.

 Example 2: If the truck in example no. 1 was

delayed by 4 hours due to the checkpoints that it
passed through, express the time t as a function
of velocity v in km/hr.?

Solution:
d
v=
t

vt=d

d
t=
v
 85
t ( v )= +4
v

 Example 3: If the distance from Manila to

Lucena is approximately 140 kilometers,
represent the function (s), where s is the speed
of travel that describes the time it takes to drive
from Manila to Baguio.

Solution:
140
s ( t )=
t

 Now for you to practice more, please take a

picture of the following problem and send your
answer to VSUEE tomorrow.

Use the problem below for the Practice Exercise:

Due to the Enhanced Community Quarantine,

Banawe Footspa temporarily stopped its
operation and to help the employees the owner
decided to split evenly its total revenue of  None so far ma’am.
₱65,000.00
 Crystal ma’am.
1. If the number of employees is represented by
x, represent a function that show the amount
each employee received.

2. If the owner held a fund-raising activity that

aimed to help the employees and collected
₱5000.00 per employee, which of the
following represents the total amount an
employee will receive?
 Yes ma’am.
 Do you have any questions class?

 Am I clear class?

 Now, let’s proceed to identifying and

differentiating rational functions, rational
equations, or rational inequality.

 But before that, let us have a short activity which

is matching type. In your activity notebook, match
column A with Column B.
  Are you done class?
 So, we will go back to this part later and check
your work.

 But before defining those terms, let us recall the

basic symbols of equality and inequalities.
EQUALITY
SYMBOL NAME
¿ equal sign

INEQUALITIES

SYMBOL NAME
≠ not equal sign
¿ greater than
¿ less than
≥ greater than or equal to
≤ less than or equal to

 Don’t forget this class because these are very Answer:

useful as we progress with our lesson.
 Now, let us proceed and define important terms 1. RE
we will encounter in this lesson.
2. RE
 Rational Expression – is an expression that can 3. NOT
be written as a ratio of two polynomials.
4. NOT
 Based from the given definition of rational
expression, let us identify if the given expression 5. RE
is rational expression or not. Say RE if you think
it is a rational expression, otherwise, say NOT.
2
x +2 x+ 1
1.
x+3

3
2. 2
5x

3.
√ x +2
2 x 2−3

x−6
4. −2
x +5
 2
x +5 x +6
5.
3

 Okay, very good!

 Now, let us differentiate rational functions,
rational equations, and rational inequality.

Rational Rational Rational

Function Equation Inequality
Definition A function in An equation An
the form of: involving inequality
rational involving
p(x ) expressions rational
f ( x )=
q (x) expressions

where p(x )
and q (x) are
polynomial
 Clear ma’am.
functions
and q (x) is
not zero
 None ma’am.
function

2
x +4
f ( x )=
x +1
5 2
5 2 2 ≤ Answer:
Example or − = x−3 x
x 3x 7
2 1. Rational Function
x +4
y=
x +1 2. Rational Inequality

3. Rational Equation
 Am I crystal class?
4. Rational Equation
 Do you have any questions regarding rational
function, rational equation, and rational
inequality? 5. None of these
(Val’s Part)
 Yes sir.
 Okay, from the discussion of ma’am Resyl. I want
you to identify, if the given expression is a
rational function, rational equation, or rational
 inequality. *Student’s will check their own work*

1
1. f ( x )= 2  Yes sir.
x + 6 x+5

2 1
2. 3 x+ <
x +3 2 x
*Students will execute the dance
exercise*
2 x
3. −3=
x 3x+4
x+ 2 2
4. =x  Thank you, sir.
x−5

3 x + √ x−1
2
5. y= 2
x −2

 Can you now differentiate rational functions,

rational equations, and rational inequalities?

 Let’s go back to the activity given by ma’am

Resyl and check you work.

 Did you all guess it right?

 Okay, well done everyone!

 Before we proceed to solving rational equation,

requesting everyone to please stand. Let’s have
an ice breaker first. So please follow the short
dance exercise I have prepared.
 Please take your seats class.

To solve a rational equation:

1. Eliminate denominators by multiplying each term
of the equation by the least common
denominator (LCD)
2. Note that eliminating denominators may
introduce extraneous solutions.
3. Check the solutions of the transformed equations
with the original equation and identify the real &
extraneous solution.

 Extraneous solutions are values that we get

when solving equations that aren't really
solutions to the equation or will make the
 equation false.
 Real solutions are values that we get when
solving equations that will satisfy the equation or
make the equation true.

x−1 2
 Example 1. Solve for x: =
15 5

Solution: The LCD of all the denominators is 15.

Multiply both sides of the equation by 15 and
solve the resulting equation.

15 ( x−1
15 ) =( ) 15
2
5

15(x−1) 30
=
15 5

x−1=6

x=7

Check:

7−1 2
=
15 5

6 2
=
15 5

2 2
=
5 5

Therefore, the solution x=7 is a real solution to the

equation.

2 3 1
 Example 2. Solve for x: − =
x 2x 5

Solution: The LCD of all the denominators is

10x. Multiply both sides of the equation by 10x
and solve the resulting equation.
 10 x ( 2x − 23x )=( 15 ) 10 x ; x ≠ 0
20 x 30 x 10 x
− =
x 2x 5

20−15=2 x

5=2 x

5 2x
=
2 2

5
=x
2

Check:

2 3 1
− =
5
2
2
5
2 ()5

2 3 1
2∙ − =
5 5 5

4 3 1
− =
5 5 5

1 1
=
5 5

5
Therefore, the solution x= is a real solution to the
2
equation.

x 1 8
 Example 3. Solve for x: x+2 − x−2 = 2
x −4

Solution:

1. Factor the denominator in the rational

expression.

x 1 8
− =
x+2 x−2 (x +2)(x−2)
 2. Multiply the LCD to both sides of the equation
to remove the denominators.

( x +2 ) ( x −2 ) [ x
−
1
x +2 x −2
= ][ 8
( x+ 2 )( x−2 ) ]
(x +2)(x−2)

x( x+2)(x−2) 1 ( x +2 ) ( x −2 ) 8 (x+ 2)(x−2)

− =
x+ 2 x−2 ( x+ 2)(x−2)

x ( x −2 )−( x +2 )=8

2
x −2 x−x−2=8
2
x −3 x−10=0

3. Upon reaching this step, we can use factoring

method.
2
x −3 x−10=0
 None sir.
( x +2 ) ( x −5 )=0

4. Apply the Zero Product Property to solve for

x.

x +2=0 x−5=0

x=−2 or x=5

Since x=−2 makes the original equation undefined,

x=5 is the only solution.

Check:
5 1 8
− = 2
5+2 5−2 (5) −4
5 1 8
− =
7 3 25−4

15−7 8
=
21 21

8 8
=
21 21
Therefore, the solution x=5 is a real solution to the
equation.

 Any questions class?

(Resyl’s Part)

 Now let us have a challenge problem.

 Mira takes 2 hours to plant 50 flower bulbs.

Francis takes 3 hours to plant 45 flower bulbs.
Working together, how long should it take them
to plant 150 bulbs? Use the work rate formula
w=rt

Solution:

Using the formula, w=rt , we derive the formula of

w
rate r = .  Yes ma’am
t

50
Mira’s rate: =25
2

45
Francis’ rate: =15
3

Combine their hourly rates and equate to the rate

that they need to work together. The equation
now becomes:

150
25+15= ; x≠0
t

Multiply both sides of the equation by the LCD

and solve the resulting equation.

25 t+ 15t=150

40 t=150
15
t=
4
 3
t=3 hrs.
4

Therefore, it should take 3 hours 45 minutes for Mira

and Francis to plant 150 bulbs together.  None ma’am/sir.

 Did you get it class?

(Val and Resyl’s part)

d. Application

 Good to hear that class. To test your

understanding on the lessons we have tackled. In
your own sit, try to answer the following:

1. Due to the inclement weather the plane slows

down the regular flying rate which results to
additional 2 hours in covering a 4000-km
distance to its regular time.
a. Write a function that expresses the time as  Yes ma’am/sir.
a function of rate during inclement weather
in travelling. Answer:

9 4 1. Rational expressions
2. Solve for x: =
3 x x+ 2
2. Rational inequality
3. In an inter-barangay basketball league, the 3. Rational Function
team from Barangay Culiat has won 12 out of
25 games, a winning percentage of 48%. How
many games should they win in a row to 4. Rational equation
improve their win percentage to 60%? (To be 5. Cross multiply and finding the LCD.
submitted in VSUEE)

 Do you have any question class regarding our

topic this afternoon?

 If there’s none, let’s have an activity.

e. Generalization
 Our activity is entitled “Pass, Pick, and Try”. I
have here a small box which contains papers
with questions regarding rational functions,
equations, and inequalities. The mechanics of
this activity is you will sing a song Up and Down
and Shake, Shake, Shake. Whoever receive the
 small box when the song stops, you will pick a
number from the small box and answer the
corresponding questions. And if you get the
correct answer, you will receive rewards.

 Are you ready class?

Questions:
1. It is an expression that can be written as a ratio
of two polynomials.
2. It is an inequality involving rational expressions
p(x )
3. It is a function in the form of f ( x )= where
q (x)
p(x ) and q (x) are polynomial functions and q (x)
is not a zero function
4. It is an equation involving rational expressions
5. What are the two ways in solving rational
equations?

f. Evaluation

 I think you are all ready for the quiz. Get 1 whole
sheet of paper and answer the following within 15
minutes.

1. What is rational function?

2. An object is to travel a distance of 10 meters.

Express velocity v as a function of travel time
t , in seconds.

3. Identify which among the given expression is

the rational function, rational equations, and
rational inequality. Write RF for rational
function, RE for rational equations, and RIE
for rational inequalities.

5 3
a. + =0
x+2 5+ x

x−4 1
b. ≥
x +2 x
2
x −4 x+3
c. f ( x )= 2
3 x −4 x +5
 x−3 x
4. Solve for x: =
x+ 5 x +2
 Good bye and thank you ma’am
8 9 Resyl and sir Val.
5. Solve for x: =
x−5 x −4

g. Assignment

 Okay class, for you to practice more, I will give

you an assignment regarding our topic for today
and you will write your answer in a one whole
sheet of paper.

1. In an organ pipe, the frequency f of

vibration of air is inversely proportional to
the length L of the pipe. Suppose that the
frequency of vibration is a 10-foot pipe is
54 vibration per second. Express f as a
function of L.

2. Jonathan can finish a job in 3 hours

working alone while Lucas can finish the
job in 5 hours working alone. How long will
it take both people to finish that job
working together? Use the work formula to
find their rate ( w=rt )

3. Write your own example of rational

function, rational equation, and rational
inequalities. (Give one each)

5 x−3
4. Solve for x: =2+
2
x + x−6 x −2

2 3 5
5. Solve for x: + =
x+1 x x−2

 That’s all for today class, thank you for listening

and participating. I hope you have learned
something. Goodbye.

 Enjoy the rest of the day. God bless everyone.

Prepared by:
Val Edmar C. Devocion
Resyl Bei S. Morales

Approved by:

Dr. Leo A. Mamolo

Cooperating Teacher

Signed by:

Dr. Ma. Rachel Kim L. Aure

Supervisor