I.

OBJECTIVE CODE: (M9AL-IIb-c-3)

At the end of the lesson, the students are expected to:
a. translate combined variations statements into mathematical equation
b. solve problems involving combined variation
c. apply the concept of combined variation in real-life situation

II. SUBJECT MATTER – Combined Variation

A. Concept/ kx zy
Ideas The statement “z varies directly as x and inversely as y” means z = or k = , where k is the
y x
constant of variation.

To solve problems involving combined variation the following steps can be followed:
1. Identify the variables present and make a representation for each variable.
2. Translate statements into mathematical equation.
3. Find the constant variation k using the first set of the given value.
4. Rewrite the formula with the value of k.
5. Solve for the value of missing variable.
6. Check the answer obtained.
B. Materials Powerpoint Presentation, strips, video presentation, activity sheets and calculator
C. References Grade 9 Math Learners and Teacher’s Guide
D. Value Focus Accuracy
E. Integration MAPEH ( Music and Arts), Science ( Physics )
III. DEVELOPMENTAL ACTIVITY
EXPECTED
TEACHERS ACTIVITY ANSWERS/TEACHER
S NOTE

Greetings!

Prayer

Checking of Attendance
.
Setting of Classroom rules
1. Listen when someone is talking.
2. Raise your hand to speak, or to get up out of the chair
3. Follow directions when first giving
4. Be respectful to each other
5. Work quietly

A. Preliminary
DRILL: SPEED TEST Indicator # 2
Activities Used a range of
Multiply:
1. 3(4) = 12 teaching strategies
that enhance learner
2. 8(-6) = -48
achievement in literacy
3. 2(-3) = -6 and numeracy skills.
4. -3(-6) = 18
5. 5(-1) = -5 Note: Teacher
Divide: enhances students’
1. 9 ÷(−3) = -3 numeracy skills
through multiplication
2. −3 ÷(−1) = 3
and division using
3. −8 ÷(−2) = 4 speed test.
4. 8 ÷(4) = 2
5. −1 ÷1 = -1

B. Review What are the types of variation that we have learned?

 Direct Variation
 Inverse Variation
 Joint Variation
What is direct variation?
 Whenever a situation produces pairs of numbers in
which their ratio is constant.
What is inverse variation?
  Whenever a situation produces pairs of number
whose product is constant.
What is joint variation?
 Closely the same with direct variation, but joint
variation involves 3 or more variables.

Identify what type of variation is involved in each group of

pictures.
A

C. Motivation
Now, I want you to watch this video entitled “Minsan from Ang
Huling El Bimbo”. Watch the video carefully because there are
questions afterwards.

What can you say about the video presentation?

 Entertaining and enjoyable
 Variety of shows

How do they perform?

 They combined singing, dancing, acting and used of
spoken dialogue in one performance.

Do you know what their performance called? What is it?

 Yes, it is a Musical theatre Indicator # 1
Applied knowledge
Musical theatre it is a form of theatrical performance that of content within and
combines songs, spoken dialogue, acting and dancing. The story across curriculum
and emotional content of a musical-humor, pathos, love anger, are teaching areas.
communicated through words, music movement and technical Note: The teacher

aspects of the entertainment as an integrated whole. integrates MAPEH,

Music Quarter 4-
In math we also have this. We can combine two types of variation Module 2
to form a single type. Are you familiar with combined variation? (MU8THIVb-h-3)

D. Today, our lesson is about Combined Variation.

Presentation

This lesson deals with another concept of variation, the Combined

Variation- is another physical relationship among variables. You
will be learning how to translate variation statement into
mathematical equation, solve problems
 involving combined variation and apply combined variation in real
life situation

Present the learning objectives.

E. Activity LEARNING BY DOING
Proper Setting Standards
Indicator # 7
1. The class will be divided into 5 groups with 10 members Established a learner-
2. Each group will be given set of questions found in Activity centered culture by
Sheet to be answered in 10 minutes. Calculator is allowed for using teaching
strategies that respond
computation. to their linguistic,
3. After the allotted time, a reporter from the randomly chosen cultural, socio
group will present their work to the class in 2-3 minutes. economic and religious
backgrounds
4. Cooperate with your group and observe silence while doing the
activity. Note: The teacher presents
5. Remember the word GROUPS. the setting of standards in
doing group activity that will
Get along
result to an interactive and
Respect other’s ideas successful cooperative
On task learning amongst students.
Use quiet voices The teacher used
differentiated instruction by
Participate
grouping the students
Stay in your group according to their strength
and uniqueness.
I. Translate the variation statement into mathematical
equation, using k as the constant of variation.
1. W varies jointly as c and the square of a and inversely as b. Indicator # 1
2
kc a Applied knowledge
W=
b of content within and
across curriculum
2. The acceleration A of a moving objects varies directly as the teaching areas.
distance d it travels and inversely as the square of the time t it Note: The teacher
travels. integrates Laws of
kd Exponents,
A¿ 2 Mathematics 7,
t
Code: M7AL-IIc-4
3. The pressure P of a gas varies directly as its temperature t and And Translating
inversely as its Volume V. English Phrases and
kt sentences to
P¿ mathematical phrases
V
and sentences.
II. Solve the following problems. Code: M7AL-IIc-1
1. If x varies directly as the square of y and inversely as z and x =
12 when y = 3 and z = 6, find x when y = 9 and z = 6.
Variables: x, y and z Solving x
ky
2 k = 8, y = 9, z = 6 2.
x= ky
2
z x=
x = 12, y = 3, z = 6 z
2
8(9)
Solving k x=
6
2
k (3) 8(81)
12¿ x=
6 6
9k 648
12¿ x=
6 6
72 = 9k x=108
k=8

Checking
2 Indicator # 2
ky
x= Used a range of
z teaching strategies
2
8 (9) that enhance learner
108= achievement in literacy
6 and numeracy skills.
8 (81)
108= Note: The teacher
6 gives a worded
648 problem to enhance
108=
6 the literacy skills of the
108=108 students as well as the
numeracy skills.
In Manapla National High School, the number of girls varies
directly as the number of boys and inversely as the number of
teachers. When there were 50 girls, there were 20 teachers and 10
boys. How Many boys were there when there were 10 girls and
100 teachers?

Scoring Guide for problem Solving

Given – 1 point
Mathematical equation- 1 point
Computation with correct answer - 2 points
Checking - 1 point
Total - 5 points

Solution no. 2 Solving for b

Variables k = 100, g = 10, t = 100
g = number of girls kb
g=
b = number of boys t
t = number of teachers 100(b)
10=
100
Equation 10 = b
kb
g=
t

Solving for k
g = 50
t = 20
b = 10
kb
g=
t
k (10)
50 =
20
(50)(20) = (10)k
1000 = (10)k
 1000 ( 10 ) k
=
10 10
100 = k

F. Analysis In Activity 1 (Translate variation statement into

Indicator # 3
mathematical equation) Apply a range of
teaching strategies to
1. How do you find the activities? develop critical and
 Answers may vary creative thinking, as
well as other higher
order thinking skills.
2. What are the variations used in the statement?
 Direct variation, inverse variation and joint variation Note: The teacher
develops critical and
creative thinking as
3. What variable represent the constant of variation?
well as other higher
 k-is the constant ratio of two quantities order thinking skills
through probing
4. How do you translate combined variation statement into question. WH-
mathematical equation? questions are being
 Identify the variable used and set up equation based on the utilized by the teacher
variation used in the statement. in formulating the
questions

In Activity 2 (Solving Problems)

1. What are the variations present in the problem?

 Direct variation, inverse variation and joint variation

2. When do you use combined variation?

 When we combine any of the variations together (direct,
inverse, and joint)

3. What are the variables/quantities present in the problem?

g = # of girls
b = # of boys
t = # of teachers

4. What is the mathematical statement to represent the 3

quantities?
 The number of girls varies directly as the number of boys
and inversely as the number of teachers.

5. What did you do to solve for the value of k?

 By Using the formula being translated out of the given
statement and identifying known variables and substitute
the given to find the k or constant of variation.

6. How did you get the value of the missing variable?

 Using the value of k then substituted to original formula
and then using the second set of the given value.
G. Abstraction/ Based from your activity: Indicator # 3
Generalization Apply a range of
teaching strategies to
A. How do you translate variation statement into mathematical develop critical and
equation? creative thinking, as
well as other higher
 Identify the variable used and set up equation based on the order thinking skills.
variation used in the statement.
Note: The teacher
B. What is combined variation? develops critical and
 This is the kind of variation that involves both direct and creative thinking as
well as other higher
inverse variation. If the statement “z varies
order thinking skills
directly as x and inversely as y” means through probing
question.
 kx zy
z= or k = , where k is the constant of variation.
y x

C. How can we solve problems involving combined variation?

1. Identify the variables present and make a representation
for each variable.
2. Translate statements into mathematical equation.
3. Find the constant variation k using the first set of the
given value.
4. Rewrite the formula with the value of k.
5. Solve for the value of missing variable.
6. Check the answer obtained.
H. Applications Indicator # 3
Applied a range of
Having developed your knowledge of the concepts in this topic teaching strategies to
you are going to apply the concepts in various situations, this time develop critical and
let us have another activity. creative thinking, as
well as other higher
order thinking skills.
Go for It!
Pair up and talk it out. Find a partner in performing the
Note: The teacher
following activity. used the teaching
strategy “PAIR UP AND
I. DV and IV Combined. Translate the variation statement into TALK IT OUT “, gave
mathematical equation, using k as the constant of variation. questions/activities to
develop the higher
order thinking skills of
1. The Volume V of gas is directly proportional to the temperature the students and asked
t, and inversely proportional to the pressure p. the students to reflect
the importance of the
topic in their daily life.
kt (refer to Valuing)
 V=
p

2. The stopping distance d of a vehicle varies directly with its

speed s, and inversely with the friction f of the road.

ks
 d=
f
Indicator # 1
II. How Well Do You Understand? Applied knowledge
1. The force of attraction F of a body varies directly as its mass m of content within and
and inversely as the square of the distance d from the body when across curriculum
m = 8 kilograms d = 5 meters, and F = 100 Newtons. Find F when teaching areas.
Note: The teacher
m = 2 kilograms and d = 15 meters.
integrates Physics-
the topic
Gravitational Force
(Science 8, Quarter 1-
Week 3, Forces and
Motion-)MELC
S8FE-Ia-16)
Solution no. 1 Solving F
Variables k = 312.5
F = force of attraction m = 2 kgs.
m = mass d = 15 meters
d = distance
(312.5)(2)
Equation F=
(15)2
km 625
F= 2 F=
d 225
Solving k F=2.78 Newton
F = 100 Newton or
m = 8 kg. 25
d = 5 meters F= Newton
9

k (8)
100¿
52
k (8)
100=
25
Indicator # 9
2500=k ( 8 ) Used strategies for
k = 312. 5 providing timely,
accurate and
constructive feedback
to improve learner
performance.
Math Challenge
Note: The teacher
If z varies directly as the square of x and inversely as the monitors the students’
square root of y, and z is 3 when x is 6 and y is 16, find z when learning progress
x is 12 and y is 25. through correcting
their work/solution in
problem solving to
The following are the values of k and z that Chad and Ken got make sure if the lesson
when they solve the problem. objectives are being
met. It is more likely to
provide students with
Who do you think got the correct values of k and z? justify a continually improving
your answer learning experience.
Providing constructive
feedback to improve
Chad’s Solution Ken’s Solution their learning
Equation of variation Equation of variation performance.
kx 2 kx 2
z= z=
√y √y
Solving k Solving k
z = 3, x = 6, y = 16 z = 3, x = 6, y = 16
2 2
k (6) k (6)
3= 3=
√16 √16
36 k 12 k
3= 3=
4 4

12=36 k 12=12 k
3=k 1=k

Solving z Solving z
k = 3, x = 12, y = 25 k = 1, x = 12, y = 25
2 2
kx kx
z= z=
√y √y
2 2
3(12) 1(12)
z= z=
√25 √25
3(144) 1(24)
z= z=
5 5
 432 24
z= z=
5 5
z=86.4 z=4.8

What do you think is the importance of this concept in our real-

life situation? Give one concrete example.

For examples is Building a house

We must consider the following
number of workers (w)
number of days (d)
expense (e)

As the numbers of workers increases the number of days

decreases, when it comes to number of expenses, as the number of
workers increases your expenses also increases.

The number of workers (w) varies inversely to the number of days

(d) to finish building the house
k
w=
d
The number of workers (w) varies directly to the expense (e)
w=ke

Building a House
The number of
The number of
workers (w)
“w varies directly with e and workers (w)
varies inversely
inversely with d” varies directly
to the number of
to the expense
days (d) ke
w= (e)
d
number of workers (w)

k
w= w=ke
d
number of days (d) expense (e)

How can we apply the concept of combined variation most Indicator # 8

especially in indigenous group? Adapted and used
culturally appropriate
teaching strategies to
The Ati of Negros and Panay (Please refer to Powerpoint address the needs of
Presentation) learners from
indigenous group.
 Note: The teacher uses
Since the Ati is known for their handicrafts, Figure below shows experiential learning or
the application of Real life situation of the combined variation learning by doing
most especially in their economy, specifically in their handicraft instead of studying it
without application
making.

For instance, you have a Business Plan (Making of handicrafts)

Refer to the figure below
Suppose you will make handicrafts for a small business, you may
consider these three things:
Number of products (p)
Number of materials (m)
Capital (c).

As the number of products increases the number of materials increases

but at the same time as the number of products increases the capital
decreases.

The number of products (p) varies directly to the number of

materials needed (m), it is a direct variation when it comes to the
number of materials and number of products. But when it comes to
the capital the number of products (p) varies inversely to the
capital (c). If we combine these two different type of variations,
we can create a what we called COMBINED VARIATION,
wherein we can say that “p varies directly with m and inversely
with c”.
km
We can interpreted p=
c
Business Plan
IV. Evaluation
I. Translate the variation statement into mathematical equation,
using k as the constant of variation.

1. P varies directly as the square of x and inversely as s.

2. The electrical resistance R of a wire varies directly as its length
l and inversely as the square of its diameter d.

II. Solve the given problem.

The number of horses varies directly as the number of goats and
inversely as the number of pigs squared. When the barnyard
contained 5 horses there were 4 pigs and only 2 goats. How many
goats were there when there were 6 pigs and 10 horses?

V. Assignment
Answer the following questions.

1. Assume that y varies directly as x and inversely as z. How will y change when x
and z are each tripled?

2. Assume that z is directly proportional to x and inversely to y 2. If x is increased by

10 % and y is decreased by 10 %, what is the percentage of increase in z?

Reflection/ Total
Number
Instructional Total Number
of Number of
Decision Number of
Year & Students Students that Mastery Instructional
of Students
Section Who Got Needs Level Decision
Students Who
75% & Remediation
in Class Took the
Above
Test
For instance, you have a Business Plan (Making of handicrafts)
Suppose you will make a handicrafts for a small business, you may consider these three things
Number of products (p)
Number of materials (m) Capital (c).
As the number of products increases the number of materials increases but at the same time as the number of products
increases the capital decreases.
The number of products (p) varies directly to the number of materials needed (m), it is a direct variation when it
comes to the number of materials and number of products. But when it comes to the capital the number of
products (p) varies inversely to the capital (c). If we combine these two different type of variations we can
create a what we called COMBINED VARIATION, wherein we can say that “p varies directly with m and
Business Plan inversely with c”.

The number of km
The number of We can interpreted p=
products (p) c
“p varies directly with m and products (p)
varies inversely
inversely with c” varies inversely
of materials
to the capital
needed (m) km
p= (c)
c
number of products (p)

p=km
k
p=
c
number of materials needed Capital (c)
(m)
Directions: Given a two-column
proof, supply the missing
part/statement to prove the Midline
Theorem.
Given: , E is the midpoint
∆ HNS , Ois the midpoint of HN

of NS

Prove: , OE ∥ HS
1
OE= HS
2

Building a house
We must consider the following
number of workers (w)
number of days (d) do we have to finish construction of a house and third will be what would be
expense (e)
As the numbers of workers increases the number of days decreases, when it comes to number of expenses, as
the number of workers increases your expenses also increases.

The number of workers (w) varies inversely to the number of days (d) to finish building the house
k
w=
d
The number of workers (w) varies directly to the expense (e) w=ke
“w

N
1

2 E
O
3 T

4
H S
varies directly with e and inversely with d”

.
 Statements Reasons
1. ∆ HNS , O is the midpoint 1. Given
of HN , E is the midpoint of
NS

2. In a ray opposite ⃗
EO, 2. In a ray, point at a
there is a point T such given distance from
that OE = ET the endpoint of the
ray.
3. EN ≅ ES 3.

4. ∠ 2 ≅ ∠ 3 4.

5. ∆ ONE ≅ ∆ TSE 5.

6. ∠ 1 ≅ ∠ 4 6.

7. HN ∥ ST
7.

8. OH ≅ ON 8.

9. ON ≅ TS 9.
10. OH ≅ ST 10.

11. Quadrilateral HOTS is 11.

a parallelogram.
12. OE ∥ HS 12.

13. OE + ET =OT 13.

14. OE +OE=OT 14.

15. 2OE=OT 15.

16. HS ≅ OT 16.

17. 2OE=HS 17.

18. 1
(The segment
OE= HS
2 18.
joining the midpoints of
two sides of a triangle is
half as long as the third
side)
For verifying
1. The group shall draw and cut a different kind
of triangle out of bond paper.

2. Choose a third side of a triangle. Mark each

point of the other two sides then connects the
midpoints to from a segment
Does the segment drawn look parallel to the
third side you chose?
______________________________________
______________________________________

3. Measure the segment drawn and the third

side you chose.
Compare the lengths of the segments drawn
and the third side you chose. What did you
observe?
______________________________________
______________________________________

4. Cut the triangle along the segment drawn.

What figures are formed after cutting the
triangle along the segment drawn?
______________________________________
5. Reconnect the triangle with the other figure
in such a way that their common vertex was a
midpoint and that congruent segments formed
by a midpoint coincide.

What new figure is formed?

______________________________________
Make a conjecture to justify the new figure
formed after doing the activity. Explain your
answer
______________________________________
______________________________________
______________________________________

What can you say about your findings in

relation to those of your classmates?
______________________________________
______________________________________

Do you think that the findings apply to all kinds

of triangles? Why?
______________________________________
______________________________________
______________________________________
Directions: Given a two-column proof, supply the missing
part/statement to prove the midline theorem.
Given: ∆ HNS , Ois the midpoint of HN , E is the midpoint of NS
1
Prove: OE ∥ HS , OE= 2 HS
 N
1

O 2 E
3 T
4
H S

Statements Reasons
1. ∆ HNS , O is the midpoint of HN , E is 1. Given
the midpoint of NS
2. In a ray opposite ⃗
EO, there is a 2. In a ray, point at a given
point T such that OE = ET distance from the endpoint of the
ray.

3. EN ≅ ES 3.
4. ∠ 2 ≅ ∠ 3 4.
5. ∆ ONE ≅ ∆ TSE 5.
6. ∠ 1 ≅ ∠ 4 6.
7. HN ∥ ST 7.
8. OH ≅ ON 8.
9. ON ≅ TS 9.
10. OH ≅ ST 10.
11. Quadrilateral HOTS is a 11.
parallelogram.
12. OE ∥ HS 12.
13. OE + ET =OT 13.
14. OE +OE=OT 14.

15. 2OE=OT 15.

16. HS ≅ OT 16.
17. 2OE=HS 17.
1 18.
18. OE= 2 HS (The segment joining the
midpoints of two sides of a triangle is
half as long as the third side)

For verifying
1. The group shall draw and cut a different kind of triangle out of bond
paper.
2. Choose a third side of a triangle. Mark each point of the other two sides
then connects the midpoints to from a segment
Does the segment drawn look parallel to the third side you chose?
_____________________________________________________________________________
_____________________________________________________________________________

3. Measure the segment drawn and the third side you chose.
Compare the lengths of the segments drawn and the third side you chose.
What did you observe?
_____________________________________________________________________________
_____________________________________________________________________________

4. Cut the triangle along the segment drawn.

What figures are formed after cutting the triangle along the segment
drawn?
_____________________________________________________________________________
_____________________________________________________________________________

5. Reconnect the triangle with the other figure in such a way that their
common vertex was a midpoint and that congruent segments formed by a
midpoint coincide.

What new figure is formed?

_____________________________________________________________________________
_____________________________________________________________________________

Make a conjecture to justify the new figure formed after doing the activity.
Explain your answer
_____________________________________________________________________________
_____________________________________________________________________________

What can you say about your findings in relation to those of your
classmates?
____________________________________________________________________________

Do you think that the findings apply to all kinds of triangles? Why?
_____________________________________________________________________________
_____________________________________________________________________________
ACTIVITY CARD
GROUP 4