The document discusses mathematical concepts related to direct and joint variation, stating that the area of a wall to be painted varies directly with the number of workers and jointly with both workers and pails. It includes examples of translating statements into mathematical equations and exercises related to joint variation. Additional information about the properties of multiplication and a summary assignment on distributing canned soup is also provided.

1. MATHEMATICS 9
ROSE ANNE C TORRES
QUARTER 2

3. The area (a) of the wall to be
painted varies directly to the
numbers of workers (w).
What will be the
mathematical equation of
the given statement above?
a = kw
What type of variation is
shown in the situation?
Direct
Variation

4. Compare the first and
second situation.
How will you describe the
relationship between the
area, worker and pails?
How can you translate the
situation into mathematical
equation?
The area (a) of the wall to
be painted varies jointly to
the number of workers (w)
and the pail (p) needed to
do the task.

5. Translate the situation into
mathematical equation
a = kwp
The area (a) of the wall to
be painted varies jointly to
the number of workers (w)
and the pail (p) needed to
do the task.

6. JOINT VARIATION

7. Illustrate situations that involve
joint variation
Identify examples of situations
that involve joint variation
Appreciate the concept of joint
variation in real-life situation

8. It occurs when a quantity
varies directly as the
product of two or more
quantities
JOINT
VARIATION y varies jointly as x and z
y is jointly proportional
to x and z

9. JOINT VARIATIONS
y varies jointly as x and z
y = kxz

10. JOINT VARIATIONS
Example 2
a = kbh
The area (a) of triangle varies jointly as its
base (b) and height (h)

11. LEARNING TASK
Translate each statement into mathematical
sentence. Use k as constant of variation.
1. p varies jointly as q and r
2. v varies jointly as l and w
3. The area a of parallelogram varies jointly as
the base b and altitude h.
4. The volume of a pyramid v varies jointly as
the area of the base b and the altitude h.
5. The force f applied to an object varies jointly
as the mass m and the acceleration a.

12. QUIZ
Read each item carefully. Write the letter of the
correct answer.
1. What kind of variation is similar to direct variation but composed of two or more
variables?
a. combined c. inverse
b. direct d. joint
2. If z varies jointly as x and y, which of the following represents this relationship?
a. c. z = kxy
b. z = kx + y d.
3. To get 14,
we add 2.

13. Relationship of
Constant of
Variation for
Joint Variation
The constant of variation is
equal to dependent/independent
variable.
y = dependent variable
x and z = independent variable
𝒌 =
𝒚
𝒙𝒛

14. Example:
SOLUTION
Find the equation of
variation where y
varies jointly as x and
z, and y = 36 when
x = 3 and z = 4
y = kxz
36 = k (3)(4)
36 = k (12)
𝟑𝟔
𝟏𝟐
=
k (12)
𝟏𝟐
k = 3
Equation of Variation:
y=3xz

15. SEATWORK NO. 3
1. y varies jointly as x and z. If y = 3 when x = 3
and z = 15, find y when x = 6 and z = 9
2. d varies jointly as h and g. If d = 15 when h = 14
and g = 5, find g when h = 21 and d = 8
3. y varies jointly as x and z. If y=60 when x=3 and
z=4, find z when y=80 and x=2
4. c varies directly as a and b. If c=45 when a =15
and b=3, find c when a=21 and b=8
5. If t varies jointly with m and b and t=80 when m=2
and b=5, find t when m=5 and b=8
Translate each statement into mathematical
sentence. Use k as constant of variation.

16. Multiplying by 1
A number multiplied by 1 will result in the
same number.

17. Multiplying by 0
Any number multiplied by 0 will result in 0.

18. References
Mathspace. ‘Properties of multiplication (10x10) | Grade 4 Math | UK Primary
(3-6)’. Accessed 7 December 2023,
https://mathspace.co/textbooks/syllabuses/Syllabus-452/topics/Topic-8347/
subtopics/Subtopic-109695/?
bookType=textbook&searchString=&activeTab=theory

19. Activity
A.
B.
Key in the answer for each letter.
C.
D.
What’s the
Passcode?
The box containing the donated
clothes is secured with a lock. Key-in
the passcode by finding the missing
values mentally.
A B C D

20. 5 1 0 8
What’s the
Passcode?
The box containing the donated
clothes is secured with a lock. Key-in
the passcode by finding the missing
values mentally.
A.
B.
A B C D
Key in the answer for each letter.
C.
D.
Answer
Key
Activity

21. Commutativity or changes in
the position of numbers will result in
the same answer.
A number multiplied by 1
will result in the number itself.
Deriving facts is getting
new information from what we
already know.
A number multiplied by 0
will result in 0.
Summary

22. 6 cans per box
Assignment
Thirty boxes of canned soup are
donated.
Each box contains 6 cans of soup. If
these are to be evenly distributed to 6
different families, how many will each
family get?
Share your solution with the class in the
next session.

23. Resource
Page
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26. 2x = 6
3x - 9 = 100
3x + 7 = 98
25 -2x = 12
2x + y = 15