The document outlines objectives and activities related to direct variation in mathematics, including definitions, examples, and real-life applications such as recycling campaigns. It provides exercises for learners to identify direct variation in different scenarios and includes tasks for translating relationships into mathematical equations. Additionally, it emphasizes the importance of understanding direct variation through practical exercises and problem-solving.

1. DIRECT
VARIATION
MATHEMATICS 9 – QUARTER
2

2. OBJECTIVES
a.Illustrate situations that involve
direct variation
b.Identify examples of direct
variation
c.Appreciate the concept of direct
variation in real-life situation

3. ___ I R ___ C ____
(adj) - extending or moving from one place to another by
the shortest way without changing direction or stopping.
V A ___ I ____ T ___ O ___
- is defined by any change in some quantity due to
change in another
- fairly simple relationships or formulas
GUESS
THE
WORD?

4. 4 TYPES OF VARIATION
• DIRECT VARIATION
• INVERSE VARIATION
• JOINT VARIATION
• COMBINED VARIATION

5. If waste papers were recycled regularly,
it would help prevent the cutting of
trees, global warming and other
adverse effects that would destroy the
environment. Do you agree?
“WILL A DECREASE IN PRODUCTION OF
PAPER CONTRIBUTE TO THE DECREASE IN
THE NUMBER OF TREES BEING CUT?”

6. The SSG of Kaong National High School launches a recycling campaign of waste materials in
order to raise students’ awareness of environmental protection and the effects of climate
change. This is in connection with the celebration of Earth Month. Every kilogram of waste
materials earns points that can be exchange for school supplies. Paper, which is the number
one waste collected, earns 6 points for every kilo.
The table below shows the points earned by Grade 9 Mangga class for every number of
kilograms of waste paper collected.

7. DIRECT VARIATION

8. Activity: Think-Pair-Share
Direction: Find a partner. Read and analyze the given situation
then answer the question that follows.
Ronnie walks a distance of 1 km per 20 minutes at a constant
rate. The table shows the distance he has walked at a particular
length of time.

9. Activity: Think-Pair-Share
Direction: Put a check (√ ) on the blank before the letter if the given
situation is a direct variation and put a cross( x ) if it is not.
_____ 1. The distance an airplane flies to the time travelling
_____ 2. The time a teacher spends in checking test papers to the
number of students
_____ 3. The number of hours to finish a job to the number of
persons working
_____ 4. The age of a used cellphone to its resale value
_____ 5. The number of persons sharing a buko pie to the size of the

10. Answer the following questions:
1.) Rudy sold 3 baskets of banana at Php35 per kg. If
a basket contains 8 kg, how much did Rudy earn?
2.) Candies are sold at Php1.50 each. How much will a
bag of 420 candies cost?
3.) A photocopy machine can finish 500 pages in 3
minutes. How many pages can the machine copy in 1
hour?

11. Direct variation
- is a type of proportionality wherein one quantity directly varies with respect to a
change in another quantity. This implies that if there is an increase in one quantity
then the other quantity will experience a proportionate increase. Similarly, if one
quantity decreases then the other quantity also decreases.
- is a linear relationship hence, the graph will be a straight line.
- This can be expressed in mathematical statement of equation as y = kx, where is
the constant of variation or constant of proportionality. These statements can be:
“ y varies directly as x”
“ y is directly proportional to x” and
“ y is proportional to x”

12. EVALUATION
1. Read and analyze the problem to answer the following questions given.
2. Every week, Kenneth puts Php15.00 in his piggy bank. In the following table,
n is the number of weeks and s is the savings in peso.
Questions:
1. What happens to Kenneth’s savings as the number of weeks doubled? tripled?
2. In how many weeks would he have saved Php360?
3. How much will be his savings after 15 weeks?
4. What mathematical statement can represent the relation?
5. Give the constant number involved in the situation.

13. ASSIGNMENT
Follow-up
1. Give at least three examples of situations illustrating
direct variation.
2. Study on how to translate into variation statement a
relationship between two quantities given by (a) table
of values; (b) a mathematical equation; (c) a graph.

14. OBJECTIVES
a.Translate into variation statement a
relationship involving direct variation
between two quantities given by a table of
values and vice versa
b.Find the unknown in a direct variation
equation
c.Appreciate the concept of direct variation
in real-life situation

15. RECALL
A. Show that distance (d) varies directly as time (t)
B. Draw the graph of d against t
C. Write an equation showing the relationship between d
and t.
D. How far Kyle travelled after 8 hour and 10.5 hours?
Preliminary Activity!
Determine if the table below express a direct variation between two quantities or not.
1.
2.
3.
4.

16. EXAMPLE
The table below shows that the distance
travelled by the motorist varies directly as
the time spent. Find the constant of
variation and the equation which describe
the relation.
Solution: Since the distance d varies directly as the time t, then d = kt.

17. EXAMPLE
Using one of the pairs of values, (2, 20), from the table, substitute the values of d and t in
d = kt and solve for k. So,
d = kt
20 = 2k
20 = 2k (To find k, divide both sides by 2)
2 2
k = 10
Therefore, the constant of variation is 10. We can see that the constant of variation can be
solved if one pair of values x and y is known. From the resulting equation, other pairs
having the same relationship can be obtained

18. TRY ME
Direction: Analyze the problem then answer the following questions.
Kaong Junk shops pay Php12.00 for every kilo of tin cans bought from
collectors. The table shows c as the cost in peso and h as the number of
kilos of tin cans:
Question:
• Write a mathematical statement that relates the two quantities n and c, formulate
the equation.
• What is the constant of variation?

19. MASTER ME
Direction: In each of the following table of values, find
the constant of variation (k) and an equation that
defines the relation.
1. 3.
2.

20. We can do it
Direction: Form 5 groups and perform the problem given.
The circumference of a circle inside the basketball court of Wasid Integrated
School varies directly as its diameter. If the circumference of a circle having a
diameter of 7 cm is 7 cm, what is the circumference of it whose diameter is 10 cm?
15 cm? 18 cm? 20 cm?
Task:
1. Write a mathematical statement that relates the two quantities involved in the
problem.
2. What is the constant of variation? Formulate the mathematical equation.
3. Solve for the constant of variation given the following relation to complete the data
needed.
4. Construct a table of values from the relation.

21. Short Quiz:
Direction: Determine if the table of values express a direct variation between two
variables. If they do, find the constant of variation and an equation that defines the
relation.
•
•
1.
2.
3.
5.
6.

22. Assignment:
1. Follow-up
Find the constant of variation and write the equation representing
the relationship between the quantities of the table of values given.
2. Study on how to translate into variation statement a relationship
between two quantities given by
(a) a mathematical equation; (b) a graph.

23. REMEMBER ME
Draw the graph of the given set of ordered pairs. Plot it on one
Cartesian plane. Label the x values as time t and the y values as
distance d

24. EXAMPLE
Let us now try to replace “=” to ≥, so it
becomes y ≥ 2x + 3. We can notice that
the points on the line are still part of the
solution therefore we will still use a solid
line. Points (-6, -2), (-3, 3), (-2, 0), (1, 7),
and (-5, -4) are also solutions to the
inequality y ≥ 2x + 3. Observe the
example below:

25. EXAMPLE

26. PRACTICE
For the given example y ≥ 2x + 3
1. What can you observe about the points
2. Identify the 5 points from the left side of the line, are
they also solutions to the inequality?
3. Take a point on the right side of the line, is it part of
the solution?
4. What can you conclude?

27. PRACTICE
For the given example y ≥ 2x + 3
1. What can you observe about the points
2. Identify the 5 points from the left side of the line, are
they also solutions to the inequality?
3. Take a point on the right side of the line, is it part of
the solution?
4. What can you conclude?

28. A. Show that distance (d) varies directly as time (t)
TIME (t) hour 1 2 3 4 5
DISTANCE (d) km 10 20 30 40 50
Ratio = = 10 =10 = 10 = 10 = 10

29. B. Draw the graph of d against t

30. C. Write an equation showing the relationship between d and t.

31. Oral Participation
1. Write an equation of the following statements.
a. The distance D travelled by a car varies directly as its speed s.
b. The weight W of an object is directly proportional to its mass
s.
c. An employee’s salary S varies directly as the number of days d
he has worked.
2. If x varies directly as y and x = 35 when y = 7, what is the value
of the constant of variation? Write the equation of variation.
3. If y = 12 when x = 4, find y when x = 12.
4. If y = -18 when x = 9, find y when x = 7.

32. Find the constant of variation for the given table of
values.

33. Given the graph, find the constant of
variation and equation of variation.

34. QUIZ
A. Write an equation for the following statements.
1. The fare F of a passenger varies directly as the distance d of
his destination.
2. The cost C of fish varies directly as its weight w in kilograms.
3. The cost of electricity C varies directly as the number of
kilowatt-hour consumption n.
4. The area A of a triangle is proportional to its height h.
5. The length L of a person’s shadow at a given time varies
directly as the height h of the person.

35. B. Given the table of values, find the constant of
variation and equation of variation.
C. Given the graph, find the constant of variation
and equation of variation.
x 1 2 3 4
y 3 6 9 12
Series1
10
20
30
40
50
Column1 Column2 Column3
1 2 3 4 5
50
40
30
20
10
0
y
x

36. D. Find the constant of variation and write an
equation of variation.
1.y = 28 when x = 7
2.y = 30 when x = 8

37. Slide Title
Product A
• Feature 1
• Feature 2
• Feature 3
Product B
• Feature 1
• Feature 2
• Feature 3