DETAILED LESSON PLAN IN GRADE IX
I. Specific Objectives
At the end of 60 minute period at least 75% of the students should be able to:



Identify the equation that expresses a direct variation
Determine the table of values and the graph that expresses a direct variation
Solve the given problem using y=kx as the mathematical equation of direct variation,
where k is a constant of variation
II. Subject Matter
A. Topic: Direct Variation
B. Reference: Mathematics Learner’s Material for Grade 9, pp. 194-203
http://www.purplemath.com/modules/variatn.htm
http://www.freemathhelp.com/direct-variation.html
C. Materials:
1 Laptop – for encoding lesson plan
1 Cartolina – for visual aid
2Pentel pen (Black and Red) – for visual aid
1 Scotch tape – for visual aid
D. Concepts:
Essential Concept:
 Direct Variation
Sub-Concepts:
 Identify the equation that expresses a direct variation
 Determine the table of values and graph that expresses a direct variation
 Solve the given problem using y=kx as the mathematical equation of direct
variation, where k is a constant of variation
Teaching Strategies:
 Demonstration/Lecture Method, Discovery and Process Approach,
Problem Method, Board work
III. Procedure
Teacher’s Activity
Student’s Activity
A. Activity
1. Pre-activity
 Prayer
 Checking of Attendance
 Greetings
2. Introductory Activity
Please see at the last page of this lesson
plan.
In the name of the Father…. Amen.
Good morning, ma’am!
3. Lesson Proper
The activity you’ve done is connected to
the topic for today which is “Direct
Variation”.
Direct Variation – when a situation
produces pairs of numbers in which their
ratio is constant.
The mathematical formula of Direct
Variation,
y=kx, where x - independent variable
y – dependent variable
k – constant of variation
can be translated in mathematical
statements;
“y varies directly as x”
“y is directly proportional to x”
“y is proportional to x”
For 2 quantities, x and y, an increase in x
causes an increase in y as well. Similarly, a
decrease in x causes a decrease in y.
Suppose we have the equations, its table of
values and graphs:
y=4x
x
-2
-1
0
1
2
y
-8
-4
0
4
8
x
-2
-1
0
1
2
y
4
2
0
-2
4
y=-2x
y=2x+3
x
-2
-1
0
1
2
y
-1
1
3
5
7
From the 3 equations which do you think
is/are the example/s of a direct variation?
Why?
y=4x and y=-2x, because it’s in the form
of y=kx, while y=2x+3 is not an example
of direct variation.
Correct! This is in the form of slopeintercept form y=mx+b, where m is the
slope and b is the y-intercept.
How about y=4x and y=-2x, is it in a form
of slope-intercept form? Why?
Yes. It is in the form of y=mx+b, but its yintercept/b is zero.
Correct! Therefore, direct variation is
simply in a slope-intercept form when b/yintercept is zero. So, in y=4x and y=-2x,
their y=intercepts are zero and their slope
are 4 and -2, similarly the slope determines
the constant of variation.
Do you get it?
Yes!
From the given 3 graphs, how do you
determine if it expresses a direct variation?
The graph should have the same slope and
always crosses the origin (0,0)
Correct!
Graph of y=4x: Expresses Direct Variation
Graph of y=-2: Expresses Direct Variation
Graph of y=2x+3: Do not expresses Direct
Variation
From the given 3 table of values, how do
you determine if it expresses a direct
variation?
Correct! But when the value of x is zero the
value of y will always be zero and its ratio
will always be zero. If you take a look in its
graph, the graph of y=4 and y=-2 crosses
the origin (0,0).
Get the ratio of y and x in each column. If
it is the same/equal then there is a constant
of variation and the table of values express
a direct variation.
y=4x
Get the ratio of y and x in each table of
values to determine if it expresses a direct
variation.
x
-2
-1
0
1
2
y
-8
-4
0
4
8
R
4
4
0
4
4
As you can see the ratio is equal to the Expresses Direct Variation
constant of variation.
y=-2x
x
-2
-1
0
1
2
y
4
2
0
-2
4
R
-2
-2
0
-2
-2
Expresses Direct Variation
y=2x+3
x
-2
-1
0
1
2
y
-1
1
3
5
7
R
-1/2
-1
∞
5
7/2
Do not expresses direct variation
Now, how do you solve the given word
problem which expresses a direct variation
Example #1
If y varies directly as x and y=24 when x=6,
find the variation constant and the equation
of variation.
Solution:
Express “y varies directly as x” as y=kx
Solve for k by substituting the given values
in the equation.
y = kx
24 = 6k
k = 4 – constant of variation
Substituting 4 to k in y=kx,
y = 4x – equation of variation
Example #2
The table below shows that the distance d
varies directly as the time t. Find the
constant of variation and the equation
which describes the relation.
Time (hr.)
1 2 3 4 5
Distance (km)
10 20 30 40 50
Solution:
Since the distance d varies directly as the
time t, then d=kt. 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.
d = kt
20 = 2k
k = 10 – constant of variation
y = kt
y = 10t – equation of the variation
Example #3
If x varies directly as y and x=35 when y=7,
what is the value of y when x=25?
Solution:
x = ky
35 = 7k
k = 5 – constant of variation
x = ky
25 = 5y
y=5
Other solution by proportion, Since x/y is a
constant, we write k=x/y, from here we can
𝑥
𝑥
obtain a proportion 𝑦1 = 𝑦2 .
1
2
x1 = 35
y1 = 7
x2 = 25
y2 = ?
𝑥1 𝑥2
=
𝑦1 𝑦2
35 25
=
7
𝑦2
35y2 = 175
y2 = 5
we obtain the same answer.
Example #4 (Drill/Board work)
I. In each of the following, y varies
directly as x. Find the values as
indicated.
1. Find the constant of a direct
variation when x = 6 and y = -30.
2. If the constant variation is -4, then
what is the value of y when x = -6?
3. If y=3 when x=15, find x when
y = 5.
4. If y = -8 when x = -2, find x when
y = 32.
II. The distance traveled varies directly
with the time of travel. If the distance
traveled is 250 meters in 25 seconds,
find the distance traveled in 60 seconds.
k = -5
y = 24
k=1/5, x=25
k=4, x=8
d = kt
250 = 25k
k = 10
d = kt
d = 10y
d = 10(60)
d = 600 meters
B. Abstraction
How do you identify
expresses direct variation?
equation
that
C. Analysis
How do you determine the table of values
that expresses a direct variation?
How do you determine the graph that
expresses a direct variation?
How do you solve the given word problem
using y=kx as the mathematical equation
of direct variation, where k is a constant
of variation
D. Application
Bella uses 20 liters of gasoline to travel 200
km. How many liters of gasoline will be
use on a trip of 700 km?
IV. Evaluation / Assessment
A. Find the constant of variation and write an equation where y varies directly as x.
1. y = 28 when x = 7
2. y = 30 when x = 8
B. In each of the following, y varies directly as x. Find the value as indicated.
3. If y = 36, when x = 4, find y when x = 7
4. If y = 10, when x = 2, find y when x = 3
C. The distance you travel varies directly with time. If you travel 147 miles in 3 hours, how
long with it take to travel 245 miles?
V. Assignment
Activity 6, pp. 200-202, B(1-3, 7-8) and D(1-5).
A. Determine if the table and graphs below express a direct variation between the variables.
If they do, find the constant of variation and an equation that defines the relation.
1.
x
1
2
3
4
y
-3
-6 -9 -12
2.
x
1
2
3
4
y
3
6
-9 -12
3.
x
1
2
3
4
y
-3
2
-4
5
4.
5.
B. In each of the following, y varies directly as x. Find the values as indicated.
1. If y = 12 when x = 4, find y when x = 12
2. If y = -18 when x = 9, find y when x = 7
3. If y = -3 when x = -4, find x when y = 2
4. If y = 3 when x = 10, find x when y = 1.2
5. If y = 2.5 when x = .25, find y when x = .75
Group Activity (10 groups, good for 10 minutes)
I. Direction: Complete the table of values using the formula of the perimeter of the square
(P=4s) then graph the points from the table of values you’ve completed.
P=4s
Side of the square (cm)
Perimeter of the square
1
2
3
4
5
6
7
Direction: Base from table of values and graph you’ve completed, answer the ff. questions.
1. What equation describes the graph?
2. What happened to the perimeter when the side is 2? 3?
,
3. What happened to the perimeter of the square as the side of the square increases?
4. Find the ratio of the perimeter of the square and the side of the square.
Side of the square (cm)
Perimeter of the square
1
2
3
4
5
6
7
𝑃
Ratio ( 𝑆 )
5. What have you observed about the ratios of perimeter of the square and the side of the
square?