PROPORTIONALITY
VARIATION
SLOPE
A constant ratio in any proportional
relationship
Really just another name for unit rate!
Remember, to be constant
means it never changes!
Find the constant of proportionality/variation/slope
between the gallons and the miles.
Miles
50
100
150
200
𝟏𝟎𝟎 ÷ 𝟒 = 𝟐𝟓
𝟏𝟓𝟎 ÷ 𝟔 = 𝟐𝟓
Gallons 2
4
÷2
50
?
=
2
1
÷2
25 miles
per gallon
6
8
This means our constant
of proportionality is 25,
so if we divide the miles
by gallons we should
always get 25.
Let’s check!
𝟐𝟎𝟎 ÷ 𝟖 = 𝟐𝟓
Find the constant of proportionality/variation/slope
between gallons and miles.
Number of
Apples
9
27
36
Cost
$3.00
$9.00
$12.00
÷3
9
?
=
3
1
÷3
Constant of
proportionality = 3
Let’s Check!
𝟐𝟕 ÷ 𝟗 = 𝟑
𝟑𝟔 ÷ 𝟏𝟐 = 𝟑
We
could
this as:
Find
thewrite
constant
Miles
of proportionality.
y=45(x)
200
180
160
140
120
(2,90)
100
80
(1,45)
60
40
(0,0)
20
0
0
1
2
(4,180)
Y
45
90
135
180
X
1
2
3
4
To find our constant of
proportionality we have to divide!
(3,135)
45 =
1
45
90 =
2
45
So, our constant of proportionality is 45.
3
Hours
4
5
135 =
3
45
180 =
4
45
In our last example we had:
We will ALWAYS be able to
write our constant of
proportionality as an
equation that looks like this:
y = 45x
y=kx
And “k” will always be our constant
of proportionality/variation/slope!
Unit Rate
Constant of Proportionality
Constant of Variation
Slope
U-Swirl Frozen Yogurt
Weight (oz)
Cost
9
$2.25
11
$2.75
13
$3.25
Cost
Weight (oz)
𝑦 = 0.25𝑥
MineCraft
Minutes
Blocks
5
80
12
192
35
560
Blocks
Minutes
𝑦 = 16𝑥
Baking
Minutes
Cookies
10
12
20
24
30
39
There is no constant
of proportionality
because there isn’t a
constant rate!
42

36
Cost ($)
30 
24

18
Cost ($)
12
Weight (lb.)

6
2
4
𝑦 = 3𝑥
6
8

10
Weight (lb.)

12

42

36
Cost ($)
30 
24

18
Cost ($)
12
Gallons

6
2
4
𝑦 = 5𝑥
6
8

10
Gallons of Gas

12

42

36
Cost ($)
30 
24
There is no constant
of proportionality
because there isn’t a
constant rate!

18
12

6
2
4
6
8

10
Gallons of Gas

12

 If two quantities are proportional, then they have a
constant ratio.
 To have a constant ratio means two quantities have the same
unit rate.
 If the ratio is not constant, the two quantities are said
to be non-proportional.
 So, the two quantities do not have the same unit rate.
Will always go through the origin on a graph. (0,0)
Graph will always be a straight line.
In order to tell if a graph is proportional the line
must go through the origin. Tell if the following
graphs represent a proportional relationships.
y
y
5
5
4
4
3
3
2
2
1
1
x
1
x
1
2
3
4
3
4
5
5
Yes
Proportional ? _________
Why?
2
Line goes through
the origin
No
Proportional ? _________
Why?
Line does not
go through the origin
Guided
Practice
Let’s
Review
Distance (ft.)
State in words the
proportional
relationship shown
here.
(There are many
correct answers!)
y


2 feet per min
x





Time (min.)


You Try
Quick
Quiz
Let’s
Review


Cost ($)
State in words the
proportional
relationship shown
here.
(There are many
correct answers!)




5oz for $2











Weight (ounces)


You try: The following chart shows how much money Alex earns for mowing
lawns. Is the amount of money he earns proportional to the number of hours
that he spends mowing?
Earnin Hours Unit $Rate
gs ($)
(h)
( hr )
1
$14
1
2
$28 $14

2
1
42
3
$42 $14

3
1
56
4
$56 $14

4
1
14
28
Since the simplified ratios were equal,
this was a proportional relationship.
We typically put time (hours) on the x-axis, and
the earnings ($) on the y-axis.
Set up the graph paper to fit the data in the chart.
Plot points (x, y) from the table.
y
Earnin
gs ($)
Point
(x, y)
56
1
14
(1, 14)
42
2
28
(2, 28)
3
42
(3, 42)
4
56
(4, 56)
Connect the points.
Describe the graph of this
proportional relationship.
Earnings
($)
Hour
s (h)
28
14
1
2
3
Hours
worked
4
5
x
Ticket Express charges $7 per movie ticket plus a $3 processing
fee per order. Is the cost of an order proportional to the number
of tickets ordered? Explain .
Cost ($)
10
17
24
31
Tickets
Ordered
1
2
3
4
cost ($)
no. of tickets
$10
1
17 $8.5

2
1
$24 $8

3
1
$31 $7.75

4
1
Since all of the simplified ratios are not equal, there is no
constant ratio, so this is NOT a proportional relationship.
It passes through the origin,
but it is not a straight line.
Tickets ordered will be on the x-axis,
and the cost ($) will be on the y-axis.
Plot points (x, y) from the table.
Earnings
($)
Point (x,
y)
0
0
(0,0)
1
10
(1, 10)
2
17
(2, 17)
3
24
(3, 24)
4
31
(4, 31)
Connect the points.
Describe the graph of this
nonproportional relationship.
y
28
Cost ($)
Tickets
32
24
20
16
12
8
4
1
2
3
4
Tickets ordered
x