Constant of proportionality
and The
Unit Rate
MATH TALK
Constant of Proportionality is a constant ratio
(unit rate) in a Proportional relationship.
Rate is called the Constant of Proportionality
Each Triceratops is twice a big as the one before it!
Example:
Constant of Proportionality
10
9
8
7
6
5
4
3
2
1
So, the scale factor is? “2” = the Rate
Here’s the Math,
1 x 2 = 2
2 x 2 = 4
4 x 2 = 8
and so on…
See how the rate is the same and it repeats?
Direct proportionality: Constant of Proportionality
Example:
1.
The circumference of a circle is proportional to its diameter, with the Constant of
Proportionality equal to pi.
9
8
7
6
5
d=1
2
3
4
Example: The circumference of a circle is proportional to its
diameter, notice that the Constant of Proportionality is equal to pi.
Diameter vs. Circumference
π
Diameter Circumference
3.14159
1
3.14159
3.14159
2
6.28318
3.14159
3
9.42477
3.14159
4
12.56636
3.14159
5
15.70795
3.14159
6
18.84954
3.14159
7
21.99113
3.14159
8
25.13272
3.14159
9
28.27431
3.14159
10
31.4159
40
30
20
10
0
Diameter
Circumference
Direct proportionality: Constant of proportionality
Example:
2.
If an object travels at a constant speed, then the distance traveled is proportional to
the time spent traveling, with the speed being the constant of proportionality.
EXAMPLE:
If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling,
with the speed being the constant of proportionality.
Let the Rate be 25 mph.
Distance = Rate x Time
Distance = Rate x Time
Rate (mph) Time (Hours)
Distance
(miles)
25
1
25
25
2
50
25
3
75
25
4
100
300
270
240
210
180
150
25
5
125
25
6
150
120
25
7
175
90
25
8
200
60
25
9
225
30
25
10
250
0
Distance (miles)
1
2
3
4
5
6
7
8
9
10
Direct proportionality: Constant of proportionality
Example:
3. The force acting on a certain object due to gravity is proportional to the object's mass;
the constant of proportionality between the mass and the force is known as
gravitational acceleration.
Direct proportionality: Constant of proportionality
Example:
4. On a map drawn to scale, the distance between any two points on the map is proportional to the distance between
the two locations times the constant of proportionality (the scale of the map).
On this the map the scale is one inch = 4 miles.
So, 7”on the map would equal 7” x 4 miles/inch = 28 miles.
The constant of proportionality is 4 mi/inch.
Now lets try and do a,
Constant of Proportionality and Unit Rate problem…
EXAMPLE:
If you are traveling in your car at a constant speed, then the distance traveled is proportional to the time spent traveling, with
the speed being the Constant of Proportionality.
Distance (miles)
30
60
?
120
Time
(hours)
1
?
3
?
Rate = Distance / Time
so, from the table above, find the missing values.
30 miles / 1 hour = 30 mi/hr
60 miles / ? hours = ? mi/hr
? miles / 3 hours = ? mi/hr
120 miles / ? hours = ? mi/hr
Constant of Proportionality = 30 mph
Distance
(miles)
30
60
90
120
Time
(hours)
1
2
3
4
Use the chart to fill the missing values for Distance and Time in the table.
Distance
(miles)
30
60
90
120
?
?
Time
(hours)
1
2
3
4
?
?
y
?
180
?
150
Distance 120
(miles)
90
60
30
x
1
2
3
4
5
6
Time (hours)
7
Constant of Proportionality = 30 mph
Distance
(miles)
30
60
90
120
150
180
Time
(hours)
1
2
3
4
5
6
EXAMPLE:
The graph and the table below represent the total number of plants (in Blue) and number of
seed packets (in Green) used. What is the Constant of Proportionality?
50
50
45
Constant of Proportionality = ?
40
Number of Seed
Packets
Number of Plants
1
10
2
20
3
30
4
40
5
50
40
35
30
30
25
20
20
15
10
10
5
1
0
2
3
4
5