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Constant rate
1. Learning Targets ….
• Students know the definition of constant rate in
varied contexts as expressed using two variables
where one is t representing a time interval.
• Students graph points on a coordinate plane related
to constant rate problems.
3. Now that we have an idea of what could go wrong
when we assume a person walks at a constant rate or
that a proportion can give us the correct answer all of
the time, let’s define what is called average speed.
Suppose a person walks a distance of d (miles) in a
given time interval t (minutes). Then, the average
speed in the given time interval is
𝑑
𝑡
in miles per
minute.
4. Suppose a person walks a distance of d (miles) in a
given time interval t (minutes). Then, the average
speed in the given time interval is
𝑑
𝑡
in miles per
minute.
With this definition we can calculate Alexxa’s average
speed: The distance that Alexxa traveled divided by the
time interval she walked is
1.1
25
miles per minute.
5. If we assume that someone can actually walk at the
same average speed over any time interval, then we say
that the person is walking at a constant speed. Suppose
the average speed of a person is the same constant C
for any given time interval. Then, we say that the
person is walking at a constant speed C.
6. If the original problem included information specifying
constant speed, then we could write the following:
Alexxa’s average speed for 25 minutes is
1.1
25
. Let y
represent the distance Alexxa walked in 10 minutes.
Then, her average speed for 10 minutes is
𝑦
10
. Since
Alexxa is walking at a constant speed of C miles per
minute, then we know that
1.1
25
= C, and
𝑦
10
= C.
7. Since both fractions are equal to C, then we can write
1.1
25
=
𝑦
10
.
With the assumption of constant speed, we now have a
proportional relationship, which would make the
answer you came up with in the beginning correct.
8. We can go one step further and write a statement in
general. If Alexxa walks y miles in x minutes, then
𝑦
𝑥
= 𝐶 and
1.1
25
=
𝑦
𝑥
.
To find how many miles y Alexxa walks in x miles, we
solve the equation for y:
9. Pauline mows a lawn at a constant rate. Suppose she
mows a 35 square foot lawn in 2.5 minutes. What area,
in square feet, can she mow in 10 minutes? t minutes?
10. Pauline mows a lawn at a constant rate. Suppose she
mows a 35 square foot lawn in 2.5 minutes. What area,
in square feet, can she mow in 10 minutes? t minutes?
What is the meaning of
35
2.5
in the equation y =
35
2.5
x?
The number
35
2.5
represents the constant rate at which
Pauline can mow a lawn.
11. We can organize the data into a table.
t (time in minutes) Linear Equation
y =
35
2.5
x
y (area in square feet)
12. On a coordinate plane, we will let the x-axis represent
time t, in minutes, and the y-axis represent the area of
mowed lawn in square feet. Then we have the following
graph.
13. In the last lesson, we learned about average speed and
constant speed. Constant speed problems are just a
special case of a larger variety of problems known as
constant rate problems.
14. In the last lesson, we learned about average speed and
constant speed. Constant speed problems are just a
special case of a larger variety of problems known as
constant rate problems.
First, we define the average rate:
• Suppose V gallons of water flow from a faucet in a
given time interval t (minutes).
• Then, the average rate of water flow in the given time
interval is
𝑉
𝑡
in gallons per minute
15. Then, we define the constant rate:
• Suppose the average rate of water flow is the same
constant C for any given time interval.
• Then, we say that the water is flowing at a constant
rate, C
16. Similarly, suppose A square feet of lawn are mowed in a
given time interval t (minutes).
Then, the average rate of lawn mowing in the given time
interval is
𝐴
𝑡
square feet per minute.
If we assume that the average rate of lawn mowing is
the same constant, C, for any given time interval, then
we say that the lawn is mowed at a constant rate, C.
17. Describe the average rate of painting a house.
Suppose A square feet of house are painted in a given
time interval t (minutes). Then the average rate of house
painting in the given time interval is
𝐴
𝑡
square feet per
minute.
18. Describe the constant rate of painting a house.
If we assume that the average rate of house painting is
the same constant, C, over any given time interval, then
we say that the wall is painted at a constant rate, C.
19. What is the difference between average rate and
constant rate?
Average rate is the rate in which something can be done
over a specific time interval. Constant rate assumes that
the average rate is the same over any time interval.
20. Water flows at a constant rate out of a faucet. Suppose
the volume of water that comes out in three minutes is
10.5 gallons. How many gallons of water comes out of
the faucet in t minutes?
21. Water flows at a constant rate out of a faucet. Suppose
the volume of water that comes out in three minutes is
10.5 gallons. How many gallons of water comes out of
the faucet in t minutes?
What is the meaning of
10.5
3
in the equation y =
10.5
3
x?
The number
10.5
3
represents the constant rate at which
water flows from a faucet.
22. Using the linear equation V =
10.5
3
𝑡.
t (time in minutes) Linear Equation V (in gallons)
0
1
2
3
4
23. On a coordinate
plane, we will let
the x-axis represent
time t in minutes
and the y-axis
represent the
volume of water.
Graph the data
from the table.
24. Using the graph, about how many gallons of water do
you think would flow after 1
1
2
minutes?
Using the graph, about how long would it take for 15
gallons of water to flow out of the faucet? Explain.
25. • Constant rate problems appear in a variety of
contexts like painting a house, typing, walking,
water flow, etc.
• We can express the constant rate as a two-variable
equation representing proportional change.
• We can graph the constant rate situation by
completing a table to compute data points.
