Main Idea and New Vocabulary
NGSSS
Example 1: Identify Linear Relationships
Example 2: Find a Constant Rate of Change
Example 3: Identify Proportional Relationships
Key Concept: Proportional Linear Relationships
Five-Minute Check
• Identify proportional and nonproportional linear
relationships by finding a constant rate of
change.
• linear relationship
• constant rate of change
MA.8.A.1.2 Interpret the slope and the x- and
y-intercepts when graphing a linear equation
for a real-world problem.
Identify Linear Relationships
BABYSITTING The amount a babysitter charges
is shown. Is the relationship between the number
of hours and the amount charged linear? If so,
find the constant rate of change. If not, explain
your reasoning.
Identify Linear Relationships
+1
+1
+8
+8
+1
+8
As the number of
hours increases by
1, the charges
increase by $8.
Answer: Since the rate of change is constant, this is
a linear relationship. The constant rate of
change is
or $8 per hour.
MUSEUM The cost for groups of people visiting
a science museum is shown. Is the relationship
between the number of people and the total cost
linear?
A. Yes; the rate of change is
$9 per person.
B. Yes; the rate of change is
$8 per person.
C. Yes; the rate of change is
$6 per person.
D. No; the rate of change is
not constant.
Find A Constant Rate of Change
TRAVEL Find the constant rate of change for the
hours traveled and miles traveled. Interpret its
meaning.
Find A Constant Rate of Change
Choose any two points on the line and find the rate of
change between them.
(2, 60)  2 hours, 60 miles
(6, 180)  6 hours, 180 miles
The miles
changed from 60
to 180 while the
hours changed
from 2 to 6.
Subtract to find
the change in the
number of miles
and hours.
Find A Constant Rate of Change
Express this rate
as a unit rate.
Answer: The rate of change is 30 miles per hour.
This means that the distance increases
30 miles each hour.
Find the constant rate of change
for the number of play tickets sold
and the amount of money
collected. Interpret its meaning.
A. $5 per ticket; the amount of money increases
by $5 for each ticket sold.
B. $5 per ticket; the amount of money decreases
by $5 for each ticket sold.
C. $25 per ticket; the amount of money
increases by $25 for each ticket sold.
D. $25 per ticket; the amount of money
decreases by $25 for each ticket sold.
Identify Proportional
Relationships
SPEED Use the graph to determine if there is a
proportional linear relationship between the
speed (meters per second) and the time since a
ball has been thrown. Explain your reasoning.
Identify Proportional
Relationships
Since the graph of the data forms a line, the
relationship between the two scales is linear. This
can also be seen in the table of values created using
the points on the graph.
+1
+1
+1
+1
+1
+1
+9.8 +9.8 +9.8 +9.8 +9.8 +9.8
Constant Rate of Change
Identify Proportional
Relationships
To determine if the two scales are proportional,
express the relationship between the speed and time
for several columns as a ratio.
Since the ratios are not the same, the relationship
between speed and time is not proportional.
Answer: There is a constant rate of change 9.8, but
the ratios are not the same. The relationship
between speed and time is not proportional.
FINANCE Use the graph to
determine if there is a proportional
linear relationship between the
savings account balance and the
number of weeks. Explain your
reasoning.
A. The relationship is linear and proportional.
B. The relationship is linear, but not proportional.
C. The relationship is proportional, but not linear.
D. The relationship is neither linear nor
proportional.
The graph shows the number of
patients that can be seen per
day at a dental office depending
on how many hygienists are
working. Find the constant rate
of change for patients per
hygienist at the dental office.
A. 15 patients/hygienist
B. 30 patients/hygienist
C. 60 patients/hygienist
D. no constant rate of change
The cost to download songs onto a listening
device is $1.15 per song. Which table contains
values that fit this situation for the total cost c
and the number of songs s?
A.
C.
B.
D.