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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.
As the number of hours increases by 1, the charges increase by $8.
Identify Linear Relationships
Answer: Since the rate of change is constant, this is
a linear relationship. The constant rate of
change is or $8 per hour.
+1
+1
+1
+8
+8
+8
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.
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?
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
Find A Constant Rate of Change
Subtract to find the change in the number of miles and hours.
The miles changed from 60 to 180 while the hours changed from 2 to 6.
Find A Constant Rate of Change
Answer: The rate of change is 30 miles per hour. This means that the distance increases 30 miles each hour.
Express this rate as a unit rate.
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.
Find the constant rate of change for the number of play tickets sold and the amount of money collected. Interpret its meaning.
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.
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.
Identify Proportional Relationships
Constant Rate of Change
+9.8 +9.8 +9.8 +9.8 +9.8 +9.8
+1 +1 +1 +1 +1 +1
To determine if the two scales are proportional, express the relationship between the speed and time for several columns as a ratio.
Identify Proportional Relationships
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.
Since the ratios are not the same, the relationship between speed and time is not proportional.
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.
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. 15 patients/hygienist
B. 30 patients/hygienist
C. 60 patients/hygienist
D. no constant rate of change
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.
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.
