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**Lesson Menu**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 Main Idea/Vocabulary**MA.8.A.1.2 Interpret the slopeand the x- and y-intercepts**when graphing a linear equation for a real-world problem. NGSSS**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. Example 1**+1**+8 +1 +8 +1 +8 Answer: Since the rate of change is constant, this is a linear relationship. The constant rate of change is or $8 per hour. Identify Linear Relationships As the number of hours increases by 1, the charges increase by $8. Example 1**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. Example 1 CYP**Find A Constant Rate of Change**TRAVEL Find the constant rate of change for the hours traveled and miles traveled. Interpret its meaning. Example 2**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 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 Example 2**Express this rate as a unit rate.**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. Example 2**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. Example 2 CYP**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. Example 3**+1**+1 +1 +1 +1 +1 +9.8 +9.8 +9.8 +9.8 +9.8 +9.8 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. Constant Rate of Change Example 3**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. Example 3**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. Example 3 CYP**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 Five Minute Check 1**A. C.**B. D. 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? Five Minute Check 2