Getting Started 
Misconception/Error The student does not understand the concept of a constant of proportionality. 
Examples of Student Work at this Level The student:
 Only examines values of one variable and concludes that there is no constant of proportionality.
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 Adds or subtracts corresponding boat lengths and fees.
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 Chooses one value from the table or number from the scale of the graph and reports it as the constant of proportionality.
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Questions Eliciting Thinking What is a constant of proportionality? How did you determine the constant?
When computing the constant, do you need to also consider the daily fee? Why or why not?
What does proportional mean? How do you determine proportionality?
What does constant mean?
Can you determine the unit rate in the table (or graph)? 
Instructional Implications Provide explicit instruction on proportionality and the multiplicative relationship between values of variables that are proportionally related. Introduce the concept of the constant of proportionality and its role in describing the relationship between variables that are proportionally related. Describe proportionality as a relationship between two variables in which the value of one variable is a constant factor of the associated value of the other variable. Point out that since there is a multiplicative relationship between values of variables that are proportionally related, division can be used to determine that constant factor. Model how to determine the constant of proportionality from a table of equivalent ratios and from a graph. Have the student generate a table of equivalent ratios and identify the constant factor. Provide the student with additional opportunities to determine the constant of proportionality using both tables and graphs. Consider implementing this task again but with a new table and graph. 
Moving Forward 
Misconception/Error The student is unable to correctly find the constant of proportionality in both the table and the graph. 
Examples of Student Work at this Level The student demonstrates an understanding of the constant of proportionality by expressing it as the ratio or quotient of the daily fee to boat length for at least one of the problems. However, the student is unable to correctly calculate the constant for one or both problems
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Additionally, the student is unable to correctly interpret the constant of proportionality. 
Questions Eliciting Thinking How were you attempting to calculate the constant of proportionality in this problem? Could you use this same approach in the other problem?
What is the relationship between each boat length, its docking fee, and the constant of proportionality?
How can you use the graph to find an example of a boat length and its associated docking fee? 
Instructional Implications Confirm that the studentâ€™s approach to finding the constant is correct (i.e., finding the quotient of a docking fee and its boat length). Assist the student with finding and correcting any calculation errors. Then guide the student to use the same approach in the other problem. If needed, help the student identify boat lengths and their associated docking fees from the graph.
Model the process of determining the constant of proportionality in a variety of realworld contexts. Describe the constant of proportionality as a unit rate and assist the student in interpreting constants in the context of problems. Help the student understand the meaning of unit rates by including the units of measure in the quotient. Then guide the student to use the language of ratios to interpret the constant (e.g., say, â€śfor each additional foot of boat length, the daily fee is increased by $2.25â€ť). 
Almost There 
Misconception/Error The student correctly calculates each constant of proportionality but is unable to interpret their meaning in the problem. 
Examples of Student Work at this Level The student determines the constant of proportionality to be 2.25 in Port Canaveral and 2.50 in Fort Lauderdale. However, the student cannot correctly explain the meaning of the constants in either problem even with teacher prompting.
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Questions Eliciting Thinking What is the unit of measure for boat length? What is the unit of measure for the daily fee? Can you determine the unit of measure for the constants you calculated?
If the constant is 2.5, what does that mean in this problem?
How did you decide at which marina it is less expensive to dock a boat? 
Instructional Implications Model the process of determining the constant of proportionality in a variety of realworld contexts. Describe the constant of proportionality as a unit rate and assist the student in interpreting constants in the context of problems. Help the student understand the meaning of unit rates by including the units of measure in the quotient. Then guide the student to use the language of ratios to interpret the constant (e.g., say, â€śfor each additional foot of boat length, the daily fee is increased by $2.25â€ť). 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student determines the constant of proportionality to be $2.25/foot in Port Canaveral and $2.50 /foot in Fort Lauderdale by dividing one of the daily fees by its associated length. The student can explain the meaning of the constants in context and can determine that it is less expensive to dock a boat at Port Canaveral.
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Questions Eliciting Thinking What would you expect to get if you calculated the constant using a different pair of associated values from the table? From the graph?
What point on the graph includes the constant as one of its coordinates?
Can you write an equation that shows the relationship between the boat length and the daily fee?
How would you determine the fee for a 20 foot boat? How about a 35foot boat?
How did you decide which marina offers the better deal?
Would a 50foot boat still be cheaper to dock in Port Canaveral? Is there any length boat that would be cheaper to dock in Fort Lauderdale?
If you were to extend the line in the graph, would it connect to the origin? Why or why not? 
Instructional Implications Give a verbal description of a proportional relationship and challenge the student to represent the relationship with a table, graph, and equation. Encourage the student to identify the constant of proportionality and to describe its meaning in context.
Challenge the student to write an equation representing the proportional relationship displayed in the table and then another equation representing the proportional relationship displayed in the graph. Consider implementing MFAS task Writing An Equation (7.RP.1.2). 