Getting Started 
Misconception/Error The student writes a ratio to represent the proportional relationship. 
Examples of Student Work at this Level The student writes 1:18 and explains, â€śFor every turn of the handle, 18 inches of the fishing line is retrieved.â€ť
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Questions Eliciting Thinking Why did you write a ratio to represent the proportional relationship?
What is an equation? Can you represent the relationship with an equation?
Can you identify the constant of proportionality?
How did you decide which point is most helpful in determining the constant of proportionality? 
Instructional Implications Provide direct instruction on writing equations to represent proportional relationships. 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. Emphasize that if two variables are proportionally related, then the constant of proportionality can be found by dividing the value of one variable by its associated value. Help the student first to identify the constant of proportionality by providing him or her with guided practice. Encourage the student to verbalize the meaning of the constant in the context of problems. Consider implementing MFAS task Finding Constant of Proportionality (7.RP.1.2).
Then, model how to write an equation that represents the relationship between the two proportionally related variables. Make explicit what the variables represent and how the constant of proportionality relates associated values of the two variables. Demonstrate how the equation can be used to determine the total inches of line retrieved for any given number of handle turns by substituting different values for x (number of handle turns). 
Moving Forward 
Misconception/Error The student writes a onevariable equation. 
Examples of Student Work at this Level The student writes 18x = 54. Upon questioning, the student can identify the constant of proportionality, 18 inches, and understands that the solution to the equation is three handle turns. However, the student does not know how to write an equation having two variables.
The student writes 18x = ?. The student can identify the constant of proportionality, 18 inches, but cannot write an equation having two variables. The student identifies the second point as most helpful because it â€śshows it keeps going in a straight line.â€ť

Questions Eliciting Thinking Explain your equation to me. What does the variable stand for in your equation? What does the 18 represent in your equation?
Can you solve your equation? What does the solution to your equation represent?
Can you write an equation using two variables? What do you think the other variable could stand for?
What is the constant of proportionality? How did you decide which point is most helpful in determining the constant of proportionality? 
Instructional Implications Provide instruction on writing equations with two variables to represent proportional relationships. 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. Emphasize that if two variables are proportionally related, then the constant of proportionality can be found by dividing the value of one variable by its associated value. Then, model how to write an equation that represents the relationship between the two proportionally related variables. Make explicit what the variables represent and how the constant of proportionality relates associated values of the two variables. Compare an equation having one variable with an equation having two variables. Point out how the former represents the relationship between a specific number of handle turns and the total inches of line retrieved; whereas, the latter represents the constant relationship between the two variables. Provide additional opportunities for the student to practice writing equations to represent proportionally related values displayed in a graph. Consider using MFAS task Identifying Constant of Proportionality in Equations (7.RP.1.2). 
Almost There 
Misconception/Error The student can write an equation to represent the proportional relationship but does not identify the point (1, 18) as the most helpful in identifying the constant of proportionality. 
Examples of Student Work at this Level The student writes a correct equation (e.g., y = 18x). However, the student says â€śthe second pointâ€ť is most helpful because it allows you to find the rate of change between â€śpoint oneâ€ť and â€śpoint two.â€ť
The student writes a correct equation. However, the student says all the points can easily be used to determine the constant of proportionality. Upon questioning, the student explains how each point can be divided () to determine the constant.

Questions Eliciting Thinking How did you decide which point is most helpful in determining the constant of proportionality? What is the constant of proportionality?
What does the 18 represent in your equation?
Can you determine which point will give you the constant without having to divide ? 
Instructional Implications Confirm the validity of the studentâ€™s reasoning, but explain why the first point is most helpful in identifying the constant of proportionality. Discuss the meaning of â€śconstant of proportionalityâ€ť and â€śunit rate.â€ť Help the student see that (1, 18) is the point which represents the unit rate and can easily be identified as the constant of proportionality without having to divide .
Change the values in this task and have the student pair with a Got It partner to work through the new problem. Have the pair discuss their reasoning and reconcile their differences. 
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 writes y = 18x. Although the variables may be different, the student identifies the independent variable (i.e., the number of handle turns) as the value that is multiplied by 18.
The student identifies the first point (1, 18) as most helpful because it represents the unit rate (18 inches/1 handle turn).
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Questions Eliciting Thinking How did you come up with your equation?
What is the constant of proportionality?
Could you find the unit rate using a different point? Explain.
Would x = 18y work? Why or why not?
If we continued the graph, would it intersect the origin, (0, 0)? What would the point (0, 0) represent in this context? 
Instructional Implications Challenge the student to use his or her equation to solve problems (e.g., â€śHow many inches of line will six handle turns retrieve?â€ť or â€śHow many handle turns will it take to retrieve nine inches of line?â€ť).Â
Present the student with a graph of proportionally related quantities where the points shown are (2, 11), (4, 22), and (6, 33). Ask the student to write an equation representing the proportional relationship and identify the constant of proportionality. Have the student explain his or her process to you.
Consider implementing MFAS task Graphs of Proportional Relationships (7.RP.1.2) to further assess the student. 