Getting Started 
Misconception/Error The student does not investigate any possible relationship between given associated values. 
Examples of Student Work at this Level The student says the time and price are not proportional because two hours equals 120 minutes, so it should cost $120. Also, Â hours is one hour plus 30 minutes, so it should cost $130.
The student thinks two hours should cost $200; Â hours should cost $150 because Â equals 1.5; and Â hour should cost $75 because equals 0.75.

Questions Eliciting Thinking Why should two hours cost $120? How much should one hour cost?
Why should two hours cost $200?
What does proportional mean? How do you determine proportionality?
Do you know what a rate is? Can you give an example of a rate?
Do you know what a ratio is? Can you give an example of a ratio? 
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. Guide the student to investigate pairs of associated values to determine if there is a constant factor that relates the values in each pair.
Help the student generate equivalent ratios, perhaps in a table, and identify the constant of proportionality. Give the student another table of ratios and encourage him or her to determine proportionality by looking for a constant of proportionality. Provide the student with additional opportunities to determine proportionality given values of variables in a table. 
Moving Forward 
Misconception/Error The student determines proportionality by calculating consecutive differences between values given in the table. 
Examples of Student Work at this Level The student says the times and prices are not proportional because the price goes up by $40 and then by $50.
The student explains that the same amount must be added each time in order to be proportional.

Questions Eliciting Thinking What does proportional mean?Â How do you determine proportionality?
Is the difference betweenÂ Â andÂ Â the same as the difference betweenÂ Â and 2?
Is this set of data proportional? Explain.
Is this set of data proportional? Explain.

Instructional Implications Describe ratios as multiplicative comparisons of two quantities and not just a pattern on one side of the table. Clarify the role of addition when generating equivalent ratios. Some students adopt addition as a method for generating equivalent ratios (e.g., if the ratio 1:3 means for every blue cube, there are three red cubes, then every time you add a blue cube, you have to add three more red cubes).
Transition the student to interpreting ratios using a multiplicative comparison (e.g., if the ratio 1:3 means for every blue cube, there are three red cubes, then there will always be three times more red cubes than blue cubes). Have the student practice writing equivalent ratios using multiplication.
Describe proportionality as a relationship between two variables in which ratios of associated values are always equivalent. Show the student a table with equivalent ratios. Erase a row and show the student that the remaining ratios are unchanged and still proportional although the difference between each row has changed. Provide the student with additional opportunities to test proportionality. 
Almost There 
Misconception/Error The student determines proportionality by comparing unit rates but does not clearly explain using rate language. 
Examples of Student Work at this Level The student says the times and prices are not proportional. The student says the numbers do not come out the same every time when each row is divided.
The student determines a unit rate, and realizes that other times and prices are not consistent with the unit rate. The student says the numbers are â€śall uneven.â€ť

Questions Eliciting Thinking How did you determine the rates are not proportional?
What do you mean by â€śyou donâ€™t get the same number every time?â€ť
Why are you dividing price by time?
You divided Â and got 90. What does 90 represent?
Why did you divide each row?
Why did you compare the two hour rate with the Â hour rate? 
Instructional Implications Help the student develop mathematical vocabulary that will enable him or her to explain clearly and precisely. Explain to the student what he or she did in the task (e.g., say, â€śIt appears that you converted each ratio in the table to a unit rate and then compared the unit rates. You observed that the unit rates were not equal and concluded the relationship between the variables is not proportional.â€ť). Give the student additional opportunities to determine if two variables are proportionally related and to justify his or her conclusions. 
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 says the times and prices are not proportional.
The student explains the unit rate (in dollars per hour) differs for each rental time, and therefore, determines the rates are not proportional.
Using $180 for two hours, the student determines the unit rate then compares it to the $90 for Â hour data. Upon questioning, the student explains clearly.

Questions Eliciting Thinking Can you determine proportionality another way?
How could you change the costs given in the chart so that the relationship is proportional?
What is the difference between dividing two hours by $180 and dividing $180 by two hours? Does it matter? Will they mean the same thing?
How did you know $90 for Â hour was not proportional to $180 for 2 hours?
How does finding the unit rate help? 
Instructional Implications Introduce the concept of the constant of proportionality. Model finding the constant of proportionality in tables and graphs. Consider using MFAS task Finding Constant of Proportionality (7.RP.1.2).
Pair the student with a Moving Forward partner. Have the student model how he or she determines proportionality. 