Getting Started 
Misconception/Error The student is unable to reason about proportionally related quantities. 
Examples of Student Work at this Level The student multiplies 60 gallons by Â hour and writes Â or multiplies 75 gallons by Â hour and writes Â and can offer no rationale for doing so.
The student determines the fountain pumps 60 gallons in 12 minutes, and then multiplies 60 by 12 getting 720 gallons per hour.
The student determines the fountain pumps 75 gallons in 15 minutes, and then multiplies 75 by 15 getting 1,125 gallons per hour.

Questions Eliciting Thinking What is a ratio or rate?
How is a unit rate different from a rate?
Why did you multiply by Â hour? What would the number that you got mean?
How did you determine 60 gallons of water per 12 minutes? What does this ratio mean? How might it help you find the unit rate? 
Instructional Implications Review the concept of ratio and encourage the student to use a ratio table to write and explore patterns in equivalent ratios. Guide the student to use multiplication (rather than repeated addition) to generate equivalent ratios. Encourage the student to use ratio language (e.g., â€śfor every,â€ť â€śfor each,â€ť â€śfor each one,â€ť â€śperâ€ť) when describing and interpreting ratios and rates.
Provide direct instruction on unit rates. Describe unit rates as a comparison of some quantity to one unit of another quantity. Emphasize that the unit of one has to be the second part of the comparison. Encourage the student to use tables, tape diagrams, and double number lines to model and explore equivalent ratios and rates.
Consider implementing CPALMS Lesson Plan Letâ€™s Rate It! (ID#: 46379).
Consider implementing MFAS task Explaining Rates (6.RP.1.2) and MFAS task Unit Rate Length (7.RP.1.1) to further assess student. 
Moving Forward 
Misconception/Error The student attempts to write a proportion to find the unit rate but does so incorrectly. 
Examples of Student Work at this Level The student determines that Â of an hour equals 12 minutes, and then sets up a proportion using minutes and hours (e.g., Â = ). 
Questions Eliciting Thinking What is a unit rate? What kind of quantities are compared in unit rates?
Why did you change Â hour to 12 minutes?
Why did you use a proportion? Is your answer per minute or per hour? Does your answer seem reasonable?
Can you think of another way to calculate the unit rate? 
Instructional Implications Encourage the student to use tables to generate equivalent rates (by multiplying or dividing) and to calculate unit rates. Guide the student to be mindful of the units of measure of associated quantities.
Describe unit rates as a comparison of some quantity to one unit of another quantity, and model the use of proportions to determine unit rates. Emphasize the placement of the unit of one when setting up a proportion. Next, compare and contrast the use of multiplication and division when determining unit rates to the use of proportions. Provide opportunities for the student to explore the similarities and differences between the two methods.
Consider implementing theÂ NCTM Illuminations Lesson titled Measuring Up (Lesson 3: Whatâ€™s Your Rate?). 
Almost There 
Misconception/Error The student makes a computational error in some aspect of his or her work. 
Examples of Student Work at this Level The student uses a ratio table to incrementally convert 75 gallons per 15 minutes to n gallons per hour, but makes an error in calculating at one step of the process.
The student converts Â to 0.5.
Note: The student recognizes his or her own mistake when questioned.

Questions Eliciting Thinking I think you made a small error. Can you find it in your work?
Why did you change fractions to decimals? What is onefifth as a decimal?
How did you determine Â of an hour is 12 minutes? 
Instructional Implications Provide the student with additional problems involving associated quantities described with fractions. Have the student work with a partner to compare answers and reconcile any differences. Encourage the partners to compare their strategies for calculating unit rates.
Consider using MFAS task Unit Rate Area (7.RP.1.1). 
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 300 gallons per hour for both, and says the fountains pump â€śat the same rate which is 300 gallons every hour.â€ť
The student explains, â€śSixty gallons is for just part of the hour, so you need all five parts to make one hour.â€ť
The student explains, â€śSeventyfive gallons is just for Â of the hour, so you need all four parts to make the whole hour.â€ť
The student divides 60 gallons by Â of an hour to determine the correct unit rate of 300 gallons per hour.

Questions Eliciting Thinking Can you write the unit rate in hours per gallon?
How much water would flow in Â hours?
How long would it take for 450 gallons of water to recirculate? 
Instructional Implications Have the student graph the data and help the student discover the unit rate using the graph. Introduce the concept ofÂ constant of proportionality and relate it to the slope of the graphed line.
Transition the student to modeling proportional relationships with equations of the form y = cx where c is the constant of proportionality. Guide the student to see the relationship between a unit rate and the constant of proportionality.
Challenge the student to determine a unit rate in a given problem in context, and then apply the unit rate in order to solve another problem in the same context (e.g., ask the student to convert Â miles in Â hours to a unit rate, and then determine how many hours it would take to travel 11 miles). 