Getting Started 
Misconception/Error The student writes ratios that are not unit rates. 
Examples of Student Work at this Level The student:
 Writes the ratio using the given values, and then reverses the order to get a second ratio (e.g., Â to Â and Â to ).
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 Changes the ratio Â to Â to decimal form, then puts the number one under each part of the gear ratio, forming the ratios Â and .
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Questions Eliciting Thinking What is a ratio? What two quantities are being compared in this problem? How are they being compared?
Do you know what a unit rate is?
What do you think the ratio Â to Â means in the context of this problem?
How would you interpret the ratio ? (Ask only if the student wrote such a ratio.) 
Instructional Implications Review the concept of ratio and point out that the associated quantities in ratios may or may not contain the same units of measure. Then, provide instruction on finding unit rates with associated whole number quantities. Describe unit rates as a comparison of one quantity to one unit of another quantity. Compare and contrast rates and unit rates. Model how to determine unit rates from given rates. Be sure the student understands the concept of unit rate and is not hindered by his or her ability to perform operations with fractions and mixed numbers.
Consider implementing CPALMS lesson plan Itâ€™s Carnival Time!Â (Resource ID: 47394), and then assessing the studentâ€™s progress with any of the following MFAS tasks: Writing Unit Rates (6.RP.1.2), Identifying Unit Rates (6.RP.1.2), Explaining Rates (6.RP.1.2), and/or Book Rates (6.RP.1.2).
Next, model finding unit rates with quantities that include fractions and mixed numbers. Review operations with fractions and mixed numbers as needed. Provide the student with additional opportunities to determine unit rates. Emphasize the meaning of the unit rate in context and model the use of ratio language when describing the meaning.
Consider implementing MFAS tasksÂ Unit Rate Area (7.RP.1.1) andÂ Unit Rate Length (7.RP.1.1). 
Moving Forward 
Misconception/Error The student is unable to determine the second unit rate. 
Examples of Student Work at this Level The student correctly writes 3 to 1 for the first unit rate, but:
 Reverses it and writes 1 to 3 for the second unit rate.
 Cannot determine the second unit rate.
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Questions Eliciting Thinking Can you tell me what the rate 3:1 means? Can you rewrite this rate so that it tells us how many times the larger gear will turn for each one turn of the smaller gear?
Do ratios and rates have to contain only whole numbers? 
Instructional Implications Clarify the definition of unit rate 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. Reinforce the meaning of unit rates in context and encourage the student to use unit rate language (e.g., â€śfor every one,â€ť â€śfor each one,â€ť â€śper oneâ€ť) when describing the meaning of unit rates in context. Using a table, tape diagram or double number line, model how to find the second unit rate. Be sure to point out that the two parts of the ratio cannot simply be reversed from a:1 to 1:a. Make it clear that rates and ratios can contain fractions.
Consider implementing MFAS tasksÂ Explaining Rates (6.RP.1.2) and Unit Rate Length (7.RP.1.1). 
Almost There 
Misconception/Error The student is unable to correctly interpret the unit rates in the context of the problem. 
Examples of Student Work at this Level The student:
 Explains the procedure used for calculating the unit rates rather than interpreting them in the context of the problem.
 Does not use ratio language when interpreting the rates and says, â€śThe smaller gear turns three times and the larger gear turns one time.â€ť

Questions Eliciting Thinking What does 3:1 mean in the context of this problem?
How are the gears related? What does 3:1 mean about the relationship between how the gears turn? 
Instructional Implications Model explaining the meaning of rates in the context of problems. Use unit rate language (e.g., â€śfor each one,â€ť â€śfor every one,â€ť and â€śper oneâ€ť) when interpreting unit rates or describing their meaning. Have the student practice writing descriptions of rates using rate and unit rate language.
When the student is ready, consider editing the worksheet to include different rational values and then implement the task again. 
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:
 WritesÂ both unit rates correctly: 3 to 1 and Â to 1 or 0.3333â€¦ to 1 and
 Explains that the smaller gear turns three times for each one turn of the larger gear and the larger gear turns Â time for each one turn of the smaller gear.

Questions Eliciting Thinking How did you determine the unit rates?
If a student said the unit rates are 3:1 and 1:3, what did that student do wrong? Why?
Why does the second part of the ratio have to be one? 
Instructional Implications Have the student use the unit rate to solve problems (e.g., â€śHow many times will the smaller gear rotateÂ when the larger gearÂ rotates four and a half times?â€ť or â€śHow many times will the larger gear rotate whenÂ the smaller gear rotates four and a half times?â€ť).
Pair the student with a Moving Forward student. Have the student explain to the Moving Forward partner how to find the unit rates and what each means.
Challenge the student with the following problem: If Gear A turns n times for every one turn of Gear B, then how many times (in terms of n) will Gear B turn for every one turn of Gear A? 