Getting Started 
Misconception/Error The student does not use ratio reasoning to calculate the homework time or percent. 
Examples of Student Work at this Level The student uses addition or subtraction to attempt to find the rate or percent. The student:
 Subtracts 60 from 75 to get 15; then subtracts 15 from 100 to get 85 or says, â€śIt takes him 15 minutes less than his max time to get his math homework done each time.â€ť
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 Adds 25 to 75 to get 100; then adds 25 to 60 to get 85.
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 Estimates a number between 60 and 100 saying, â€śIf 75 increases to 100 then 60 will have to increase too, so maybe 80 or 90.â€ť
 Writes each of the given numbers as a percent, e.g., =75% or =60%.
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 Uses a combination of operations with the given numbers, showing no clear strategy.
 Attempts to generate equivalent ratios by successively adding 15 to 60 and 75.
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Questions Eliciting Thinking Can you check to see if the ratio you wrote is equivalent to 60 out of 75?
Can you explain how to write a ratio that is equivalent to a given ratio? 
Instructional Implications Review what it means for ratios to be equivalent. Be sure the student understands the multiplicative relationship between equivalent ratios. Assist the student in devising strategies for determining when ratios are equivalent, such as converting each ratio to a unit rate or testing for a constant of proportionality. Model generating ratios equivalent to a given ratio by multiplying or dividing both parts of the ratio by the same value. Then have the student test the ratios to determine if they are equivalent. Provide additional opportunities to write ratios equivalent to a given ratio and to determine if a set of given ratios are equivalent.
Guide the student to write a set of ratios equivalent to the ratio given in this problem. Assist the student in expressing 60 out of 75 as a unit rate and as a rate out of 100. Explain the relationship between rates out of 100 and percents. Use a double number line to show the relationship between parts of 75 and their corresponding percents.
Provide additional opportunities to convert ratios to percents. 
Moving Forward 
Misconception/Error The student uses ratio reasoning incorrectly to calculate the homework time or percent. 
Examples of Student Work at this Level To find the proportional amount of time, the student:
 Attempts to write a proportion but does so incorrectly.
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 Writes a proportion but is unable to correctly solve it.
To find the percent, the student divides 80 by 100.
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Questions Eliciting Thinking Can you explain how you wrote this proportion? How did you decide where to put the values 60, 75, 100, and x? What does x represent in your proportion?
Can you explain how you solved your proportion? Can you solve it without trying to crossmultiply? Is there another way that makes sense?
Can you use your answer, 80, to write a ratio that is equivalent to 65 out of 75? What does this ratio have in common with a percent? Can you express it as a percent? 
Instructional Implications Guide the student to think about equivalent ratios in terms of their multiplicative relationship. Model a variety of strategies for solving problems involving equivalent ratios and percent. Encourage the student to use ratio tables, tape diagrams, and double number lines as an alternative to writing and solving proportions. For example, show the student that 60 out of 75 is equivalent to 4 out of 5 since 60 Ă· 15 = 4 and 75 Ă· 15 = 5. Record these ratios in a table and challenge the student to find other equivalent ratios by multiplying or dividing both parts of the ratio by the same value.
Review the meaning of percent as a quantity out of 100. Explain that 80 out of 100 is the same as 80%. Give the student other ratios that can easily be converted to quantities out of 100, such as 17 out of 20 or 9 out of 25. Ask the student to complete the conversions and write the ratios as percents. 
Almost There 
Misconception/Error The student is unable to adequately explain how a rate out of 100 is related to a percent. 
Examples of Student Work at this Level The student:
 Explains that the answers are related because they both have 80 in them or refer to the amount of math homework time.
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 Does not attempt an explanation. Additionally, the student may make a calculation error.
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Questions Eliciting Thinking What does percent mean? How would you write a percentÂ (e.g., 25%) as a ratio?
What does 80 out of 100 have in common with a percent? 
Instructional Implications Review the meaning of percent as a quantity out of 100. Explain that 80 out of 100 is the same as 80%. Give the student other ratios that can easily be converted to quantities out of 100, such as 17 out of 20 or 9 out of 25. Ask the student to complete the conversions and write the ratios as percents. 
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 that Carlos would spend 80 minutes on math out of 100 minutes total,
 Says this is equivalent to 80%, and
 Explains that any rate out of 100 is equivalent to a percent, since percent means â€śout of 100.â€ť
The student may use equivalent fractions, a proportion, a ratio table, or other ratio reasoning.
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Questions Eliciting Thinking Suppose Carlos spent 30 out of 120 minutes on math. How would you determine the percentage of time he spent on math?
Suppose he spent 80% of 2 hours on math. How many minutes did he spend on math? 
Instructional Implications Provide opportunities to solve percent problems given in context. Vary the unknown quantity so that the student must find the percent, the part and the whole in various reallife contexts. Encourage the student to be flexible in the use of strategies and to avoid procedures such as crossmultiplication, until the student has had the opportunity to deepen his or her understanding of equivalent ratios and percent by working extensively with ratio tables, tape diagrams, and double number lines.
Consider implementing other tasks to provide further experience with rates, ratios and percents. 