Getting Started 
Misconception/Error The student does not use ratio reasoning to solve the percent problem. 
Examples of Student Work at this Level The student:
 Multiplies or divides 470 by 20 without any clear overall strategy.
 Attempts a sequence of operations that make no mathematical sense.
 Calculates 20% of 470.

Questions Eliciting Thinking Is 80% the same thing as 80 milligrams?
What does 100% mean compared to 20%? Which is more? How could you find out how many times more 100% is than 20%?
Why did you multiply/divide? Can you draw a diagram or make a table to show what you're thinking? How could you show steps from 20% to 100%?
Will the amount of salt that is 100% be more or less than the amount that equals 20%?
What would twice the percentage and twice the amount of salt be? How can that help you solve this problem? 
Instructional Implications Present the concept of percent in the context of rates and proportional relationships. Guide the student to use ratio reasoning to solve percent problems. For example, assist the student in observing that 100% is five times 20%. Since 470 is 20% of some quantity, to find 100% of that quantity, 470 can be multiplied by five as well. Guide the student to find other percents of the whole in a similar manner, e.g., ask the student to find 40%, 60%, 80% and 120% , and to record the values in a table such as:
Challenge the student to use the values in the table to find other percents of 2350 such as 5%, 10% and 50%.
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. Ask the student to calculate a constant of proportionality for the values given in the table above and to use this value (23.5) to verify that the table represents a proportional relationship.
Provide additional opportunities to solve percent problems using ratio reasoning. 
Moving Forward 
Misconception/Error The student uses ratio reasoning to calculate the daily recommended quantity of sodium but makes an error. 
Examples of Student Work at this Level The student:
 Attempts to multiply 470 by five or divide 470 by 0.2 but does so incorrectly.
Â Â Â
 Attempts to write a proportion but does so incorrectly.
 Writes a proportion but is unable to correctly solve it.
 Attempts to scale up 470 but multiplies it by four instead of five.

Questions Eliciting Thinking Can you explain how you wrote this proportion? How did you decide where to put the values 20, 470, 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?
How did you decide to scale up 470 by a factor of four? How many times more is 100% than 20%?
How did you decide to multiply 20% and 470? What are you actually finding when you multiply these two numbers?
Is 470 the total amount for the day or part of the dayâ€™s total? 
Instructional Implications Guide the student to think about percent in terms of equivalent ratios and their multiplicative relationship. Model a variety of strategies for solving problems involving 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 20 out of 100 is equivalent to 1 out of 5 since 20 Ă· 20 = 1 and 100 Ă· 20 = 5. Record this ratio in a table and ask the student to find other equivalent ratios by multiplying both parts of the ratio by the same value. Challenge the student to find an appropriate scale factor so that 1 out of 5 can be scaled up to 470 out of a total.
Guide the student to write an equation to show the relationship between a pair of equivalent ratios. Model the use of proportion language to read the equation, (e.g., 470 is to x as 20% is to 100%). Encourage the student at this level to use strategies based on an understanding of equivalent ratios to find missing values rather than techniques such as crossmultiplying.
If needed, help the student identify whether the given amount is the part or the whole. Encourage the student to consider if the answer makes sense in the context of the problem and to revisit the problem if it does not. 
Almost There 
Misconception/Error The student makes errors in showing work or in expressing the final answer. 
Examples of Student Work at this Level The student correctly calculates the daily recommended quantity of sodium using ratio reasoning but:
 Gives no units or writes the answer as a percent.
Â Â Â
Â
 Writes mathematically incorrect or unclear statements.
Â Â Â
Â

Questions Eliciting Thinking What is the unit of measure of the value you found?
Is 2350 a percent? What did you actually find?
Can you reread what you wrote and see if it accurately describes what you did?
Can you explain what this work on your paper means? 
Instructional Implications Provide feedback and allow the student to revise his or her work. Show the student examples of appropriately written work of other classmates. Provide more opportunities for the student to solve percent problems and encourage the student to show work appropriately. 
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 uses ratio reasoning to correctly calculate the daily recommended amount of sodium as 2350 mg.

Questions Eliciting Thinking How can you check your answer?
Can you find 10% of the total? 
Instructional Implications Provide additional 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. 