Getting Started 
Misconception/Error The student does not understand the relationships among the quantities described in the problem. 
Examples of Student Work at this Level The student calculates 15% of 3.40 and says the final answer is 0.51. Upon questioning, the student does not indicate an understanding of the relationship between last yearâ€™s and this yearâ€™s gas price and the significance of the 15% given in the problem.

Questions Eliciting Thinking Can you restate the problem for me?
When was the price of gas higher â€“ this year or last year?
What does it mean for a quantity to be 15% greater than another quantity? 
Instructional Implications Guide the student to identify important quantities in the problem (e.g., the price of gas last year and the price of gas this year) and to verbally describe the relationship between them (e.g., this yearâ€™s gas price is 15% higher than last yearâ€™s gas price). Assist the student in mathematically modeling the relationship by writing and solving an equation (e.g., 3.40 = 1.15x or Â = ) where x is last yearâ€™s gas price. Provide additional opportunities to solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals). Initially focus on identifying important quantities and verbally describing their relationship. Then ask the student to write an equation that models this relationship. Provide additional review, as needed, on solving linear equations in one variable.
Provide instruction on solving problems involving percent. Present more than one approach to modeling percent problem relationships (e.g., by writing an equation or a proportion). Allow the student to use the approach of choice.
Address any errors the student might make with writing amounts of money, (e.g., writing $0.51 as 0.51Â˘). 
Moving Forward 
Misconception/Error The student does not have an effective strategy for working with percent. 
Examples of Student Work at this Level The student understands that 3.40 is 15% greater than a previous price but:
 Calculates 15% of 3.40 and then subtracts this quantity from 3.40.
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 Attempts to write a proportion to solve the problem but writes noncorresponding ratios. For example, the student writes Â = Â where x is last yearâ€™s gas price.

Questions Eliciting Thinking What does it mean for a quantity to be 15% greater than another quantity?
If something costs $1.00 now and then increases in price by 15%, how much is the increase? What is the new price? Is 15% of $1.15 equal to $0.15?
Explain your proportion in the context of the problem. What do the values of the numerators represent? What do the values of the denominators represent? 
Instructional Implications Provide direct feedback to the student concerning any error in his or her approach. (E.g., If the student subtracts 15% of 3.40 from 3.40, explain that 15% of the increased price is not equal to 15% of the original price.) Demonstrate this by calculating 15% of the quantity the student determined to be the original price and comparing it to 15% of 3.40 (or $0.51). Review solving problems involving percent. Present more than one approach to modeling percent problem relationships (e.g., by writing an equation or a proportion). Allow the student to use the approach of choice. Provide additional opportunities to solve multistep realworld and mathematical problems involving percents.
Address any issues the student has with communicating mathematical work (e.g., writing Â = 0.034 x 15 = 0.51). Be sure the student understands that the first expression in this equation is not equal to the second or third expressions. 
Almost There 
Misconception/Error The student makes an error in implementing an effective strategy for solving the problem. 
Examples of Student Work at this Level The student writes an equation such as 3.40 = 1.15x or Â = Â where x is last yearâ€™s gas price but:
 Makes a calculation error when dividing 3.40 by 1.15.
 Rounds the final answer incorrectly.

Questions Eliciting Thinking I think you may have made a calculation error. Can you check your work to see if you can find it?
What are the conventions for rounding? If rounding to the nearest hundredth, what should 2.956â€¦ be rounded to? 
Instructional Implications Guide the student to locate and correct his or her error. Have the student exchange papers with other Almost There students to analyze each otherâ€™s original work and to determine and correct any errors. Provide additional opportunities to solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals). 
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 and correctly solves an equation such as 3.40 = 1.15x or Â = Â where x is last yearâ€™s gas price.

Questions Eliciting Thinking How could you check your answer?
Another student solved the problem by writing this equation: x = 3.40 â€“ .15(3.40) to find last yearâ€™s gas price. Can you find the mistake this student made? 
Instructional Implications Provide additional opportunities to solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals). 