Getting Started 
Misconception/Error The student is unable to correctly solve the given inequality. 
Examples of Student Work at this Level The student:
 When dividing both sides of the inequality by 0.99, neglects to divide 3.99 by 0.99.
 Unnecessarily changes the direction of the inequality symbol when subtracting 3.99.
 Subtracts 0.99 from 21.01 rather than dividing 21.01 by 0.99.

Questions Eliciting Thinking If this were an equation, what steps would you use to solve it? Can you follow that same process to solve the inequality?
It appears that you were dividing each side of the equation by 0.99. Is there any reason that you did not divide 3.99 by 0.99?
Why did you change the direction of the inequality symbol? 
Instructional Implications Review the differences between equations and inequalities with regard to the potential numbers of solutions and considerations in solving (e.g., multiplying or dividing each side by a negative value). Begin by asking the student to write and solve onestep inequalities to solve word problems. Ask the student to interpret solutions in context and provide feedback. Eventually introduce inequalities of the form px + q > r, varying the inequality symbol. Also vary the side of the inequality on which the constant appears and help the student develop flexibility in interpreting inequality statements written in a variety of ways.
Note: The Gift Card Inequality worksheet is editable and can be rewritten with new coefficients, operations, inequalities, and context for later assessment. 
Moving Forward 
Misconception/Error The student is unable to explain the meaning of the solution of the inequality in context. 
Examples of Student Work at this Level The student misinterprets the:
 Meaning of the variable, saying the answer represents money (e.g., cost per song, total amount of money spent, money left over, or simply writes $21.22).
 Inequality symbol, saying Aaron can buy â€śmore thanâ€ť or â€śat leastâ€ť 21 songs.
The student describes:
 The inequality, xÂ 21.22â€¦, in words without offering a contextual explanation.
 How to solve the inequality rather than explaining the solution in context.

Questions Eliciting Thinking What does the variable in the inequality represent?
Can you read your solution to me? What does the inequality symbol mean? How is it read?
What does the solution xÂ 21.22â€¦ mean in the context of this problem? 
Instructional Implications Ask the student to review the explanation of the inequality to find specifically what the variable represents. Guide the student to consider what 0.99x and 0.99x + 3.99 represent and why the latter quantity must be less than or equal to 25. Then, assist the student in developing an interpretation of the solution, xÂ 21.22â€¦. Guide the student to consider the context of the problem and the meaning of the variable in identifying the kinds of numbers (e.g., whole numbers, integers, or rational numbers) represented by the solution set. Then, ask the student to determine every possible value in the solution set paying particular attention to the maximum value and considerations in determining it.
Have the student develop a list of key words/phrases that lead to the use of each inequality symbol. This list might include:
 > More than, exceeding, above, greater than.Â
 At least, no fewer than, not under, no less than.Â
 No more than, not above, no greater than, does not exceed, at most.
 < Fewer than, below, less than.
Provide additional opportunities to write, solve, and interpret the solutions of inequalities.

Almost There 
Misconception/Error The student makes minor errors in the mathematical solution or the contextual explanation of the answer to the inequality. 
Examples of Student Work at this Level The student:
 Makes a minor calculation error while solving the inequality but all other work and explanations are correct.
 Writes the solution as xÂ = 21 and interprets it as Aaron can â€śbuy 21â€ť or â€śabout 21â€ť rather than interpreting it as a range of values such as â€śat most 21â€ť songs.
 Rounds incorrectly (e.g., rounds 21.222â€¦ as 20, 21.23, or 22).

Questions Eliciting Thinking I think you may have made a small error. Can you go back and review your work to look for the error?
Does x actually equal 21? What other values might x equal?
What kinds of numbers does x represent? Can x equal a rational number such as 18.5?
Can you explain how you rounded 21.222â€¦? 
Instructional Implications Review the studentâ€™s error and provide feedback. Allow the student to revise his or her work. Provide additional opportunities to write, solve, and interpret solutions of inequalities. Pair the student with a classmate to compare work and reconcile any differences.
Provide the student with a completed problem that contains errors. Have the student identify and correct the errors. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level A. The solution is xÂ 21.22... or 21.22â€¦ Â x.
B. The student explains that Aaron can buy no more than 21 songs (or â€ś21 or fewerâ€ť). The student may clarify that Aaron can buy between zero and 21 songs after buying the game for $3.99.
The student may add clarifying comments such as:
 It has to be greater than zero because you canâ€™t buy negative songs.
 It would have to be whole numbers, because you canâ€™t buy parts of songs.

Questions Eliciting Thinking Why canâ€™t Aaron buy 22 songs or 21.22 songs?
What if the solution had been xÂ 21.8? Would that change the maximum number of songs that Aaron can afford?
If the price per song increased, what would happen to the number of songs he could purchase? How would that change the inequality?
If he did not buy a game and only bought songs, how would that change the original inequality? 
Instructional Implications Ask the student to list all values in the solution set and graph the solution set on a number line. Challenge the student to identify a variable that is better represented by rational (or real) numbers (e.g., time, weight, or length) and to write a word problem containing this variable that can be modeled by an inequality. Ask the student to write an inequality to model the problem, solve the inequality, and graph its solutions on a number line. 